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I favor constructivism, btw. I just don't find this particular example to be very convincing for the cause. Take the square root of 2 -- it's a real number that

by maaku 12y ago

I favor constructivism, btw. I just don't find this particular example to be very convincing for the cause. Take the square root of 2 -- it's a real number that can never be written down with any finite pencil and paper. Except for tricks that define the problem away like using a sqrt-2 base, I will only be able to manipulate approximations of that number. But I can still hold in my hand a square object and know that the diagonal is sqrt-2 times the length of a side. If you want to make objections about the square not really being square due to physical tolerances, then you could look to quantum and atomic physics, where such numbers show up and are measurable to whatever accuracy our experimental apparatus provides.

btilly 12y ago
You are talking about something very different than I was. To a constructivist, sqrt(2) is a perfectly reasonable number. I can write down an algorithm to approximate it to any desired degree. I can describe its properties. I can find ways in which it arises naturally. I can compare it to rational numbers. It is quite reasonable to talk about its existence. What constructivists object to are pure existence proofs. Take the numbers that I was describing. A classical proof of their existence would be to demonstrate that the set of such numbers has measure greater than 0. Therefore they both exist, and there are lots of them. But now we're stuck. What can we say that is sensible about them? There are lots of them! There are some over here. There are some over there. How about some that start off with 1.2345678910111213...? Nope, we can compress the start of that one too easily. How about some that start off with 3.14159265359...? Nope, we can compress the start of that one too easily. How about ones that start off with 0.45399480855940464218372115598800974602442341777409214271555307732203389646276632569332825355032120233072728272048609801012859425779656167823065863748297...? Hrm, kind of hard to tell, we'd need to enumerate the output of a ton of programs and we might not be able to figure out whether some of them halt so if there are, we can't find out. But there are lots of them! Promise! This is a fairly strong example demonstrating how existence to a classical mathematician does not agree with what a constructivist means.

by maaku 12y ago

btilly 12y ago

You are talking about something very different than I was. To a constructivist, sqrt(2) is a perfectly reasonable number. I can write down an algorithm to approximate it to any desired degree. I can describe its properties. I can find ways in which it arises naturally. I can compare it to rational numbers. It is quite reasonable to talk about its existence. What constructivists object to are pure existence proofs. Take the numbers that I was describing. A classical proof of their existence would be to demonstrate that the set of such numbers has measure greater than 0. Therefore they both exist, and there are lots of them. But now we're stuck. What can we say that is sensible about them? There are lots of them! There are some over here. There are some over there. How about some that start off with 1.2345678910111213...? Nope, we can compress the start of that one too easily. How about some that start off with 3.14159265359...? Nope, we can compress the start of that one too easily. How about ones that start off with 0.45399480855940464218372115598800974602442341777409214271555307732203389646276632569332825355032120233072728272048609801012859425779656167823065863748297...? Hrm, kind of hard to tell, we'd need to enumerate the output of a ton of programs and we might not be able to figure out whether some of them halt so if there are, we can't find out. But there are lots of them! Promise! This is a fairly strong example demonstrating how existence to a classical mathematician does not agree with what a constructivist means.