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Saturday, June 25th 2016

20:43

The marvels go on and on.

  • STATE OF EXISTENCE: good

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:

probability

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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