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Friday, May 15th 2020


Camus, The Plague, and God


The other day, my friend Steve Hays posted the following question on my Facebook timeline:

A while back you said you were reading The Plague in the original French. As we know, Camus poses a dilemma: if a plague is sent by God, is it impious to fight the plague? Are we fighting God by fighting a heaven-sent plague? If you feel up to it, how would you respond to the (false) dilemma?

3 Musketeers

As far as reading in French goes, I'm having fun with it. It's slow going right now. However, I understand that learning or working in a different language is supposed to keep one's brain from aging. I followed up with The Stranger, but got bored with it halfway through. Now I'm amusing myself with Les Trois Mousqetairesand am having a riot with it. Any number of witticisms in French get lost in translations. It's a long book, though ...

The question, like many others touching on the problem of evil, needs two kinds of answers, one confessional and one apologetic, viz. what I believe and why I think that Camus is wrong.

For the first part, the Heidelberg Confession expresses my thoughts rather nicely. I'm not Reformed in the denominational sense, but I agree with a number of their doctrines with regard to the sovereignty of God. Here are questions 27 and 28, as shared by the website of Westminster Theological Seminar

Article 27

Q. What do you understand by the providence of God?

A.God's providence is His almighty and ever present power,[1] whereby, as with His hand, He still upholds heaven and earth and all creatures,[2] and so governs them that leaf and blade, rain and drought, fruitful and barren years, food and drink, health and sickness, riches and poverty,[3] indeed, all things, come not by chance[4] but by His fatherly hand.[5]

[1] Jer. 23:23, 24; Acts 17:24-28. [2] Heb. 1:3. [3] Jer. 5:24; Acts 14:15-17; John 9:3; Prov. 22:2. [4] Prov. 16:33. [5] Matt. 10:29.

Article 28

Q. What does it benefit us to know that God has created all things and still upholds them by His providence?

A. We can be patient in adversity,[1] thankful in prosperity,[2] and with a view to the future we can have a firm confidence in our faithful God and Father that no creature shall separate us from His love;[3] for all creatures are so completely in His hand that without His will they cannot so much as move.[4]

[1] Job. 1:21, 22; Ps. 39:10; James 1:3. [2] Deut. 8:10; I Thess. 5:18. [3] Ps. 55:22; Rom. 5:3-5; 8:38, 39. [4] Job 1:12; 2:6; Prov. 21:1; Acts 17:24-28.

I cannot believe that God is ever out of control at any time in any place. Let me qualify, though, that you can hold to a view of God being in charge of all things without necessarily expressing it as strongly as Heidelberg.

Undoubtedly, if I stated my view to Camus, he would respond that I'm only aggravating the problem by defending the views expressed by Father Paneloux in theThe Plague.

In his first sermon, the priest interprets the plague as a judgment from God, calling on us to repent from our sins. Camus is surprisingly subtle in his depiction at that point. He pictures Paneloux as stern and dogmatic, but not uncaring. The priest is greatly troubled by the plague but sees the cause in our sins and the cure in our repentance. My reaction to the idea of the plague as judgment is that 1) we find such an interpretation represented in the Bible multiple times, and, thus cannot rule it out, but 2) unless I were to get direct word from God that some catastrophe is meant as a judgment, as the OT prophets did, I would be very, very careful in expressing such a claim.

The Plague

Challenged by Dr. Rieux, Fr. Paneloux joins the team to care for plague sufferers. After he witnesses the death of a young boy, he starts to change his mind. In his second sermon, he still claims that the plague has been sent to us by God, but as a test of our faith, not as a judgment. God's reasons may be mysterious to us, but we must not question him. Our obligation is to have faith in him, regardless of whether we understand his ways.

Dr. Rieux does not accept this view. To be honest, my first reaction to such a blanket objection is not to resort to my arguments concerning the problem of evil, but to shrug my shoulders and say that, if Dr. Rieux doesn't want to believe in God or divine providence, that's his option. I get bored with internet atheists who proclaim that they can no longer believe in God because of all the evil in the world and then resort to irrelevant rabbit trailing, which frequently includes imposing on theists a description of God that neither Jew nor Muslim nor Christian nor any other theist would recognize as the God whom we worship [footnote to Marilyn Adams, Horrendous Evil and the Goodness of God.] If you think that you cant believe in the God of the Bible, then don't. Or, if you care to, listen to some reasons why you should, but don't flaunt childish unbelief as the triumph of rationality, let alone think that other people must stop believing because Camus says we shouldn't.

Ooh, that's harsh.

So, I'll go to chapter 7 of No Doubt About It and try to explain to M. Camus that if God is omnipotent, omniscient, and omnibenevolent, then the most plausible explanation for the present evil and suffering in this world is that God is in the process of eliminating it, and that this process is necessary because it brings about a higher good than would be possible without evil. I won't bother here with the lengthier argument to that effect because Camus does not really accept my supposition that God is o & o & o. If God truly possesses all of the properties that a genuine theist attributes to him, then the most reasonable conclusion is the one I mentioned above. God is in control and will abolish suffering in line with his goal for all of creation. It doesn't make any sense to invent a description of God, tell theists that they must accept that description, and then criticize the theist for holding an unsound belief.

However, Camus and many other people, will label my position as naïve, if not downright wicked, because it allegedly misleads people. Still, Camus does not follow the atheist's route of affirming the non-existence of God. He questions my affirmation that an o&o&o God has a good reason for allowing evil. As far as he is concerned if there is a God, then he is not good, and human beings must fight against this God who is robbing us of our freedom. He elaborated on this position in L'omme Revolté, but in doing so, Camus argues himself into a corner.

If God's allowing the plague entails that God is evil, then any sane person would presumably fight the plague as sent by God and in the process visualize himself as fighting God. Of course, what the person is doing is actually no more than fighting the plague. If he wants to imagine himself as fighting God in the process, there's nothing to stop him, except that he is endowing his work with futility. Assuming that God would still be omniscient and omnipotent, neither Camus nor his followers would have any chance to succeed, and the human's task would, indeed, be as non-productive as that of Sisyphus.

Albert Camus

The question rears its head as to whether the thing that Camus wishes to fight is still reasonably called "God." God, as believers think of him, is good and trustworthy. The idea of an entity that has all of the attributes of God other than goodness, whether it be deemed evil or just uncaring does not respond to what a theist would call "God." Camus and a few others who deny God's goodness may have created a "Super-Satan," and undoubtedly everyone should resist him, or, as the Bible tells us, to flee him.

So, there's an equivocation at work in considering God as not good. The term "God" does not include an evil nature. In order to conceptualize the kind of God Camus proposes for us to accept, we would have to revise our vocabulary. But there would be no motivation for doing so unless there were no plausible answer as to why an o&o&o being would permit evil. I'm quite sure that there is (again please see ch. 7 of NDAI for an elaboration).

Let us think for a moment about a specific evil, such as the plague. It comes to us, as Heidelberg summarizes, by God's fatherly hands. How can a loving heavenly father send (or permit, if you want to soften things a bit), a plague? Let me make sure we understand each other here: I do not consider a plague to be a good thing.

And neither does God.

A theist (and I'm most qualified to speak as a Christian), whether embracing Heidelberg or not, knows that God has given us the ability to recognize evil for what it is. We do not share the conclusion frequently expressed by atheists that there is no such thing as objective morality. A question that Camus never could answer if why some things are evil and others aren't. In the final analysis, he did not promote an existentialism as his friend Sartre did--not that Sartre really had a better answers--but he ultimately resorted to a rather weak and unconvincing humanism [Walter Kaufmann, Existentialism, Religion and Death(New York: Signet, 1967), pp. 205 ff]. Nevertheless, Camus was convinced that we should fight evil (and I'm not sure how, or even if, he actually personally fought against evil other than by joining various groups and signing petitions).

So, if God permits a plague, what are we supposed to do? There's no question here for us because, in contrast to Camus, we do have a basis for an objective morality.

We're supposed to fight the plague, right alongside Dr. Rieux with his foreclosed attitude. But we're not fighting against God; we're doing what he has called us to do. And if it is God who has called us to do something, we can have confidence that, no matter how far from us the result may appear, God will bring it to a perfect end.

1 Corinthians 15:58: Therefore, my dear brothers, be steadfast, immovable, always excelling in the Lord's work, knowing that your labor in the Lord is not in vain.
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Wednesday, December 5th 2018


Chinese Birthdays

  • STATE OF EXISTENCE: Not reported
  • IN THE BACKDROP: furnace noises

So, yes, you're eyes are not deceiving you. I'm going to give the old blog another try. Sometimes it's easier to develop a thought without the confinement of little narrow comment windows. Also, sometimes I want to exhume the creativity I used to draw on in the good old days of my blog.

I have accidentally deleted a number of entries from 2017, not that there were all that many. Regardless, once more, here's an attempt at a new start. For now, if I stick with it, the rebeginning will basically be simple answers to good questions, but who knows where it may go once I get into it. 

Chinese Birthdays

In a recent post on FB, I alluded to the Chinese way of counting birthdays, and a dear friend told me that she found it confusing. I don't blame her because it involves a different way of numbering years than we non-Chinese are used to. However, once you make one particular switch, it does make sense.

Insofar as I may speak for us Westerners, we count years in the same way as we count socks. We start with zero socks, and then every successive sock gets its number. So, this would be five socks, or, in the spirit of the season, Christmas stockings:

5 socks

I can't really imagine how or why we might chance upon half a Christmas stocking, but if that would occur, we would not have six stockings, but only 5 1/2.

Five and a half socks

We Westerners use the common "counting numbers," or cardinal numbers for collections and ages. As long as you don't have the entirety of something, you can't give it credit with a whole number. A boy may proudly inform his grandmother that he's no longer 5 years old, but 5 1/2, and that's fine. But in our system he would be wrong if he said that he was 6.

However, when Chinese people talk about their age they express it in terms of chronological rankings. Someone who has just been born has entered the first year of their life. The difference is though that, according to the Chinese way of counting, this means that the infant is one year old. That takes getting used to. Sticking with the Christmas stocking motif, this is how it goes.

5socks with ordinal numbers

Then, when we add the mysterious half-sock, we have a sixth one, and, even though it is not yet present in its entirety, it is still number six and should be counted as number 6. In application, if the child is 5 1/2 years old by our standard, he is in his sixth year and therefore is considered to be six according to Chinese traditional reckoning.

6 socks Chinese style

Furthermore, according to Chinese tradition, people don't get a year older on the annual repetition of their birthday. It's much easier here. Everybody can count themselves a year older on Chinese New Year. The New Year of the Chinese lunar calendar falls into late January or early February of our solar calendar. There are numerous traditions connected to it. They vary regionally, including gift-giving, visiting relatives, and eating special foods. I had the privilege one year to partake of "wax duck," which tasted just about the way it sounds, viz. "waxy" and "ducky." For our purposes here, the most important custom is that everyone can count themselves a year older come New Year's celebration.

So, the following scenario is not at all unlikely. Let's say that a baby will be born today on December 5, 2018. It's her first year of life, so the answer to the question, "How old are you?" would be "1" if she could speak. Well, New Year's is just around the corner; on February 16, 2019, the present "year of the dog" will be finished, and the "year of the pig" will commence. Our little baby girl, along with the rest of the traditional Chinese population will be deemed one year older. So, she will have been one year old from December 5 of this year through February 15 next year and then will turn two on February 16.

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Monday, December 4th 2017


... and miles to go before I sleep

  • STATE OF EXISTENCE: flying solo

...and miles to go before I sleep.

But sleep I will. And I will wake up alongside "Buggy,"
and all the tears and regrets and loneliness will be gone.

Look! God's dwelling is with humanity,
and He will live with them.
They will be His people,
and God Himself will be with them
and be their God.
He will wipe away every tear from their eyes.
Death will no longer exist;
grief, crying, and pain will exist no longer,

because the previous things have passed away.

Rev. 21:4,5

For those blog readers who may be clueless as to what I'm writing about, my dear wife, June, passed away on October 1 of this year. It had turned out that her increasing weakness and incapacity was due to her breast cancer of 10 years ago. It had never returned to her breasts, thereby yielding clean breast biopsies, but had metastasized throughout her skeleton. You can go back to find my Facebook postings over the last year or so if you're interested in more details.

I have not yet sorted out all of my feelings sufficiently to write a helpful, let alone inspiring, account of my thoughts, doubts, and discoveries during that time. Suffice it to say that a part of it was hellish, and that I am incredibly grateful for everyone who was there for me and for June, as well as those people who would have been if it had been a possibility. -- And for the people who reassured me about the relevance of God's presence at that time.

Here is the little video about June's life I played at her memorial service and then posted on Facebook and YouTube.

Also, before I leave the topic, please let me once more express my gratitude to all those wonderful people who provided financial help so as to keep this personal tragedy from being compounded into a financial disaster. For example, the nursing home's rate for room and board (which Medicare paid for only partially) was around $7,000 a month, so you can be sure that every last penny I received went to June's care.


For the time being, at least, I figured that the only way I could get back into doing my blog was simply to start rather than first provide a detailed prolegomena of the last fourteen months. So, I'm going to jump right into the last couple of weeks and talk about my ten days in Alaska.

When June passed away, my older brother Wolf gave me a choice: Either he could come out to the memorial service, or he would be happy to spend the same amount of money for me to come out to see him and his clan in Sitka.

That wasn't much of a choice for me. Obviously, spending ten days in Sitka with him and his wife, as well as other relatives and relations-by-marriage, was highly preferable to having a short encounter along with a bunch of other visitors around the time of the service. So, I began my trip there on November 20 and started the return on December 1. If you're going to try to calculate the times, please take into account that Alaska time is four hours behind Eastern time.

Sitka Panorama

A panorama view (i.e. actually a semi-circle) of Sitka with Mt. Edgecombe on the left.
Towards the right you see the steeple of St. Michael the Archangel Russian Orthodox Church.

Here's another shot, focusing on St. Michael's through a bit of forested area.

Sitka framed

So, presumably you can tell that Sitka, located on Baranoff Island just off the southern end of the Alaskan panhandle, is a wonderful place aesthetically. In case you're wondering about the weather, it pretty much hung around the upper thirties and lower forties (above freezing) the entire time, and it doesn't get as cold there as it does here in Indiana.

The trip out was already beset by long layovers: four hours in Seattle, five hours or something like that in Anchorage. The flight from Anchorage to Sitka was supposed to include a quick stop in Juneau to drop off and pick up passengers, but that one took the longest time of all. Things seemed to be going smoothly on that last leg, though for some reason, the captain had to abort the first approach. A flight attendant let everyone know via the intercom that the captain would talk to us shortly, a little bit of an ominous-sounding announcement. After a few minutes he did come on the air and, in that casually reassuring tone that one gets accustomed to from the flight deck, apologized for the somewhat "botched approach." He casually mentioned that there had been a little problem with the landing flaps, and that we should be making another approach shortly. Once again we found ourselves above the clouds.

However, I noticed something weird. The landing flaps weren't out as far as they were supposed to be. Furthermore, while we were making a big circle around Juneau, the captain steadily kept up the nose of the plane at a unusually high angle. That's something you do in order to slow down the aircraft massively, as I had learned back when I used to spend time on my EASports flight simulator. A pilot with Air Alaska who happened to be seated in front of me confirmed that my suspicions were right. The flaps were stuck. He also mentioned that this was something highly unusual, which he had never encountered before in his decades of flying, except in the flight simulator. I was pleased to learn that he had made a safe landing on the simulator each time, and he seemed pretty sure that the present captain would most likely do the same, but somehow I didn't feel entirely reassured.

Nonetheless, we did get into Juneau safely. I think we came in just a little hot, but it was smooth, and--like I said--the runway was quite long. Obviously, we couldn't go on with that plane, and the lay-over in Juneau came to about 8 hours. There's not much to do in the gate area of Juneau airport, except to take some pictures of the gorgeous surroundings.

Pines around Juneau airportmountain around juneau

Finally, Air Alaska found another aircraft and flew all of us who were Sitka-bound to our destination. I made suret to listen to the landing flaps come out.

So, I got into Sitka in the evening rather than in the morning. Of course, at this time of year, there's not much daylight in between. The sun rose around 9 am and set around 4 pm.

I added up all of the actual time I had spent on the trip, beginning with leaving my house Monday afternoon to drive to the airport and walking into Wolf and Yvonne's house on Tuesday evening, and it came to 32 hours.

We'll pick up here with the next entry.

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Wednesday, December 7th 2016


June's Adventures in Hospital Land

  • STATE OF EXISTENCE: good, though a bit tired
  • IN THE BACKDROP: Property Brothers in New Orleans

June’s Adventures in Hospital Land

Enough people have asked how June's tests went, that I'm writing a single narrative here. Thanks for your concern. Nothing really happened, but, as many people know, I enjoy giving my P.G. Wodehouse-inspired side a chance to show off. The really important thing will be the results, and we have no idea when we'll hear anything. Much of what I’m writing here is based on June’s ongoing reports to me, and I’m pretty sure I remember everything accurately. However, she is not responsible for any hyperbolic descriptions subsequently forced on me by my irresistible need for creative expression..

First, she somehow disappeared through a mirror, and then she met this really big white rabbit. And when she saw Humpty Dumpty sitting on a wall, things really got weird. No, wait … Sorry.  Different person, different events.

Obviously, June’s condition has not changed because nothing has yet been diagnosed or treated. Yesterday (Tuesday) and today (Wednesday) we were at the hospital for cardio testing: a cardio-echogram and a Cardiolite radioactive stress test. In case you’re interested in some of the chemical aspects of the latter, I put together a summary about it in my blog from May of last year, when I had that test (and passed it with flying colors). It’s a two-day test, consisting of an injection of radioactive 43Technecium (99mTc), a CT-like nuclear tracing on day one, followed by more injections, a treadmill test, and another session with the nuclear folks on the second day. The cardio-echogram was supposed to happen after the radiation test on the first day. Sounds relatively simple. But …

Day 1: After a short time in the waiting area, June was called into the cardio quarters and injected with the radioactive brew. No, she didn’t glow, but as I said on the blog back then, if there had been a Geiger Counter around her, it would have been ticking. Right after the injection, the ultra-sound lady, apparently not acquainted with other items on the schedule, swooped down on her and proceeded with the cardio-echo. June has had ultra-sounds before, but it never had been painful before. This lady apparently was just a little abrupt in her manners, and her technique was not exactly gentle, as she virtually stabbed her wand between her ribs. (I warned you about hyperbolies.)  Afterwards, she told June she could go home.

So, June came back to the waiting area where I was huddling and started to put on her coat. Fortunately, I knew that the radio-active test was far from finished, so, after a couple of minutes of all-around confusion and a teensy-weensy bit of assertiveness on my part, things got resolved. Not too long thereafter, we were off to the nuclear medicine dungeon. This time I went in with her, not for any reason other than to keep June company and because I like that creepy place. It’s actually a perfectly normal x-ray-looking facility, except that there are bodies lying motionless on cots surrounded by technological apparati. The scene reminded me a little bit of "The Matrix" or other films with similar components.

June went through her immobile period in grand fashion. Parmenides and Zeno, the ancient Greek philosophers who asserted that there was no such thing as motion, would have been proud of her. When she was done, the nuclear medicine technician came over and confessed that, alas, a piece of her intestine had blocked the machine’s view of her heart.

Lunch time. We returned to the nuclear reactor, woops, I mean department. This time everything worked. Home. For someone who is suffering from extreme exhaustion, unsurprisingly, June was extremely exhausted.

Day 2: Today started with some more injections and the treadmill test. The friendly woman who served as treadmill administrator informed June that she would just spend a few minutes walking on the rolling band, which would increase its incline every minute. Just a short time earlier, when we had entered the hospital, June had some difficulty walking up a slanted hallway, needing to stop several times to catch her breath. (Don’t ask me why there is such a thing as an uphill hallway in a hospital.) So, June responded to the treadmill lady that she was certainly happy to walk on the treadmill, but that she couldn’t guarantee that her knees wouldn’t buckle in the process.

