35 ms·

There is a distinction between analytically correct expositions, and ones which build on naive intuition to teach students. It is inappropriate to expect begin

by btilly 8y ago

There is a distinction between analytically correct expositions, and ones which build on naive intuition to teach students. It is inappropriate to expect beginning students to follow a logically rigorous exposition. That said, there are philosophical questions about truth, infinity, unknowable statements and the like. Mathematicians have by and large settled on a set of answers to these. Every statement is true or false, regardless of whether we know the answer, or even whether we can know the answer. Infinite sets exist, and are described by a known set of axioms called ZFC. Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?) All of these statements are part of classical mathematics. Almost every elementary exposition will implicitly assume that they are try. Yet they can all be questioned, and their truth can never be settled in any absolute sense. However woe betide the student who dares question these in a math class.

AlexCoventry 8y ago
> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. I don't think many people see that as a matter of philosophy.
- btilly 8y ago
  Most people are uninformed. This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy. To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics. More precisely any contradiction found in Constructivism necessarily will lead to a contradiction in classical mathematics. The converse is only partially true. Gödel did prove that a logical contradiction in the classical handling of infinity will lead to a contradiction in Constructivism. But a flaw in a specific set of classical axioms, such as ZFC, need not lead to a flaw in usual Constructivism.
  - AlexCoventry 8y ago
    I think you needed to be more precise. Most people, would read "almost all" in the context of the reals as dependent on the existence of an uncountable set, to make sense.
    - btilly 8y ago
      I was perfectly precise. In real analysis you learn that "almost all" means that the exceptions are a set of measure 0. Since all countable sets have measure 0, the result is trivially true in classical mathematics. In the constructible universe, you again have measure theory. Almost all still has a perfectly well-defined meaning. And all sets with enumerations again have measure zero, just like in classical mathematics. But "uncountable" now is a statement about self-referential complexity, not size. Next, "the set of all numbers with finite definitions" is not a well-defined set. And numbers without concrete definitions do not exist.
hnuser355 8y ago
Well I don’t mean to be rude but many logicians and set theorists I know will be sad to find themselves out of a job if what you said about these questions being settled is true
- btilly 8y ago
  I disbelieve your claim. In particular I maintain that any competent logician or set theorist should be able to explain to you the sense in which the statements that I made cannot logically be settled, and also explain to you the extent to which the rest of mainstream mathematics accepts these statements as true.
jacobolus 8y ago
> Infinite sets exist, and are described by a known set of axioms called ZFC This is the assertion of many mathematicians, but the justification for it is “this is convenient” and/or “we take this as an article of faith” (or often “I never really thought about it, but it doesn’t much affect my work day to day one way or the other”). There is no way to prove that infinite sets “exist” by reasonable definitions of “exist”. Indeed, by a conventional definition of “exist” infinite sets pretty clearly don’t qualify. Instead, mathematicians have redefined the words “exist” and “true”. In mathematics it now means something like “if we accept a particular set of non-obvious and rather handwavey premises, we will also accept any conclusions that result from symbolic manipulations thereof following our established formal rules.” [This is not a full or precise definition of mathematical existence; folks interested can do a search for those keywords and find piles of material.] * * * Personally I am happy to accept ZFC or the like because it is convenient and I can’t be bothered to work up an alternative system from scratch and carefully examine all of the conclusions that might follow from that, and whether ZFC is “true” or not doesn’t really affect me. It seems intuitively wrong to me, but I remain agnostic.
- harf 8y ago
  (i.e. one of the standard set theory axioms amounts to "an infinite set exists")
- fmap 8y ago
  > “I never really thought about it, but it doesn’t much affect my work day to day one way or the other” In my experience, people won't come out and say it, but this seems to be what everyone is thinking. :) The problem with this is that it is wrong. Classical ZFC in particular is a very strong and specific set of assumptions* with a very tenuous link to any practical application. If you actually want to develop a useful bit of mathematics it makes sense to consider the "foundations" as a moving piece. It's a part of the design space for modeling your problem domain, not some god-given notion of truth. You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things. --- *) I mean that second-order ZFC has basically no non-trivial models, so there is no real way of extending ZFC to talk about domain specific aspects of your problems.
roenxi 8y ago
> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?) A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down. In a sense, it seems like the uncomputable reals are an artifact of assuming continuity, ie, between any two numbers located on a line there must be more numbers. Part of the reason it is so unintuitive is we don't have any real lines to play with at the physical human scale, they fall apart at the atomic level and turn out to be non-continuous approximations.
- umanwizard 8y ago
  I don't think "between any two numbers located on a line there must be more numbers" is the property you're looking for, since it holds for the rationals (all of which are, of course, computable). Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 < 2, and B is the set with x^2 > 2, then the rationals are the union of A and B, and there is a "hole where sqrt(2) should be", so the rationals are disconnected. It's possible to define the word "connected" in a way that makes this notion precise. A related notion is what's called "(sequential) completeness". The infinite sequence whose terms are (2, 2 + 1/2, 2 + 1/2 + 1/6, 2 + 1/2 + 1/6 + 1/24, ...), where the nth term is obtained by adding 1/(n!) to the previous term, intuitively "should" converge, since its elements get arbitrarily close together as n gets arbitrarily large. Any such sequence converges to a real value (this one converges to the exponential constant "e"). But if our number system is only countably infinite, there must be some sequences that get arbitrarily close together but don't converge. For example, if we restrict ourselves to rational numbers, this is a valid infinite sequence (every element is rational), and its terms get arbitrarily close together as "n" is large, but it doesn't converge to anything.
  - deleted 8y ago
    [deleted]

by btilly 8y ago

AlexCoventry 8y ago

> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. I don't think many people see that as a matter of philosophy.

