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Everyone has gotten distracted by the infinite expansion and has missed the point of my question. Let me try again, with no infinite expansion. The constructi

Everyone has gotten distracted by the infinite expansion and has missed the point of my question. Let me try again, with no infinite expansion. The constructivist position is to refuse to accept the statement "every real number is either rational or irrational" because they're not in possession of proofs, for each possible real-valued expression, that its value is rational or, alternatively, a proof that its value is irrational. And yet, according to the quoted paragraph, they accept the statement "every integer is either even or odd", quote, "because we have an effective means of determining which alternative holds for any integer". My argument is that they should reject this statement on the same grounds: I named an integer whose value is not known, and therefore it is not known whether it is even or odd. There is not, in fact, an effective procedure for determining the evenness or oddness of the value of an arbitrary integer-valued expression. It's only after one actually knows the value that answering the question becomes easy.

dllthomas 11y ago
IANAM, but I think the difference comes down to the fact that we know the integer you are producing is one we possess a proof for. We do not know that for reals.
plonh 11y ago
Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to even ask of "it" is odd. Infinite vs finite is an important distinction. A fact that takes infinite time to prove is not constructive and is likewise not computable.
pavpanchekha 11y ago
Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one more than any member of that set. Since the set is a constructive set, if you have anything you claim to be an integer, you must be able to either prove that it is 0 or one more than an integer. The number you named isn't zero, so presumably it must be more than some integer ("one less than the number you named"). But if I then ask you whether that number is one more than an integer, and then the same about that one, and so on, at some point you will be unable to answer me. Since you can't provide the evidence that qualifies something as an integer, there's no reason for me to think it is an integer.
- ScottBurson 11y ago
  I see. The difference, then, between that situation and the situation where I provide a real number named as the sum of an infinite series is that in the latter case I may be able to provide also the modulus of convergence, thus proving that the number is a constructive real number [0]. This is different from a situation where I have a number defined non-constructively and no series is known that converges on the number. [0] https://en.wikipedia.org/wiki/Constructivism_%28mathematics%29 https://en.wikipedia.org/wiki/Constructivism_%28mathematics%...

dllthomas 11y ago

IANAM, but I think the difference comes down to the fact that we know the integer you are producing is one we possess a proof for. We do not know that for reals.

plonh 11y ago

Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to even ask of "it" is odd. Infinite vs finite is an important distinction. A fact that takes infinite time to prove is not constructive and is likewise not computable.

pavpanchekha 11y ago

Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one more than any member of that set. Since the set is a constructive set, if you have anything you claim to be an integer, you must be able to either prove that it is 0 or one more than an integer. The number you named isn't zero, so presumably it must be more than some integer ("one less than the number you named"). But if I then ask you whether that number is one more than an integer, and then the same about that one, and so on, at some point you will be unable to answer me. Since you can't provide the evidence that qualifies something as an integer, there's no reason for me to think it is an integer.

ScottBurson 11y ago
I see. The difference, then, between that situation and the situation where I provide a real number named as the sum of an infinite series is that in the latter case I may be able to provide also the modulus of convergence, thus proving that the number is a constructive real number [0]. This is different from a situation where I have a number defined non-constructively and no series is known that converges on the number. [0] https://en.wikipedia.org/wiki/Constructivism_%28mathematics%29 https://en.wikipedia.org/wiki/Constructivism_%28mathematics%...