I appreciate you explaining intuitionism. I was aware of the ideas, but getting confirmation of what the important/foundational ones were was nice. (Also, that implication about physical laws was really interesting.)
Short:
So we can understand the idea of an infinity of natural numbers by thinking of them as a sequence of reflection upon temporal moments that goes on and on without end, 1, 2, 3, 4…. For Intuitionists, there are only potential infinities, not actual infinities. What that means is that we cannot regard an infinity as something that exists as a completed object (it never exists fully at any moment in time). Instead, when we talk about an infinite set, like “the set of all natural numbers” what we should mean is, “the temporal process that generates each natural number with no limit on how long it may continue to run”.
So “infinite” just means:
The process does not terminate itself.
The list cannot fully be created, because it is generated by a process without an (inherent) end.
Long:
In finite time, my computer will only ever print out a certain finite subset of the natural numbers. The process is infinite only in the sense that it has no natural stopping point (we are not considering contingent stopping points like my computer crashing or breaking down or losing power).
Assuming your computer is normal, there is a limit to how big a number it can express, much less contain (relative to a given method). (This amount can be calculated from how much “space” there is on your computer, though if you used up all the space, it might not be able to handle displaying the number, or make it particularly easy to access.)
In contrast to Intuitionism, classical mathematics regards infinite sets as well-defined abstract objects.
With some debates concerning their relative sizes?
et, “A or not A” may becometrue at a later point in time, once we have produced a proof of one of the two sides.
Causality seems a better model, but time works well enough. Another way of handling this would be ‘the truth value of “A or not A” is not defined until you choose A’.
I see the truth value of “A or not A” as ‘defined in the abstract as true’ as a consequence of having defined ‘truth’, “or”, and deciding that I will give A a value of true, or a value of false. (Alternatively, I can leave it as a function.)
So, the rules of intuitionistic logic and the account of the essentially temporal nature of numbers are mutually consistent.
If one motivated (or was cherry picked to justify) the other this doesn’t seem like it should be a surprise?
In contrast, classical mathematics (which has been used to develop all our current theories of physics) assumes that every real number possesses an infinite degree of precision at every moment in time, and this is consistent with classical logic’s endorsement of the law of the excluded middle.
In the same sense that for in turing machine, it must either halt or not halt—whether or not we have any way of knowing that.
“there exists arbitrarily precise approximations of such and such a real number.”
This is equivalent to “we can construct as many digits of this number as we want, though it might be infinitely long after the decimal”.
I appreciate you explaining intuitionism. I was aware of the ideas, but getting confirmation of what the important/foundational ones were was nice. (Also, that implication about physical laws was really interesting.)
Short:
So “infinite” just means:
The process does not terminate itself.
The list cannot fully be created, because it is generated by a process without an (inherent) end.
Long:
Assuming your computer is normal, there is a limit to how big a number it can express, much less contain (relative to a given method). (This amount can be calculated from how much “space” there is on your computer, though if you used up all the space, it might not be able to handle displaying the number, or make it particularly easy to access.)
With some debates concerning their relative sizes?
Causality seems a better model, but time works well enough. Another way of handling this would be ‘the truth value of “A or not A” is not defined until you choose A’.
I see the truth value of “A or not A” as ‘defined in the abstract as true’ as a consequence of having defined ‘truth’, “or”, and deciding that I will give A a value of true, or a value of false. (Alternatively, I can leave it as a function.)
If one motivated (or was cherry picked to justify) the other this doesn’t seem like it should be a surprise?
In the same sense that for in turing machine, it must either halt or not halt—whether or not we have any way of knowing that.
This is equivalent to “we can construct as many digits of this number as we want, though it might be infinitely long after the decimal”.