by Edward Nelson, Princeton University
Mathematics and religion—strange bedfellows indeed! But perhaps if one goes deeply enough into any subject, one encounters religion. I want to focus on one question, regarded from three points of view. The question is this: does there exist a completed infinity consisting of all numbers 0, 1, 2, 3, ...?
First consider the question from the perspective of monotheistic faith. As I understand it, such faith regards everything in creation as contingent; God is not bound by necessity. Are we to believe that the truths of number theory, such as Fermat's last theorem (for n greater than 2 there is no solution in positive numbers of x to the n plus y to the n equals z to the n), could have been different had God chosen to make them so? The 19th century mathematician Leopold Kronecker famously said, "God created the integers; all else is the work of Man." It is hard to imagine this act of creation. To quote from my book Predicative Arithmetic, "Nowhere in the book of Genesis do we find the passage: And God said let there be numbers, and there were numbers; odd and even created he them, and he said unto them, be fruitful and multiply; and he commanded them to keep the laws of induction." But if the numbers were not created, do they exist in their infinite magnitude by necessity? The point I am trying to make is that there appears to be a problem for anyone who subscribes to monotheistic faith and also subscribes to faith in the existence of a completed infinity of all numbers.
I was greatly intrigued by Max Tegmark's presentation during the Mathematics and Religion roundtable at the Philoctetes Center. He is an exponent of pure Pythagoreanism: all is number. To maintain that there is no basic difference between existence in physical reality and existence as mathematical possibility is a challenging and thought-provoking position. I hope Max will not be offended if I confess to difficulty in imagining what a creature somewhere in the multiverse with aleph 17 toes would look like. (Aleph 17 is one of Georg Cantor's infinite cardinals.) Historically, the Pythagorean Society was a religious group, and it also created mathematics as a deductive discipline—mathematics as mathematicians understand mathematics. So although the topic of mathematics and religion may seem strange to us, mathematics has a religious origin.
I would guess that this is an accident of our planet and that when we encounter some intelligent extraterrestrials, they will not have any pure mathematics—just a very advanced Babylonian mathematics tied to the everyday world. Incidentally, the Pythagorean Society is often called the Pythagorean Brotherhood, but one of the few things we know about them with some degree of confidence is that women were members with equal status. The belief that all is number, that the universe is permeated by the music of the spheres, is a beautiful religious belief. And Pythagoreanism is the most harmless of all religions. But I have not been converted.
If we reject the notion that the completed infinity of numbers was divinely created, and if we reject the notion that this completed infinity exists uncreated and by necessity, as it was in the beginning, is now, and ever shall be, what is left that we can accept? Just this: that the notion is a human fabrication. The idea of a completed infinity is an abstract notion, but it is a terrible reality that abstract notions have concrete consequences. The example of the abstract notion of the Aryan race suffices to make the point.
Mathematical activity is a concrete human activity. Mathematicians prove theorems and the community of mathematicians agrees, after sufficient study, as to whether or not the proof is correct. It is just a matter of checking. This is an astounding consensus covering the globe and millennia of work. No other field of human endeavor matches this. The chief concern of mathematicians in practice is proof—correct deductions from axioms (thanks to Pythagoras!). The notion of truth in mathematics, however, is a matter of dispute among mathematicians; truth in mathematics is an abstract notion. Andrew Wiles proved Fermat's last theorem, and those competent to judge agree that the proof is a correct deduction from the axioms. The theorem has some concrete content—no one will ever find n, x, y, and z that falsify the theorem. This is due to the fact that the theorem has a simple logical structure. It is of the form: for all numbers, something concrete holds.
But consider a more complicated assertion of the form: for all numbers, there exists a number such that something concrete holds. An example is the twin primes conjecture: for all numbers n there exists a number p with p greater than n, such that both p and p + 2 are primes. This is an open problem. The notion of truth for this problem is entirely abstract&mdashit involves a hypothetical and impossible search through all numbers. What mathematicians hope for is that someone will find a proof (a concrete object) either of the conjecture or of its negation. And this leads to the final thing I want to say.
The question as to whether number theory (Peano Arithmetic) is consistent or not is a question similar to Fermat's last theorem. If number theory is inconsistent, there is a concrete proof of a contradiction from the axioms. I am working on showing that this is indeed the case. The almost universally held belief that this is impossible is based on the abstract notion of truth in number theory, a belief in the existence of the set of all numbers as a completed infinity.
This paper was commissioned by the Philoctetes Center as part of a grant from The John Templeton Foundation. This and three other commissioned pieces are featured in a special issue of Dialog, the newsletter of the Philoctetes Center.