Cardinality of points on a line

[Indistinguishable]

Yes, it’s terminological unfortunateness; “one-to-one correspondence” is usually used to mean “For every x in X there corresponds exactly one y in Y and for every y in Y there corresponds exactly one x in X”, but “one-to-one function” is usually used to mean just “For every x in X there corresponds exactly one y in Y and for every y in Y there corresponds at most one x in X”. C’est la vie.

I shy away from using “one-to-one” in either sense myself, and just say “isomorphism” or “bijection” in the first case, and “injection” in the second case.

[DrCube]

Who uses intuitionist math? It always seemed like hogwash to me.

[Thudlow_Boink]

@Indistinguishable

Yes, it’s terminological unfortunateness; “one-to-one correspondence” is usually used to mean “For every x in X there corresponds exactly one y in Y and for every y in Y there corresponds exactly one x in X”, but “one-to-one function” is usually used to mean just “For every x in X there corresponds exactly one y in Y and for every y in Y there corresponds at most one x in X”. C’est la vie.

I shy away from using “one-to-one” in either sense myself, and just say “isomorphism” or “bijection” in the first case, and “injection” in the second case.

Wikipedia spells it out pretty well. The adjective “one-to-one” describes a function which is injective, but the term “one-to-one correspondance” means a function that is bijective.
[QUOTE=One of my grad school profs]
Originally this was called an “equijection,” which sounds like somebody was thrown from a horse.
[/QUOTE]

[Indistinguishable]

@DrCube

Who uses intuitionist math? It always seemed like hogwash to me.

I do. I also use classical math. (Much in the same way that I sometimes use non-abelian groups and I also sometimes use abelian groups).

Hogwash? In what way could intuitionistic mathematics be hogwash? “Well, clearly, the law of the excluded middle is true. It’s hogwash to deny it.” This is a bit like saying “Well, clearly, commutativity is true. It’s hogwash to deny it.” or “Well, clearly, the non-existence of a square root of -1 is true. It’s hogwash to deny it.” (And the fellow committed to the other point of view could say, just as dogmatically, “Well, clearly, the principle of universal continuity is true. It’s hogwash to deny it.”)

As always, one can trade off assumptions for generality… Intuitionistic logic is more general than classical logic. By not assuming the law of the excluded middle, one obtains results which hold more widely. For example, the sheaves over a topological space form a model of intuitionistic set theory, one which will almost never be classical (for the open sets form a Heyting algebra, not a Boolean algebra). Intuitionistic mathematics works very well for describing the “internal logic of the computable universe” (the law of the excluded middle fails in this context because not all properties are computably decidable), and, of course, given the remark which started this whole sidetrack, for describing the “internal logic of the continuous universe” as well. Forcing arguments in set theory are perhaps best understood as involving passage through an intermediate intuitionistic construction (cf. Cohen’s notion of “strong forcing”). The Curry-Howard correspondence reveals intuitionistic logic in the type systems of programming languages. Mathematics brims with contexts where intuitionistic logic can be usefully applied.

You can always talk about these things without the language of intuitionistic mathematics, of course, but the language is convenient. It helpfully points out the analogies that exist to classical mathematics, while also demarcating where those analogies end.

And intuitionistic mathematics can be fun. It’s fun to work in intuitionistic mathematics, and avail oneself of possibilities which would classically be barred. Smooth infinitesimal analysis (where R contains infinitesimals and all functions from R to R are infinitely differentiable, agreeing exactly with their first-order approximations on infinitesimal inputs) is fun. If you don’t want to worry about hassles of non-smoothness, you don’t have to. An intuitionistic framework can be more convenient than a classical one, depending on what you’re doing.

And all the same for other logics as well… linear logic, quantum logic, non-commutative logics. Every nice rule system is worth thinking about. Of course, no one person has time to study everything, but everything is worth studying.

[Indistinguishable]

No more edit window, but…

@Indistinguishable

For example, the sheaves over a topological space form a model of intuitionistic set theory

I’d like to replace the words “set theory” here with “mathematics”, not because there’s any formal difference in what I would mean by that, but just for connotational reasons… You don’t have to be a set theorist to appreciate the connection between topology and intuitionistic mathematics.

[Learjeff]

Thanks for clearing up that one-to-one business. I was always confused about the diff between “one-to-one” and “one-to-one and onto”.

[Learjeff]

BTW, back in the 70’s I knew a math genius (and prof) who told me that they could map a line to a plane, all but one point on the plane. But that was off the top of his head when we were discussing this kind of thing (cardinality) and might have been incorrect. Still, it seemed pretty amusing. This infinity is that infinity, plus one.

[Indistinguishable]

You can easily put a plane and a line in bijection, but not continuously (interweave two coordinates’ decimal representations into one number’s, with some care about non-uniqueness of decimal representations). You can also continuously surject from a line onto the plane, but not injectively (space-filling curves). I don’t know what the “all but one point on the plane” business is about, though.

