4 ms·

But why would anyone assume that n horses, for arbitrary n, are the same color? That's absurd. Anyone who's ever seen a field with horses in it would triviall

by na85 5y ago

But why would anyone assume that n horses, for arbitrary n, are the same color? That's absurd. Anyone who's ever seen a field with horses in it would trivially know that this is a faulty assumption to make. You might as well say "We can prove that coconuts can migrate. First, assume that coconuts can migrate. Thus, coconuts migrate". It's equally valid logic.

mannerheim 5y ago
So it's worth noting that the inductive hypothesis isn't false for all values of n - namely, it is true for n = 0,1. But beyond that, in mathematical logic, for p -> q, you can still prove this to be true, even if p is false. In fact, if you can prove p is always false, p -> q is 'vacuously true' - it's true because you can never come up with an example of 'p and not q'. So mathematically, there's no problem with assuming an induction hypothesis that's actually false here. The real problem occurs is that the result doesn't follow from the assumption because the induction step of the 'proof' sneakily assumes that n >= 3, i.e., the induction step simply doesn't work for the n = 2 case.
ajuc 5y ago
> But why would anyone assume that n horses, for arbitrary n, are the same color? because that's how inductive proofs work. You assume something is true for N and show that this implies that it's also true for N+1. This combined with base case (for example for N=1) proves that it's true for all N >= 1. The assumption is just a tool used to check if you can prove the implication, the real "meat" of the proof is in the implication. Are you aware of proofs by contradiction? A: "assume N is the largest natural number". If that's true then there shouldn't be any natural numbers larger than N, but we can create N+1 and show it's larger AND natural, so assumption A leads to contradiction, so there is no such thing as the largest natural number. We used false assumption in our proof, but the proof was correct. It's a similar idea with inductive proofs - you make an assumption and see where it leads you. You don't use the assumption for the proof, you use the implication A(N)=> A(N+1) for the proof, the assumption A is just a way to see if you can prove the implication. > You might as well say "We can prove that coconuts can migrate. First, assume that coconuts can migrate. Thus, coconuts migrate". It's equally valid logic. The logic isn't actually circular, your coconut analogy is wrong. Correct analogy would be: "when we assume N coconuts can migrate we can formally show that it implies N+1 coconuts can migrate" + "we can show that 1 coconut can migrate". You only need these 2 facts to prove all coconuts can migrate.
- na85 5y ago
  >because that's how inductive proofs work. Well, anyone who's seen a field of horses knows that "assume any set of n horses has the same color" is not valid. So if what you say is true, then that implies the entire field of inductive reasoning is horse shit. But I don't think that's the case, so something's missing.
  - ajuc 5y ago
    What is missing is understanding that in math the sentence "if X = 3 then 2 * X = 6" is true even when X = 5. For the proof to work the implication must be true, not the assumption.
    - na85 5y ago
      But you're not trying to prove X = 3 and starting out by assuming X = 3. There wasn't even an attempt to disprove the assumption in TFA.
      - ajuc 5y ago
        > But you're not trying to prove X = 3 and starting out by assuming X = 3. No, you're trying to prove for all N>=1: A(N) is true and start by proving that IF A(N) is true THEN A(N+1) is true This implication can be true even if A(N) is false, and your final proof will only use the implication, not the assumption used when proving that implication. Assumptions in math proofs can have scopes. For example in many proofs you split the domain into subsets and prove that something is true assuming X>=0 and X<0 separately. Naively you would say that we cannot assume X>=0 and then assume X<0 in the same proof because that's a contradicion. But these assumptions were used in different scopes so there's no contradiction.
        na85 5y ago
        We're talking past each other. I give up.
  - 5y ago

by na85 5y ago

mannerheim 5y ago

So it's worth noting that the inductive hypothesis isn't false for all values of n - namely, it is true for n = 0,1. But beyond that, in mathematical logic, for p -> q, you can still prove this to be true, even if p is false. In fact, if you can prove p is always false, p -> q is 'vacuously true' - it's true because you can never come up with an example of 'p and not q'. So mathematically, there's no problem with assuming an induction hypothesis that's actually false here. The real problem occurs is that the result doesn't follow from the assumption because the induction step of the 'proof' sneakily assumes that n >= 3, i.e., the induction step simply doesn't work for the n = 2 case.

ajuc 5y ago

> But why would anyone assume that n horses, for arbitrary n, are the same color? because that's how inductive proofs work. You assume something is true for N and show that this implies that it's also true for N+1. This combined with base case (for example for N=1) proves that it's true for all N >= 1. The assumption is just a tool used to check if you can prove the implication, the real "meat" of the proof is in the implication. Are you aware of proofs by contradiction? A: "assume N is the largest natural number". If that's true then there shouldn't be any natural numbers larger than N, but we can create N+1 and show it's larger AND natural, so assumption A leads to contradiction, so there is no such thing as the largest natural number. We used false assumption in our proof, but the proof was correct. It's a similar idea with inductive proofs - you make an assumption and see where it leads you. You don't use the assumption for the proof, you use the implication A(N)=> A(N+1) for the proof, the assumption A is just a way to see if you can prove the implication. > You might as well say "We can prove that coconuts can migrate. First, assume that coconuts can migrate. Thus, coconuts migrate". It's equally valid logic. The logic isn't actually circular, your coconut analogy is wrong. Correct analogy would be: "when we assume N coconuts can migrate we can formally show that it implies N+1 coconuts can migrate" + "we can show that 1 coconut can migrate". You only need these 2 facts to prove all coconuts can migrate.

na85 5y ago
>because that's how inductive proofs work. Well, anyone who's seen a field of horses knows that "assume any set of n horses has the same color" is not valid. So if what you say is true, then that implies the entire field of inductive reasoning is horse shit. But I don't think that's the case, so something's missing.
- ajuc 5y ago
  What is missing is understanding that in math the sentence "if X = 3 then 2 * X = 6" is true even when X = 5. For the proof to work the implication must be true, not the assumption.
  - na85 5y ago
    But you're not trying to prove X = 3 and starting out by assuming X = 3. There wasn't even an attempt to disprove the assumption in TFA.
    - ajuc 5y ago
      > But you're not trying to prove X = 3 and starting out by assuming X = 3. No, you're trying to prove for all N>=1: A(N) is true and start by proving that IF A(N) is true THEN A(N+1) is true This implication can be true even if A(N) is false, and your final proof will only use the implication, not the assumption used when proving that implication. Assumptions in math proofs can have scopes. For example in many proofs you split the domain into subsets and prove that something is true assuming X>=0 and X<0 separately. Naively you would say that we cannot assume X>=0 and then assume X<0 in the same proof because that's a contradicion. But these assumptions were used in different scopes so there's no contradiction.
      - na85 5y ago
        We're talking past each other. I give up.
- 5y ago