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Comment by joshdick | Hacker News Reader

I wonder how many false conjectures could pass muster using this sort of probabilistic argument.

I guess every false conjecture can be made to pass it. The trick is to make the set of items searched in large enough.

For example, to show that no elephants exist, start with the (infinite) set of all possible chromosome sets. The proportion of them that produces an elephant is zero. QED.

Examples from mathematics:

The number 42 does not exist (logic: pick an integer. The probability that it equals 42 is zero. QED)

There are no even primes.

There are no primes.

There are no integers.

There are no rational numbers.

All numbers are transcendental.

Continuous functions do not exist.

There are no regular polygons.

Several of these don't work, as it's assumed you tackle smaller values in another way, before your -> infinity method kicks in, so most of these would be easily knocked off.

not sure I follow

"the integers don't exist, because the percent of real numbers that are integers is effectively 0" is what the parent post is implying, which "works", in the sense that using this proof methodology "works"

But it's not the proof "methodology" given by Feynman. Fenyman's result also gives a moderate-sized probability for small N, then eliminates those through known mathematics.

Naw most of these wouldn't work when actually written out as Feynman did. Example: you can easily give an upper bound to the "chance that N is a prime" that goes to zero as N increases. But you would also need to show that it's sum from 0 to infinity over all N also goes to zero.

In fact, there's the classic result that this probability is about 1/log(N) [1], which diverges towards infinity. Hence you would probabilistically expect infinite primes and would be correct.

[1] https://en.wikipedia.org/wiki/Prime_number_theorem

That's because you start with a set (the integers) that contains disproportionally many primes. Just pick a larger set of numbers to compare things with. If you look at the reals, the probability of finding a rational already is zero, and there aren't primes that aren't rational numbers.

Mathematicians of course, would not do that. They do use the technique of estimating the solution size to get a feeling for the difficulty of a problem, but always try to pick a reference set that is such that the exercise teaches them anything, and starting with all reals doesn't (how do they know that? Intuition, or they may do it anyway, but realize half-way through that it is silly, or even publish it, and, eventually, get corrected)

Feynman's solution has more good math, but still makes the fatal mistake of stating that "measure zero implies does not occur" (although he probably knew, since he states it was good enough for him)

One can 'prove' the non-existence of any countable infinite set of numbers this way.

Your arguments are just fundamentally different from Feynman's. We're trying to estimate whether something exists. If you are using a counting measure on the integers, say, then whether or not something exists is whether or not the measure of that set is zero. If you're dealing with, say, a Lebesgue measure on the reals then the measure being zero tells you nothing about whether the set is empty. Put the counting measure on the reals and then you can work, but then the answers you get won't be zero.

Whoa, hang on. Statistical arguments for unproven conjectures are bad, but this counterargument is as bad or worse, especially when you start talking about infinity. Just to address your first example:

> The number 42 does not exist (logic: pick an integer. The probability that it equals 42 is zero. QED)

I object! What is your probability distribution function over the integers? Your phrasing sort of implies a uniform distribution, but there is no such thing as a uniform distribution on an infinite set, and as soon as you pick a plausible pdf the argument stops working.

It is easy to address your objection. On a uniform distribution on [n] := {0, 1, 2, ...n}, P(X=42)->0 as n->infty. This is similar to Feynmann's argument.

You're forgetting to integrate the probability over the domain 0 to n. If you do, you always get 1 for n at least 42.

Those are two separate events. I'm talking about the event n = 42, which makes sense for all [n]. You're talking about the event n >= 42, which is a very different event.

No, I'm not talking about integrating the probability that n >= 42.

What's the chance that a random number in [n] is 42? It's 1/n, if n is at least 42. Sum that over all x in [n] and you get 1, i.e. the probability that there is a number in [n] that is 42.

You're still talking about different events. The event that there is 42 in [n] is different from the even that x = 42.

Then I guess I just don't understand what you're trying to conclude from this argument.

Richard Feynmann was trying to argue probabilistically about the possible existence of a counterexample to FLT. To do this, he more or less tries to put a probability value on the event "a counterexample exists". The problem is that his kind of reasoning is similar to the probability of an event such as "x = 42". Counterexamples to FLT may, a priori, be as rare as the number 42 is within the integers, i.e. with probability tending to zero as you consider larger and larger sets of integers. But a very small probability of a counterexample, even a probability that tends to zero, does not mean a counterexample doesn't exist anymore than it says that 42 doesn't exist.

See I can't see the analogy from your argument to Feynman's for the following reason: you are computing the probability that x=42 for an individual x and saying that it is going to 0 as x->infinity. Feynman first computes the probability that N is x^n + y^n but then integrates that over all N and n in order to get the probability that there is a solution anywhere. His argument takes into account the fact that there could be just one single lone solution in a infinite sea of non-solutions, since his probability estimate is for "there is a solution anywhere" not "this particular number is a solution". And I think that when you add that part to your estimate, you do get a probability of 1 that 42 exists somewhere, even if individually you get a low probability of it being at any given large x.

This detail doesn't matter. He's still computing a tiny nonzero probability that is only zero in the limit. We came up with other examples of tiny nonzero probabilities that are zero in the limit that can lead to the wrong conclusion, such as saying that 42 doesn't exist.

Feynmann's argument is clumsy and wrong. This kind of approximation is ok for a physicist but it doesn't work for mathematics.

> to show that no elephants exist, start with the (infinite) set of all possible chromosome sets. The proportion of them that produces an elephant is zero

This is not obvious. If you're going to postulate an infinite number of possible chromosome sets, you're also going to have to admit that an infinite subset of them all produce elephants.

For example, if you show genome A which does not produce an elephant, perhaps I could show genome A', consisting of (1) genome A; (2) a reference elephant genome; (3) some chromosomes that have the effect of disabling genome A.

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