5 ms·

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

Someone 10y ago
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.
- CJefferson 10y ago
  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.
  - aji 10y ago
    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"
    - tgb 10y ago
      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.
- tgb 10y ago
  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 https://en.wikipedia.org/wiki/Prime_number_theorem
  - Someone 10y ago
    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.
    - tgb 10y ago
      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.
- Analemma_ 10y ago
  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.
  - jordigh 10y ago
    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.
    - tgb 10y ago
      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.
      - jordigh 10y ago
        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.
        tgb 10y ago
        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.
        jordigh 10y ago
        You're still talking about different events. The event that there is 42 in [n] is different from the even that x = 42.
- thaumasiotes 10y ago
  > 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.

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.

CJefferson 10y ago
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.
- aji 10y ago
  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"
  - tgb 10y ago
    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.
tgb 10y ago
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 https://en.wikipedia.org/wiki/Prime_number_theorem
- Someone 10y ago
  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.
  - tgb 10y ago
    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.
Analemma_ 10y ago
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.
- jordigh 10y ago
  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.
  - tgb 10y ago
    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.
    - jordigh 10y ago
      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.
      - tgb 10y ago
        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.
        jordigh 10y ago
        You're still talking about different events. The event that there is 42 in [n] is different from the even that x = 42.
thaumasiotes 10y ago
> 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.