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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. p

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?
- darkmighty 12y ago
  Thanks for the reply, really enjoying the discussion. What I had in mind is a particular case I heard of, Chaitin's Omega, which can't even be approximated arbitrarily by a fixed algorithm. But still that number seems useful to study it's properties as a number. Expanding further, that's an issue I've had with definitions of computability also. Certain problems like the halting problem are extremely interesting but their non-computability sort of turns them dead. Approximating them even to a finite degree and studying them profoundly seems very worthwhile, from a practical and theoretical standpoint.

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?

darkmighty 12y ago
Thanks for the reply, really enjoying the discussion. What I had in mind is a particular case I heard of, Chaitin's Omega, which can't even be approximated arbitrarily by a fixed algorithm. But still that number seems useful to study it's properties as a number. Expanding further, that's an issue I've had with definitions of computability also. Certain problems like the halting problem are extremely interesting but their non-computability sort of turns them dead. Approximating them even to a finite degree and studying them profoundly seems very worthwhile, from a practical and theoretical standpoint.