Constructive Mathematiccs

The mathematical sense of the word "Constructive" has to do with the nature of existence. Roughly seaking, something exists if there is an effective way for constructing, or finding it. The nonconstructivists are happy to have things exist that we may never be able to construct or find explicitly. That's the fundamental difference between the two camps. My sympathies lie with the constructivists, but the reader should be warned that constructivists are a minority among mathematicians, and most first-year calculus courses teach nonconstructive calculus.

One of the simplest points of difference is the comparison of real numbers. A nonconstructivist is happy to assert, for every real numbers a and b, either a < b, a = b, or a > b.

But the constructivist rejeects this, because there is no general way of distinguishing these cases that always works. Let's say you have two streams of digits. One is the digits of a, the other is the digits of b.
a starts off: 4.6725749274922...
and b starts off 4.6725749274922...
At this point you can't tell which, if either, is bigger. You'll just have to look at more digits. And if they're still the same, still more digits. And more, and more. If a < b or b < a, at some point they will differ, and you'll know. But as long as they stay the same, you'll never know. You'll say, maybe they're equal, but maybe if I compute still more digits, I'll find out otherwise. You can't know whether a and b are the same unless you can look at *all* the digits. But that's an infinite process, and will take forever. Mind you, for lots of pairs of numbers, sucn as 3.246 and the square root of two, it's easy to decide (these happen to be different)You just can't claim that it's possible to decide for *every* pair of numbers

Now if you have access to the mechanisms generating these streams of digits (ssuming there is one, of course), you might be able to analyse the machinery and determine that yes, they will always produce the same outputs. But there's no guarantee that that will be possible. Mechanisms, even simple mechanisms, can have extremely complex behaviour, and for most interesting questions about them, it can be shown there is no effective way of answering them conclusively in all cases.

(Example: there's are methods similar to long division, but more complicated, for calculating cube roots and square roots. You could try calculating the square root of two this way, and you could try calculating the cube root of the square root of eight. Lots of work later, you'd have the first few digits of both numbers, say, 1.4142 , and you might suspect they are the same numbers, but you wouldn't *know* without an insight such as that the cube root of the square root is the same as the square root of the cube root. But with this insight, you can prove that the two numbers you are calculating are the same. Just don't expect that there will *always* be a method of proving equality if you haven't been able to prove inequality/

You cannot assert constructively that in general always a < b or a = b or a > b. To constructively assert a disjunction you must be able to find out which of the alternatives is true.

The nonconstructivist has no such restriction: he'll say that one of the three alternatives has to be true even if he can *never* know which one. I don't consider this a very useful result. It's not even clear to me what it would mean in concrete terms.

Now philosophically, you could ask which of these two views of mathematics is true? Do mathematical objects exist independently of whather we can find them? Or is mathematics inherently about what we can construct/find/compute? And if this isn't mathematics, what *is* it? Entire books have been written about these issues.

I more-or-less understand what constructive existence means; I find it utterly obscure what a nonconstructive means by the word.

Now calculus was originally invented to calculate things. Where costructive mathamatics stands out is that it is a theory of what you can calculate. If you've proved a theorem constructively, you know you can calculate the things it's about. Nonconstructively, you can prove things exist that you cannot find.

There are mathematcians that consider it desirable the ability to prove that things can exist even though it's impossible to find them. They say that this gives them extra deductive power (they can prove more theorems). But this extra deductive power may sometimes give you an easier way to provee some theorems, but it may just delude you into wasting time trying to calculate things that are impossible.

Most practical calculations (here I'm talking exclusively about things that *can* be calculated) aren't affected much by the decision to think constructively. There are differences in the precise conditions that have to be satisfied for various theorems to hold. And constructivists often use terminology differently from nonconstructivists. For example what the constructivist calls "continuity" the nonconstructivist calls "uniform continuity". But pretty well all functions normally used in calculations that the nonconstructivist calls continuous are in fact also what he calls "uniformly continuous". There's a subtle distinction there, true, and it profoundly affects the way theorems have to be formulated and proved, but it isn't all that important in applications.

Because I'd like to recommend the book by Gabriel Stolzenberg et al.

(question: Are arithmetic operations on the algebraic numbers computable?)