btilly 8y ago
Most people are uninformed. This question strikes at the heart of the debate between Constructivism and Formalism. A debate about what it means for things to exist, statements to be true, and so on. This is very much a matter of philosophy. To a Constructivist, most of classical mathematics is nonsense. And Constructivism is at least as logically consistent as classical mathematics. More precisely any contradiction found in Constructivism necessarily will lead to a contradiction in classical mathematics. The converse is only partially true. Gödel did prove that a logical contradiction in the classical handling of infinity will lead to a contradiction in Constructivism. But a flaw in a specific set of classical axioms, such as ZFC, need not lead to a flaw in usual Constructivism.
- AlexCoventry 8y ago
  I think you needed to be more precise. Most people, would read "almost all" in the context of the reals as dependent on the existence of an uncountable set, to make sense.
  - btilly 8y ago
    I was perfectly precise. In real analysis you learn that "almost all" means that the exceptions are a set of measure 0. Since all countable sets have measure 0, the result is trivially true in classical mathematics. In the constructible universe, you again have measure theory. Almost all still has a perfectly well-defined meaning. And all sets with enumerations again have measure zero, just like in classical mathematics. But "uncountable" now is a statement about self-referential complexity, not size. Next, "the set of all numbers with finite definitions" is not a well-defined set. And numbers without concrete definitions do not exist.

hnuser355 8y ago

Well I don’t mean to be rude but many logicians and set theorists I know will be sad to find themselves out of a job if what you said about these questions being settled is true

btilly 8y ago
I disbelieve your claim. In particular I maintain that any competent logician or set theorist should be able to explain to you the sense in which the statements that I made cannot logically be settled, and also explain to you the extent to which the rest of mainstream mathematics accepts these statements as true.

jacobolus 8y ago

> Infinite sets exist, and are described by a known set of axioms called ZFC This is the assertion of many mathematicians, but the justification for it is “this is convenient” and/or “we take this as an article of faith” (or often “I never really thought about it, but it doesn’t much affect my work day to day one way or the other”). There is no way to prove that infinite sets “exist” by reasonable definitions of “exist”. Indeed, by a conventional definition of “exist” infinite sets pretty clearly don’t qualify. Instead, mathematicians have redefined the words “exist” and “true”. In mathematics it now means something like “if we accept a particular set of non-obvious and rather handwavey premises, we will also accept any conclusions that result from symbolic manipulations thereof following our established formal rules.” [This is not a full or precise definition of mathematical existence; folks interested can do a search for those keywords and find piles of material.] * * * Personally I am happy to accept ZFC or the like because it is convenient and I can’t be bothered to work up an alternative system from scratch and carefully examine all of the conclusions that might follow from that, and whether ZFC is “true” or not doesn’t really affect me. It seems intuitively wrong to me, but I remain agnostic.

harf 8y ago
(i.e. one of the standard set theory axioms amounts to "an infinite set exists")
fmap 8y ago
> “I never really thought about it, but it doesn’t much affect my work day to day one way or the other” In my experience, people won't come out and say it, but this seems to be what everyone is thinking. :) The problem with this is that it is wrong. Classical ZFC in particular is a very strong and specific set of assumptions* with a very tenuous link to any practical application. If you actually want to develop a useful bit of mathematics it makes sense to consider the "foundations" as a moving piece. It's a part of the design space for modeling your problem domain, not some god-given notion of truth. You can translate between different logical theories by building a model of one in another, so it's not like you loose anything. But it's cooky to insist that we should start with ZFC of all things. --- *) I mean that second-order ZFC has basically no non-trivial models, so there is no real way of extending ZFC to talk about domain specific aspects of your problems.

roenxi 8y ago

> Almost all real numbers that exist can never, even in principle, be written down or described in any meaningful way. (In what sense do they exist again?) A profound question to reflect on; and interesting to contrast with modern science as a philosophical foundation that there might not be any continuous objects present in reality. Might be discrete all the way down. In a sense, it seems like the uncomputable reals are an artifact of assuming continuity, ie, between any two numbers located on a line there must be more numbers. Part of the reason it is so unintuitive is we don't have any real lines to play with at the physical human scale, they fall apart at the atomic level and turn out to be non-continuous approximations.

umanwizard 8y ago
I don't think "between any two numbers located on a line there must be more numbers" is the property you're looking for, since it holds for the rationals (all of which are, of course, computable). Maybe "connectedness" is the notion you're trying to get at -- the real numbers are topologically connected, but the rationals aren't. If "A" is the set of rational numbers x with x^2 < 2, and B is the set with x^2 > 2, then the rationals are the union of A and B, and there is a "hole where sqrt(2) should be", so the rationals are disconnected. It's possible to define the word "connected" in a way that makes this notion precise. A related notion is what's called "(sequential) completeness". The infinite sequence whose terms are (2, 2 + 1/2, 2 + 1/2 + 1/6, 2 + 1/2 + 1/6 + 1/24, ...), where the nth term is obtained by adding 1/(n!) to the previous term, intuitively "should" converge, since its elements get arbitrarily close together as n gets arbitrarily large. Any such sequence converges to a real value (this one converges to the exponential constant "e"). But if our number system is only countably infinite, there must be some sequences that get arbitrarily close together but don't converge. For example, if we restrict ourselves to rational numbers, this is a valid infinite sequence (every element is rational), and its terms get arbitrarily close together as "n" is large, but it doesn't converge to anything.
- deleted 8y ago
  [deleted]