There is a standard alternative. Instead of walking on the treadmill, the patient’s heart rate can be elevated chemically by way of an adenosine injection. After some consultation, it was decided that they would do this with June. Apparently, it was not totally routine, though, since the room was suddenly filled with personnel from various departments, just in case something would not go as hoped for. June did not show evidence of a bad reaction to the adenosine, and everything went fine, not as they had planned, but as I had figured.

Now back to the nuclear department for another round of motionless scanning. Well, make that two rounds, with a time for lunch in between. Once again, June’s insides blocked the camera’s view the first time, but settled sufficiently downward the second time around.

That’s it. Nothing really dramatic, just some unexpected little contingencies adding interest to what could have been a purely routine set of medical tests. Now we're waiting for the results.

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Tuesday, July 12th 2016


The Golden Ratio and the Perception of Beauty


This blog entry is coming to you from our little hide-out south of Smalltown, USA. It's been a good two days so far. Originally, we thought of driving further, but at this time that doesn't make much sense. I get to do two of my favorite things every day: riding horses and swimming. Today I got to ride on Whiskey again, my favorite, as some readers may remember. June has very little energy, and we'll get into that some later time. Right now there are so many folks hurting and in urgent need of prayer, so we'll wait our turn. Does that make sense?  

Before getting back to phi and the golden ratio, just a word in general in the context of Christian apologetics. There's a line of argumentation that we all would do well to avoid. I've seen it used by Christians in conversation with people of other religions, and I've just now experienced it myself. The argument runs this way:

Person A: My religion/world view is X.

Person B: If your religion is X, then you also believe Y, and Y is absurd. 

Person A: But I don't believe Y

Person B: But you must believe Y if you're X, and so, either you don't know your own religion, or you hold to an absurd belief. 

Person A: But I really don't believe in and it doesn't fit with what else I actually believe. 

Person B: Too bad. I know that you are obligated to believe Y, and you're a fool for believing such nonsense. 

Person A (slaps himself on forehead): Okay, I see. You're right. How could I not accept the obvious nonsense that you say is a necessary a part of my worldview/religion? Thank you for showing me my duty to believe something absurd so that you can take pot shots at me.

Maybe Person A is inconsistent; maybe Person B is uninformed. Either way, it doesn't make much sense to criticize a person for a belief that he or she doesn't hold. This dialogue would be funny if it  came from Abbot and Costello. But in the real world, it's not exactly the best approach.

Typical examples from Christians:

Jews are obligated to believe the Old Testament and are required to expect the re-institution of temple sacrifices. 

Hindus and Buddhists must be pantheists if they only understood their religion completely.

From a non-Christian to me:

As an evangelical Christian, I must accept a young earth theory of origins, which I don't. But it doesn't matter that I say I don't; if I were consistent with my belief in the inspiration and inerrancy of the Bible (as defined in a bizarre way that I hadn't I heard of in a long time, if ever), I would believe in the young earth, and, thus, I'm an idiot for believing it. 

This foolishness has to stop. 


And now back to our contemplation on the golden ratio and phi. The general belief is that the golden ratio in some inexplicable way catalyzes our appreciation of beauty. Certain experiments have supposedly shown that this idea cannot be documented with evidence. I suspect that this matter is similar to a court case where each side brings in their expert witnesses to support their case. We're talking about a very narrow window. 1 : 1 1/2 is too little; 1 : 1 2/3 is too much. So personally, I would be surprised if eventually there were some clear and objective proof for the aesthetic appeal of phi in art, but it's not something that I can be or want to be dogmatic about. The one thing I do want to caution us about is that, as I have insisted all along, there is a lot of beauty in phi, but its appeal, if any, is not due to some supernatural numerological  power . 

The table below the video contains a picture of a "Cherokee maiden" that I took ...

[Queue up Bob Wills, Merle Haggard, or Asleep at the Wheel!]

... a couple of years ago. Each one of these pictures has a different ratio of its sides; one of them is in the golden ratio. Is there one that strikes you as more beautiful than others? I sure hope so because some of them are pretty badly distorted. Which one do you like best? I shall disclose which one is in the golden ratio at some future time. In the meantime, have fun with it, assuming that you find this kind of thing amusing or interesting. 

1 2 3
4 5 6

We're almost done with this series. Next time or so, I'll bolster the section on the Fibonacci numbers in nature. 

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Monday, July 11th 2016


The Parthenon and the Taj Mahal

  • STATE OF EXISTENCE: ready for a break

The Golden Ratio in Ancient Architecture

golden triangleIt’s time to come back to the point of this series. Its topic is the beauty of the number that is usually called phi (ϕ). Given all of my stacking and popping, I think a little recap is in order:

Contrary to the things you frequently may read, phi is not derived from the series of Fibonacci numbers, though they do converge to the value of ϕ: 1.61803…. Its origin lies in the geometry of a pentagon from which we can derive a “golden triangle,” which is distinguished by the fact that the ratio of one side to its base, ...

side over base

... is equal to the ratio of the side and base combined to a side to the base.

side plus base

So that:

golden ratio.

If we want to express this relationship with regard to a straight line, we can say that there is a line, connecting points A and C, running through point B,

golden ratio

and the ratio of BC to AB is the same as the ratio of AC to B, namely the famous phi: 1.61803 ….

I recounted some fascinating properties of phi, and showed you a few interesting features of phi in relationship to the Fibonacci series. After an excursion on some wonders in π, I asserted that, for Christians, the presence of the Fibonacci numbers in the universe and the beauty within the world of numbers itself should lead us, without hesitation, to affirm the wonderful hand of God displayed in his creation. Then I set out on a long excursion on some scientists and mathematicians who see the world of numbers as divine, but, not wanting to acknowledge the God of the Bible, find God in some unexplained and inexplicable manner within his creation itself.

Given the length and seriousness that this series has taken on, I will flesh out the earlier section on the Fibonacci series in nature. For now, I would like to address two questions that seem to go hand in hand.


We find beauty in many work of art. Some works may be more beautiful than others. Many people say that the perception of beauty may in certain cases be due to the fact that artists have incorporated the golden ratio in their creations. Thus we have two considerations to address:

1. Do we actually find the golden ratio in some of the works that are usually cited as examples of manifesting ϕ?

2. Does the presence of the golden ratio actually trigger our response to consider some things as beautiful?

We need to address the first question first because that's inherent in being the first question.

Artists are free to incorporate the golden ratio in their works to their hearts’ content. And if we find beauty in their production, so much the better. A good example is Salvador Dali’s painting, “The Sacrament of the Last Supper.” Its proportions in internet reproductions appear to be pretty close to ϕ, and I can’t be sure how much may have been lost in either trimming or framing. In this case, a tolerance of a couple of millimeters or pixels can be taken allowed.

Salvador Dali Last Supper

For the reproduction I have picked out of the many on the web, this one comes out at 1 : 1.58. Dali leaves no doubt about his intentions, seeing that he inscribed his painting with a dodecahedron, whose twelve sides consist of pentagons.

Livio Golden RatioApparently the two most frequently used illustrations of the golden ratio are the chambered nautilus and the Parthenon, the ancient Greek temple devoted to the virgin Athena, located on the Acropolis. We already mentioned that the nautilus increases his cells in the shape of a logarithmic spiral, though the ratio is not ϕ. Nevertheless, it is frequently used. Even the front cover of Livio’s book, in which he clearly states that the nautilus is not an example of the golden ratio, greets us with a representation of this misunderstood mollusk. You see that authors are frequently less in control of their books than one might imagine.

The Parthenon is often used to illustrates how an architect employed the golden ratio to endow his work with beauty. It appears to be almost a given that you can find the golden rectangle in the Parthenon. But we must ask, where exactly in the Parthenon do we find the golden ratio? People making this claim usually illustrate it, and the lines that are drawn differ from person to person. Precision is important. When it comes to pictures of buildings, we’re not dealing with millimeters, but with much larger entities.

Here's one example from the webpage culturacolectiva.com. It finds not just one, but two golden rectangles in the building, situated next to each other, inscribed by a golden spiral. Note how the smaller one on the left goes down to an arbitrary line on the ground.


We can contrast that depiction with what we see on the site Design by Day™ Everything you need to know about the Golden Ratio in Graphic and Design.

Angela Roche Graphic

As most of these pictures do, it extrapolates to the top of the pointed façade, which is lost. The bottom goes to somewhere in the rubble at the bottom of the stairs. The two sides drop in alignment with the eaves and connect to the base in no-where’s land, cutting the platform where the rectangle requires it, but where there is no architectural indication for it.

Similarly, a site on Greek and Roman art, seems to give priority to the rectangle at the expense of architectural detail.


The location of points A and C in this picture are crucial to finding an additional golden triangle, but don’t seem to play a role in the actual construction of the building. Here is another picture on the same site. Note the pronounced leftward shift of the rectangle.


On Pinterest I ran across this little gem:

From Pinterest

The angle at the top is a little more obtuse than in some of the other pictures that extended the lines, and so the rectangle does not need to be as tall to meet the proportions. The base line is located above the stairs. The line is straight, but the placement of the columns is not, except for the front two. And again, the location of the two bottom corners is established by the geometry of the golden rectangle, not by any mark in the building.

We can find the following entry in the Nexus Network JournalTM:

From Nexus Journal

If I read the context correctly, this diagram is intended to make the point that there is no single way of inscribing the golden rectangle in the Parthenon. And that observation obviously negates the idea that phi is incorporated into the Parthenon, and that we see its beauty because of the golden ratio.

Finally, Gary Meisner, on his website ϕ = Phi ≈ 1.61803 comes up with yet another placement of the rectangle, cutting off the eaves.

Gary Meisner

But he expresses some hesitancy about imposing the golden ratio on the temple front as a whole, because it seems to require too many arbitrary decisions. He then seeks for it in parts of the structure, but, even if that should work, it's out of keeping with the conventional belief that the entire facade attracts us with the golden proportion, and so we'll skip that exercise. 

I think that my point should be pretty obvious by now: Lots of people agree that the Parthenon receives its beauty at least partially from the golden rectangle. But there is no unanimous agreement where precisely it is located, a fact that at a minimum should make us a little doubtful concerning this idea. I don't think that beauty is entirely in the eye of the beholder, but the golden rectangle might be. 

Actually, a couple of weeks ago, when I was thinking about this topic as a future entry, I decided to try my hand at the process without knowing what to expect. I knew what would be coming with the Parthenon, but I could not recollect ever seeing any pictures of the Taj Mahal in connection to the golden ratio. There are many of them; I had just not seen them. Well, I have a few good pictures of the Taj Mahal that I took in 2006, so I started to select various rectangles on the building and measured their lines in pixels to see if I could get close to a proportion of phi anywhere. The obvious place to start was with the entryway, trying to be as careful as I could, but also being open to the possibility that I could make an error in placement. Nothing I tried there worked out, and neither did the window-like openings.

Taj Mahal August 2006



Taj Mahal Taj Mahal
Taj Mahal Taj Mahal

I finally decided that I could not impose the golden rectangle anywhere on the Taj Mahal without fudging.

Then, in the course of looking for good pictures of the Parthenon, I came across my first Taj Mahal/Golden Ration depiction. I did not write down the URL at the time (or I misplaced it), but I remember exactly what it looked like, particularly because it was indicated in exactly the area I had been searching. My measurements confirmed that here was, indeed, a genuine golden rectangle--or so it seemed.

Taj golden

But then a closer look revealed that the person who posted that picture had been creative in placing the rectangle without taking the decorations into account. His or her top line is clearly above the somewhat lighter frame, but then the sides run along its inside. I thinned out the line a little bit compared to what I originally posted last night and highlighted the frame in order to make the obvious fudge more clearer for anyone who might have not seen it. I think I'm just as happy I couldn't find my way back to that particular URL because that person might just be angry with me. Some people ascribe mystical, supernatural powers to phi and the golden ratio, and probably aren't happy when an obvious fudge comes to light. (If I do find it, I will post it, so you don't have to take my word for it. )

Taj goldenTaj golden

The story of phi and the Taj Mahal resembles that of the Parthenon. "Everybody" insists that it's there. "Nobody" can agree where exactly it is, and there is a lot of fudging going on. I'm not saying that there are no instances of finding the golden ratio in art. But I'm skeptical about how strongly it is represented in some of the traditional supposed examples. I have maintained that there is beauty in numbers; I'm not so sure that works of art depend on numerical values for their beauty. More on that next time.

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Saturday, July 9th 2016


The Dogmatic Agnostic


T. H. Huxley

"Agnosticism" was a word invented by T. H. Huxley (1825-95) to describe his attitude that he did not know whether there is a God. He took the label of the ancient movement called Gnosticism, whose members prided themselves on their esoteric knowledge, and put an a in front of it, taking pride in what he claimed he didn't know. It was also Albert Einstein's favorite label for his religious outlook, as much as I can make out.

I'm going to repeat a few thoughts here that I brought up in No Doubt About It, 88-89. There are two (or more) ways of construing the term "agnostic": constructive and destructive.

An atheist who reviewed the book a number of years ago really took offense at my using the terms "benign" and "malignant." I hope that by now he has gotten over it, though the words are not inappropriate. Bottom line: the constructive form of agnosticism can lead to a growth in faith; the destructive one is irrational and can lead to a person's spiritual death.

Constructive agnosticism can be summarized with the statement, "I don't know if God exists." Well, if you don't, then you don't. For many people coming to this realization is the best thing that ever happened to them. Someone may recognize that this issue is a whole lot more important than, say, whether Pluto is a planet, and start to search for an answer. That's a good thing. For some young people who have grown up within a Christian environment, it can even be a necessary step to make the faith with which they have grown up their own.

However, frequently people use the language of "I don't know" when they really mean "you can't know" or "nobody can know." At that point the agnosticism has become destructive and does not really differ from atheism for all practical purposes. Someone claiming this view must assume that all possible ways of knowing about God have been tried and failed, and, furthermore, that he or she has personal knowledge of all of these tests and their outcomes. This is, of course, not possible. That's why I said that destructive agnosticism is irrational. It presumes an omniscience, which has not been granted to any human being.

Here's the third YouTube video, this one is about Albert Einstein's religion.

There also is a separate Wikipedia article on the religious views of Albert Einstein. It does, indeed, take a lengthy article to compile all the relevant information, and I'm just going to give you a few highlights. The video, some internet sites, and biographical accounts in books are my main sources. I'm just using Einstein as an example and not pretending to put forward a full biography.

1. Einstein called himself an agnostic. I appears to me that many times his use of the term was a genuine expression of humility. He never failed to acknowledge his limits and the limits on knowledge that all human beings share--even when he was simultaneously trespassing them.

2. This humility is one reason why Einstein eschewed atheism and did not have kind words for those who promoted that position. He saw atheism as a destructive world view that robbed people of the transcendence that human beings need.

3. Einstein definitely did not believe in a personal God, including what he thought was the God of the Bible. There was no shortage of people who held that fact against him, as though he had a greater obligation to believe in God than other people. Incredibly, there were prominent Christians who exhorted him that, since he was Jewish, he was giving Judaism a bad name by not accepting the God of the Old Testament. But Einstein, much like Kaku now, was not open to a "God of Intervention." He considered the idea of a personal God, as found in Judaism and Christianity, a childish fantasy used to instill fear in people so as to make them behave. Einstein had some Christian teachings as a part of his early education, and, for all that I know, his teachers at that time may have been using God as a bogey man figure to frighten children into being good. Needless to say (I hope), we're once again looking at a caricature of the biblical God, and--as far as I can tell with my very limited research--he never pursued educating himself further on a more mature understanding of God. Once he was done with it, all that was left was a patronizing smile from his allegedly more rational point of view.

4. Apparently there were several occasions when some media outlet declared that Einstein believed in a personal God, and these reports made him furious.

5. As mentioned in the last entry, Einstein frequently averred a particular preference for the pantheistic God, as described by Spinoza. Let me just reiterate that his admiration of Spinoza's view requires the romantic glasses through which Spinoza was being read starting in the early 19th century. The things that intrigued Einstein about the universe and the possibility of some kind of a deity are the very things that Spinoza reduced to a flat, undifferentiated monism. To be sure, Spinoza's God was rational and orderly, but the beauty and elegance that mesmerized Einstein are not found in Spinoza's Ethics.

6. Just as we saw with Kaku, it is a whole lot easier to compile what Einstein did not believe than what he did believe. As soon as we try to look more closely at the positive side, we see expectations, beginnings, and inconsistencies without resolution. Here are two quotes that I took out of the Wikipedia article. This is the first one:

"God is a mystery. But a comprehensible mystery. I have nothing but awe when I observe the laws of nature. There are not laws without a lawgiver, but how does this lawgiver look? Certainly not like a man magnified."

Einstein is in awe of the laws of nature, an attitude we should applaud. He goes one step further and asserts that these are laws that must have been legislated by a lawgiver. Once again, we're running up against a highly unusual understanding of the laws of nature, at least as it shows up in the phrasing. The laws of nature are not commandments given by God in the way a government makes laws for its citizens. To repeat something I said earlier, the laws of nature are descriptions and perhaps statistical generalizations. Some of them appear to us to be ironclad and unrevisable. They are discovered by people working in the sciences. So, the idea of a prescriptive divine Lawmaker is somewhat odd for the laws of nature, and it was probably not what Einstein was really saying. I think we can agree that what Einstein meant was that the law of nature are such that one is driven to see an intentionality underneath them.  But please note also that Einstein overstepped his professed agnosticism in this statement. He ruled out any anthropomorphic understanding of God, but then he must know something about the one who is beyond our knowledge. If he is truly in the dark about God, then he is not in a position to set up rules as to God's true nature.

This point is important because the descriptions of God that Einstein called "anthropomorphisms" are the attributes of God that lead us to understand him as a personal being. From time to time Einstein used anthropomorphic language about God as well.

Der liebe Gott würfelt nicht.
"God does not play with dice."

Does that assertion mean that Einstein pictured God as a person with hands who performs various actions, but refuses to roll a pair of dice. Of course not. Einstein was using an anthropomorphic image to make a greater point about a being that does not literally have hands or could be tempted to entertain himself with dice. And that is the proper method for understanding the Christian notion of a personal God as well. What's sauce for the goose is sauce for the gander. Our language is limited by our earthly environment, and we cannot express what God has revealed to us about himself without using terms from a finite context and applying them to the infinite. (Some of the best philosophers of religion place the language of religion under the technical label of "analogy.") No, God is not a magnified human being, as the atheist Ludwig Feuerbach proclaimed in his Essence of Christianity. But the only way in which we can talk about God is with human language. The alternatives are either to say nothing at all or to utter something meaningless. Einstein did not choose the former option, and his dictates about the nature of God seem short on meaning. One simply cannot declare the ineffable, let alone make up rules for it.

7. We see the same confusion in the second quotation:

A knowledge of the existence of something we cannot penetrate, of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms — it is this knowledge and this emotion that constitute the truly religious attitude; in this sense, and in this alone, I am a deeply religious man.

I am reminded of the controversy of thirty or so years ago with regard to the work of art produced by Andres Serrano. I'm not going to describe it here because the details don't matter. Suffice it to say, it was rightly considered blasphemous and offensive by Christians, particularly Roman Catholics. However, I ran across one review [Art News 89 (April 1990):163], in which the author defended Serrano. Specifically he said that critics were misunderstanding him; he was not an opponent of religion. In fact, he was a deeply spiritual man who was devoted to a religion focused on bodily fluids. That case certainly is crassly different from Einstein and his beliefs, but the same logic appears to be at work. Whatever Einstein declares to be a religion, regardless of whether it violates any reasonable understanding of religion, is a religion.

religion ≡ whatever Einstein declares to be a religion (def)

Einstein confesses to the existence of something that is almost entirely inconceivable and ineffable. That recognition makes him a deeply religious man, he claims. Once more we are allowed only a scanty look, and the picture can't be filled in because there isn't even a sufficient outline. All we have is a vague apperception that overwhelms us as we study the cosmos. If that's sufficient to be called a religion, then Einstein was, indeed, a deeply religious person. I must admit, however, that I think we're looking at something extremely thin here.

