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Thursday, June 30th 2016

14:23

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.

Comments

 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     

 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 …  

√2

[irrationals]

 

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      ⇒

2i

[complex]

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. 

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  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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