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Logical difficulties in modern mathematics (2012)

njwildberger.com

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Re: Logical difficulties in modern mathematics (2012)

#91

Post by pron »

MaxBarraclough wrote: post #90
pron wrote: post #89

Earlier quoted context omitted.

> Some philosophical ideas seem increasingly unsustainable in the face of mounting scientific knowledge. Vitalism, for instance. OK, but the philosophy of mathematics is a more slippery beast. > But if you do include it, don't you have to commit arbitrarily to either AC or ¬AC? No. You can have a theory that's consistent with either. > Isn't this 'just' a question of knowing that the correspondence is valid? Correspo…

> No. You can have a theory that's consistent with either. I think I see your point. If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all. > Are those objects platonic ideals? Are they abstractions of physical reality? My intuition is to favour the former, as the latter seems a much stronger claim. It seems reasonable to say that…

> If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all.

That's not what I meant. I meant that choice can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false.

> My intuition is to favour the former, as the latter seems a much stronger claim.

Well, Platonism is a popular philosophy of mathematics among mathematicians, but many strongly reject it.

Re: Logical difficulties in modern mathematics (2012)

#92

Post by MaxBarraclough »

pron wrote: post #91
MaxBarraclough wrote: post #90

Earlier quoted context omitted.

> No. You can have a theory that's consistent with either. I think I see your point. If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all. > Are those objects platonic ideals? Are they abstractions of physical reality? My intuition is to favour the former, as the latter seems a much stronger claim. It seems reasonable to say that…

> If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all. That's not what I meant. I meant that choice can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false. > My intuition is to favour the former, as t…

> can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false

So it's similar to, but not the same thing as, modifying the LEM to permit Unknowable as a third category (beyond the proposition is true and its negation is true). Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.

Re: Logical difficulties in modern mathematics (2012)

#93

Post by pron »

MaxBarraclough wrote: post #92
pron wrote: post #91

Earlier quoted context omitted.

> If your theory has no way to express the question Is the AC true? then the LEM is no issue, as there's no 'proposition' to worry about at all. That's not what I meant. I meant that choice can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false. > My intuition is to favour the former, as t…

> can be neither proven nor disproven. The philosophical fact that, due to LEM, it must either be true or not is irrelevant, because you cannot rely on it being true or on it being false So it's similar to, but not the same thing as, modifying the LEM to permit Unknowable as a third category (beyond the proposition is true and its negation is true ). Instead we maintain that it has a truth value one way or the other,…

> modifying the LEM to permit Unknowable as a third category

LEM doesn't need to be modified, and it's not a third category. In classical logic (with LEM), unlike in constructive/intuitionistic logic (no LEM), truth and provability are simply not the same -- in fact, it is LEM that creates that difference.

> Instead we maintain that it has a truth value one way or the other, but that it happens to be both unknowable and of no consequence.

Precisely! That is the difference between classical and constructive logic. In constructive logic truth and provability are the same, and something is neither true nor false until it has been proven or refuted. But before you conclude that this kind of logic, without LEM, is obviously better (philosophically), you should know that any logic has weirdnesses. What happens without LEM, for example, is that it is not true that every subset of a finite set is necessarily finite. ¯\_(ツ)_/¯

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