7 ms·

There does exist real numbers that cannot be specified in (finite) space or time...

by maaku 12y ago

sukilot 12y ago
This is what btilly is talking about. To parahrase: "there exist entities that cannot ever be constructed or named in any Universe like ours"
btilly 12y ago
According to classical mathematics, this is true. I can make the statement even stronger. There are real numbers which not only cannot be specified, but are strongly not compressible. That is, for the first n digits, there is no significantly more efficient way to specify what they are than to just list them. But we are now talking about the existence of something that cannot be specified or written down in any useful way, shape, or form. What does it actually mean to say that this exists? We've demonstrated that no direct evidence of existence can ever be found. This is pretty much the opposite of what a lot of people would actually want "existence" to mean. From a classical point of view what is wrong is your notion of existence. But it is perfectly reasonable to instead conclude that it is classical mathematics which has gone off the rails and they are now using words like "existence" in a nonsensical way. Both points of view are perfectly defensible. In fact it was proven by Goedel that any inconsistency in one leads to an inconsistency in the other. Therefore neither is actually able to claim to have a better grounding in logical consistency than the other. Historically the debate was that classical mathematics is more convenient to work with, while constructivism actually means something more reasonable. But once the set paradoxes faded, and with no difference in consistency, the simplicity of classical mathematics won.
- darkmighty 12y ago
  So what's constructivism useful for anyway? Why change the basis of logic instead of proving non-constructability within the reasonable existing logics? (E.g. prove a real number isn't constructable instead of foregoing it's existence) There are also cases of interest beyond constructivism: what if a number is non-constructible but can be approximated? Saying it doesn't exist doesn't sound particularly useful.
  - btilly 12y ago
    So what's constructivism useful for anyway? Not much. The philosophy of math is not noted for being particularly useful. :-) That said, historically constructivism arose during a period when math was having trouble creating solid foundations. Today we can talk about "the reasonable existing logics". But a century ago you couldn't. Attempts to create such a thing had repeatedly resulted in paradoxes, and an inability to find solid foundations had caused repeated mathematical crises for a century. In that light, it was reasonable to doubt lines of reasoning that proved the existence of things in infinite sets but provided no way to actually find them. This smelled too much of the kinds of things that had lead to paradox. Hence the appeal of a philosophy that insisted that things are only known to exist when we have constructions for them. Proofs are only valid when they follow lines of reasoning that can, at least in principle, actually be followed in finite time. These days this is mostly a historical footnote. But an important one. There are also cases of interest beyond constructivism: what if a number is non-constructible but can be approximated? Um, actually a constructivist would say that that number exists. To understand, you have to ask what a real number is. One definition of a real number is a Cauchy sequence of rational numbers. Which is to say an infinite series of rational numbers that converges. Constructivists make this more precise. Rather than talking about an infinite series of rational numbers, they talk about a algorithm that generates a series of rational numbers. Rather than saying "for every epsilon greater than 0 there exists an N..." they insist that there is an algorithm which can take in epsilon and spit out N. Combining these two, from a valid specification of a real number, you can generate an algorithm that given an epsilon > 0 can generate a rational number within epsilon of the real number. And an algorithm that does that can easily be turned into a Cauchy sequence. Thus to a constructivist, being able to approximate a number arbitrarily well is perfectly fine as a definition of said number. Things get odd in that you can have two different constructions of real numbers but you don't always know whether they are specifying the same number. But you quickly get used to this. Saying it doesn't exist doesn't sound particularly useful. Let me turn that around. What is the use of asserting the existence of things that you cannot construct? What are you going to do with something that you cannot find in an infinite haystack? A thing that may be impossible to find, even in principle?
    - 12y ago
- 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.

There does exist real numbers that cannot be specified in (finite) space or time...

by maaku 12y ago

There does exist real numbers that cannot be specified in (finite) space or time...

This is what btilly is talking about. To parahrase: "there exist entities that cannot ever be constructed or named in any Universe like ours"

btilly 12y ago

According to classical mathematics, this is true. I can make the statement even stronger. There are real numbers which not only cannot be specified, but are strongly not compressible. That is, for the first n digits, there is no significantly more efficient way to specify what they are than to just list them. But we are now talking about the existence of something that cannot be specified or written down in any useful way, shape, or form. What does it actually mean to say that this exists? We've demonstrated that no direct evidence of existence can ever be found. This is pretty much the opposite of what a lot of people would actually want "existence" to mean. From a classical point of view what is wrong is your notion of existence. But it is perfectly reasonable to instead conclude that it is classical mathematics which has gone off the rails and they are now using words like "existence" in a nonsensical way. Both points of view are perfectly defensible. In fact it was proven by Goedel that any inconsistency in one leads to an inconsistency in the other. Therefore neither is actually able to claim to have a better grounding in logical consistency than the other. Historically the debate was that classical mathematics is more convenient to work with, while constructivism actually means something more reasonable. But once the set paradoxes faded, and with no difference in consistency, the simplicity of classical mathematics won.

darkmighty 12y ago
So what's constructivism useful for anyway? Why change the basis of logic instead of proving non-constructability within the reasonable existing logics? (E.g. prove a real number isn't constructable instead of foregoing it's existence) There are also cases of interest beyond constructivism: what if a number is non-constructible but can be approximated? Saying it doesn't exist doesn't sound particularly useful.
- btilly 12y ago
  So what's constructivism useful for anyway? Not much. The philosophy of math is not noted for being particularly useful. :-) That said, historically constructivism arose during a period when math was having trouble creating solid foundations. Today we can talk about "the reasonable existing logics". But a century ago you couldn't. Attempts to create such a thing had repeatedly resulted in paradoxes, and an inability to find solid foundations had caused repeated mathematical crises for a century. In that light, it was reasonable to doubt lines of reasoning that proved the existence of things in infinite sets but provided no way to actually find them. This smelled too much of the kinds of things that had lead to paradox. Hence the appeal of a philosophy that insisted that things are only known to exist when we have constructions for them. Proofs are only valid when they follow lines of reasoning that can, at least in principle, actually be followed in finite time. These days this is mostly a historical footnote. But an important one. There are also cases of interest beyond constructivism: what if a number is non-constructible but can be approximated? Um, actually a constructivist would say that that number exists. To understand, you have to ask what a real number is. One definition of a real number is a Cauchy sequence of rational numbers. Which is to say an infinite series of rational numbers that converges. Constructivists make this more precise. Rather than talking about an infinite series of rational numbers, they talk about a algorithm that generates a series of rational numbers. Rather than saying "for every epsilon greater than 0 there exists an N..." they insist that there is an algorithm which can take in epsilon and spit out N. Combining these two, from a valid specification of a real number, you can generate an algorithm that given an epsilon > 0 can generate a rational number within epsilon of the real number. And an algorithm that does that can easily be turned into a Cauchy sequence. Thus to a constructivist, being able to approximate a number arbitrarily well is perfectly fine as a definition of said number. Things get odd in that you can have two different constructions of real numbers but you don't always know whether they are specifying the same number. But you quickly get used to this. Saying it doesn't exist doesn't sound particularly useful. Let me turn that around. What is the use of asserting the existence of things that you cannot construct? What are you going to do with something that you cannot find in an infinite haystack? A thing that may be impossible to find, even in principle?
  - 12y ago
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.