So what? someone may respond. If Einstein was content with that description as his "religion," why is that my concern? Why am I picking on Einstein (and on Michio Kaku earlier)?

If you are asking that question, please read that quotation again and take cognizance of the claim that he is describing the "truly religious attitude." Albert Einstein is not just asserting that this recognition of something beyond is a religion in its own right; he is making an exclusive claim for its truth and superiority. Consequently, that quote does challenge us to interact with it. I cannot help but see Einstein asking the world to emulate his own highly underdeveloped notion of spirituality, inspired by reason but beyond rationality.

I'm going to add one more statement that is going to come across as pretty harsh, I'm afraid. One could say that, even taking all of the above into account, we should pay greater attention to what Einstein is saying in the realm of religion and philosophy because he was, after all, such a wise person. At least some of my present readers will remember that in many ways I am a great fan of Einstein, and I'm enamored with his theories--even though he was wrong at times. It's easy to second-guess his contributions to the Manhattan Project, which for a time broke his life-long pacifism, and I can't judge him on that in either direction. But whatever else Albert Einstein was, he was not a man of wisdom. He was very intelligent and well-spoken, and it would be absurd for me or anyone else to diminish the contributions he has made. Unfortunately, those attributes alone do not make for wisdom. Wisdom includes the application of knowledge in one's life. An extremely smart person may make some terribly unwise decisions, and a biography of Einstein will disclose to you pretty quickly that he was a man with some serious faults.

Yes, yes, we all have serious faults, but we're not asking to be given an exemption in order to be considered prophets of an irrational mysticism. One other item that Einstein stressed frequently was that the God who legislated the laws of nature did not legislate morality and that morality was a purely human creation. Sadly, his life confirmed  that his pseudo-religion did not impact his personal conduct.

Oh, how I wish he had seen the Creator's hand within creation!

Next time: Back to the Golden Ratio!

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Tuesday, July 5th 2016


A False Dilemma

June and I were both really wound up when we came home last night after the family fireworks. Plus I really wanted to get that previous entry posted. What I thought would take about fifteen minutes took 2 hours, thanks to formatting glitches, so that set my brain twirling. (And once again, I'm back to writing my code on Notepad, and then copying and posting it to Bravenet.) We finally settled down around 4:30 am, but it was several more hours before I actually fell asleep for a few hours. June tells me that at one point during my sleep I started to sing out loud, which woke her up for a while. But she didn't remember (or recognize) the song. We've both been pretty exhausted all day today. 

Please be aware of the fact that all of my previous posts in this serious are collected on my site: Phi--Let's Get it Right

Let us move on to a second video in which the eminent theoretical physicist, Michio Kaku, declares his opinions about God.

(1) Let's ignore the gushing interviewer's comments about the "spirituality" of most cosmologists. If I were doing a one-on-one interview with someone of Dr. Kaku's stature, I, too, would treat him as the celebrity that he is. She self-edited her first question as she went along, and I think I have been able to assemble the bare bones of it without the asides and qualifications:

Q: "What is your view on life, and where is it? What are we doing when we teleport life?"

A: "Well, if I had an answer to life, I would have an inside track up there." He's pointing straight up. 

Response: Kaku's answer has several dimensions (though not ten or eleven). For one thing, this answer may lead us to infer that Kaku holds to a view of God as transcendent--beyond the world and unknowable. In what follows, however, it becomes clear that Kaku's conception of God is of one who is entirely immanent, who is one with the world and its structure. Another observation I must make is that he seems to have no room for the common belief among those who believe in a transcendent God, that he has the ability to reveal himself and that he has done so. As a Christian, I can't say that I have an "inside track" to God, whatever that could mean. But I know his message and his will because he has disclosed himself to us, not in all of his infinity, of course, but in a way that becomes intelligible to us. Furthermore, what I know about him is available to anyone. When I read the Bible, I do get information about the meaning of life, not exhaustively, but enough that I'm not in the total dark about it. Obviously, non-Christians don't see it that way, and there is no need to make a big point out of that. My main objective is to help you understand what I think Kaku is trying to say, and doing so involves exposing the inadequacies of his theological concepts. If some things that he says don't make sense to you, don't blame yourself. Right here he seems to be referring to an all-transcendent, but essentially unknowable, being. But he doesn't stick to that script for very long. 

(2) Prof. Kaku continues to expound his views of God. He refers to Einstein and his answer to the question of whether he believes in God. As a physicist one must, of course, be scientific in dealing with that question, and he clearly takes great pleasure in his elitist standing in such matters. The perception he seems to want to create is that he is more rational and informed than your average poorly educated believer in God. Consequently, he avers, we need to have a clear definition of what we mean by God. One cannot help but see the subtext that people who are not scientists do not live up to his supposed strictly rational analysis.  There are two ways of thinking of God, he claims. (I'm treating this part of the exchange in three sections: the limit of two options, the God whom he dismisses, and the God whom he believes in.)

Response: In informal logic, we encounter a fallacy known as a false dilemma. One commits this fallacy when one presents an issue as though it had only two options, and one of them is so unacceptable that one must embrace the other one.

Your only choices are A and B. A is wonderful, while B is totally unacceptable.

"Do you want to be a benevolent pacifist and abandon all physical violence, or do you
choose to be the kind of monster who thinks that we should nuke
all the countries that we don't like and not worry about the consequences?

Hopefully, rational people understand that there are many more options between absolute pacifism and extreme war-monger-ism. That example is obviously not our topic, but simply serves as an illustration of the kind of polemic that Prof. Kaku engages in. To be sure, there may be times in our lives when one may have only two options to choose from, and one of them may be so bad that it really doesn't deserve consideration. Then there is no fallacy. The actual fallacy occurs when one imposes such a scheme on an issue while either ignoring or being ignorant of other options. My paraphrase:

There are two ways of looking at God: One comes with the clout of being held by scientific geniuses,
while the other one is childish and can be dismissed with a gentle chuckle.

If Prof. Kaku only knew what he doesn't know! There are many more ways of understanding God than a laughable cartoon version and his allegedly scientific one. What grates as much as anything is the air of superiority with which Kaku spreads his peacock feathers of ignorance to the world. He, after all, is a scientist, so he is in a far better position to understand the idea of God than the huddled masses. This is pure arrogance, particularly in light of the fact that, when we're all done we'll see that he doesn't even have any coherent understanding of God. Just to mention a few options in the conceptualization of God of which doesn't seem to be aware: personal religious theism, impersonal philosophical theism, trinitarian theism, unitarian theism, personal pantheism, impersonal pantheism, deism, panentheism of various forms (e.g., Hegel's transcendental panentheism or Whitehead's process panentheism), polytheism, henotheism, and others, not to mention combinations of some of them. I am tempted to say that before one proceeds to declare that there are only two options, one ought to have acquainted oneself with all of the options. But I cannot make that statement because undoubtedly there are many more ways of understanding God that I'm not aware of--an observation that underscores my point. But there are many models well accessible to study and learn about, and to present a picture of only two choices, one infantile and the other unintelligible, clearly demonstrates a lack of learning. 

Now, one can come up with a good rebuttal to what I just presented. One could say that, regardless of the model of God one proposes, each of them clearly falls into one of two fundamental categories. But that's not what Kaku is doing. He's not giving us two classes of gods into which we can divide all the available models. He is having us choose between two specific models of God, the "God of Intervention" and the "God of Order." And that's why I must say that he is trying to entice us with a false dilemma.

(3)The first option for understanding God is as a God of intervention who answers all of our prayers, who parts the waters, who kills Philistines on our behalf as per our requests. Einstein, Prof. Kaku tells us, had a hard time believing in that kind of God.

God and toys

Response: So do I. What Einstein and Kaku are dismissing is the "Santa Claus" version of God that maturing Christians should learn to grow out of. It's not the God of the Bible. Yes, the God of the Bible is a personal God who can and does act within the world he has created. But he is not a god out of, say, Greek or Hindu mythology, who changes his mind frequently in order to accommodate our demands so as to continue to be liked by us.

(4) Kaku endorses the idea of a God of order, harmony, beauty, simplicity, and elegance. A look at the physical universe bedazzles us with those qualities. God would not have needed to create the universe in that way. The God whom both Einstein and Kaku are promoting is the "God of Spinoza," a God of order and regularity.

Response: I cannot say whether either of these two men has ever actually studied Baruch "Benedict" Spinoza (163-1677) or, if I may be just a little crass, whether they understood what they were reading. To say that "Einstein read Spinoza's Ethics" does not tell us much. I shall skip the virtually mandated biography of Spinoza and give you just a bit of insight about the core of his system. He set up his philosophy in a form that resembled Euclid's Elements, drawing up axioms, theorems, and corollaries, a procedure that obviously holds a certain attraction for these scientists. Conceptually speaking, he began with René Descartes' understanding of substances, namely that substances do not have properties in themselves, but that the properties are added to an underlying substrate, which is the substance. I like to call this the "pin cushion" theory of substances. The properties are the pins that are inserted into the cushion. They can be removed or exchanged, but the underlying cushion, i.e. the true substance remains the same. Spinoza took off from the notion that substances per se have no intrinsic properties. So, we can consider one substance and declare that it is a substance, and that we know so by a direct rational intuition rather than by the properties it displays. Then we can look at a second substance and again recognize the fact that it has no intrinsic properties.

But wait! How do we know that we're not looking at the same substance for a second time? If a substance has no discernible properties that distinguish it from other substances, it is not possible for us to recognize more than one substance. There is only one substance. So then, all diversity of substances, actions, change, motion, etc. are not truly real, but are mere modalities in which the true Substance manifests itself. It's not necessary here to go through a lengthy argument demonstrating the unresolvable contradiction that runs between identifying infinite substance with finite things. The point I'm making is much simpler than that. In Spinoza's philosophy, the details of the universe, i.e. the beauty, harmony, elegance, and order of the world are swallowed up by simplicity. In fact, this simplicity is an all-absorbing one-ness. In short, as Kaku presents his picture of God, it is not the God of Spinoza. Isabelle T. mentioned on Facebook that underlying his thought may be the Buddhist concept of sunyata, the all-pervasive emptiness of all.That's a good analysis. But still, that's not the God of Einstein or Spinoza, and in Buddhism there is no such thing as a Creator who made a universe that displays beauty, order, simplicity, elegance, and harmony. Either way, Kaku's God is severely ill-defined and self-inconsistent.

(5) The attribute of simplicity is of great importance to Kaku. All the equations of physics can be written down on one sheet of paper. Better yet, the fundamental formula of string theory (Kaku's contribution, he is careful to add) is extremely short. The shorter, the more elegant. The more elegant, the more beautiful. Kaku's formula captures the essence of God, it would appear.

Response: Kaku may claim that his belief in this strange God is based on observing the order, beauty, elegance, etc. of the universe. But his conceptualization of God has yet to attain order, let alone beauty, or elegance. One feels as though there is something else that Kaku knows (and maybe Einstein did as well) that is hidden from the rest of us. One wishes for a genuine definition or description; he started out by gloating over the assumption that as a scientist he is duty-bound to start out with a clear definition of God [my paraphrase]. But such a definition is not forthcoming. I, for one would be delighted if we could start with comprehensibility, not of God, but of what Kaku is saying. Labeling his deity as the "God of Spinoza" doesn't do it because Spinoza's God is clearly not Kaku's God. Displaying his formula to the world doesn't convey anything to most of us, except that he loves simplicity, elegance, etc. But that's where we started, so that equation doesn't tell us any more about the nature and function of this God-thing.  We're pretty clear on what Kaku does not believe, namely a caricature of God as a cosmic Santa Claus. The only thing we can say definitively is that we still don't know what he really means by God and that this God supposedly differs drastically from all other conceptions of God. There seems to be an inside track, after all, though the path does not appear open to most of us. 

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Monday, July 4th 2016


Prophet and Priest: Michio Kaku's Role in the Cosmos


Happy Fourth of July to all relatives, friends, acquaintances, fans, colleagues and work associates of the past, present, and future. Whether you’re celebrating or not, the day is on your calendar, and I hope it is a good one for you. The question here in Smalltown USA and environs is whether the weather will clear up for fireworks tonight. Just like yesterday, it’s cold (for July), dark, cloudy, and drizzly outside. The hour-by-hour forecast has it clearing up by around 9 pm tonight, which would be the time that it started to get dark enough for our incendiary displays. We’ll see.

Late addition: The weatherman hit the proverbial nail on the head. June and I just got back from a wonderful evening with Nick & Meghan and Seth & Amber at the former’s house in the country. N&M had everything ready for hoboes for supper and strawberry shortcake for dessert, and we had some awesome fireworks. Obviously, Sunako was there, since she lives there, and S&A brought Misha, the aging whippet, and Evey, the baby Great Dane.

I have added all of the previous posts relating to this series to the Phi website. Let’s get ourselves caught up on where we are in this long and drawn-out series, beset by many a tangent. I’ve stated that a Christian, scientist or not, should see God’s hand in the beauty and order in the world that God has created, and that such beauty should extend to the world of numbers with a beauty all its own. I expressed my disappointment with those Christians who are too eager to maintain standing in the so-called community of scientists to acknowledge that even the question of whether there is scientific evidence of intelligent design in the cosmos is a legitimate one. The subtopic at this point is those scientific luminaries who appear to go in the opposite direction and seem to be finding God in the mathematical structure of the universe itself.

Michio Kaku (1947– ) is known as a successful physicist who invented string theory and some of its subsequent developments, such as super string theory. He is also an adept populariser of physics and mathematics, and, consequently, has become well known; he quite obviously enjoys talking to the world’s population at large about science and himself, not reluctant to express opinions outside of his area of expertise.  As promised in the last entry, here is his YouTube video, entitled “Is God a Mathematician?”

I’m going to follow Professor Kaku fairly closely in this round. Don’t worry; I’m not going to challenge him on the validity of string theory. There are three major figures mentioned in this short talk: Newton, Einstein and Kaku. In the actual video he does not address the question whether God is a mathematician, but of what good math actually is.

Note: I have tried several ways of formatting this discussion, and I found it to work best for me if I include my responses and critiques right along with the summaries.

(1) Isaac Newton. Kaku uses Isaac Newton as an example of a time when innovations in mathematics and physics came up together.

(a) Kaku applauds Newton for asking the question: “If an apple falls, does the moon also fall?” He classifies this question as among the greatest that any member of homo sapiens has ever asked in the 6 million years since we departed from the apes.

Response: Along with all other members of our species, Kaku is entitled to his opinion, as long as we are clear that it is nothing more than his opinion. Personally, I have a problem with his evolutionary assumption, and even if I didn’t, I’m pretty sure that I would not include Newton’s concern among the top questions ever asked by humankind. But that’s a matter of perspective.

(b) Kaku’s narrative goes on: Newton’s answer to his own question was that, yes, the moon also does fall (viz. the moon is attracted to the earth by the pull of gravity). However, Newton lacked the mathematical tools to explain this idea. So, he invented calculus in order to make it possible.

Response: Is that how it happened? I don’t think so. Newton described gravity with a famous equation, oftentimes referred to as the Inverse Square Law (ISL), where G is called the gravitational constant.

Inverse Square Law

The force of gravity between two objects is obtained by multiplying the mass of the two objects, dividing that number by the square of the distance between them, and multiplying the result by the gravitational constant.

I do not see any calculus here. This formula was included in Newton’s Principia Mathematica, which came out in 1687, and which did not contain any calculus. His first publication on calculus did not appear until 1693.

(c) Kaku then summarizes Newton’s conclusion: “The moon falls because of the Inverse Square Law. So does the apple.”

Response: I replayed that segment of the video several times in order to make sure I heard correctly, and the closed captions also bore me out.

“The moon falls because of the ISL.”

Scientists don’t usually say such things. The laws of science are descriptions. Often they are statistical generalizations, and frequently they appear so ironclad that there appears to be no way around them. But there is one thing that they never are, namely causal agents. The ISL describes the earth/apple and earth/moon attractions mathematically. It tells us how to calculate the gravitational force, but it does not explain the presence of a gravitational force. Both the moon and apple would still be falling, regardless of whether we had the ISL or not.

Is this just verbal nit-picking? Should I just dismiss Kaku’s phrasing as a momentary and careless slip of the tongue, something that happens to all of us from time to time? Usually I would say so, but in light of the status that Kaku eventually gives to the laws of nature a little later on, I can’t rule out that he looks at the ISL—among others laws of nature—as more than just a description. Also, as I said above, scientists are usually far too careful to use such careless phrasing, especially, I would think, in a video published to the entire world (at least the part covered by YouTube), so I can’t help but see some intentionality in Kaku’s remark.

(2) Albert Einstein. Albert Einstein is Kaku’s example of a case where a physicist was able to draw on a mathematical method that was already available prior to making an innovation in physics.

(a) Kaku tells us that Einstein came up with a new way of looking at gravity. The key to Einstein’s general theory of relativity was that he envisioned space as curved, and gravity is due to that curvature, not a force of attraction. The fact that he remains in a chair, Kaku assures us, is not because gravity pulls him, but because curved space pushes him down.

Response: I will just say that I have never heard it put that way. More commonly the explanation is that mass and energy create curvatures in space, so that my sitting in a chair causes a little protuberance in space into which I have sunk myself. Kaku seems to substitute one mysterious force for another, i.e. pushing instead of pulling, and I’m just a little confused on why he would put it that way. Still, I’m not going to argue with Kaku over matters that are purely physics. Maybe someone else can comment (nicely) on that description.

(b) According to Kaku, Einstein was also in need of a mathematical means to describe his innovation in physics. But he did not have to come up with anything new; he could simply avail himself of “differential calculus,” which already was in place. After all, he reminds us, studying calculus usually begins with the motion of falling objects, i.e. gravity.

Response: The last part of that little summary is true. Among other things, differential calculus analyzes the rate of change in the velocity of an object in motion at a particular time, e.g., the acceleration of a moving object, or even the acceleration of the acceleration. This is one part of calculus that Isaac Newton and Leibniz had already invented, and if Einstein would first have had to learn differential calculus at that point in time, he would not have been much of a physicist. (The other part of calculus is called integral calculus.)

Einstein did not need differential calculus to make his theory of space rigorous.  Professor Kaku's statement implies that we begin the study of calculus with differentiation (aka “finding derivatives”), and that would most likely be true for pretty much everyone, I would think. But Einstein’s math goes far, far beyond “differential calculus.” Yes, there is calculus involved, but so are addition, subtraction, multiplication, and division. Surely all of these mathematical procedures become trivial characterizations in connection with Einstein’s work. If we were to try to cover Einstein’s math with short expressions, we might want to say that it was “tensor calculus” for the special theory of relativity and “Riemann field geometry” for the general theory. I don’t understand why Kaku trivialized Einstein’s work with the expression he used. 

Could this just be a sloppy choice of words, and am I maybe nit-picking again? Once again, I doubt it. Who, given just a small amount of understanding of what is involved, would say that Einstein resorted to “differential calculus” for his general theory? This is not just careless, but misleading. One of the world’s foremost scientists must know better than to make such a mistake. My hunches: Kaku is trying to put us at ease by using a term that is not going to frighten us away. After all, we’ve all studied calculus, right?  Could there be a further reasons to minimize Einstein and, thereby, establish a contrast to himself?

To get a flavor of relativity math—or maybe even to learn it—may I suggest Peter Collier, A Most Incomprehensible Thing: Notes toward a (very) gentle introduction to the mathematics of relativity (Incomprehensible Books, 2012). I must say, though, that Collier and I might just disagree on the meaning of “very gentle.” I suspect that there actually is no gentle way of learning this material.

(3) Michio Kaku. As you can see in the video, Professor Kaku is not exactly humble about his accomplishments, and I, for one, want to congratulate him on his life’s work. Even if string theory, super-string theory, M-theory, etc. are not going to hold up in the long run (and I am definitely in no position to judge that), his ideas have been significant in keeping the search for ultimate physical reality moving ahead.  

