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In this blog post, we find the following passage:

This connects with something Thomas Forster said, when he rightly highlighted the distinctively modern conception of a function as any old pairing of inputs and outputs, whether we can define it or not — this is the ‘abstract nonsense’, as Thomas called it [...]

But isn't that definition of a function still productive (theoretically), even if it encompasses functions that we may not be able to construct? After all, it seems at first that anything we will say about functions defined that way will stay true even for functions we could not construct? Or is the difficulty that for functions that we could not construct (for example because they would require an infinite extensional enumeration that would be suspicious), that definition may lead to further difficulties? What is really the issue with that definition?

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  • What does "construct" mean?
    – Daron
    Feb 18, 2023 at 22:31
  • @Daron let's say that "constructible" means "computable" as in en.wikipedia.org/wiki/Computable_function.
    – Frank
    Feb 18, 2023 at 23:04
  • The issue is that from a constructive point of view a function is defined when we have a rule (specified by a finite number of instructions) to compute the output for an input whatever. Feb 19, 2023 at 9:47
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    @Daron "But isn't that definition of a function still productive (theoretically), even if it encompasses functions that we may not be able to construct? After all, it seems at first that anything we will say about functions defined that way will stay true even for functions we could not construct?" I am wondering if that style of definition will run into difficulties later. It is pretty clear.
    – Frank
    Feb 19, 2023 at 15:10
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    See Constructive Mathematics: "Constructive mathematics is distinguished from its traditional counterpart, classical mathematics, by the strict interpretation of the phrase “there exists” as “we can construct”." Feb 20, 2023 at 7:38

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This connects with something Thomas Forster said, when he rightly highlighted the distinctively modern conception of a function as any old pairing of inputs and outputs, whether we can define it or not — this is the ‘abstract nonsense’, as Thomas called it.

‘Abstract nonsense’ is a phrase that emerged in mathematical culture in reaction to category theory, but it was cheekily appropriated by category theorists themselves who use it to describe their own work with an ironic championing of extremely high levels of abstraction. (Citation for my claim needed.)

I can agree that the idea that a function is nothing more than “an association between the elements of two sets” is a relatively new idea in the history of mathematics. I forget what I read about when this happened, but it may have been as late as the 20th century. I think people found that mathematics “works nicer” when you use this definition.

But isn't that definition of a function still productive (theoretically)?

Extremely. It is hard for me to imagine doing math without that definition, since it has become universal. It is not an advanced way of looking at functions, but is the standard way they are defined, in undergraduate mathematics courses. It is arguably extremely productive since I estimate most people doing work in mathematics are using that definition. But your question opens the fascinating question about modern alternative definitions of “functions”, or if anyone has suggested discriminating between functions defined as pairs of elements from sets, vs. functions as some kind of sequence of “transformation” of input data. (The problem is that operations in modern math are functions, which are defined as associations, so I think you either have to accept functions as a primitive notion, or try to define them in terms of something else than functions).

Even if it encompasses functions that we may not be able to construct?

The idea of non-constructible functions is a pretty advanced topic that I don’t know much about. I guess you’re asking about how theorems about functions change in constructive vs. nonconstructive mathematics.

After all, it seems at first that anything we will say about functions defined that way will stay true even for functions we could not construct?

Not necessarily. In fact, that’s kind of the main emphasis in constructive vs. non-constructive math: the theorems change between the two. Some theorems are true in one and false in the other.

So I do hope to follow up and give you more concrete detail soon. The simplest answer is that actually no, you would not necessarily have the exact same mathematical behavior, if you modified the exact definition of function in that way.

History of the concept of functions

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  • Sounds quite promising Mar 17 at 18:03
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    +1 As a side note, everything that is true in constructive mathematics is true in “regular” mathematics, but the reverse fails. Mar 17 at 21:28
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Constructibility is a Way of Life

Correct me if I am wrong, but people interested in only constructible functions tend to be interested in constructible Mathematics in general.

They would not recognise a proof something exists unless you can write down an explicit counterexample. A proof like

Assume suchandsuch does not exist, yadda yadda yadda, axiom of choice, hence we arrive at a contradiction.

Would not be recognised by a constructibist. They want a proof like

We claim $g(x) = 2X log(x) / 5 + $ is the desired counterexample. We must prove it maps every point to the codomain, and has the desired growth rate. To that end. . . .

Then proceed constructibly through each part of the proof.

What is the point of all this? It is about your comment on what we can say about functions in general

After all, it seems at first that anything we will say about functions defined that way will stay true even for functions we could not construct?

From a constructibilist point-of-view, some claims about functions in general are not true, since they have not been proved constructibily yet.

For example the well-known statement that every function is surjective if and only if it has a right inverse relies on the axiom of choice to prove. The axiom of choice is non-constructive. The proof does not fly from a constructibility point-of view.

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    OK, so it's the same issue all over again as usual: AC and infinity.
    – Frank
    Feb 19, 2023 at 14:27
  • @Frank AC is just the first example that springs to mind. Nothing about infinity so far. Perhaps you can find an example.
    – Daron
    Feb 19, 2023 at 14:47
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    "Infinity" because for finite cases, AC is not an issue.
    – Frank
    Feb 19, 2023 at 15:09
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Actually, @PeterSmith (the author of your linked post) occasionally pops his head by here. I will summarize the points leading up to your question, then discuss it in context:

(a): The main question is this: Does math need philosophy?

(b): In one sense (that of broad strokes), perhaps not. Phil math is most concerned with, say, platonism (small p) and its rivals. Any good philosophy of math will (i) validate math as commonly practiced (ii) fit with our best theory of knowledge, abstract objects, etc... Since (i) must hold true, and (ii) is not mathematical work, mathematicians can leave it to the philosophers.

(c): Contra (b), Smith brings up predicativism, one of the (bigger) schools of phil math. The predicativists (Russell, Poincare, Weyl) took impredicative definitions to be problematic, the paradigmatic case is Russell's paradox. One way to cash out impredicative definitions is as follows: they define E by quantifying over a domain which includes E (classic example: inf X, which is defined as the greatest lower bound, must quantify over lower bounds, which inf X is part of). Ramsey noted that such impredicative defintions were fine- in a realist context. (example: the tallest man in the room, impredicative since it quantifies over men in the room, of which the tallest man is one of). Since modern math uses impredicative definitions all the time, it is (perhaps) implicitly assuming that one of the big schools is wrong. Hence, in affirmative to (a), math as practiced currently does rely on a particular philosophy.

Now, to discuss your question: the modern notion of function is very productive mathematically. The issue, in context, is that many functions are defined or constructed impredicatively* ( the classic is sup x, or if one insists on a function, given a sequence of functions consider \sup_{f_n}(x)- this is an instance of a more general case, see @Andrej's answer here: https://mathoverflow.net/questions/26220/impredicativity). So the problem in context is philosophical, although @Carl's answer on the same linked post discusses some of the mathematical consequences as well.

*Predicativism has many different exact characterizations. Historically, predicativism is taken to lie somewhere between intuitionism and platonism, Weyl himself took the natural numbers as given and Poincare defended some Kantian notion of intuition in math. The linked post offers some different ways as to how its used in math today.

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