I Am a Strange Loop

divisor. This contradiction came out of our assumption that there was a Finite, Closed Club of Primes, gloriously crowned by P, and so we have no choice but to go back and erase that whole amusing, suspect vision.

There cannot be a “Great Last Prime in the Sky”; there cannot be a “Finite, Closed Club of All Primes”. These are fictions. The truth, as we have just demonstrated, is that the list of primes goes on without end. We will never, ever “fall off the Earth”, no matter how far out we go. Of that we now are assured by flawless reasoning, in a way that no finite amount of computational sailing among seas of numbers could ever have assured us.

If, perchance, coming to understand why there is no last prime (as opposed to merely knowing that it is the case) was a new experience to you, I hope you savored it as much as a piece of chocolate or of music. And just like such experiences, following this proof is a source of pleasure that one can come back to and dip into many times, finding it refreshing each new time. Moreover, this proof is a rich source of other proofs — Variations on a Theme by Euclid (though we will not explore them here).

The Mathematician’s Credo

We have just seen up close a lovely example of what I call the “Mathematician’s Credo”, which I will summarize as follows:

X is true because there is a proof of X;

X is true and so there is a proof of X.

Notice that this is a two-way street. The first half of the Credo asserts that proofs are guarantors of truth, and the second half asserts that where there is a regularity, there is a reason. Of course we ourselves may not uncover the hidden reason, but we firmly and unquestioningly believe that it exists and in principle could someday be found by someone.

To doubt either half of the Credo would be unthinkable to a mathematician. To doubt the first line would be to imagine that a proved statement could nonetheless be false, which would make a mockery of the notion of “proof ”, while to doubt the second line would be to imagine that within mathematics there could be perfect, exceptionless patterns that go on forever, yet that do so with no rhyme or reason. To mathematicians, this idea of flawless but reasonless structure makes no sense at all. In that regard, mathematicians are all cousins of Albert Einstein, who famously declared, “God does not play dice.” What Einstein meant is that nothing in nature happens without a cause, and for mathematicians, that there is always one unifying, underlying cause is an unshakable article of faith.

No Such Thing as an Infinite Coincidence

We now return to Class A versus Class B primes, because we had not quite reached our revelation, had not yet experienced that mystical frisson I spoke of. To refresh your memory, we had noticed that each line was characterized by differences of the form 4n — that is, 4, 8, 12, and so forth. We didn’t prove this fact, but we observed it often enough that we conjectured it.

The lower line in our display starts out with 3, so our conjecture would imply that all the other numbers in that line are gotten by adding various multiples of 4 to 3, and consequently, that every number in that line is of the form 4n + 3. Likewise (if we ignore the initial misfit of 2), the first number in the upper line is 5, so if our conjecture is true, then every subsequent number in that line is of the form 4n + 1.

Well, well — our conjecture has suggested a remarkably simple pattern to us: Primes of the form 4n + 1 can be represented as sums of two squares, while primes of the form 4n + 3 cannot. If this guess is correct, it establishes a beautiful, spectacular link between primes and squares (two classes of numbers that a priori would seem to have nothing to do with each other), one that catches us completely off guard. This is a glimpse of pure magic — the kind of magic that mathematicians live for.

And yet for a mathematician, this flash of joy is only the beginning of the story. It is like a murder mystery: we have found out someone is dead, but whodunnit? There always has to be an explanation. It may not be easy to find or easy to understand, but it has to exist.

Here, we know (or at least we strongly suspect) that there is a beautiful infinite pattern, but for what reason? The bedrock assumption is that there is a reason here — that our pattern, far from being an “infinite coincidence”, comes from one single compelling, underlying reason; that behind all these infinitely many “independent” facts lies just one phenomenon.

As it happens, there is actually much more to the pattern we have glimpsed. Not only are primes of the form 4n + 3 never the sum of two squares (proving this is easy), but also it turns out that every prime number of the form 4n + 1 has one and only one way of being the sum of two squares. Take 101, for example. Not only does 101 equal 100 + 1, but there is no other sum of two squares that yields 101. Finally, it turns out that in the limit, as one goes further and further out, the ratio of the number of Class A primes to the number of Class B primes grows ever closer to 1. This means that the delicate balance that we observed in the primes below 100 and conjectured would continue ad infinitum is rigorously provable.

Although I will not go further into this particular case study, I will state that many textbooks of number theory prove this theorem (it is far from trivial), thus supplementing a pattern with a proof. As I said earlier, X is true because X has a proof, and conversely, X is true and so X has a proof.

The Long Search for Proofs, and for their Nature

I mentioned above that the question “Which numbers are sums of two primes?”, posed almost 300 years ago, has never been fully solved. Mathematicians are dogged searchers, however, and their search for a proof may go on for centuries, even millennia. They are not discouraged by eons of failure to find a proof of a mathematical pattern that, from numerical trends, seems likely to go on and on forever. Indeed, extensive empirical confirmation of a mathematical conjecture, which would satisfy most people, only makes mathematicians more ardent and more frustrated. They want a proof as good as Euclid’s, not just lots of spot checks! And they are driven by their belief that a proof has to exist — in other words, that if no proof existed, then the pattern in question would have to be false.

This, then, constitutes the flip side of the Mathematician’s Credo:

X is false because there is no proof of X;

X is false and so there is no proof of X.

In a word, just as provability and truth are the same thing for a mathematician, so are nonprovability and falsity. They are synonymous.

During the centuries following the Renaissance, mathematics branched out into many subdisciplines, and proofs of many sorts were found in all the different branches. Once in a while, however, results that were clearly absurd seemed to have been rigorously proven, yet no one could pinpoint where things had gone awry. As stranger and stranger results turned up, the uncertainty about the nature of proofs became increasingly disquieting, until finally, in the middle of the nineteenth century, a powerful movement arose whose goal was to specify just what reasoning really was, and to bond it forever with mathematics, fusing the two into one.

Many philosophers and mathematicians contributed to this noble goal, and around the turn of the twentieth century it appeared that the goal was coming into sight. Mathematical reasoning seemed to have been precisely characterized as the repeated use of certain basic rules of logic, dubbed rules of inference, such as modus ponens: If you have proven a result X and you have also proven X ? Y (where the arrow represents the concept of implication, so that the line means “If X is