(a) The holy grail, so to speak, of theoretical physics is a unified field theory, an explanation for everything. Einstein spent the majority of his life in a futile quest for it. Kaku believes that he has found it with his super-string theory. Such a theory should be able to be captured in a short equation, not more than an inch or so long. He has accomplished it, and here it is:

Kaku's Equation

Response: I have no idea what this equation says, and am in no way qualified to evaluate it. It represents the vibrations of really tiny strings in the 10th and 11th dimensions, and I understand that the high degree of difficulty in understanding it is not just limited to us amateurs who like to read books about math.

(b) The math for Kaku’s string theory is analogous to that of topology, much of which was first formulated towards the end of the nineteenth century. It is an extension of algebraic geometry, reaching into higher and higher dimensions. As Kaku tells it, those who contributed to the development of topology took great delight in the fact that this was math at its best, namely, math that would never have any practical applications since it includes multiple dimensions beyond our usual three (or four if you count time).

Response: The more dimensions you give yourself to work with, the more problems you can solve. If you only had two spatial dimensions, you could have a flat depiction of the eye of a needle and some thread, but you could never thread the needle. To do that, it takes three dimensions. Topology uses that principle and develops some truly bizarre models of algebraic objects in greater spatial dimensions. I’ll leave it to Professor Kaku to judge to what extent the mathematicians delighted in the so-called lack of applicability.

(c) Along came Michio Kaku, who used 10 and 11 dimensions in order to come up with his super-string theory. Mathematicians were surprised, if not shocked and appalled.

Response: As well they should be. You’ve got to be in awe of Kaku’s work.

(d) The vibration of these strings constitute the “music of the spheres,” and the very voice of God. He, Michio Kaku, has had a glimpse of the core of the universe, where he finds God communicating to him through the equation that he, Professor Kaku, has constructed.

Response: It would appear that Michio Kaku has endowed himself with the offices of prophet and priest in the religion of super-string physics. He has departed from physics and is trying to do theology, and the result is dubious. 

We need to leave the more specific content of this religion for next time.



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Thursday, June 30th 2016


Which Numbers did God Create?

  • STATE OF EXISTENCE: pretty good

The “God” of some Scientists

In rereading some of my blog entries over the last 11 or so years (the anniversary will be July 3) , it seems to me that about a quarter of them begin with some kind apology for having skipped so many days, dragging out a series, or whatever. I have a feeling that, just by itself, that gets tiresome to read; I know it is for me. So, I’ll try to restrain myself. In addition to posting this entry, I do need to connect it to all of the previous ones of this series that began with the contemplation of phi (ϕ).

Yesterday, Wednesday, was a beautiful day with temperatures in the 70’s, sunshine, and a slight breeze. June and I both worked in the yard, June more than me. I didn’t get much done because I spent over an hour in a losing battle trying to get “weed-eater” string fit properly onto a spool. It’s been a source of frustration for June, with her physical limitations due to her fibro and the weather oscillating between lots of heat and thunderstorms, that she hasn’t been able to work with her flowers as much as she wants to (and as the flowers want her to), I’m really glad that she had quite a bit of time for it yesterday, though today she has no physical reserves left. I continued my current practice of trying to go to the pool, and I raised my personal best at the pool to 10 lanes in a row. 

On Monday, I went to Menard’s to buy a new door for the basement to the outside. That will be my major summer time project for the next couple of weeks: Take out the old basement door and replace it with the new one. The door is pre-framed, and so it sounds easy-peasy, and you may think that a couple of weeks to install a door is awfully slow. Well, yes it is. But there are two reasons. 1) The frame has to be fastened to the inside of the door opening, i.e. into a larger frame. I think I will probably have to replace that one, too, but I won’t know for sure until I can take a good look at it with the old frame gone. Then I’ll know what, if any, lumber to buy and mount it. That’ll probably mean a couple more trips to the store, unless I find some suitable leftovers here in the garage. 2) I’m slow. June and I have discovered over the last few years that, if we try to start and finish a project in one day, it’ll probably take longer to get it done due to the ­inevitable crash and exhaustion than if we space out a project over a number of days.

In what follows, I’m going to refer to three videos, and—if at all possible—I’ll try to embed them here next time. They are about the concept of God, as espoused by Albert Einstein and Michio Kaku. There’s undoubtedly little or no need for me to introduce you to Albert Einstein. Michio Kaku is becoming quite well known as well as a theoretical physicist, popularizer of what is trendy in subatomic speculations, and the originator of string theory. He is also outspoken on his beliefs concerning God.

I’m obviously not going to provide a line-by-line commentary on all of that’s being said. My choice of details is simply governed by what peaked my interest.

Let’s start with the video that is entitled “Is God a Mathematician”?

We need to interpret that question in such a way that it makes sense. Obviously, God does not do long division; he already knows the square roots of all prime numbers, and it will be of no news to him when (or if ever) we finally find out whether "all of the significant zeros of the zeta function have real part ½. " That last phrase is the celebrated Riemann hypothesis, and mathematicians have struggled for over a century now, trying to find either a proof or a disproof for it. In the meantime, God already knows the outcome; he just has chosen not to reveal it. (If he always immediately solved our puzzles for us, life would get awfully boring.) The real meaning of the question, as I would read it, is whether God intentionally created the universe with the mathematical orderliness and symmetry that mathematicians discover in it, including the interesting features of the numbers themselves.

Whoops! I just put myself into the position of having to add yet another level of stacking. But really, my patient readers, one cannot write about God’s creation of numbers without mentioning the view held Leopold Kronecker (1823-1891). It would be like writing about art museums in Paris and making no mention of the Louvre. I must begin by saying that Kronecker made numerous serious contributions to math and was by no means a crackpot. However, his philosophy on the nature of numbers did pervade his work, and the active way in which he promoted his views made for sour relationships with many of his colleagues. If he is at all known among sideline math fans, such as your bloggist, it is probably for the alleged statement:

"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk."

The dear God made the whole numbers; everything else is the work of humans.

First, about the quote itself. It is frequently translated as “God made the integers,” in which case he could have been referring to the various numbers that classified as ℤ, which includes zero and negative numbers. I believe, though, that he was really restricting himself to ℕ, the positive integers. Regardless, there is not much sense in exegeting the quote since it has come down to us third hand as a report of what he had supposedly said to someone else in a personal conversation, and so we can hardly be sure of the exact wording. Still, the sentiment fits in well with his outlook on math.

Another, probably apocryphal, statement is associated with the occasion when a mathematician named Ferdinand von Lindemann (1852-1939) had published a significant proof with regard to the nature of π and received well-deserved acclaim. Kronecker supposedly asked him the rhetorical/polemical question: "Of what use is your beautiful proof, since π does not exist?"

Whether Kronecker actually made either of these utterances has been questioned. Nonetheless, they do represent his basic outlook on the reality of numbers. If you don’t recall the meaning of the 𝕙𝕠𝕝𝕝𝕠𝕨 letters, here’s a quick summary. If you totally missed that earlier episode, you can go back and read the descriptions in greater detail. 

Natural numbers: The positive integers beginning with 1.

Integers: adds zero and negative numbers.

Rational numbers: adds fractions and closed decimals.

Real numbers: adds “irrational numbers,” including ϕ, π, and √2.

Complex numbers: adds √–1, called i.


In Kronecker’s view, only the numbers that God created, ℕ--identified with or maybe also ℤ--have any reality. Thus, he excluded fractions (ℚ), the so-called “real numbers” (ℝ), and, quite obviously the complex numbers (ℂ), which are still designated with i, the "imaginary" square root of  -1.   π lives in ℝ, of course, along with his fellow celebrities, ϕ, e, √2, etc.  Kronecker was convinced that ultimately the only real objects in math were these “whole numbers,” and theorems and proofs that went beyond that perimeter were useless. Thus he very strongly objected to many of the innovations put forward by some of the most recognized mathematicians of his day. For example, in his position as editor of an important journal, he refused to publish several papers by Georg Cantor, the father of set theory, who had concluded that the infinite set of real numbers (ℝ) was larger than the infinite set of natural numbers (ℕ).

Cantor's idea does sound just a little crazy when you first hear it, doesn’t it? And one can understand that Kronecker might have believed that Cantor had lost his common sense. Nevertheless, I would still have to go with Cantor, if my vote were to mean anything. So-called common sense is not necessarily in a position to judge uncommon objects or situations. Regardless, Kronecker had little use for his sets.  In our contemporary world of math, Kronecker has received an echo in the work of Doron Zeilberger of Rutgers University.

Approaching this point of view from the perspective of a philosopher, it seems to me that you can’t have one or the other. If God created the natural numbers ℕ, then he also created the rest of the number systems that stem from ℕ.


 4 + 5 = 9





Within ℕ there’s no limit as to how many numbers you can add.

 4 – 5 = –1              

–1  [integers]




But when you start subtracting you may need to leave ℕ and resort to ℤ. (“ℕ is not closed.”)

 4 – 4 = 0     





ℕ may also require 0 as an integer, again needing help from ℤ.


16 ÷ –4 = –4




Within ℤ, more calculations are possible than in ℕ, e.g., using negative numbers

4 ÷ 16 = 1/4


1/4 [fractions]



But ℤ is also not closed for division when a solution requires a fraction. Then we must take recourse to ℚ.

2 × 2 = 4        

4 ÷ 2 = 2

√4 = 2



Everything seems to be going nicely in ℕ, ℤ, and ℚ when we let 4 and a couple of 2s play together. In ℚ we can describe 2 as the square root of 4, thus square roots have a legitimate place in ℚ,



√2 = 1.4142 …,   –1.4142 …  




But what worked so nicely for 4, does not work for 2. Now ℚ needs ℝ, the set of “real” numbers, which includes the irrational ones.




√–2      ⇒



If –2 is a legitimate number from ℤ on, and it is “legal” to take square roots since ℚ, then we need to accommodate square roots of negative numbers. We do so by making use of i and the complex numbers.


I trust that my point is clear and, hopefully, plausible. If you start with ℕ and proceed courageously to unpack what you have and what you can do with it, sooner or later you should wind up with ℂ. [See Robert and Ellen Kaplan, The Art of the Infinite (Oxford, 2003) for an extended rhapsody on this theme]. You can’t pick and choose. If God created the natural counting numbers of ℕ, then the implications and sound theorems that follow from basically logical operations on them are also a part of his creation. If fractions, let alone the real numbers, are merely human inventions that are not real, then the natural numbers that make them up can’t be real either. If 3 and 4 are real, dividing 3 by 4 does not make them unreal, and thus ¾ must may also partake of whatever reality numbers have.

Let me quickly add this. There are three major lines of understanding the reality of numbers:​

1, The “Platonic” view, which holds that numbers have genuine reality. It would seem that Kronecker holds that view for the “whole numbers,” but only for them.  Doron Zeilberger , whom I mentioned above as an echo of Kronecker, stated explicitly:

I am a [P]latonist, and I believe that finite integers, finite sets of integers, and all finite combinatorial structures have an existence of their own, regardless of humans (or computers). I also believe that symbols have an independent existence.

I am not sure if Zeilberger at the time of that presentation in 2001 was aware that Kurt Gödel, who becomes one of his foils, was a Platonist and that his “incompleteness theorem” was intended as a reductio ad absurdum of non-Platonic views. (Please see my article with Michael Anderson, "The Strawman Strikes Back: When Gödel's Theorem is Misused").  Not-so-by-the-way, in this context, the word “Platonic” somehow has become the convenient catch-all term for the idea that numbers have their own existence, but one does not need to subscribe to Plato’s Forms to hold to this view. 

  3. The “formalist view,” according to which numbers do not have independent existence, but are discovered and expanded upon by human thinkers. There is a formal, logical structure that mathematical objects must follow, but one should not think of those objects as independent of human discovery. Insofar as it makes sense to talk about their reality, they don’t have reality before someone has found them. At that point, one can find out more of their properties, and their reality does not depend on their or their implications being proven. E.g., the Riemann’s Zeta function is real, regardless of whether Riemann’s hypothesis concerning it may be found true or not.
  4. The “intuitionist” school, which sees numbers and mathematical objects purely as the product of human effort and allows them only to have pseudo-reality insofar as they can be proven to be proven. It even attempts to dismiss the law of excluded middle. Kronecker’s assessment of the numbers beyond ℤ could be seen as foreshadowing intuitionism, but his understanding of the “whole numbers” as created by God and, therefore, real rules out labeling him as an intuitionist.

Looks like discussing the videos will have to wait for the next entry. 

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Saturday, June 25th 2016


The marvels go on and on.


No. 14 Tractor

Eight lanes back and forth at the pool Thursday. Only four on Friday, but I was pretty exhausted before I ever got into the water. On Thursday I had taken the red No. 14 Tony Stewart and A.J. Foyt Sears Craftsman Silberpfeil tractor out to cut the grass and wound up with a flat rear tire. Friday, I tried to fix it, but each stage—removing the wheel, jacking up the tractor, pumping air into the tire, etc., not necessarily in that order—came with its own sub-problems to the point where I finally decided that on Monday I’ll take the wheel to Troy Greer’s and ask them to, please, fix it. I'm hoping that they won't just inform me that the rim is bent. Today (Saturday) the pool was too crowded to do a lot of uninterrupted laps.

In line with my normal practice, I’m not making a whole lot of progress on this series as a whole because I keep running across information that I find to be either necessary, helpful, or just plain cool. So, this post also does not go nearly as far as I wanted it to, but at least I have started to raise the theological questions.

The ubiquity of the Fibonacci numbers and their steady companion, ϕ, has become the occasion for much thought and writing about the mathematical regularity and apparent intentionality of the universe. In fact, the closer we look, the more astounding things become.

Again, because in this series I’m trying to stress the beauty of the numbers themselves, let me give you another example that strikes me as really astounding. We all know about ϕ’s big brother pi (π) mostly for his part in the formula for the area of a circle:

A =πr2

But π also shows up in places where we would not expect to see him. Skip down if you like.

Jakob BernoulliLet us look at an interesting problem, known as the "Basel Problem" (Derbyshire 63) because it was in Basel, Switzerland, that it was posed. The problem, as originally stated by Jakob Bernoulli (1655-1705), went as follows.

Here is a convergent series, which means that it eventually approaches the limit of a specific number rather than diverging to infinity:

Basel Problem

As usual, we can come closer and closer to the number, but will never quite reach it. A precise enough approximation of the number is

1.64493406693 …

Are you getting tired of those three little dots ( ), which indicate that you’ll never truly get to the end of calculating the formula’s value? Apparently Bernoulli was. In a publication on various related topics, he included this series and asked for input from other mathematicians about this question. Would anyone be able to state it in a form that did not lose itself in the forever of infinity, but in a “closed” formula that could be substituted for it?

Leonhard Euler (1707-1783), whose original home also happened to have been Basel, came up with a solution:

Basel Soution

I don't think I'd be going out on a limb when I say that most of us are rather surprised to see pi popping up there.

Very quickly let's switch to a different topic, this time probability theory. Suppose you are asked to pick two numbers at random from a set of integers. What is the probability that they will not have any common divisors? E.g., 6 and 8 would share a divisor (2), but 4 and 9 would not. The probability of picking two numbers without a common divisor is:


The actual number is ~0.6079 or ~60.79%, but that’s obviously not what strikes us. This probability calculation turns out to be the reciprocal of Euler's solution to the Basel problem! The University of Illinois has set up an interactive webpage that illustrates this formula with some input from you.

The marvels go on and on.

Ben Stein Expelled

Christians, Jews, and others who believe in a personal God look at these wonders of the universe and see in them the magnificent hand of its Creator. How could they not? I’m not offering that sentiment as a piece of apologetic on behalf of the created-ness of the world or the existence of God, but as a report on what is (or should be) an obvious part of what theists believe. Or at least should. Ben Stein, whom I was privileged to count among my blog readers for a while, made the film “Expelled: No Intelligence Allowed” to publicize the fact that the scientific world on the whole (including atheists and purported Christians) has slammed the door shut on even asking the question of whether the cosmos manifests intelligent design. The entire movie is available on YouTube There are exceptions, such as William Dembski, who has not allowed his views to be dictated by the strong arm of a supposed scientific consensus, at the expense of his ongoing career. As Mr. Stein's movie displays, even atheists who allow the question to be raised may find their jobs terminated.

There is another group of scientists, however, including some of the brightest people of the last 100 years, who have not been afraid to express belief in God on the basis of the mathematical coherence of the universe, although they do not subscribe to a traditional understanding of God. The concept of God, as they have been espousing it, is not that of a personal Creator, and it doesn’t even really fit into any other traditional categories. It is not pantheism in which God and the universe are considered to be identical; let alone deism according to which God created the universe and then abandoned it to run according to the program he implanted in it. In some ways it’s reminiscent of the God of process theology who is persuading the world to follow his directives, from the cohesion of molecules on up. Still, there is far too much metaphysics associated with process theism to suit these celebrity believers. It appears to be the very regularity of the universe, the math behind the phenomena, so to speak, that constitutes God for them. Next time I’ll continue with these observations on Albert Einstein and Michio Kaku, and then return to phi.

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Wednesday, June 22nd 2016


Excursus: A Corn Stack


Woops! I meant to go on talking about ϕ, but got caught up in the agriculture of Indiana by way of a remark on the weather. That's why I like to describe my blog as "mercurial." Long-time readers will know that there have been other times when I was surprised at what I wrote about rather than what I intended to write about.

For the last few days I had to go without a certain medication due to some snafus beyond my control, and I’m very glad to have it back. It was not good, and that's all I can--or maybe want to--say about that.

A couple of cloudy, rainy days, but the temperatures still are definitely in the summer range. If you should drive, bike, or walk to the outskirts of Smalltown, USA, you will find yourself facing corn fields, and you might just hear the corn growing. The standard expectation is that the corn should be “knee high by the Fourth of July.” Many of the fields are knee high already, and will be close to “person high” by Independence Day, I should think. The fields definitely will dominate the view for the next few months.

It looks to me as though the timing worked out quite well for planting corn (aka maize, btw) this year. The big question is always when the fields will be dry enough for tractors and machinery not to sink in the mud. Around March or April, after the snow has melted and we’ve gotten our usual overdose of rain, many of the fields look like lakes. The farmer has to wait for the soil to dry out, and appearances can be deceiving in that regard. An early unusually warm spell can make the fields look dry, but that part may only be a crust with a lot of mud underneath. This is not helpful, for one thing because the farmer’s tractor might get stuck, and for another because such a crust also stands in the way of further evaporation of the water underneath the crust.

Around here, for corn to have enough time to reach full maturity, it needs to be “in” by “Decoration Day,” as I once overheard a farmer say.

  • Excursion: That’s “Memorial Day” for most of us. Among the rural folks here whose social life centers on the “Post” or the “Legion,” it is indeed still “Decoration Day.” Ruby, a sprightly 94-year-old (at the time) lady at one of the churches where I used to preach, would go to the local cemetery first thing in the morning of Memorial Day and make sure there was an American flag planted on the grave of each former service man or woman. Furthermore, just in case she had accidentally omitted someone, she would place two crossed flags at the edge of the cemetery so that they would be included as well. – I find these kinds of observances intriguing, particularly in light of their analogues in other cultures.

Anyway, I’m pretty sure that everyone got their corn planted before Decoration Day. One must realize, of course, that there are different kinds of corn with different requirements. Overwhelmingly, the corn grown here is “horse corn” earmarked to become animal feed or silage. It lacks the appeal of the traditional “sweet corn” which is a different species altogether. Horse corn stays out in the field until it's all dried out. You wouldn't want to eat an ear of sweet corn in that condition.

Van Buren, Indiana Speaking of corn, let’s say that you want to watch a movie, and you think a bowl of popcorn would really go well with it. Great! Just be sure you have real popping corn. If you just take a handful of any old type of corn, put it in your microwave or other appliance, and stand by, ready with salt and butter, you may find that it doesn’t work that way. The kernels are not going to spring to life, and you can put the salt and butter back in the cupboard. Popcorn is a different kind of corn from the others mentioned so far. It is distinguished by the fact that, when the kernels have matured, they surround a tiny little drop of water. The heat brings that little bit of moisture to the boiling point, which increases the internal pressure, and the kernel, along with most of its colleagues in the bag, explodes into the white treat we’re all familiar with. So, popcorn is also among the agricultural items that farmers plant around here. About half an hour’s drive from here there is the little town of Van Buren, Indiana, population 864 as of 2010 (Compare to Smalltown, USA,’s population of 5,145. Both are shrinking.) It declares itself to be the Popcorn Capital of the World and celebrates an annual popcorn festival.

