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Sewing-Machine comments on Syntacticism - Less Wrong Discussion

-3 Post author: ec429 23 September 2011 06:49AM

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Comment author: [deleted] 24 September 2011 02:42:41AM *  2 points [-]

But why should feasibility matter? Sure, the more steps it takes to prove a proposition, the less likely you are to be able to find a proof

Incidentally but importantly, lengthiness is not expected to be the only obstacle to finding a proof. Cryptography depends on this.

As to why feasibility matters: it's because we have limited resources. You are trying to reason about reality from the point of view of a hypothetical entity that has infinite resources. If you wish to convince people to be less skeptical of infinity (your stated intention), you will have to take feasibility into account or else make a circular argument.

But saying that things are true only by virtue of their proof being feasible... is disturbing, to say the least. If we build a faster computer, do some propositions suddenly become true, because we now have the computing power to prove them?

I am certainly not saying that feasible proofs cause things to be true. Our previous slow computer and our new fast computer cause exactly the same number of important things to be true: none at all. That is the formalist position, anyway.

Similarly, no matter how low your prior probability for "PA is consistent", so long as that probability is not 0, learning that I have proved a theorem should cause you to decrease your estimate of the probability that you will prove its negation.

Not so. If I have P(PA will be shown inconsistent in fewer than m minutes) = p, then I also have P(I will prove the negation of your theorem in fewer than m+1 minutes) = p. Your ability to prove things doesn't enter into it.

Comment author: ec429 24 September 2011 03:57:54AM 0 points [-]

lengthiness is not expected to be the only obstacle to finding a proof

True; stick a ceteris paribus in there somewhere.

You are trying to reason about reality from the point of view of a hypothetical entity that has infinite resources.

Not so; I am reasoning about reality in terms of what it is theoretically possible we might conclude with finite resources. It is just that enumerating the collection of things it is theoretically possible we might conclude with finite resources requires infinite resources (and may not be possible even then). Fortunately I do not require an enumeration of this collection.

I am certainly not saying that feasible proofs cause things to be true. Our previous slow computer and our new fast computer cause exactly the same number of important things to be true: none at all. That is the formalist position, anyway.

So either things that are unfeasible to prove can nonetheless be true, or nothing is true. So why does feasibility matter again?

P(I will prove the negation of your theorem in fewer than m+1 minutes) = p

No, it is > p. P(I will prove 1=0 in fewer than m+1 minutes) = p + epsilon. P(I will prove 1+1=2 in fewer than m+1 minues) = nearly 1. This is because you don't know whether my proof was correct.

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