Local & Necessary Truths In Mathematical Proofs — And Probability

This video requires little math beyond what used to be taught in college. Its speaker is Terry Tao, who some say is the world’s best living mathematician. Its (fun—truly) subject is gaps between primes. Most of the video is taken up on the Twin Prime Conjecture.

This is the claim that for primes p_n+1 and p_n, p_n+1 – p_n = 2 infinitely often. It’s easy to find these twin primes. Examples: 5 and 7, 11 and 13, 17 and 19 and on and on. Probably forever.

That “on and on” hasn’t been proved. Everybody believes it will be proved. Or, to be strict, as we should be, most are fairly certain it will be proved someday. The level of this certainty is the point of this article.

Starting at about 3 minutes is the story of Yitang Zhang, who discovered a bound of the lowest gap which occurs infinitely often, which (ignoring the screwy number 2, and gap between 2 and 3 is 1) is thought to be 2 (the gap). Zhang originally proved this gap is bounded by p_n+1 – p_n ≤ 70,000,000, which is larger than 2, but, relatively speaking, is still a small number.

This 70 million was successively shrunk, first to 4,680, then 600, then 246. This was the best, Tao says, that can be done using a particular method. This method relies on what is called the Elliott–Halberstam conjecture, another proposition nobody knows if it is true or false. But, like the Riemann Hypothesis, it is generally thought it might be true.

Are you still with me?

Let me state it another way. It is true that the lowest bound (using Zhang’s method) is less than or equal to 246 if the Elliott–Halberstam conjecture is true. Or, rather, assuming it is true.

It doesn’t make any difference whether the EH conjecture is false. It will still be true that 246 is this lowest bound assuming the EH conjecture is true—even if it is false.

We can call “246”, then, a local truth.

Mathematicians use these kinds of tools all the time. Local truths abound. The hope is to turn then into universal or necessary truths. If, for instance, the EH conjecture is proved, it itself is turned into a necessary truth. It being a necessary or universal truth, the 246 then becomes a universal truth.

Be careful about what we mean here. It is not that the EH conjecture is turned from false to true by any of our actions. It simply is true or false, but we don’t yet know. It is our knowledge of the conjecture’s truth or falsity that is changed by the proof.

This is so for every proof. That which we seek to prove is true or false independently of us. It’s truth or falsity is only unknown. Any proof only changes our knowledge. All proofs, local or universal, rely on long strings of argument, where each premise in the argument, and each working step or tool, is known itself to be true by the same means. (Much more on all this in Uncertainty!)

Proofs, then, are epistemological. And not only that. All proofs are conditional, as we just saw.

In other words, math and probability are the same.

Right now we have that

     Pr(Riemann Hypothesis | all that is known about math) = large.

It’s not quantified, because it’s not clear precisely just how to tie “all that is known about math” to an exact number. But any mathematician would agree to this statement—though perhaps not to classing it in probabilistic terms, because some still labor under the belief that probabilities are frequencies or are subjective. Neither is true. Probability, like math, is pure logic.

Here’s another that’s easier:

     Pr(2 + 2 = 4 | all that is known about math) = 1.

Which is the same in philosophy as this:

     Pr( That 246 twin prime bound | EH conjecture ) = 1.

Which is not the same as this:

     Pr(That 246 twin prime bound | all that is known about math) = large.

This is because all that is known about math acknowledges the EH conjecture is not known to be true, but it is likely true given all that is known (stay with me, here).

All that “Pr(*|*)” business is just shorthand for the long, but always incomplete, proofs mathematicians write. Those proofs are incomplete because no mathematician (or logician) writes every premise out, explicit and implicit. Indeed, implicit premises are the bulk of premises used is most proofs. These include knowledge of what the symbols mean, the logical steps used, and so on.

It’s easy to see why probability is not frequency, because there’s no frequency of the EH conjecture: It’s true or it isn’t. And probability is NOT subjective, because there’s no room for doubt in necessary truths.

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