But, as another saying goes, “There’s more than corn in Indiana.” Beans (i.e. soy beans, though nobody call them that around here), go in about a week or two later than corn. There are also a few occasional wheat fields; it’s “winter wheat,” which means that it was planted way back in last October or so. You don’t see anything of it all winter, but these fields become some of the earliest and brightest displays of green when spring arrives. By now the wheat has attained its golden color, and it should be ready for harvesting pretty soon. The farmers are also getting ready for the first round of mowing hay.

As long as I’m pursuing this subject, you will also see occasional tomato fields in Indiana and surrounding states. According to the Wikipedia's statistics, if you’re in Indiana and you see a tomato field, there’s a 95% chance that the harvest will be processed into one of the many kinds of canned tomato products at the Red Gold plant, just a short three miles west of here in tiny Orestes, Indiana (pop. 414, not shrinking). Red Gold also has contracts with 80% of the tomato growers in the rest of the Midwest.

  • Excursus: Red Gold still likes to present itself as a mom & pop organization, and maybe it is. The daughter or granddaughter of the owners, I don’t remember which, was in Seth’s 4-H chapter, and went along on at least one fossil and mineral hunting trip with us. By “us” I mean a few 4-H’ers, the leader (Archie H.) who knew all of the best quarries in the state (and all of their managers, and all of their children, and all of their names), and yours truly driving. Still, when I tell you that in 1993 “mom and pop” acquired Stokely-Van Camps’ tomato division, you may see that they’re a major player.

I might mention that, despite a strong heritage of sorghum (aka milo) production in Indiana, it is no longer a major crop here. The Hoosier State had been the nation’s leader in this regard at one time, but it became unprofitable. Nevertheless, there still are a number of sorghum fields, even if the US department of agriculture doesn’t acknowledge their existence. In its early stages, corn and sorghum look a lot alike; the distinction being that sorghum plants are a little shorter than corn and its leaves are spikier. Later on, though, sorghum clearly stands out with much larger and wider tassels than you’ll see on corn.


Pictures courtesy of Sorghum: The Smart Choice--All About Sorghum


Some efforts are being made to reintroduce sorghum as a profitable crop on both public and private levels.

Next time, if I can keep my mind from running off into different directions, the perennial question: Is God a Mathematician?

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Monday, June 20th 2016


Bull's Eye (almost)


Most of this was written originally on Saturday evening, but needed some refinement and a lot of uploading of formulas, plus adding enough cute stuff to give it appeal to a general public. Hope you like the Staypuft Marshmallow man gif I concocted.

The weather continuous to be quite nice, and definitely appropriate for summer. We are heading into the high nineties again today. A couple of weeks ago I procured my annual season pass to the Beulah Park Pool, here in Smalltown, USA, and I’m making good use of it, every day if I at all possible. It’s funny (so to speak): on the first day I went out there this year, I felt like I had lost an enormous amount of strength in my arms and legs, and I wasn’t surprised. I couldn’t even manage to swim one entire lane without taking a break. However, by this past Friday, after only about a week, I managed four, and as of yesterday and today, I’m up to six already. That’s definitely also been a surprise, and obviously a good one, for which I’m grateful. On most days, after swimming one set, I usually take a break and visit the water slide once or twice to satisfy my need for speed. Then I do another set, casually and without trying to go to any limit.

Well, let’s get back to the numbers.

StayPuff Marshmallow Man

There are several mathematical surprises connected to the Fibonacci series in its own right, apart from its involvement with ϕ. For example, even though ϕ is definitely an irrational number, it does manifest a certain amount of regularity, such as certain digits recurring at precise intervals. As intriguing as those things are, I need to refer you to Livio for details. If I don’t observe some limits, this series would still be going a year from now, and—who knows—even the spammers might be scared away by then. So I’m going to continue to focus on ϕ as much as I can, and talk some more specifically about the relationship between the Fibonacci series and ϕ.

I’m going to introduce a new feature in this entry. The post would become meaningless if you or I just skipped all of the details, but I think I’ll keep more readers interested if I show them what they can skip without missing out on all of the content. So, I will introduce the problem for the day, but then give you the option of clicking on a link that will take you right to the conclusion without going through all of the steps to take you there. You will see it shortly below as a green rectangle.

There is a formula known as “Binet’s Equation,” named after Jacques Phillipe Binet (1786-1856), who was actually not the first person to discover it, but apparently created more interest in it than other famous mathematicians before him. Now, I’m going to trot out Binet’s equation, and you may just be horrified when you first see it. It looks like a huge monster, perhaps reminiscent of one such as the StayPuft Marshmallow Man ® from “Ghost Busters.” We’re going to describe Binet’s equation, and pare it down until it becomes one simple calculation. Then we’ll use it to find out what the 12th Fibonacci number is.

[Added on Monday: When I was working on this on Saturday night I had no guarantee that it would actually work. To be more specific, I assumed that the formula was right, but I did not know if I was interpreting and using it correctly. If it didn’t work out, I had no one to ask, at least immediately, what I had done wrong. Thus, my emotional reaction at the end was totally authentic. I couldn’t believe that I had brought of both stages: understanding the formula (a good first step to clarifying it for my readers) and applying it in order to get the right answer. ]

Skip to ConclusionHere is the promised link that allows you to skip over the material that looks intimidating.


Here goes. This is Binet’s Equation:

Binet's Equation

Intimidated? You should be—but only until we’ve taken it apart and seen the simplicity that’s behind the apparent complexity.

Seriously, I’m going to go extremely slowly in clarifying this equation and highlighting all steps. Maybe you will realize that working with such formulas one does have to be careful, but it does not have to be “*cough* laborious *cough*, if I may tease John of LRT one more time (all in good fun, I hope he understands if he should ever read it). One of the nicest compliments I ever received on my writing was with regard to my commentary on 1 and 2 Chronicles, when the editor at B&H told me: “Win, you have made Chronicles come alive.” I don’t know if it’s a comparable challenge, but I shall try to make Binet’s formula come alive as well.

Binet's Equation

“Fn” stands for which Fibonacci number in the chain you wish to calculate, i.e. the “nth” number. This is totally of your own choosing. For now, we’ll try to use the formula to figure out the 12th Fibonacci number, a number we can easily verify by just adding up the Fibonaccis. Maybe later, then, we’ll try our hands at calculating a much larger number, perhaps F95, but first we’ll have to see if we make it through this trial.

Binet's Equation

The expression “for n ϵ ℤ” reminds us that n has to be an integers. The Fibonacci numbers per se are not irrational, and are inhabitants of ℤ. The irrationality of phi and squareroot of 5 comes into play insofar our answer may need rounding up or down to keep the solution within the ℤ circle as well. From here on out, we're going to dispense with that little tail.

Binet's Equation

Wow! What are we going to do with that messy clutter highlighted in red? Actually, you may know already what we’re looking at here once I let it stand by itself. It’s the positive root of our good old equation for ϕ:



Binet's Equation

So, we can substitute ϕ in for that formula, and life is beginning to look a lot simpler already. We now know that, when the time comes for actual calculations, we can insert 1.61803 as the value of ϕ.

Binet's Equation

A similar thing is true for the next packet that I’ve highlighted in red.

Binet's Equation

This is the negative solution for phi’s equation, and you may or may not recall that it’s value is the same as the negative conjugate of ϕ, but terminology aside, you may remember that it amounts to -0.61803. Rather than writing “-1/ϕ”, which would mean that we're building up fractions again, I’ll use an expression that’s equivalent, but a little easier on the eyes, ϕ-1.

Binet's Equation

Let’s not forget that, when we eventually subtract that negative number, we can substitute a “plus” sign for the two negatives—if we ever have to do so. The last thing I can point out before we actually install some numbers is that both ϕ and its negative reciprocal will be raised to the power of n.

Binet's Equation

Now we’re looking at the formula that we want to instantiate.

Binet's Equation

I know; I know. You’re probably still not buying into my claim towards simplicity. Okay, I have one more trick up my sleeve to try to persuade you, and it's a good one. There’s a corner we may cut without doing ourselves any serious harm (just don’t run with the scissors, please). That whole –(-ϕn) business is going to get extremely tiny, yea infinitesimal, so quickly that it can be pretty much treated as negligible for our practical purpose. I’m allowed to say that because this formula does have a practical purpose, namely to find a specific number in the Fibonacci series. (Alright, different definitions of “practical” perhaps.) If we give ourselves permission to leave off the negative part, we can actually calculate the “nth” Fibonacci number of the series with the simple formula: One multiplication (ϕ raised to the nth power) followed by one division, dividing the product by the square root of five.

Binet's Equation

We set out to find the 12th Fibonacci number, and now we’re at the point where we can put some numbers into that simplified formula. If our result comes out really skewed we’ll have to work a little harder. Let’s substitute 12 for n:

Binet's Equation

If you’re using a sophisticated program, you can just plug in the symbols. I don’t expect that too many of us carry a value for ϕ12 in our heads, but Wolfram/Alpha has given me an answer of 321.987… . We can use a value of 2.236 for √5. So we can take that number and divide it by the number that Wolfram has given us. The crucial moment is drawing nigh:

Binet's Equation

We do the division and get …


Let me tell you how nervous I am right now. As I said above, I have not done this before with all of the appropriate details. Before starting to write down anything formal, I gave it a rough once-over trying just to see if it might work, but now I’m really wondering. I’m sure you’re in suspense, too, about whether I worked it out so as to get a plausible result. Of course, it’s possible that it didn’t. I am fallible and prone to small mistakes that generate drastic changes in the outcome. That reminds me of a time when …

Stop Babbling

Okay, okay, I’ll stop dawdling. It’s taking me a bit to get past this approach/avoidance dilemma. Let’s take it to Wolfram then and have it do the last division for us. The expected answer was 144. Here is the result of applying the truncated form of Binet’s equation:

144.0015 …

I can’t believe it. I really, honestly did not expect anything nearly as close to 144. I’m stunned and almost emotional. This was a long and winding road. Thank you to those of you who stayed with me through the entire length of this entry. I know it was demanding. But isn’t that result a whole can of high-octane awesome-sauce‼ Back in my childhood days in Gymnasium, the math teacher, Frau Dr. von Borke, had us memorize the multiplication table for up to 20. Thus I realize that 144 is not only the 12th Fibonacci number, but also 122. We can’t derive a rule from that fact, but we can see another piece of that beautiful mosaic of numbers. When God built in numbers onto the universe he created, he not only gave it regularity, but he infused into it a beauty that we ignore to our own loss.

It’s too bad that it has become almost fashionable these days to promote yourself as someone who doesn’t get along with numbers, wearing that self-deprivation almost like a badge of machismo. Look at what you’re missing!

I wish all of my readers a day filled with beauty on many different levels.

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Friday, June 17th 2016


Making Progress

  • STATE OF EXISTENCE: Waiting with a happy face.

Thanks to everyone who tried the Tripod sites. They are working now. I still can't get into my file manager, but for the moment there are lots of ways of getting around that, and I shall be patient. Come to think of it, being patient is about all I can do. I can wait with a happy face or with a frowning face, but wait I must. 

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Friday, June 17th 2016


Lycos.com and Tripod.com in a Coma. Requesting Some Assistance!

  • STATE OF EXISTENCE: Somewhat Frustrated in a Pickle.

For some unknown reason Tripod/Lycos has been inaccessible for the last several days. I house most of my on-line pictures with them, as well the majority of my subdirectorysites, including the presently growing series on PHI  and the all-encompassing www.wincorduan.com,  I also cannot get to their administrative sites, and my e-mails seem to vanish in the ether. From what I've been able to test, the problem lies with neither my computer nor any browser.  I have no idea what's going on with them, and I hope that the issue will be fixed pretty quickly. 

In the meantime, a lot of material is missing, such as quite a few pictures in this blog. I will try to transfer to Bravenet as much a I can as soon as I can; much of it is stored in my "local" files that I can re-upload, but some things are in their appropriate form only on Tripod. Obviously, I can't transfer those until I get access to them.  I guess that if I do get access to them again, I won't need to transfer them, but it may not be a bad idea to have important thinks saved in several places. 

So, please, be patient with me about the absence of certain illustrations and some resource sites. And, if you happen to think of it, would you please from time to time check either one of these addresses, both of which will send you to slightly different versions of my home page:

 www.wincorduan.com or win_corduan.tripod.com,

and let me know if you actually got through.

I will do the same of course, but someone else may be the first to find that we're hooked up with Tripod again. 

Mr. Lycos or Mr. Tripod (or whatever your names are), would you please 1) acknowledge my existence, 2) fix this mess, 3) show me what I can do--if anything--to help fix this mess, and 4) give me a prorated refund for the sizable amount of money I spend on you. 

Thanks, everyone!

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Wednesday, June 15th 2016



  • STATE OF EXISTENCE: bleary-eyed

Examples of the occurrence of the Fibonacci series in nature abound, and so do their descriptions. Therefore, apart from referring you to some websites and simply mentioning a few examples, I will not focus on that aspect. It’s been done often and it’s been done well. It’s the mathematical relationships that I’m trying to clarify.

“John” (no further name seems to be available) wrote an interesting article for the website, Let’s Reason Together, “It’s All in the Numbers.” He begins with the confession:

I’m not a mathematician. In fact, I generally have an aversion to numbers. They’re restrictive, require systematic, step-by-step (*cough* laborious *cough*) methods to manipulate, are terribly predictable, and generally unresponsive to creativity—at least, the right-brained sort (my sort) of creativity. 

I’m taking what “John” said there as a somewhat humorous expression of his humility. Part of what I’m trying to show here is that, even though working with numbers may require systematic step by step analysis, doing so properly also needs “right-brain” creativity. Furthermore, its laboriousness is not any more tedious than what I see many serious artists practice on a daily basis. Producing a work of beauty may necessitate long hours of intense labor. Take the Taj Mahal, for instance … Furthermore, if one wants to write on a phenomenon in math, one should ideally be on good terms with the subject, and I’m sure that most people would agree with that sentiment.

Taj Mahal“John’s” article mentions four areas of nature in which the Fibonacci numbers play an organizing role:​

1. the sections of pine cones and pineapples, which are arranged in a Fibonacci spiral;
2. the spiral arrangement of seeds within a sunflower;
3. the genealogical pattern of the male population within a beehive;
4. the relationship of the phalanges and metacarpal bones of the human hand.

We can add other examples, such as

5. the number of petals in a rose.

These are realities that Christians can point to and say that there must have been some intelligence involved in constituting the world in this marvelous way. Sure, we can find reasons why these arrangements are in the pattern of the Fibonacci series insofar as they contribute to the well-being of a plant, an animal, a geological formation, etc., but that observation only strengthens the argument that these examples are not random aesthetic or mechanical coincidences. If they serve a purpose, then there’s all the more reason to believe in an intelligent Creator behind the construction of the universe.

Sadly, once again I need to recite my litany about wishing that some Christian apologists were a little more careful when they assemble their evidence on behalf of God and the Bible. For example, the article by Fred Willson of the Institute for Creation Research, “Shapes, Numbers, Patterns, and the Divine Proportion in God’s Creation” [Impact: Vital Articles on Science/Creation,” #354 (December, 2002)], quite uncritically compiles what appears to be every claim for the golden ratio ever made, including some that are dubious at best, thereby undermining the credibility of what could have been a good case. E.g., his very first example consists of a correct description of the chambered nautilus, which he then labels, incorrectly, as displaying the “golden spiral.” (Note my earlier point that the mollusk in question does manifest a logarithmic spiral, but it does not fit the golden ratio as usually understood. At the time I just referred to this error in general; I had not yet seen Willson’s article. )  

I trust that John of CLR will indulge me as I return to indulge myself in the mathematical aspects of the Fibonacci numbers and the number phi. Theoretically, there need not have been a Fibonacci series in order for there to be a phi, since, as we have shown, the actual birth of phi occurred in geometry and number theory. The Fibonacci series can be described with an equation, in which Fn stands for the rank of a given Fibonacci number. What is the “nth” Fibonacci number?

(Fn = Fn-1 + Fn-2)

e.g., the 9th Fibonacci number = the 8th Fibonacci number plus the 7th Fibonacci number

So, we can just remember the sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, … and fill in the slots.

34 = 21 + 13

This procedure is indeed lame, predictiable, and extremely slim on creativity. The matter of far greater interest is, of course, that the further we go, the closer the ratio of one number to its predecessor converges to phi. I realize that I have heretofore spoken of this relationship between Fibonacci and phi in a somewhat deprecatory manner, but, since you now realize that phi is not directly derived from the Fibonacci series, I can go back and endow the idea of convergence with the respect that it actually deserves. By “convergence” we refer to the properties of an ongoing series of numbbers whose numerical value comes increasingly closer to a particular number, ultimately coming so close to that number that the difference appears trivial.

John Derbyshire [Prime Obsession (Washington, DC: Joseph Henry Press, 2003)] presents us with a good illustration of convergence by contrasting it with its opposite result, called divergence. Most of this entry from this point on summarizes and maybe clarifies (if necessary) Derbyshire’s exposition. “Divergence” means that the value of the series keeps growing and, thus, eventually could be said to be infinity (∞).  He illustrates divergence by using the so-called harmonic series, which consists of the reciprocals of regular counting numbers (those that inhabit ℤ).

harmonic series

A close relative of the harmonic series does, in fact, converge. It’s as simple as re-creating the harmonic series, but using exponential powers of 2 as the denominators:

reciprocals of powers of 2 series

For a moment there you might think that, similar to the harmonic series, this one will also extend to infinity, but it doesn’t work out that way. To be sure, the number does keep growing; however, the rate of growth declines rapidly so that, as you go further along the best you can get is near-identity with a finite number. Let’s see what happens when we add it up this far:

fib ratio series

That’s pretty close to 2, and the further we go, the closer we will get, though we’ll never truly reach it. (Is anyone else reminded of the sizes of wrenches and sockets in an American-style set of tools?) We can say that the formula approaches 2, and that’s good enough for many purposes.

fib ratio series

Similarly, the ongoing ratio of the Fibonacci series approaches phi.

fib ratio series

and that, too, is good enough for many purposes.  

One such purpose is to help us determine the “nth” Fibonacci number when it’s a long way up the chain, say the 95th, without having to memorize the entire set of Fibonacci numbers or go through the tedious and (*cough*) laborious (*cough*) method of generating the lengthy chain of 95 links. But the clock, my energy, and your patience all lead me to realize that I better save that excitement for the next entry.

(Also, this is the first time that I’ve availed myself of MS-Word’s equation feature, which is pretty nifty. However, I wonder to what extent these formulas are going to translate into html. If I thought that thumb crossing had any value, I’d ask you to figure out how exactly one crosses one’s thumbs and then maintain your digits in that position.  -- Alas, it didn't work, so I needed to convert the equations to pictures for now.)

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Friday, June 10th 2016


Bring on the Rabbits!

  • IN THE BACKDROP: The hum of a new air conditioning system

Well, I've played my little game long enough, and I thank you for your patience. But now the time has come to start counting lagomorphs.

I hope that in my preceding discussion I have made it clear that the so-called golden ratio number φ (phi), has a life of its own, apart from the Fibonacci series. If you have caught on to that point, we are now ready to talk about the Fibonacci numbers and how they are intertwined with phi. (If you still say that phi is derived from the Fibonaccis, you're probably only teasing me, or you didn't read the preceding posts on the topic.)

Leonardo Bonacci (1170?-1250?) made a number of important contributions to math, the most significant of which is undoubtedly his promotion of the Indian/Arabic numerals in Europe, which made life a lot easier to anyone having to undertake any calculations. If you've heard of Leonardo at all, it was probably under his nickname, Fibonacci, and most likely in connection with the series of numbers that is designated after him.

