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> If in fact the mind relies on a continuum for its cognitive functions, this would explain why cognitive functions cannot be inherited and why our minds can de
by chriswarbo 4y ago
> If in fact the mind relies on a continuum for its cognitive functions, this would explain why cognitive functions cannot be inherited and why our minds can deal with real numbers, despite the paradoxes.
I disagree with the assertion that "our minds can deal with real numbers". AFAIK there's no way to distinguish an uncomputable number from a computable one, using a finite set of computations (i.e. there's always some computable model which gives the same answers as our uncomputable number, even if it's highly convoluted); so we're still only learning about the countable, computable sub-set.
- Nomentatus 4y ago"Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think): You can show that some functions that give real numbers are computable. (Let's say, computing that number digit by digit. Binary digit or base-ten, you choose.) You can't show that all Turing machines halt, so you can't use the mere existence of a Turing machine as proof of a computable function. Therefore, we can't know or compute the set of all real numbers because we can't compute the set of irrational numbers; there will always be some Turing machine that, unbenownst to us, calculates a (likely irrational) function that spits out a real number. If we can't describe even a way to sort functions into two piles: "computes a real number (digit by digit)" and "doesn't compute a real"; then we can't be said to "deal with" the set of real numbers at all well. This rather presupposes the (now pretty much accepted) concept that numbers "exist" if they can be computed (whether they have been computed, or conceived of, or not.) "Intuitionism" in other words, IIRC.