In a previous essay I discussed the elegance of the laws of physics. It is a topic that deserves a book – so the least I could do is write another essay.
Is Elegance Surprising?
Maybe we shouldn’t be surprised if the mathematics expressed in our universe is beautiful, because mathematics in general is beautiful. I could pull all sorts of quotes from the internet from famous mathematicians who agree, like from here. I could elaborate on examples of such beauty, like Euler’s identity or Penrose tilings or the Mandelbrot set (rest in peace Dr. Mandelbrot) or the distribution of the prime numbers. You ought to Google these examples if you haven’t heard of them. I think I will elaborate on the first one because it’s frankly incredible.Euler’s Identity
Euler’s identity is e i π + 1 = 0. Amazingly, this equation relates the five most important numbers in mathematics using each of the most important operators exactly once.(The operators are exponentiation, multiplication, and addition, which are fundamental to our basic number system. The numbers are as follows: 0 and 1 are obviously the most important concepts in mathematics without which we couldn’t do anything. The number e, a little over 2.71, is the base of the natural logarithm and appears all over the place in statistics and physics. The number π [pi], about 3.14, is the ratio of a circle’s circumference to its diameter and also appears often in analysis. The numberi is defined as the square root of −1, which opens a whole branch of mathematics called complex analysis that is very important in physics.)
That they all come together so simply is amazing. It’s all the more amazing when you consider e and π are transcendental numbers, which means basically that they are deeply irrational (their irrationality is deeper than that of the square root of two, for example, which is irrational but not transcendental).
One would not expect two seemingly unrelated transcendental numbers to come together so simply to produce 1 and 0. There are some nice quotations about the identity’s beauty on Wikiquote, such as this one from Benjamin Peirce: “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.” The reason the identity is proven is that it is a special case of Euler’s formula,e i θ = cos θ + i sin θ, where θ is set to π. Euler’s formula is also amazing in the way it unifies branches of mathematics. There are a several cool proofs for it in its Wikipedia article. Physicist Richard Feynman called the formula the “jewel” of physics because it is so astonishing yet so useful.
Math, Beauty, and Truth
So that’s how we know this amazing equation is true. The question that’s impossible to answer is why it is true. Why is it true? It just is. It couldn’t notbe true. Once you define the operators and the numbers, the equation is automatically true, just like 1 + 1 = 2 is true. If aliens appeared from a parallel universe that operates in a fashion fundamentally different from ours, and we asked them for the value of e i π + 1, we can’t comprehend them answering anything other than, “Ah yes, that’s equal to 0.” This beautiful equation is inherently equal. You might then say that beauty is inherent to truth.The next question then is what role God might have played in “creating” mathematics. On the one hand, it seems that the ideas of mathematics are inherently true given the axioms. On the other hand, saying God is bound to the logic of mathematics would seem to be a limitation on God’s omnipotence. It’s a paradox I guess. I like the quote from Leopold Kronecker, “God created the integers, all else is the work of man.” Kronecker meant it in opposition to certain mathematical developments in his era that are now accepted; but the idea itself—God established the axioms, and the rest worked itself out—is still very deep, and very debatable. (Kronecker was a Jew, actually. So between him and Mandelbrot I’ve referenced two famous Jewish mathematicians who were also born in Poland. I don’t think that happens very often. Also incidentally, God Created the Integers is the title of another book by Stephen Hawking.)
As for Rabbinic philosophers on this issue, the only source I’ve seen is from the Rambam, who says in the Guide I, 75 that God is in fact bound by the rules of mathematics, yet that in no way is a defect in His perfection. If anyone knows of other sources or a discussion somewhere on this topic I would be grateful to hear about it. I might otherwise suggest that God is beyond logic, like “your thoughts are not My thoughts,” and so the logic and mathematics of our universe is a kind of tzimtzum like the rest of reality (I believe these kinds of ideas exist in kabbala).
As an analogy, we’re like people in Flatland trying to imagine objects of higher dimensions, while God is beyond dimensionality. These are deep metaphysical questions, of course, and I don’t pretend to have made any sort of start at fully asking them, let alone answering them. But I think it’s clear that the answer won’t be given by Stephen Hawking with M-Theory.
Anyway, my purpose at the moment is to point out that mathematics has inherent beauty.
Back to Physics
So, if mathematics itself has inherent beauty, what do we make of the beauty in our universe’s laws of physics? Can it still point to the creativity of God? Or is it “just” a manifestation of beauty that was “already” part of the math?Well, first of all, it’s amazing that physics follows mathematics at all. There’s this famous essay by a Nobel-winning physicist (also Jewish!) on that topic, or even better, the ideas of this profound philosopher (not Jewish).
The next point is that while mathematics has a lot of true statements that are beautiful, it also has a lot of statements that are trivial. Again, the difference between beauty and triviality is in the eye of the beholder, but I don’t think many would disagree that e i π + 1 = 0 is beautiful and 1 + 1 = 2 is trivial. Even once we get past the fact that reality obeys mathematical equations, it’s still amazing that it illustrates so many of the beautiful equations so vividly. As Feynman said, you can’t “change the laws much” without losing a lot of the elegance.
Maybe the following analogy can work: The world of mathematics is like a library of musical scores. There are simple compositions, like straightforward addition, and there are sophisticated ones, like Euler’s formula. Then, physics is the orchestra hall where the music is actually performed. And it appears that God chose some of the most beautiful melodies to play in His symphony, and arranged them together to produce spine-tingling harmonies.