The series is embedded in a puzzle that Fibonacci posed to his reading audience (Livio, 96):

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Please follow along with the picture as I walk us through the beginning of solving this puzzle. I might clarify, by the way, that we are counting the mature pairs only, just because that's how it's presented. The outcome would be no different if we counted immature pairs alongside the mature ones, except we would have only one numeral 1 at the outset, rather than two of them.

Fibonacci's Rabbits

  1. During the first month we have only the 1 original pair.
  2. In the second month the original pair will have produced one pair of offspring, which is not yet mature enough to produce any baby rabbits, so we are still holding at 1 mature pair.
  3. By the the time of the third month, the original pair will have given us another immature pair, and the earlier pair will now have matured, but not yet yielded any new rabbits. Thus the number we can put down for the third month for mature rabbits is 2.
  4. Come month 4, there is more offspring: one pair each from the original pair and the second generations pair. Also, the offspring from the newer pair will have matured. So the talley of mature rabbits for that month is 3.
  5. In the fifth month we get 5 mature rabbits, and each of the three that had matured in the previous month have contributed an as-yet immature pair.
  6. In the sixth month, the three previously immature ones will have matured and can get added to the previous five. Thus we have 8 mature pairs and five pairs that are only a month old--but probably can't wait until they are grown sufficiently to add to the pool.

I trust that you can see the pattern that is developing. Month by month the collection of rabbits is increasing. The numbers of mature pairs that we have now for the first six months are:

1 1 2 3 5 8

Each new number is created by taking the last two and adding them. So, let's continue this pattern until we reach the twelfth month, which will give us the answer to the specific puzzle that Fibonacci posed:

1 1 2 3 4 8 13 21 34 55

So, there will be 55 pairs of rabbits (110 individuals) living in the man's enclosure after a year. That doesn't sound too bad, but keep in mind that we are now entering a phase when the numbers will grow quite rapidly. Give it another three months, and the number will be 233 pairs, which is to say, of course, 466 grown rabbits. Then there will also be another 144 immature pairs (288 young rabbits), giving us a total of 754 rabbit hopping around, looking for lettuce and carrots.

As fans of the Fibonacci series know, and as I have intimated already, there is a close relationship between the Fibonacci numbers and φ. Take a number in the series and divided by the previous one, and you'll get some number that is in the vicinity of phi. So, let's look at two of such ratios.

1) The number of the twelfth month (55), divided by that of the eleventh (34): 55/34 = 1.617647059 ...

2) Since I brought up the fifteenth month, let's see what we get when we divide 233 by 144. The answer is 1.618055556 ...

Let us recall the approximate numerical value of phi: 1.618033988 ... . We can see that the longer we go on with the process, the closer we will get to phi. But there's something else that may not jump out at you immediately because of the many digits. Let us put these three numbers into numerical order, lower to higher:

1.617647059 -- 1.618033988 -- 1.618055556

The ratio of the 12th number to the 11th was below phi, whereas the estimate of the 15th to the 14th was a bit higher. I knew that would be the case, even before I tested it, just to make double sure. This is one of the interesting aspects of the relation of the Fibonacci series to φ: the different ratios of one number to its predecessor will always be an "approximation," though they get extremely close. But the direction of the error will alternate. The ratios with even numbers will be slightly lower than an more accurate rendering of φ, while the odd ones will exceed it just a little. Needless to say, as you have seen for yourself if you did the little exercise at the outset of this series, even with non-Fibonacci numbers the approximation gets very close, how much more within the Fibonacci series!

Next time: some more cool stuff about the Fib. numbers, as well as some theologial reflections.

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Thursday, June 9th 2016


Phi in the Sky

  • IN THE BACKDROP: Debra Whyte on StreetJelly.com

Yesterday was a gorgeous day in Hoosierland! Blue sky, sun shine, no need for an air conditioner at the moment. Today not so much. It’s supposed to get very hot again, so I’m glad we were able to make the arrangements to fund a new system. D. has already removed the old unit, will purchase the new machinery tomorrow on our behalf, and have it in place, hopefully, by the end of the week. --- Well, today that didn't happen. The funding is not yet showing up in our bank account. Hopefully it will be there tomorrow, so we can get the show on the road.


I’m going to continue with my discussion of phi. If this is the first time you happen upon my blog in a while, you may be a little confused as to the topic, particularly coming into the middle of an ongoing series. So, please know that the first four entries, which go all the way back to last year, are collated into a single document, to which I will add this one and the last two as soon as I can, so that it’s all in one place, and you don’t have to find your way backwards through my blog archive in order to read what came before.

Allow me to revisit what we’re doing here. We’re examining the nature of phi (1.61803…), the golden ratio constant, and for the moment we’re doing so as far away as we can get from the Fibonacci series and its exemplifications in art or nature. My point is to try to demonstrate its inherent elegance, which stems from its Creator just as much as the ­arrangement of flower petals and other “fibonaccied” items in nature.

Let’s go back to the question of two days ago: How contrived is the golden ratio? For example, take the golden triangle, which includes phi as an important ratio. Is it something that has been concocted just to show off phi, or is the golden triangle something that we can uncover and discover in other places. A partial response has been that it is included in the formation of a regular pentagon, and, thus, can be discovered without being guilty of fabricating an artificial instantiation just to smuggle in phi.

Phi is also found apart from geometry in number theory, which we can combine with a little algebra. Remember that phi is just one member of the infinity of all real numbers, which, as demonstrated by Georg Cantor, has turned out to be larger than some other infinities. However, it stands out from this uncountable crowd, right next to p, e, i, √2, and a few others, due to its special properties or notoriety. 

Here are two ways of finding phi without geometry. We will take a look at a couple of somewhat unusual-looking formulas and turn them into equations, which will resolve into phi. Here is the key: We came up with the formula for phi by setting the length of the base of a golden triangle as 1, and one of its sides as x. We were able to manipulate those numbers into a quadratic equation

x2 -x -1 = 0

The method for finding phi tonight is going to consist of finding several equations that can also be rearranged into the same quadratic equations. I found the following two equations in Livio.

Let’s start with a formula that consists of an endless fusion of square roots. Given:

 square root cluster

By the way, this thing is a formula, not an equation. You can only have an equation if there are two (or more things) that are considered to be equal, as, e.g., in

 square root cluster

It would be extremely tedious to work out a value for x from this equation if we want to continue to add the square roots of square roots, etc. But there is an easier way to do so.

We can square both sides. Squaring x gives us x2, and squaring the formula to the right resolves the first square root into a 1.

 square root cluster

Now it is apparent that the collection of square roots that follows after “1 +” is still identical to the original given formula since both extend to infinity,

 square root cluster

and, what’s more, we have already designated it as x above.

square root cluster

Then, substituting x for that nest of square roots, we get

which we can reformulate to fit the pattern we had looked for as

This is, of course, the formula which has phi as one of its solutions. Just think: we have derived it from that unwieldy collection of square roots.

So, now you feel like the Lone Ranger and want to save the world from more confusing formulas. That’s great. I’m with you all the way. Let’s try this monstrosity, which in mathematical argot is counted among a large group of “continuing fractions.” We start with a 1 and add the fraction, which has 1 for its first numerator and is followed by a never-ending, always-repeating denominator. Given:

Again, before we do anything else we need to turn this formula into an equation and label it with the variable x.

As you seek something wonderful in this equation, may I call your attention to the section underneath the topmost numerator of 1.

As in the previous equation, what you find there is actually identical to the originally given formula. There is no difference because both of them can be extended to infinity. And thus, we can safely apply the same letter variable, x to this new continuing fraction, which is also the old one. Remember now, that the area we have marked out is the denominator, and that the numerator of 1 still stands as before.

So, once we have substituted x for all of that clumsy denominator, we get a refreshingly simple equation.

Let’s get rid of the fraction 1/x by multiplying every term by x,

Then we have

and once more we can rearrange this equation into our favorite configuration:

Once again, we have turned a beast (the continuing fraction) into a beauty (phi). We have stumbled on yet another way of deriving phi without getting out our rulers or measuring tapes.


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Tuesday, June 7th 2016


Golden Rectangles


I’ll never be able to maintain a regular blog as well as take care of other things if I keep making myself include all kinds of diagrams and drawings. Still, it is fun, and for most of us a picture paints a thousand words—which is not to say that people who draw a picture necessarily forego an additional thousand words (ca. 1050 for this post). I started this entry yesterday (Monday), while still in Pokagon State Park. It was a good day, too. I had my first trail ride on horseback of the year, and June and I spent lots of time outside and swimming in the pool. Now we just got back to Smalltown, USA, and we need to see what we can do to get the new a/c system installed. Also, of course, I’m trying to finish this entry.

Pokagon Swan

Before returning to phi allow me to mention that my next StreetJelly date will be my regular set on Thursday evening at 9 pm EDT. I had to cancel last week due to circumstances that weren’t entirely beyond my control. After all, I was not forced to go to the dentist. Regardless, I see no impedance for this week. I’m planning on doing a show of some of my original songs. That’s at StreetJelly.com.

Please let me remind you that all of the earlier discussions on phi (except the last one—for now) are collected in a single site so that you can read the previous sections in a sequence that makes sense.

I left off last night by showing that phi manifests itself when you bisect two adjacent angles of a pentagon and, thereby, create a “golden triangle.” Then we were able to give birth to more and more golden triangles of diminishing size by bisecting one of their base angles each time. There is also a process that gives rise to new generations of golden rectangles, as we shall see below.

golden rectangle

By now I’m sure you have figured out why this rectangle should be golden: the ratio of the longer side AB to the shorter side AD is the same as the combined sides AB+AD to the larger side AB.

Now, we can lop of a square of length AD from the one of the ends rectangle, and we have a smaller rectangle left. I have placed the square on the left side of the rectangle. There is no rule governing that placement, nor can there be, since one can always flip the figure without doing it any damage. I’m placing my squares so that I can use the ongoing generation of golden rectangles to make a specific point in a short while.  

golden rectangle

The remaining rectangle (EBCF) now has the golden proportions. Let’s continue the process and remove another square designated by EBHG, and we have produced yet another golden rectangle, answering to the name of GHFC.

golden rectangle

Are we done now? Only if you want to be. We can remove another square and enjoy the sight of golden rectangle GIJF.

golden rectangle

And let’s do one more and call it GILK.

golden rectangle

And so forth … This is another unique treat that phi brings to us: We can go on and on bringing out golden rectangles by removing squares from one of its side.

Let us now reverse this process by starting out with the smallest golden rectangle and adding squares to it so as to create a newer, larger one, which will yield another golden rectangle by means of the same procedure.

golden rectangle

It is at this point that the placement of the square takes on significance. If I were to continue the enlargement procedure indefinitely according to my pattern, my arrangement will give us a spiral. In order to turn the tiniest of our rectangles into the next largest size, we’ll put a square underneath it. To reach the next size, we can place our square to its right. Moving on the next larger one, we can place the square on top of the one we have. Finally, to reach the largest size with which we began, we can expand it by means of a square on the left. Again, there is no point at which we have to stop, except for intrusions into our mathematical world, such as lack of available bandwidth, old age, or boredom. What you see is the beginning of a spiral. If we were to continue the process, the sequence can continue with the pattern of adding squares: down, right, up, left. Each time we get a new rectangle, it’s a golden one, and each one stands in proportion to all of the other by multiples of phi, and the spiral, that we call a "Golden Spiral" grows.

Here is a golden spiral from Wolfram Alpha.  I chopped it up and turned it into an animation.

growth of golden spiral

Are we still in math or, more specifically, geometry? Yes, we are. However, having shown you these constructions, I cannot forego mentioning one example of where we some people believe that they can see such a building process in action in nature, viz. an increase by a factor of phi in each particular stage.

The item in question is an animal belonging to a group of mollusks called "cephalopods"  [“feet on the head”], known among his friends and family (e.g. the octopus) as the chambered nautilus. This animal sets out its life in a rather small shell, but as it grows older and bigger, it manufactures larger chambers for its comfort and convenience—modern living for mollusks! A common belief is that this growth occurs according to the golden ratio. Unfortunately for ardent enthusiasts of the golden ratio, this is not so. It’s really a shame because once upon a time even your bloggist, who holds a B.S. in zoology, had been a victim of the same misconception. Gary B. Meisner, who calls himself the “phi-guy” and maintains a quite sizable website devoted to the golden ratio, explores a number of ways in which phi could be found as part of the proportions in the chambered nautilus, and leaves us with the ambivalent answer that, as yet, no nautilus that fits the pattern has been discovered, though it could not be ruled out that it will. For my purposes (and Meisner’s, if I read him correctly), that’s not enough to marvel at the golden ratio in nautili, though it leaves plenty of other things about which to marvel.

Chambered Nautilus

There are other, much better, examples in nature, but I’m once again postponing writing about those. There’s still too much I want to share with you about phi itself and where it shows up in its own world, the cosmos of numbers and formulas.


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Monday, June 6th 2016


Direct from the Pentagon


This entry comes to you from a state park in northern Indiana, where we are spending a few days. In order to make sure I get this installment out and start to establish a regular pattern again, I’ll save our adventures here so far for a later time.

Some Quick Catching Up

With regard to our health in general it’s been up and down for both of us, slightly down for June due to her fibromyalgia, slightly up for me because the expected neurological symptoms are not progressing as expected, and I’m extremely thankful for that fact.

If you’ve been seeing my various entries on Facebook, you know that I’ve been putting in a lot of time making music on StreetJelly.com as well as honing my drawing skills a little bit, the latter mostly for the sake of various websites. Long-time readers know that some of the websites I work on require use of pseudonyms to protect our readers (I’ll be happy to explain that idea if it doesn’t make sense to you), so I’m afraid that I can’t just provide a link from here under my name in order to show off my creativity, such as it is.  

I might mention that, during this blog hiatus I have made one new video, namely the Buddhist story of “The Death of Kisa Gotami’s Son.” 

I’m in the process of producing a Christian commentary video to go with that narrative. The graphics for it are already pretty much in place, and I just need to put together its narration.

After writing the entries on militant Islam, I was really beginning to feel burned out with the blog. I had no intention of dropping it then, and I still don’t, but I felt that I really needed a lengthy hiatus. Contributing to that sentiment was the fact that the various issues in the air, e.g., the presidential races and other social hot buttons have been matters on which I’d just as soon not get too caught up in the never-ending discussions, except for the occasional zinger on FB. (Please note: I didn’t say “mindless chatter,” because not all of it is, but much of the current discussions run in vicious circles that seem to do little more than to generate a lot of unwanted centrifugal force.)

On Thursday our air conditioning went out. Apparently, the motor fused, the wires burned out, and there’s nothing much to salvage on the outside unit, which is more than 20 years old. Since the temperatures have been around 90, this was not a good thing at all. D. of Riteway Heating and Cooling has informed us that replacing the outside unit (which is unavoidable) involves replacing the entire system, due to the prevailing rules of the EPA.  So, as mentioned, at this moment we are in a state park in Indiana, this time a couple of hours north of Smalltown, USA, waiting for the good folks of RHC to find a good replacement at a do-able price.


And now back to phi. Wow! That was a long time ago. Yes, indeed, but I had never intended to leave off where I did. I collected the previous entries on this topic, going all the way back to last November, in one single website, entitled PHI—Let’s Get It Right­­­­, and I highly recommend that you at least look very quickly at what I said earlier on the topic, so that this new addition will make more sense to you. I left off with this question: It’s all very well to stipulate a “golden triangle,” viz. one in which the proportion of one of its sides to its base is equal to the proportion of the sum of the base and the side to the side. 

golden triangle proportion

And it’s great that this proportion is precisely phi with all of the properties that phi has to offer. But is this geometric figure anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings? Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry? Let's go to the Pentagon to find out!

Let me issue an insincere apology to anyone who may have been misled by the term “Pentagon.” I was, of course, not referring to the building that houses the U.S. Department of Defense, and so I sup­­pose I should not have capitalized “pentagon.” Could it be that I was deliberately creating an ambiguity to get people to click on my blog?


Not this Pentagon!

As long as I’m pretending to issue apologies, let me hand out another one together with a rain check for the Fibonacci series, which figures so prominently in discussions on this topic. It has its own formula, Fn = Fn-1 + Fn-2, which is just a compact definition of the Fibonacci numbers and will tell us nothing new once I’ve explained the series. There also is a formula that we can use to determine directly the value of a specific Fibonacci number at any given point in the series, but to do so, we already need to know the value of phi and its negative reciprocal. As I continue to insist, as amazing as the Fibonacci series it, it yields phi only by convergence, not directly. Nothing wrong with convergence, but it’s not what I’m after at this time.

So, instead, let’s look at a pentagon in geometry. It’s easy to draw one with modern software programs; let’s remember, though, that in geometry anything that we draw will always be approximations. On the one hand, that means that you can’t just solve a geometric problem by measuring the lines. On the other hand, we need not attempt to go beyond normal human abilities in drawing our figures. And if­­­­ they turn out a little lop-sided, that’s okay, they’re only attempts to illustrate something that some aspects of reality cannot be reproduced by a sketch with paper & pencil sketch or a computer program.

So, let us think of a regular pentagon, viz. a two-dimensional figure with five equal sides.

pentagon/golden triangle

Now let us bisect two side-by-side angles. For convenience, we’ll go with the bottom two in the drawing, those that are located at points C and D. They converge at point A.

pentagon/golden triangle

Surprise! We have found the triangle ADC, and it turns out to be a golden triangle. (I’m skipping how that fact is derived from calculating the angles involved. See Livio, pp.  if you’d like more on that topic.)

pentagon/golden triangle

golden ratio

which comes out to phi on both sides of the “equals” sign. But that’s not all. Let’s take one of the triangle’s base angles, bisect it and run a straight line to the edge of the new triangle. We get a new triangle, DGC, and it, too, has the coveted “golden” proportions, as indicated by the number phi.

pentagon/golden triangle


No reason to stop there. Go ahead and bisect another base angle of the latest triangle, and welcome another golden triangle (GHC) into the family.

pentagon/golden triangle

We could go on and continue the process, but let’s go no further and celebrate our new discoveries. 1) We have found that a golden triangle is not just something created ad hoc for the sake of accommodating phi, but it is a direct property of a regular pentagon. 2) A golden triangle gives rise to further, but smaller, golden triangles each time one draws a straight line bisecting one of the base angles.

It’s not just the golden triangle that has the property of reproducing similar smaller versions of itself. Next time, we’ll look at the golden rectangle and take our first outing into the physical world by considering the chambered nautilus.


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Sunday, May 1st 2016


A Festschrift for Dr. Norman L. Geisler


Time to get back to the old blog to talk about some things too long for a Facebook post.

The catalyst this time is that yesterday (Saturday) I received in the mail a copy of the long-awaited Festschrift in honor of Dr. Norman L. Geisler.

I Am Put Here

What is a “Festschrift?” you ask. That’s a German term referring to a­­n anthology of essays and recollections in honor of a scholar who has made great contributions over a long career. More often than not, it’s tied to a particular point in time, such as the person’s 65th birthday, or whatever. It’s a good thing that this one wasn’t attached to any special event because it took a whole lot longer to see the light of print than was anticipated by any of the contributors, let alone by Terry Miethe, who edited this tome.

Actually, this is the second book that has been called a “Festschrift” for Norm. The first one, To Everyone an Answer: A Case for the Christian Worldview (IVP, 2004), was edited by Frank Beckwith, Bill Craig, and J. P. Moreland. It bore the secondary subtitle Essays in Honor of Norman L. Geisler, and it contains good articles by good people. There are four of us who have made contributions to both books, Ravi Zacharias, the Brothers Howe (Thomas and Richard), and your humble bloggist. There are a lot of names on the front cover, including mine, though not as Win Corduan. For anthologies my pen name is “and others,” and so I show up on a lot of book covers.

As I’m poking just a little bit of fun, please know that, my silliness aside, in terms of content, the 2004 book is truly excellent. Also, the occasion of its release actually got me a one-time invitation to take part in one of Bill Craig’s annual apologetics conferences. The title of my article in the 2004 volume was “Miracles,” not entirely the focus of my scholarship, but a topic that I do enjoy writing on from time to time.

