Statistics

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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Categories: Statistics

19 replies »

  1. And our esteemed host just disproved the Copenhagen interpretation of quantum physics. Nicely done.
    Notice that quantum computing is dependent upon Copenhagen being true. Which it is not. Gotta love a good science scam.

  2. I just watched ‘Havana’ (1990) starring Robert Redford as a professional gambler in the casinos of Havana in the run-up to the revolution. At one point you see he has two books on his nightstand: ‘The Laws of Probability’ and ‘A Theory of Numbers’. Most likely this was some prop-master’s idea for items that suggest that the reason he is a successful poker player is that he studies math —- with ‘a theory of numbers’ being misunderstood as a synonym for math as a whole. Alternatively, however, it could be that this professional gambler was researching prime number distribution in his spare time !

  3. William: Or maybe Redford’s character was just an expert at reading faces. Poker is a very large percentage “bluff”, which I don’t recall occurring in math, though it might now.

  4. My comment only applies to live poker. I forget how many people think playing all day with a computer is somehow real poker (or real anything for that matter).

  5. “Examples: 9 and 11, 11 and 13, 17 and 19, and on and on. Probably forever.”

    Forgive me if I’m speaking in ignorance, but 9 isn’t a prime number; it’s divisible by 3.

  6. Shecky, you just reminded me I have to write a Derisive Empty Laugh Fallacy chapter in my new book.

    Your comments will supply all the examples I need.

    Thanks!

  7. “It simply is true or false, but we don’t yet know. ” And it may be true and unproveable, per Kurt Godel (though, admittedly, not likely). Also, “given the Peano Axioms (or similar) as a basis.”

    “…70,000,000, which is larger than 2, but, relatively speaking, is still a small number.” Any number you can name or define is, relatively speaking, a small number. Whatever number you name or define, almost all numbers are larger.

    “Pr(Riemann Hypothesis | all that is known about math) = large” There is a compelling probabilistic argument involving the apparent “randomness” (I know, I know) of the primes that the Riemann hypothesis is true with probability 1. On the other hand, the fact that all know non-trivial zeros of the Riemann zeta function are on the Re=1/2 line in the complex plane and that (last I heard) the imaginary part of the largest known zero has the absolute value of something like 1.3×10^22 gives little comfort since we are looking at functions involving log log and such functions grow ridiculously slowly.

  8. Hitler dropped a load of lead in my shorts, and all Briggs cares about are some odd numbers. Oh, the humanity!

  9. Brought to mind this exposition on Fermat finding a conjecture that MIGHT be easier to prove together with an “unknown” but small conjecture which would bridge the “POSSIBLE” conjecture with Fermat

    It would be left to prove the conjecture which had a greater possibility of proof

    https://www.youtube.com/watch?v=nUN4NDVIfVI

  10. I just finished your excellent book. Got it from Amazon last Wednesday. Seems to me that Fauci and the media report mainly local truths about Coronadoom. Each local truth building upon the next leading to galactic terror. Nobody is liable for spreading any of the falsehoods because each is plausibly true given some assumptions. Unfortunately, the pile of local truths refuses to fall when one of the bricks crumbles into falsehood. Same can be said for Climate Change and many other beliefs.

  11. You say: “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.”

    Yes, nicely said! Lewis Carroll said approximately the same thing more elegantly (though not so concisely) some years ago.
    http://www.ditext.com/carroll/tortoise.html

  12. “It’s easy to see why probability is not frequency, because there’s no frequency of the EH conjecture:”

    It is easy to see that a frequentist wouldn’t define that as probability in the first place, since probability is defined as converging relative frequency irrelevant to the starting point (randomness).

    Justin

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