What do Constructivists think of Integrals?

["pi"mp]

So if we regard something as only being well defined if we can construct it, does this somehow affect what we think about integrals? The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull an anti-derivative out of thin air. So it seems like constructivists might take serious issue with an integral. Maybe not in and of itself, but certainly if you actually want to evaluate it.

It seems like every single physical quantity expressed as an integral, we have absolutely no business knowing. Unless we happen to be clever in that particular case. I'm curious what the standard line of thought here is. Is this something people commonly take issue with?

 

[pwsnafu]

The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull an anti-derivative out of thin air.

What do you mean by this? You can prove from the definition of Riemann integral that
$\int_0^x t \, dt = \frac{x^2}{2}$.
You can prove from the definition of stochastic integral that
$\int_0^x W_t \, dW_t = \frac{W_x^2-x}{2}$.
You can prove prove from the definition of Lebesgue integral that
$\int_0^x I_{\mathbb{Q}}(t) \, dt = 0$.

There are also plenty of important integrals that don't have easy closed form expressions. But we have plenty of numerical techniques to use.

Unless we happen to be clever in that particular case

Depends on what you mean by clever. I guess the Risch algorithm (and its extensions) are clever.

 

["pi"mp]

But in general, unlike differentiation, there doesn't seem to be an algorithm for integrating functions. It's just a patchwork of different techniques that only apply to certain cases, along with approximations, asymptotics, and numerics. Maybe I'm being dumb, but all those seem to me to be incredibly inelegant ways of finding something that we have no business knowing in general.

 

[pwsnafu]

But in general, unlike differentiation, there doesn't seem to be an algorithm for integrating functions. It's just a patchwork of different techniques that only apply to certain cases, along with approximations, asymptotics, and numerics. Maybe I'm being dumb, but all those seem to me to be incredibly inelegant ways of finding something that we have no business knowing in general.

And that's a problem, because? We don't have elegant ways to factorize either, and that's a good thing.

But in general, unlike differentiation, there doesn't seem to be an algorithm for integrating functions

Does there exist an algorithm which calculates the derivative if it exists or returns "no" in finite time? Because there appears to exist computable functions with uncomputable derivative.

 

["pi"mp]

And that's a problem, because? We don't have elegant ways to factorize either, and that's a good thing.

Yeah, I see your point, but we can, at least in principle, factor a huge number by hand provided we have enough patience. I just find it very unsettling that important numerical quantities for the universe are embedded in these integrals which we can't do without making approximations or applying numerics.

Does there exist an algorithm which calculates the derivative if it exists or returns "no" in finite time? Because there appears to exist computable functions with uncomputable derivative.

Hmm, this is interesting. Even for functions of a single real variable there are some whose derivative would take infinite time to compute?

 

[pwsnafu]

Yeah, I see your point, but we can, at least in principle, factor a huge number by hand provided we have enough patience.

In theory, there is no difference between theory and practice. In practice, there is.

Hmm, this is interesting. Even for functions of a single real variable there are some whose derivative would take infinite time to compute?

Looks like it. I didn't know this until 15 mins ago. Google search for "computable function with uncomputable derivative" gives an exercise sheet with the content

EXERCISE 5: The lecture constructed a computable function f ∈C1[0;1] with uncomputable derivative

and if you do a book search we get this index entry...from a journal I've never heard of. Hopefully someone on this forum can give us more insight.

 

[micromass]

So if we regard something as only being well defined if we can construct it, does this somehow affect what we think about integrals? The way I understand it, there is absolutely nothing in mathematics that tells you how to actually do an integral. Fundamentally, all we can do is cleverly pull an anti-derivative out of thin air. So it seems like constructivists might take serious issue with an integral. Maybe not in and of itself, but certainly if you actually want to evaluate it.

It seems like every single physical quantity expressed as an integral, we have absolutely no business knowing. Unless we happen to be clever in that particular case. I'm curious what the standard line of thought here is. Is this something people commonly take issue with?

In constructivism, there is absolutely no problem with the integral. The definition of the integral as the limit of Riemann sums is perfectly fine and allow us to approximate the integral to any degree of accuracy. We can then prove the fundamental theorem of calculus constructively and it yields that if $g(x) = \int_a^x f(t)dt$, then $g^\prime = f$ for continuous $f$, and also that it is up to constant the unique function that does this. The details can be found in Bishop's "Constructive Analysis".

 

["pi"mp]

But the Fundamental Theorem of Calculus doesn't tell you how to actually *do* an integral, unless you're first clever enough to guess an anti-derivative. Someone pointed out earlier that we don't have an elegant way to factor, but there is at least a fool proof algorithm for factoring, it just might take a ridiculous amount of time. There is (to my knowledge) no algorithm anywhere that tells us how to integrate a function in general.

But you're saying, this is not an issue in constructivism? They are happy with the perfectly well-defined notion of an integral and aren't worried about evaluating them?

 

[micromass]

Constructivism gives a way to evaluate integrals. It gives you a way how to compute (in principle) $\int_a^x f(t)dt$. That is, for any point $x$ that you choose, you get a specific formula for that function.

 

["pi"mp]

Wow, that's really interesting. I had never heard of that. Thanks!

 

[micromass]

It really sounds better than it is. The constructivist proofs give no easy way to find an antiderivative, and certainly not in closed terms. For example, the constructivst proofs can be used to evaluate $\int_0^x t^2 dt$ for every $x$. For example, let us evaluate it for $x=1$, the evaluation is defined as the following limit:
$$\lim_{n\rightarrow +\infty} \sum_{k=0}^n \left(\frac{k}{n}\right)^2 \frac{1}{n}$$
It can be shown constructively that this limit exists, so the constructivists give a way to evaluate its value in principle. But note that we do not get a nice closed form expression of the limit this way. Indeed, a real number in constructivism is defined as a Cauchy sequence of rational numbers. So to specify a constructivist real number one needs to constructively specify all the terms in the Cauchy sequence (and then show it is a Cauchy sequence). In this case, it is easy, we get the cauchy sequence
$$y_n = \sum_{k=0}^n \left(\frac{k}{n}\right)^2 \frac{1}{n}$$
What does this mean in practice? That we can approximate $\int_0^1 t^2 dt$ to any degree of accuracy. This is the constructivist way of defining and evaluating integrals. Of course, one can say that the antiderivative of $t^2$ is $\frac{1}{3}t^3$ and thus that the integral is equal to $1/3$. This is true in constructivism, but the statement is "IF $g$ an antiderivative, then ...". So no easy way of finding the antiderivative is given. You always have one, but it won't be an easy expression.

 

["pi"mp]

So for an arbitrary f(t) there's a straightforward way to write the analog of y_n you have above? But it will be an un-illuminating, unwieldy mess?

 

[micromass]

Right. Constructivism does not care about efficient calculations, it only concerns with whether an approximation can be done in principle given finite (but possibly large) time.

 

["pi"mp]

I see. I had no idea such a thing existed. Thank you. So I guess it is like the factoring example; there is a fool-proof algorithm but it is entirely useless in all but certain simple cases.

 

[FactChecker]

I see. I had no idea such a thing existed. Thank you. So I guess it is like the factoring example; there is a fool-proof algorithm but it is entirely useless in all but certain simple cases.

I wouldn't say that. There are numerical methods to calculate integrals that are used very often when the closed form solutions is not known. Factoring can be much harder.