Part of that enjoyment comes from the predictable knee jerk reactions one gets from internet atheists who get outraged by my conviction that they’re not qualified to determine if, when, or how a miracle may have occurred. After all, they’re atheists. They don’t think that God exists; in fact, many of them are not at all hesitant to tell us in so many words that people who believe in God are intellectual Neanderthals. But surely, if that’s their position, then they would be really bad internet atheists if they thought they could decide whether God had performed a miracle at a certain time and place. You can’t have miracles without a God who does them. So my duty is to shield internet-atheists from violating whatever oath or other formality they may have taken to ridicule Christians and other religious people. Maybe they pledge their allegiance to Richard Carrier or the well-known subject mentioned in the opening verse of Psalm 14 . I wouldn’t know. Nobody likes an inconsistent atheist, and if they try to declare what Christians should believe about miracles, we owe it to them to clarify that their opinions are irrelevant and hopefully thereby keep them from looking silly. If you don’t believe in God, you may perhaps believe in magic, but not miracles. (Note: not all atheists are “internet atheists.”)

Sorry, I'm still stacking as always. Pop!

Alas, as good as the 2004 anthology is, it did not contain many of the features that typically come with a genuine Festschrift, such as personal memories of the person honored. Terry Miethe decided that Norm deserved something that really fit the typical pattern and took the initiative to edit a book that conformed much more to the usual expectations for a Festschrift. Terry and I both go back to the golden years of Trinity Evangelical Divinity School when its faculty was a veritable Who’s Who in evangelical scholarship. We both got our MA’s the same year, with Norm as our thesis director. Terry went on to get his first [sic!] Ph.D. at St. Louis University only three years later and not too soon thereafter a second one at USC. He also bears the official title of “DPhil Oxon cand.,” which means he has done doctoral studies at Oxford University. You know, that old school in England.

I probably still have Terry’s e-mail somewhere, in which he invited various folks who have had close associations with Norm to contribute to his effort. I won’t check it now, and I can’t even remember right now what year that was, but his words for the project were something along the line of “I will get this done or die trying.” It turned out that he came close to the second option. He saw it through during a time when his wife, Beverly, was fighting cancer, and he also has come down with health issues. If I understand things correctly, he’s had to take early retirement on disability shortly after taking an unpaid semester-long leave to work on this project. Adding the virtual inability to find a publisher for what turned out to be a 441-page tome added a lot of stress to his life, which he did not need. Amazingly, with God’s help, he made it! And we are thankful to Wipf & Stock for publishing the book.

I think I understand now why Craig, Beckwith, and Moreland got their volume published with apparently little difficulty, namely because they did not include much of the usual Festschrift materials, but basically stuck with the collection of essays. At this juncture in time, publishers apparently are leery of investing in Festschriften because they see them as having ephemeral significance. Once the occasion is over, people have a copy of the book as quasi-souvenirs, and there may be no further call for it. In your bloggist’s opinion, the potential publishers are probably right for many cases, but in the case of Norm Geisler, may have been just a bit short-sighted. I’m pretty sure that this anthology will have a long life. Norm Geisler has left a large legacy, and—let’s face it—there are some really good essays in that book.  

Speaking of the essays, different readers will have varying opinions of them, and I expect that some of the reviews will reflect their personal opinions on “Stormin’ Norman” as much as, or perhaps even more than, the value of the contributions. Many of the essays are framed by the authors’ reminiscences of how Norm affected their lives. Terry did me the honor of taking my remembrance and moving it into the first section, entitled “Tributes to Norman L. Geisler,” under the heading of “Biographical Reflections” (xxxv-xxxvi). There I recount how … Oh, did you really think I was just going to give it away here?

Aquinas by Scrivelli

For my article I chose a somewhat improved version of a paper that I wrote for Norm as part of an independent study in the philosophy of St. Thomas Aquinas entitled “Some Features of Finite Being in St. Thomas Aquinas” (169-191). He had taught a course on Aquinas a couple of years earlier, and it would not be on the schedule again for a while, so I worked through the Angelic Doctor’s writings relying on various good resources and the occasional consultation with Norm. This would not be the last time that I would be grateful for all of the years of studying Latin (which I would eventually even teach), and it certainly came in handy there as well as for my thesis and—much later—my Ph.D. dissertation.

I might mention here parenthetically that for my first year of teaching at Taylor U, my department head and the dean had arranged that my teaching load would be just a little lower than normal (at the time 28 hrs. a year), but I gave up the load reduction for the spring semester because the philosophy majors asked me to, please, give them a course on Aquinas. Like I always say, Taylor students have been and probably still are simply the best! (Yet another stack! :) )

Anyway, I remember vividly writing that paper. It would have been at the end of the first quarter of my second year at Trinity and, thereby, when June and I had been married for just a few months. We lived in Kenosha, WI, because the cost of subsistence in Wisconsin was so much lower than in Illinois and it was worth the longish commute. At the time the MA program in philosophy required of 72 quarter hours of class work (usually two full years) plus a thesis, theoretically written during the second year while also taking a full load of classes. Many of us squeezed the two years into three.

In my typical fashion I waited to write the paper until the very last moment, specifically two days before it was due. First, I needed copies of some of Aquinas’ works that were only available at the nearby St. Mary’s Seminary in Mundelein, IL. (The most amazing library I have ever been in, right out of a medieval setting.) Since I wanted to take the books home, inter-library loan was the means to procure them for a time, but that would usually take several days. So, I devised a method that caused the raising of a few eye brows. “We’ve never done things this way before” was the response I got on both Trinity’s and St. Mary’s ends. I suspect they may not have allowed it after my machinations either, but the surprise factor probably helped me out. With an honest-looking guy (with beard and long hair, of course) standing there staring innocently and pleadingly into the librarian’s eyes, I got approval on both ends. I asked the inter-library loan person at Trinity if she would sign the request forms for a few volumes, e.g. De Potentia Dei, then personally took the forms to St. Mary’s and picked up the books in the name of inter-library loan. I was warned on both ends that if I screwed up (undoubtedly not the expression they used), my days of utilizing library services at St. Mary’s were over, but I was good and returned the books before they were due back. (Nowadays all those works are on-line.)

Then I went home, stacked the books on one side of the desk and started typing the final text right then and there; no note cards, outlines, or rough drafts. I do not recommend that method, but 1) there was no time for any preliminaries, 2) I had a clear outline in my head, and, thus, I knew exactly where I was going and which books contained the content that I needed, and 3) I was a good typist (and I guess I still am, though word processing has taken the adventure out of it), so I knew that slow and inaccurate typing would not be a serious impediment.

The bottom line: Norm liked the paper, and wrote some comments on the front that I am far too humble to disclose. With its publication, I hope that further readers will also enjoy it and learn from it. If not mine, then surely some of the other essays.

Other than Norm’s name, Terry’s is the only one on the front cover. And that’s how it should be. No “stars,” no special mentions, and, best of all, no “and others.” Thank you, Terry, for your work on this project and for pouring so much of yourself into it.

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Wednesday, December 9th 2015


Islam Material Gathered

  • STATE OF EXISTENCE: Tired of this topic.

This post is merely intended to let you know that all of the blog material on Islam from the last month or so has now been gathered at one site. You may find it more convenient to read the entries first to last, rather than having to backtrack blog-style. 

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Wednesday, December 9th 2015


Isis and it Expansion, part 6

  • STATE OF EXISTENCE: Happy to be done with this project (if, indeed I am)

This entry is the straightforward continuation of what I posted just a few minutes ago (part 5). It won’t make any sense without reading that entry first. (I don’t know that it will with it, but I know it won’t without it. :)  I will provide further formatting in a little while. 

The Early Situation in General

After a period away from large population Boko Haram established itself in Maiduguri, the capital city of the state of Borno. Its adherents probably came primarily from Muslims in northern Nigeria, who perceived that their condition in life, frequently abject poverty, was due to having to live in a non-Muslim (many would say “anti-Muslim”) society. However, as Cook astutely points out (2014, 5-6), if so, it would still be an unwarranted stretch to ascribe rampant poverty as the cause or raison d'être for Boko Haram as an organization. The means for recruiting members frequently do not dovetail with the overall goals of a group. I don't know to what extent Yusuf used this strategy intentionally, but it is certainly often easier to rally people together under the hatred of a common perceived enemy than under a common cause to attain a positive goal. 

Ever since 2002 Boko Haram carried out armed raids, frequently in order to obtain weapons as spoils in order to arm themselves and to bolster their often-meagre finances, but also to create fear among the population as they destroyed products of "Western Learning," such as schools, churches, medical facilities, medical storage sites, and, yes, quite a few mosques.

Inadequate Responses

Inside of Nigeria resistance was provided almost entirely by the Nigerian police, aka Security Forces, and occasionally vigilante groups (Cook, 2014, 14). Because Boko Haram frequently crossed the borders into Chad and Niger, either for raids or to seek concealment, the governments of those two countries used their armies to keep Boko Haram out of their land and hair. But they could not have an effect in stopping Boko Haram's advances in northern Nigeria. The intended goal of the establishment of northern Nigeria (at a minimum) as a separate Islamic state was slowly starting to take on reality. Only one crucial military unit, in fact, the most essential one, did not get involved, and that is--no, with all due respect, not the United States Marines--the army of Nigeria itself.

Cook points out some apparently inexplicable aspects in this set of events. One such was the fact that the Nigerian government did not act to protect schools.

What is most interesting for the outside observer is that there does not appear to have been any serious security measures in place in any of these locations. For Boko Haram’s attacks to be defeated, there needs to be a system for guarding and alarm for isolated schools. It is unclear, when Boko Haram has set educational institutions as its target, why the Nigerian government and military have not responded with setting up the appropriate security measures (Cook, 2014, 12).

However, the most glaring enigma is the more general one included in the above quotation, that the Nigerian government just did not get involved in earnest until finally in the spring of 2015, i.e. this year. Cook asserts in 2014 that “the track record of the Nigerian military to counter Boko Haram has been a miserable one” (2014, 14), and I’m afraid that he made a vast understatement when he wrote those lines.

The news agency al-Jazeera carried an article recounting the experience of a captain in the Nigerian army who had been sent out with about thirty men to rid a town of about 200 Boko Haram recruits. They received their orders and weapons (mortars/grenade launchers) with little notice and headed out to carry out their mission. When they were sufficiently entrenched and the combat commenced, the Nigerian soldiers to a man realized that their equipment was broken and the ammunition was out of date and spoiled. On the other hand, the Boko Haram fighters were relatively well-equipped and definitely trained in the use of modern weaponry. (I'll come back to that point below.) Somehow Captain X and his men prevailed by using their handguns and actually receiving reinforcements in response to their plea. However, that's as far as the Nigerian military was willing to take their mission, a few pro forma interventions here and there, but no significant effort, while the police and foreign military fought to contain Boko Haram as much as they were able. Cook describes the period from about 2010 to 2014 as a civil war in which only one side was fighting (2014, 29).

Some Significant Events

I am not going to detail the many inhumane acts carried out by the group over the last few years. Please see Cook's papers and other various articles on and off line. I must seriously caution you that if you're prone to physical reactions from descriptions of torture and other violations, such as the abduction of hundreds of 10-12 year old girls and their subsequent ordeals, you should not pursue the events in a lot of detail. Still, reality is what it is, and I do need to highlight a couple of occurrences over the years.

2009: The Execution of Muhammad Yusuf

The original leader of Boko Haram, as mentioned, was one Muhammad Yusuf. It became an open secret that, whereas he was expecting all of his followers to live a highly frugal existence, he enjoyed a life of luxury and indulgence (another enigma, but one that is widespread across many different cultures and religions). On July 25, 2009, the federal police in Maiduguri captured members of Boko Haram, which enabled them to track and arrest Yusuf. Cook (2011, 11-12) provides a translated excerpt of his interrogation in the police headquarters, which reads almost like a segment from Abbot and Costello. Yusuf was manipulative, mendacious, and verging on being insolent in his initial responses. Predictably, though not excusably, his captors reacted with increased anger. They humiliated and intimidated him, and eventually simply killed him without any further process of justice. A video of what happened is posted on YouTube, but I'm not going to link to it. There’s no need for us to get desensitized to brutality. If you really need to see it, e.g. for writing a term paper, it’s easy enough to find.

Needless to say, the execution of Yusuf caused a short pause in Boko Haram's activities, only to be renewed with increased vigor and a sizable increase in membership 2010 under the leadership of one Abubakar Shekau, a person who has made up for his apparent lack of knowledge both inside and outside of Islam by displaying unparalleled cruelty in word and deed.

Sidelight: It is interesting to me now that  in early November of 2009 I had a short conversation with a Nigerian Senator that touched on religion in his country. At the time I had no idea of what had happed a few months earlier, and my new acquaintance, clearly a serious Christian, obviously saw no need to bring it up. I attach no significance to that fact, except that I now realize how close I was to someone whose opinion I would have valued, had I only known about the situation and been tactless enough to ask.

Declaring a Caliphate

“Boko Haram” has always been a nickname, though its members and leaders used it themselves. However, in 2014, they changed their official designation to ISWA, Islamic State of West Africa, and stated that from that time on they constituted a caliphate. When I first read about that announcement, I tried to figure out whether it meant that ISWA was establishing its own caliphate under Shekau or aligning itself with the caliphate of Caliph Ibrahim of ISIS. Then I realized that much of the rest of the world also did not know what this new claim entailed either. I suspect that initially Shekau may have thought of himself as the ideal candidate, but since then ISWA has extended a request to ISIS to become a part of its team, and ISIS has accepted the offer.

Sidelight: One can find a number of good articles debunking the idea that the office of "caliph" truly has the great significance that many Muslims and Western analysts tend to ascribe to the term. From my standpoint as a scholar interested in precision, I appreciate such corrections; as a scholar interested in the role of religions in the contemporary world, once again I need to fall back on the fact that perceptions are frequently more important than theoretical accuracy in understanding the actions of people. (See, for example, Madawi al-Rasheed, Carool Kersten, & Marat Shterin, eds., Demystifying the Caliphate (Oxford: Oxford University Press, 2015).

A Bit of Fresh Air

In the short report on his studies of Boko Haram in the aforementioned alumni magazine, Professor David Cook, has finally felt free to sound a hopeful, if not optimistic note. After the elections on March 25 of this year, the government finally authorized the military to carry out a full-scale mission against Boko Haram, and has succeeded more so than many people had hoped for. To be more specific, as I understand it, the organization, which by that point was establishing territorial control over parts of northern Nigeria has been driven underground again. "Boko Haram's reign of terror is coming to a close at this time, thanks to a decisive response just in the nick of time," Cook rejoices and adds a little later that "it has been profoundly satisfying to hear that Boko Haram is finally on the run." Given the horrendous state prior to army intervention, one cannot help but to share his joy. Much oppression has been ended, and captives--particularly some of the abducted girls and women--have been returned, though bearing serious scars.

But Cook realizes that the hostilities are not over for good, and that Boko Haram still exists. Just as I thought about writing on this topic, I came across a report of a Boko Haram sponsored bombing in northern Cameroon. Furthermore, the Wikipedia article on the group contains a rather lengthy list of acts of terror ascribed to them this year, so after the momentary joy of some success, I'm afraid that we need to be more sober (it not somber) again.

Going Further

Writing in the context of exploring the options in American foreign policy in reaction to the events of Nigeria, Cook has suggested a rather cautious approach (2011, 23ff, & 2014, 24ff). Since clearly my opinion is far less informed than Prof. Cook’s, the fact that I concur may not carry a whole lot of weight. Our military resources are limited, and we do not have sufficient national interest in the fate of Nigeria as a political entity to get involved in one of their civil wars, even if we find one side quite repugnant. We cannot be the policeman of the world. A veritable crusade to West Africa to right all wrongs and establish a permanent Western-style government and economy would, without a doubt, wind up as disastrous as its counterparts in the Middle Ages. Nevertheless, there are some very good reasons to keep a close eye on what is going to happen in Nigeria in the near future.

1. Boko Haram, now ISWA, still exists, not just as an idea, but as a group of people who continue to pursue their goal, as made clear by the ongoing acts of terror.

2. Anti-government armies in Africa often are made up of a large number of mercenaries who move from country to country, offering their services to various causes. They provide experience in war in addition to bolstering the numbers of insurgent combatants who are capable with modern weaponry. The experience of Captain X, according to the article I mentioned above, is an apparent case in point. Thus, at least theoretically, the number of potential warriors on behalf of ISWA could actually exceed the number of Nigerian members of the group. Given sufficient inducement, ISWA could make a large-scale comeback, in which case the United States may have to get involved simply to continue the “war on terror.” Obviously, I have to leave it with “could” and “may.” Despite the statements of our country’s highest leader in foreign policy, we have yet to neutralize al-Qaeda, making it meaningless to say that we’re going to “do” ISIS in a shorter time frame. Any statements concerning ISWA suffer from even greater uncertainty.

3. We have begun to feel the impact of ISIS in the Western world, including even on our shores. I began this lengthy essay with the observation that ISIS is at the moment territorially restrained, but that it is also still expanding and increasing its influence (and President Obama recognizes both points, at least implicitly). Hopefully you may now be able to see one key to understand this conundrum. ISIS is expanding geographically because other jihadi organizations may (explicitly or implicitly) seek to make common cause with them.

I have tried to give an intelligible account of how this method has worked out in one West African country, and there's no good reason to expect it to end soon if it turns out that potentially new members of the intended global IS are receiving benefits from claiming affiliation with ISIS.

4. In the meantime, ISIS definitely gains from such expressions of support. Let me clarify.

The most effective strategy for guerilla warfare consists of spending much time in hiding, launching surprise attacks on the enemy so as to inflict maximum damage in a minimal amount of time, and returning to concealment quickly enough that the enemy has little chance to respond. To do so, one must have bases from which one can launch these sorties, and groups such as Boko Haram, even with few facilities, can provide just that kind of opportunity.

Is Boko Haram, given its recent losses, going to carry out terrorist attacks in Western Europe or the United States in the foreseeable future? Probably not. In contrast to ISIS, they appear to be short on resources and often look for financial support by means of plain old bank robbery.  

However, if they remain a viable, albeit underground, organization, they can still be of great assistance to ISIS, which has now displayed its capability for destruction on a world-wide level.

In short, in my debatable opinion Boko Haram/ISWA considered by itself should not be seen a threat to us here in the U.S. However, since they are probably once again at a point of desperation and presumably continuing to crave a role on the world stage, we need to take it seriously as a relatively new arm of the movement towards a world-wide Islamic State, as currently led by ISIS. Given the rhetoric of the last six or more years, they may very well be available to ISIS as a second-ranked group that is willing to do virtually anything in order to receive acceptance among other Islamic militant groups.

Do I mean, then, that the U.S. should send troops along with drones and bombs to Nigeria in order to root out what remains of Boko Haram? Of course not. I don't see where we could contribute anything other than creating a lot of bad will among all concerned parties—governments and insurgents alike—with only a minimal probability of success. If I can make a suggestion, it would only be the obvious one of keeping a close eye on northern Nigeria with a view towards exposing, if possible, significant contacts with other Neo-Kharijite and jihadi groups or a visible influx of potential mercenaries.

All of those prognostications aside, my overall purpose here has not been to advise what should be done, but merely to explain one means by which ISIS can increase its power as a global threat. I hope my efforts towards providing a little bit of further understanding have not been entirely in vain. 

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Wednesday, December 9th 2015


Isis and its Expansion, part 5

This is the first of two entries that I’m uploading pretty much simultaneously. Once both of these two posts are up, I will start gathering all of the recent material into one website.

Introduction: Gratitude from an Ancient “Owl”

Barring any unexpected events or questions that merit a lengthy public reply, this entry should be the last one directly dealing with Islam. Exploring Boko Haram, its background, and its potential implications, has been a learning experience for me. And it has definitely not led me to marvel at the inherent goodness of human nature.