[Senegoid]

@Indistinguishable

You can easily put a plane and a line in bijection, but not continuously (interweave two coordinates’ decimal representations into one number’s, with some care about non-uniqueness of decimal representations). You can also continuously surject from a line onto the plane, but not injectively (space-filling curves). I don’t know what the “all but one point on the plane” business is about, though.

Perhaps confusing the case with the sorts of situations where some transformation of a surface onto itself necessarily entails having a singularity somewhere?

(Cite is to a current column in NYT describing singularities in the whorl in the hair on your head, or in fingerprints.)

[Hari_Seldon]

In defense of intuitionism:

Let me say, I am not an intuitionist. But I do see the point. Any function from the reals to the reals that can be calculated by a machine is continuous. To explain this, let me explain what it means to calculate a function. First, a “number” is given by an infinite sequence of approximations. You might want to think of its decimal expansion but there are numbers I can describe in the first sense that I cannot give even one decimal place of. I give an example later. But the crucial thing about computability is that from any finite sequence of approximations of the number you must be able to calculate an increasingly good approximation of the value of the functions. All ordinary functions such as polynomials, trig functions, exponentials, etc., have this property. But you cannot now tell me whether the number x that I am about to describe is either positive or negative. So the function that is 1 at all numbers except 0, where it is 0, cannot be computed in this sense. The number 1 + x might start with lots of 9s if x < 0 or lots of 0s if x >= 0.

Here is x. Assume that you have already computed that the n-th approximation is 0. To get the next, look at all the digits of pi starting with the (n+1)st. If that is the first of a trillion consecutive 0s and n+1 is even, let x = 10^{-(n+1)}; if odd, let x = -10^{-(n+1)}; otherwise continue to add 0 digits. We cannot, given current knowledge, say whether x is positive or negative (it is unlikely to be 0), but we approximate it is finely as we like. If it is 0, we can never know it.

So intuitionist mathematics is the mathematics of what is calculable. Not to be dismissed.

[President_Johnny_Gentle]

@Indistinguishable

For example, the sheaves over a topological space form a model of intuitionistic set theory, one which will almost never be classical (for the open sets form a Heyting algebra, not a Boolean algebra).

Since it’s not worth discussing this point in further detail here, is there a decent reference (preferably online) for this? I’m familiar with sheaf theory (however I haven’t used them since grad school), but I know next to nothing about intuitionist math.

@Senegoid

Perhaps confusing the case with the sorts of situations where some transformation of a surface onto itself necessarily entails having a singularity somewhere?

(Cite is to a current column in NYT describing singularities in the whorl in the hair on your head, or in fingerprints.)

The word “singularity” really bugs me for this usage. A singularity IMO ought to be reserved for occasions when you’re dealing with undefinable quantities. For the “singularities” in the posting, we’re not confronting something undefinable. Rather, we’re confronting something that is precisely zero.

I would say the two situations have some relation, but I think of them as fairly distinct.

In the case of the discussion in this thread, comparing (-1,1) to [-1,1], the two intervals both have the same number of points, but the geometry is distinctly different. In the closed interval, the interval has a distinct end in either direction. This interval, as indicated earlier, is considered to be compact. (Compact can be considered as a way of saying that a space is “small”.) The open interval, despite at first glance appearing to be smaller (since it has fewer points), is not compact, which essentially means that its geometry is “larger” than that of [-1,1].

A continuous function is essentially one that preserves the concept of “nearness”. Since (-1,1) is geometrically far larger than [-1,1], there is no continuous one-to-one correspondence, or else both spaces would be either compact or not.

The issue confronted in the NYT posting involves vector fields on spaces, which is a question that assumes continuity in the first place. (And, quite often, also assumes compactness as well.) This involves a deeper property of the space in question, namely its Euler characteristic. Wikipedia has a decent short treatment of the subject.

[Indistinguishable]

@President_Johnny_Gentle

Since it’s not worth discussing this point in further detail here, is there a decent reference (preferably online) for this? I’m familiar with sheaf theory (however I haven’t used them since grad school), but I know next to nothing about intuitionist math.

I suppose the standard reference is “Sheaves in Geometry and Logic” by MacLane and Moerdijk, although it isn’t online (unless it happened to be floating around in pirated form, hint hint…).

You could also try reading various articles at the nLab, though I suspect it’s not particularly friendly as an introduction. (Also, it appears to be down at the moment…)

[President_Johnny_Gentle]

@Indistinguishable

I suppose the standard reference is “Sheaves in Geometry and Logic” by MacLane and Moerdijk, although it isn’t online (unless it happened to be floating around in pirated form, hint hint…).

You could also try reading various articles at the nLab, though I suspect it’s not particularly friendly as an introduction. (Also, it appears to be down at the moment…)

Thanks!