[Tainnor]

You're skimping on some of the complexity by equating a definition with a particular way of computing it, but that's completely inadequate for mathematics, as there are many things we can't compute (either in theory or in practice).

In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong.

[konjin]

I find it odd that you pretend a meta-mathematical question is a mathematical one.

You are dismissing constructivism as not mathematics, and you ignore that the halting problem is a way to deal with a class of results which include non-existence proofs by running an algorithm forever.

I can easily create calculations that will never return results which classical mathematics say are impossible by the simple fact they never return any results at all.

The Risch algorithm being a complex example, finding the square root of two in the rational number domain being a simple one. I can still deal with them as though they return results in all calculations though, without the need for baroque semi-mystical notation. Unless you want to claim some sort of divine human essence not present in Turing Machines and Lambda Calculus which lets us transcend their computational capabilities?

[Tainnor]

I literally don't understand what the hell you are talking about.

But this has nothing to do with constructivism. Even if you only allow constructive definitions and proofs, there is still the world of a difference between the definition of an integral and the result you get from evaluating it.

Yeah, sure, in theory you can represent an integral as a function that takes another function and two boundary points and returns a value...

But first, it may not be possible to determine the value of the integral exactly because there is no known method of doing so (the Risch algorithm, apart from it basically being so complex that it's implemented almost nowhere fully, only works for elementary functions!).

And second, if integrals are "just functions", you lose the ability to manipulate them according to known theorems, e.g. additivity, triangle inequality, Cauchy-Schwarz, convergence theorems, ...

So yeah, here's where I get the feeling that some people should do some more maths and spend good parts of their days proving theorems and playing with definitions before they start complaining about how dumb its language is.

[konjin]

>So yeah, here's where I get the feeling that some people should do some more maths and spend good parts of their days proving theorems and playing with definitions before they start complaining about how dumb its language is.

Some people have a PhD in mathematical physics and wrote the higher function code of axiom. I guess those people would be difficult to understand for non-experts.

[Tainnor]

Ok, I misjudged your experience apparently, but you could still do a better job actually engaging with the arguments.

[konjin]

You sound like a third year maths student who has just been taught the Lebesgue integral and has decided that it is the _real_ definition of definite integration. Quite frankly I don't have the energy or inclination to have adversarial arguments with people who don't understand what I'm saying. Maybe talk to your professors about the generalizations of integration and why none of them are the 'real' way to integrate a function.

Also the integral procedure defined at the top isn't a function, it's an operator. It returns functions as results, not values.

[afiori]

I don't know lisp so maybe there where more nuances in your code, but often your want to analyse an integral in symbolic terms (for whatever integration definition you are using)

The expression of the integral operator as a function in code is contrary to that with how people usually think about functions and code.

The only language I know that properly manages to represent integrals as code is Wolfram Mathematica by using rich rewrite systems.

That is Integrate(f,a,b) is not code but a data structure to be interpreted by an external (and customizable) integration context that defines numerical types, algorithms, lazyness, etc.

From the links I know of Wolfram Mathematica and lisp this could well be what you meant, but it is quite different from giving a single integration algorithm.

[JadeNB]

> In particular, you definition of integral assumes that integrable functions always have an antiderivative, which is wrong.

I was objecting at the same time as you were, but I don't think this is the right objection. It's true that not every integrable function has an elementary anti-derivative, but every integrable function f does have an anti-derivative F, at least in the sense that F is almost everywhere differentiable, and the derivative is almost everywhere equal to f. (And, of course, if f is continuous, then F is everywhere differentiable, and its derivative everywhere equals f.)

[Tainnor]

If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics. Even then, I think your claim that every integrable function has a "generalized antiderivative" is also only true for the Riemann integral. The Dirichlet function is Lebesgue integrable, but it doesn't have an antiderivative even in this weaker sense.[^1] And mathematicians generally prefer the Lebesgue integral.

I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition.

And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all.

[^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though.

[afiori]

My favourite anti-derivatives of the constant zero and Dirichlet function are monotonically increasing.

https://en.wikipedia.org/wiki/Cantor_function

[JadeNB]

> If you say "at least in the sense that F is almost everywhere differentiable", you're already redefining "anti-derivative" to some extent, I feel. But this is just arguing over semantics.

It is, but let's! Before we re-define the anti-derivative, we'd have to define it. A sensible definition is: a function F is an anti-derivative of a function f if F is everywhere differentiable, and if F' = f everywhere. By this definition, not every integrable function has an anti-derivative.

On the other hand, we could also just choose to define—not re-define!—an anti-derivative of f to be a function F that is almost everywhere differentiable, and such that F' = f almost everywhere. This definition is more complicated, but also more inclusive; and it handles everything the old definition could.

In this respect it is, and it's no accident, exactly like the Lebesgue integral vis a vis the Riemann integral. Lebesgue's integral has a more complicated definition than Riemann's, and we could call it a re-definition; but, since it handles everything that Riemann's does (with the same answer), we could say in retrospect that Lebesgue's was the correct definition, and Riemann's was just the special case we happened to discover first.

> I think the more important insight here is that integration fundamentally isn't defined through the anti-derivative, and that the two notions are actually related is a deep theorem, rather than just a definition.

Certainly I agree with this!

> And the fact that non-elementary antiderivatives exist is interesting in theory, but in practice you can't use them directly for anything. In particular, in practical situations you will often use numerical methods to integrate a function which will not be based on any notion of anti-derivative at all.

Here again I'd argue over semantics, though I'd concede it's much more a matter of personal preference than my argument above, which I think has mathematical weight behind it. Namely, I'd argue that the numerical integration is doing something directly with the non-elementary anti-derivative, namely, evaluating it at a point—just like we call reading off the value of, say, the sine of an angle from our calculator doing something directly with the sine, even though what we're really doing is summing sufficiently many terms in a Taylor-series approximation.

> [^1] Edit: I think I was wrong here. If you take the function identically zero, then its derivative is identically zero and as such equal to the Dirichlet function almost everywhere. So this is not a counterexample. I still think it's weird to call than an "antiderivative" though.

I agree that it's not a counterexample for the reason you say, and there's no arguing with perceptions of something being weird; it certainly is counter to intuition built out of Riemann integrals. And yet, if we didn't steel ourselves to handle this weirdness, we'd have to say that it didn't have an anti-derivative at all; and why artificially restrict our theorems to match our intuition, rather than expanding our intuition to meet our theorems?

[Tainnor]

As to your first point: In the sense that it's useful to say "every integrable function f has some antiderivative F so that you may compute the integral by computing F at the endpoints", yes, your definition can be useful. On the other hand, it's also an important question to consider "which functions can be derivatives?" and in that sense, the definition is less useful. But definitions are definitions; the most we could objectively argue about is which one is the more standard one.

> Here again I'd argue over semantics, though I'd concede it's much more a matter of personal preference than my argument above, which I think has mathematical weight behind it. Namely, I'd argue that the numerical integration is doing something directly with the non-elementary anti-derivative, namely, evaluating it at a point—just like we call reading off the value of, say, the sine of an angle from our calculator doing something directly with the sine, even though what we're really doing is summing sufficiently many terms in a Taylor-series approximation.

Fundamentally, at a mathematical level, yes. That's what it means for two definitions to be equivalent. But on an algorithmic level, the process of evaluating an integral numerically and the process of finding an antiderivative (especially symbolically) are quite different things.

But in the end, it doesn't seem that we fundamentally disagree.