I mentioned Professor David Cook of the Baker Institute at Rice University before. Actually it was reading his summary in the glossy pages of an alumni pamphlet from the religion department that alerted me to the subject and its implications. [David Cook, “Boko Haram” Religion Matters 2 (Fall 2015):2-3.] ---Obviously, this is not the first time that I have profited from the teaching from a professor at this fabulous institution.---The insights he provided sent me off to pursue this track, and I’m glad that he motivated me to do so. As it is, the best I can produce here is a quick snapshot, when it really should take an entire movie of the length of “Gone with the Wind” to get a decent understanding of Islam in sub-Saharan Africa.  I still love learning, and I picked up more than I could accommodate in these entries, but I hope that what I have done will be helpful to you to see this dimension of Islam outside of the Middle East.

One of the things that I appreciate particularly about Prof. Cook’s little summary in the alumni magazine is that he did not seek to hide his feelings about the subject: the frustrations, the disgust, the disillusionment, and, at the time that he wrote the piece, the exultation over the fact that things were apparently starting to head in the right direction. Sometimes the subjective reactions become a part of the facts without which an objective account is incomplete. (It was at Rice that I really learned to appreciate the value of phenomenology.) Prof. Cook’s accounts consist of two pdf articles under the aegis of the Baker Institute: Boko Haram: A Prognosis, 2011 and Boko Haram: A New State in West Africa, 2014. They are available as pdf articles, so it is easier, once one has downloaded them, to refer to them simply by their year and page number, rather than to insert hyperlinks each time. To a certain extent they are the backbone of my summary, supplemented by other books and articles. However, and I don’t mean this just as the traditional formality, whatever errors I have committed or am committing are truly my own fault, and, as always, I am happy to receive kindly worded constructive critiques.  

The Protean Identity of Boko Haram

In the last entry I brought up the idea, promoted by a number of scholars, that establishing shari'a (Islamic jurisprudence) in the northern states of Nigeria was unsuccessful partially because it was unenforceable on local levels as well as unenforced by the federal government. Of course, a group seeking to attain that goal could possibly defy the government and take enforcement into its own hands. The group that has become known as Boko Haram is a case in point. 

Aside: If you pronounce the name as “bokoe haraam,” you’re close enough to an acceptable way of saying it. Haram is indeed the same word as “harem,” that is used to describe traditional women’s quarters, and I don’t advocate changing how we say that word in English, but we should make an effort to get closer in this context. In a video to which I make reference below, I heard it pronounced in this way.

So, who is this group? What are their core beliefs? What scholarly traditions of Qur’anic interpretation do they follow? We cannot say as much as we would like to in answer to the last two questions, not because we don’t know, but because there seems to be little to know. It appears to me that, beyond a rigid commitment to Islam as an unclarified exclusivist ideology and the drive to eliminate everyone who does not accept them as the only true Muslims in Nigeria, Boko Haram has shown itself to be a group in search of a permanent identity. If we try to understand its rather thin ideology, we can only do so by, first of all, recognizing that perceptions usually outstrip reality in radical movements, religious and otherwise.

A general classification of this group places them amidst other groups that are called Salafi-jihadi (Cook, 2011, 2).  The second part of the label, "jihadi," is clear enough: they do not shy away from using armed violence to promote their cause. "Salafi," as I have mentioned in the past, is not as easily accommodated. The basic meaning of the term is based on a particular perception of the first four caliphs (Abu Bakr, Umar, Uthman, and Ali ibn Talid), who were known as the “Companions" (of Muhammad). They have been considered to be "rightly guided," and their teaching is treated as the only true and necessary understanding of Islam. But the content derived from their teaching and role modeling is not necessarily the same for groups that use it.

I noted in chapter 5 of the second edition of Neighboring Faiths that the Muslims of Sau'di Arabia reject being called Wahhabis and prefer to be called Salafis. Although I usually have little problem referring to groups of people by their preferred appellation, I said that in the case of the Wahhabis, the term Salafi is neither helpful nor really appropriate since there is no denying that the origin of their version of Islam lies with Muhammad ibn Abd-al-Wahhab (1703-1792)  and that, for that matter, they appear to follow the Hanbalite school of sharia. Here we see another reason: the term "Salafi" has been given to various other neo-Kharijite groups, including Boko Haram, who actually reckon Wahhabis with those who should be killed since they are tied to current power structures. The term Salafi simply does not say as much as might have been intended. In looking at whatever declarations and transcripts of interrogation I have been able to find, it strikes me that the leadership of Boko Haram has been steeped in folk Islam, but does not have a strong background in the study of the Qur'an, let alone its history of interpretation. I don't think that it is necessarily going too far to say that Boko Haram is driven by the desire to be seen as a major player in the Islamic world and might just be willing to accept any label that treats them as a major Islamic terrorist group to be reckoned with.

Even more than other Muslim groups, Boko Haram’s biggest target has been "Western learning," pejoratively called BokoHaram means forbidden, so this group became known under the negative label, "Western learning is forbidden.” Its original name was probably Jamā'atu Ahli is-Sunnah lid-Da'wati wal-Jihād, "People Committed to the Prophet's Teachings for Propagation and Jihad.” Cook (2011, 8) transliterates the name as Jama`at ahl al-sunna li-da `wa wal-l-jihad and clarifies that this choice of name (i.e. Jama`at) would place them alongside other similar groups in various parts of the world. The overall strategy of a jama group is to postpone overt violence until the movement has spread itself throughout an entire country over a long period of time. Since Boko Haram’s existence began almost immediately with violence, this formal name does not truly reveal the nature of the group. We need to consider another name that the group adopted a little further below.

In the absence of finding much material concerning the conceptual basis of Boko Haram in its early years, it seems to have become customary—if not obligatory—to present the following statement made by Muhammad Yusuf in a rare interview on the BBC:

There are prominent Islamic preachers who have seen and understood that the present Western-style education is mixed with issues that run contrary to our beliefs in Islam. ---Like rain. We believe it is a creation of God rather than an evaporation caused by the sun that condenses and becomes rain. --- Like saying the world is a sphere. If it runs contrary to the teachings of Allah, we reject it. We also reject the theory of Darwinism. (Cook, 2011, 8)

Hardly the stuff to go to war over, we might say. But things do run a little deeper than that. Abu Zayd, a frequent spokesman for Abubakar Shekau (Cook, 2014, 4), the group’s new leader after Yusuf’s death, explained that “Western learning” actually means "Western civilization," the perceived acme of everything that is false and perverted, in contrast to the purity of Islamic learning and Islamic civilization. Well, as Christians, we might be tempted to try to build a bridge and respond that we, too, do not like the immoral aspects that have made themselves at home in contemporary Western culture. But the term "boko" includes some items that we might want to acclaim as some of the better advances in the history of humanity, such as government by democracy. As far as I can tell, from Boko Haram's perspective democracy is not just a form of government, but also a deliberate means rationalize away the need for an Islamic  . And, furthermore, unfortunately a bridge is not of much use if it does not connect both shores of a river, and Boko Haram’s mission can be visualized as doing away with any potential bridge heads. Their policy, called takfiri, falls in line with other neo-Karijite groups, as it refers to cleansing Islam of impurity by use of the sword, but exceeds most of them in their brutality and lack of discernment. 

Concerning present rulers, Abu Zayd put forward the following statement, which astute students of logic could use as an example of the fallacy of composition and division.

This is a government that is not Islamic. Therefore, all of its employees, Muslims and non-Muslims, are Infidels (Cook, 2011, 11).

Since the United States is (rightly or wrongly) seen by many people as the strongest bulwark of Western democracy, it is the archenemy, even if it does not have a strong palpable presence in Nigeria. The Qutbi doctrine of the illegitimacy of any contemporary government seems to be highly visible. In fact, several times Abu Zayd stated as a goal of Boko Haram to render the country “ungovernable,” (Cook, 2011, 11 & 18) so that then the solution of a genuine Islamic government will be the obvious one. However, their recent support of a present caliphate does not fit in with the true agenda of Qutbism, and, with all of this ambiguity, I'm not sure that any presently available word (Salafi, neo-Kharijites, Qutbi, etc.) really captures their nature.

Aside: As my ever-faithful readers know, I think that the term "fundamentalist," particularly if it is used to lump together evangelical Christians, Hassidic Jews, and Islamic terrorists, combines far too many cultural and religious forms to be anything but an empty label. Someone somewhere decided to use the term beyond its original setting, journalists and scholars picked it up uncritically, and they now debate its true meaning as an overall category as though “being fundamentalist” were an objective attribute claimed by all the of the groups in question. I would suggest that a better alternative is to drop using it outside of its original Christian setting where it has a very concrete meaning derived from the book series called The Fundamentals, published from 1910-1915. 

All-in all, it's probably not very helpful to place Boko Haram into any neatly designed conceptual box. Pace Hegel, in this case the real and the rational are not identical; or, more specifically for this case, the perception of the real and the rational do not necessarily coincide. 

The final installment should be up in just a few minutes. 

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Friday, December 4th 2015


Isis and its Expansion, part 4

  • STATE OF EXISTENCE: Tuckered out, as usual :)

The Latest Attack

Yet another terrorist attack, this one on our soil in San Bernadino, CA, and apparently carried out by a lovely husband-and-wife team—their names are Syed Rizwan Farook and Tashfeen Malik, but I’ll just call them Mr. and Mrs. Killer—who first made sure that their baby was safe before embarking on their rampage. What a beautiful role model for progressive parenting in the twenty-first century! At this point it appears to be clear that they acted on behalf of ISIS. Their target was the state agency that had been the source of Mr. Killer’s bi-weekly paycheck; among its tasks was to provide help to families with special-needs children. And the time of their attack was, of all things, a “holiday celebration”—you know, as in peace on earth and good will to all.

Mr. Killer did not make it to the office party; I imagine that his fellow employees reasoned that, since he was a Muslim, he did not celebrate Christmas, Hanukah, or Kwanza. Little did they know that he would show up late, accompanied by his wife, and that together they would create a veritable apocalypse. One gets the impression from the reports that the Killers simply fired round after round into the assembled folks, not aiming at any particular person, just making sure that they would leave as many casualties as they could before departing

Prayer in San Bernadino
People praying on an adjacent golf course.
Picture hosted by CNN

It took the FBI a while to decide whether to classify this event as an act of terror, which may have struck you as a bit irrational with more than a dozen people dead. But there often are significant reasons not to make that decision on the spur of the moment. In this instance, since this was Mr. Killer’s place of work, he might just have had a grievance against his employer, which, he felt, could only be rectified by means of a mass shooting of his colleagues. In that case, he and Mrs. Killer would still be homicidal maniacs, but not terrorists. In this particular occurrence, since the couple was killed, the immediate consequences of the label are not as visible as if they had been taken into custody. Just keep in mind the fact that terrorists are considered enemy combatants who receive neither Miranda rights nor habeas corpus nor pro bono legal representation from some hot shot lawyer who wants to make a name for himself by defending the indefensible. Moreover, the follow-up is going to be very different now since it has been established that ISIS was behind the event, and that it wasn’t just the case that Mr. Killer was angry because he had a private grievance against his employers.

A Suggestion on Our Response

I would hope that it is impossible for anyone who hears or reads of this incident not to feel a great amount of anger. Let me underscore once again that Islamic acts of terrorism should be considered acts of war. In 2001, some people were upset with President Bush for using the language of war with regard to 9/11. However, that particular matter had already been settled by Osama bin Laden himself in 1994 (a long time before 9/11/2001) when he declared a fatwa that all true Muslims should kill as many Americans as possible in the context of an ongoing war. He described the US as the head of a large association of anti-Islamic nations, and he asserted that eliminating the head would cause the body to fall apart as a result. Keep that image in mind when you hear about people in the sands of the Sahara, who may never have had contact with any Americans, declaim how much they hate America in general and each American in particular.

Still, I pray that, as we respond to these world events, we will keep from becoming like our enemies. I confess that at times I have allowed myself to dehumanize them in my rhetoric, but doing so is not good because it can turn a travesty into a caricature. The people we are looking at are neither sub-humans, nor beasts, nor demons, nor “monsters thinly disguised as humans,” as I put it recently. If they were beasts, demons, monsters, or other sub-humans, it would presumably be in their nature to commit atrocities. But terrorists (and there’s no need to limit ourselves to Islamic ones) are human beings, and by their actions they show that they are evil human beings, surpassing others in their capacity to do evil even among our fallen race. If we say that they cannot help but go around killing other people because they are Muslims, we are not only stating something that’s as untrue as it is mindless, but we are also implicitly excusing their actions. After all, people shouldn’t be held responsible for something that they are programmed to do. It is the fact that they are human and that they violate their humanity in the most despicable manner that makes them such evil persons. The gravity and the moral decrepitude that we see exhibited here is so devastating because these are human beings, like you and me in many aspects, but with a putrid, rotten, decadent sense of right and wrong.

Now, having had this attack occur on US soil makes it all the more urgent that we get a grip on what has transpired in far-away Nigeria. ­­­Eventually the connection will become evident.

Uthman dan Fodio: A Historical Precedent

Uthman dan FodioIslam came to Nigeria more than 1,000 years ago and settled in various areas, particularly in the north where it makes contact with the Sahara desert. As is true with any religion in any region, over time the actual faithfulness of the governments and of the people to Islam varied greatly. There were times and places of great commitment, and there were also times and places where Islam was practiced quite superficially. As a result, from time to time there were reformers who attempted to purify Islam, and sometimes doing so would take the form of a jihad, a “holy war.”

A very prominent reformer and amazingly prolific writer was a man named Uthman dan Fodio (1754-1817; be aware of different transliterations of his given name) who lived in Northern Nigeria. After his message of calling for greater obedience to the Qur'an was rejected, he became the leader of a jihad that ended up in establishing a caliphate in the region of Sokoto. At that time it bordered on the Sultanate of Bornu, another strongly Muslim state. The caliphate of Sokoto lasted from about 1815 to 1904, when it was abolished by Great Britain in the process of turning Nigeria into one of its colonies. Actually, to be more accurate, they allowed the caliph to continue in a role as spiritual leader, but stripped him of political authority. By that time Uthman dan Fodio had become a hero for many Nigerian Muslims for bringing about this allegedly true Muslim state almost a hundred years earlier.

Sokoto Caliphate

Furthermore, in the eyes of many Muslims in Nigeria, dan Fodio’s jihad and the resulting Islamic state represent the ideal that should be repeated and re-attained in Nigeria. They state accurately that the British had clipped their wings, and they exhort each other to work towards the goal of turning all of Nigeria into a Muslim state as exemplified historically in Sokoto.

Religious Governance: It's Not Easy

Before continuing this brief narrative, I need to make a point concerning the difficulty of setting up a religious state. There's a perennial issue surrounding any attempt to establish a country under the umbrella of a particular religion. Exactly how will one implement such a plan? One can make rules, but can one bring about devotion and piety by force? The Old Testament gives us a good case study. God’s Law was the foundational document for the Hebrew people, and it included many dimensions, all of them obligatory: criminal law, civil law, laws on worship and sacrifices, laws on ritual purity, and so on. A person not only needed to be righteous and considerate, but also to meet the strict requirements connected with the worship of Yahweh, which specified when, where, and how to do so. And yet God’s own people strayed from him time and again. [Please allow me at this point to skip the opportunity for stacking and to forego a digression on the topic of the United States as a “Christian nation.”]

Islam is in a different position than some other religions because one of its underlying concepts is the mandate to establish an Islamic state, as I tried to show in my last entry. Consequently, it has a clearer set of requirements as to what would constitute an acceptable government. --- Please keep in mind that pure Qutbism, as exemplified perhaps by Osama bin Laden, aims for the immediate abolition of all governments, so what I'm writing here applies to al-Qaeda only with some with some qualifications. --- First of all, the government must be constituted by Muslims. Cooperative jizya-paying Jews and Christians are allowed to reside in the state and practice their religion insofar as it does not interfere with the supremacy of Islam in any significant way. So far so good, but where do we go from there? Presumably one could mandate prayers and pilgrimages, allow only halal (“permitted”) food to be eaten, and proceed to spell out a specific list of what else is halal and what is haram (“forbidden”). Or, one could simplify the process and decree in one fell sweep that the country, state, or region shall be governed by shari’a—Islamic jurisprudence.

Government by Shari'a

Please note that I said “jurisprudence,” i.e. the process by which legal cases are addressed and settled. It is not a set of crimes and their appropriate punishments per se. Some American Christians are haunted by the specter of shari’a laws being imposed on this country due to an increase of the Muslim population, not a likely scenario. Still, frequently such expressions of unease are accompanied by examples, such as adulterous women being stoned, or thieves having their hands cut off. But that’s not how shari’a works. By itself it is a method, not a book of laws. Those examples, as alienating as they are to us, are supposed to be outcomes of judicial deliberation, not overarching rules. In fact, some of the most frightful instances, e.g. the scenario in which an innocent girl was raped and subsequently executed by her brother, are not the product of shari’a at all, but miscarriages of justice that should never emerge when shari’a is practiced conscientiously. Shari’a is a rather complicated matter, and it is not surprising that there are four Sunni schools of law, which I have outlined as a part of my site on “Groups of Islam.” There’s no point in my repeating what I wrote there, but, please, read that particular page or get the information from some other source before you say anything else about shari’a. And, I might mention that fatwas issued without a preceding trial by such Muslim luminaries as the Ayatollah Khomeini of Iran or Usama bin Laden are as contrary to shari’a as a sentence being handed down without trial in our legal system would be. Shari’a can be a problem for non-Muslims, but evil people who will perpetrate injustice will not be constrained by any system, and they are the real problem. I think they would be just as evil under British-style common law, Roman/Napoleonic law, or African elder/ancestor-based traditional law as under shari'a. 

When the British left Nigeria around 1960, the newly founded federation faced decades of instability, and, sad to say, the right balance has still not been found. Democracies and dictatorships have come and gone, and corruption has been commonplace. We’ve already mentioned the strife between Muslims and Christians as well as the agitation between different groups of Islam, particularly between the conservative Sunnis and other groups. By the last decade of the twentieth century, finally a consensus seemed to have emerged among Nigerian Muslims, namely to unite in bringing about the one item that could theoretically engender a truly Islamic society: Muslim states should be governed on the basis of shari’a. I don’t need to point out that Christians and other non-Muslim residing in these states were quite unhappy with the idea. In some cases there may have been attempts at a dual-track systems whereby Christians were judged by a civil magistrate and Muslims by a Khadi, utilizing shari’a. David Cook published a map of those northern states in Nigeria that helps us visualize the apparent (but only apparent) solidarity of the Muslim population of northern Nigeria in agreeing to shari’a jurisprudence.  

David Cook Nigeria

It didn’t work. Among the reasons that Cook adduces for this failure, two stand out: 1) Nigerian states do not have their own police forces. All police units report directly to the federal government, and the federal government was not interested in enforcing shari’a, especially since some of its conclusions were at odds with federal laws. So, the absence of any power to uphold shari’a in those states that tried to apply it pretty much guaranteed its failure. 2) The Sunnis of northern Nigeria adhere to the Malaki school of shari’a, and this school, the second most conservative of the four, does, indeed, prescribe stoning for adulteresses. Several cases in which women were prosecuted under that category received international attention, and, in light of the pressure from outside of the Nigerian states concerned, the fatwas could not be implemented. Of course, once one aspect of the law showed itself weak enough to be overturned, there was no good reason to expect the others to be any stronger.

Cook summarized in 2011:

With the failure of essentially any possibility of enforcing the adultery laws, and the very obvious failure of  most northern Nigerian states to enforce bans on alcohol and other non-shari’a activities, 10 years after the initial implementation of shari'a it is clear that no ideal Muslim society has resulted (Cook, “Boko Haram Prognosis,” 2011, p. 8).

And he added the ominous words:

Most probably the frustration felt by the Muslims as a result of that fact has led to the rise of Boko Haram, first in Maidiguri (the capital of Borno) and then throughout the Northeast.

You may never have heard of Boko Haram before, but they, too, have made the United States one of their targets of destruction. And here’s the thing we’ll pick up on next time: They have some powerful friends. 

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