The Math of God
Posted by Anthony Mannucci
This post is inspired by “The Character of Physical Law” by Feynman. It is also generally inspired by the “spirit” of Feynman, who always inspired me.
One can now view his lecture “The Character of Physical Law” on YouTube. He eloquently expresses the importance of mathematics to modern science. This makes science impenetrable to many, since mathematical skill is reserved for the few. Suppose one tried to express physical law without mathematics. Would it appear as religious reasoning?
Feynman makes the excellent point that we are talking about a certain level of abstract mathematics here. The math used by physicists is not a variant of simple arithmetic. I admit this is a point I may have missed in my book. In the book, I declare that mathematics is a form of logic, and leave it at that. Feynman makes the correct point that, if math is a form of logic, it is of a very special kind. To do the math that physics requires, you really need to write down different kinds of symbols than are required by logic. Math that physicists use has to “look mathematical” to work.
If physical law (nature) is so closely tied to mathematical reasoning, and math “unlocks” many secrets of nature, what are the implications for those trying to understand nature before math was invented? Isn’t it possible that the search to understand God was essentially an attempt to understand nature, but without the math? Without math, nature cannot be described as effectively as we do it today. I use the word describe advisedly. With math, we can describe how nature behaves, but we cannot say why. (Feynman makes this point also in his lecture).
Is it possible then that religion is a branch of science? Both are concerned with the “almighty”. In one case the almighty is natural law, in the other, something more poorly defined. I am not sure how to define God “in general”. Am I suggesting that religion + math = science? Perhaps, at least in a certain sense.
Why abstract math is so important to understanding nature is a mystery, even to scientists. Back to my mistake in the book: math is a form of logic, so all we need is logic to understand nature. Logic is, at least in theory, something that everyone understands.
Feynman pointed out my error. Math, particularly advanced math, is really quite different than logic, although logic is an important element. Perhaps I am most convinced by the following: the importance of i. Not i as in “me” (but lowercase). i as in the “square root of -1”. Most of you have heard of i, but it does not matter if you have not. I will explain a little.
i is a number that, multiplied by itself, gives -1. Try using simple arithmetic to multiply a number by itself and get a negative number. It cannot be done. That’s because, using simple arithmetic, two negative numbers multiplied by each other always yield a positive number. Since simple arithmetic really is a form of logic, it would seem that if “logic rules nature”, then nature ought to follow the rules of arithmetic. That means nature should avoid i. And so it seemed to until the 1920s, when quantum mechanics (QM) was invented.
Although before QM, scientists sometimes used i as a mathematical convenience, it did not have a fundamental role. One could express the laws of nature strictly in terms of “real” numbers, that is, numbers that do not contain i. Thus, it appeared that nature was logical, mathematics was logical, and the reason math was useful for describing nature is that math was logical. The “reality” of the world was logic, not math.
Feynman set me straight. The reason is that, with the advent of QM, i was no longer a convenient mathematical trick, but it entered into the fundamental physics equations themselves. Before QM was known, using i was unnecessary (although I am sure there were clever theorists who could jam it in there. But it was not really needed). With the advent of QM, i became indispensable. We know of no way to write the fundamental laws without i. It appears that i itself is fundamental.
Since nature and logic are so closely related, one would expect that such a number as i, although possible to invent mathematically, would have very little to do with nature. Nature is supremely logical, but would have no use for a crazy mathematical construct such as i. Yet, we appear stuck with it. What does this mean?
To me, it means that nature is not as logical as I once thought. Nature is more mathematical than logical. Whereas logic is staid and solid, mathematics can be weird indeed. For example, mathematicians have learned to do arithmetic with infinity. They have defined shapes that are smooth, but have no slope. And they have defined shapes that always have a well-defined slope, but are very “jaggy” and jumpy. Mathematicians define worlds with infinite dimensions, and with fractional dimensions, none of which can be visualized or really grasped in an ordinary way.
Certainly, mathematical constructs, even the weird ones, must embody logic. These weird things that mathematicians invent have an underlying logical structure. But they are so much more than that, and so different than the logic that underlies the simple numbers.
Which gets me back to God. Understanding nature requires much more than using logic. It requires inventing weird objects such as i. Without this invention, we would have no way to describe how nature behaves. Nature is supremely logical, but it is also supremely weird. This is something physicists have understood for some time now. These days, the search for new physics is really the search for a new kind of math. I don’t know why that must be.
I believe nature is like a supreme being. It rules all. How nature rules seems to be impenetrable to us, despite all the scientific progress we’ve made. Thus, our understanding of science today, that leads us to reject religion as “foolish” and “arbitrary”, is somehow at a crossroads. The closer nature is to being impenetrable, the closer science is to religion. I am not suggesting that science and religion are the same, but that they become closer to each other as scientific understanding becomes ever more abstract and mathematically weird.
Such may be the math of God.