The boundaries of mathematics?

Question

• There are five possible answers to a multiple-choice question. Given that the student does not know the answer, what is the probability that the student chooses the first answer?

That is not a well formed math problem, to say the least, but I once told a student that it is not actually a math problem and he couldn't understand why, and thought it was obviously a math problem.

But I wonder if our understanding of what constitutes a mathematics problem should be expanded somewhat beyond what mathematicians conventionally think it is, in order to include instances of reasoning that can be done by mathematicians and is unknown to others, to the extent to which those others are not mathematicians.

Obviously this calls for examples, and at this moment I have only two:

• Clearly Regiomontanus's angle maximization problem is a mathematics problem. One posits a painting hanging on a wall with its lowest and highest edges above your eye level. Given those two heights above your eye level, how far from the wall should you stand to maximize the angle whose vertex is at your eye and whose rays are incident to the top and bottom of the painting? (This has a solution by elementary geometry not involving calculus, another solution by a somewhat unconventional way of completing the square, and another by calculus, in which perhaps the most efficient way is to directly maximize the tangent of the angle.) However, often people phrase this incorrectly: they ask what distance from the wall gives you the best view of the painting. So let us suppose that I announce the following "theorem":

  The distance from the wall that maximizes the angle does not generally coincide with the distance that gives you the best view of the painting.

  If this is a theorem then it has a proof. Here it is:

  First suppose that the lower edge of the painting is exactly at your eye level. Then you would maximize the angle by placing your eyelid in contact with the lower edge of the painting. As you approach that point, the angle approaches $90^\circ,$ and cannot get more than that. But this is clearly not the best possible view. Next, suppose the lower edge is a tenth of a millimeter above your eye level. If the top of the painting were one meter above eye level, then the angle is maximized by making the distance one centimeter. Clearly still not the best view. The quality of the view varies continuously with the distance, and the angle also varies continuously with the distance, so the angle cannot suddenly coincide with the quality as you slowly back away. Quod erat demonstrandum.

• Suppose cheese and chalk are two commodities that you are in the habit of buying. If I steal some of your cheese or some of your chalk, I leave you worse off than you were, and if you gain some of either your lot is improved. Now suppose if you have $20$ units of cheese and $30$ units of chalk, you are just as well off as if you have $50$ units of cheese and $20$ of chalk. If $f(x,y)$ is your utility of $x$ units of cheese and $y$ of chalk, then the level set of $f$ passing through the two points just described could have any of many shapes; this is an "indifference curve". Discovering these curves is an empirical matter, not a matter of mathematics, and economists have found that in realistic situations they may have any of many shapes. None of that is a mathematical theorem; rather it is an empirical finding. But now suppose if you invest your wealth in a certain way, then the probability distribution of the amounts of cheese and chalk you will have next February 30th is thus-and-so, but if you invest your wealth in another way, you have a different probability distribution. Suppose you are indifferent between those two probability distributions. Now we state a "theorem":

  The indifference curve between your two equally valuable probability distributions must be a straight line.

  Proof: Since you are indifferent between them, you don't care if I throw dice to decide which one you get. Therefore any weighted average of the two distributions has the same utility as either of them. Q.E.D.

Why should we regard these "theorems" and "proofs" as belonging within mathematics rather than being located in some other region of the intellectual realm? To which I reply: Because only an understanding of mathematics can make it possible to understand them.

This is not a position of which I am convinced, but I do think the foundations of the subject are not as well understood as many feel they are.

Questions: Do such examples bear upon the question of where the boundaries of the discipline are?

Answer 1

You seem to be asking two related questions: (1) are the boundaries of mathematics known? (2) have the foundations of mathematics been well understood?

Rather than answering these from first philosophical principles such as the realism/nominalism dichotomy, I would suggest that the answer lies in an examination of the history of mathematics. At the turn of the (19th) century mathematics was understood as arithmetized analysis as practiced by Weierstrass and his students. In 1900 in his famous lecture at the world congress of mathematicians in Paris, David Hilbert turned the tables on the traditional view by offering a broad scope of problems including set theory, logic, and axiomatisation of physics (only a minority of which relates to arithmetized analysis). This is of course not the only but perhaps one of the most famous instances when the boundaries of the subject have changed dramatically over time.

If the subject matter of mathematics evolves surely its foundations necessarily evolve as well, as is illustrated by the historical development of category theory starting in the 1950s, challenging the traditional set-theoretic foundations.

An important aspect of the picture is the fact that the boundaries of mathematics are not always growing; sometimes they are shrinking. Thus, infinitesimals were part and parcel of the landscape of mathematical analysis for about two centuries: 1670-1870. Sometime after 1870 a more-or-less official ban against them was issued (that was not uniformly followed however) that held for about 90 years. Around 1960 infinitesimals were restored to respectability though some still seek to enforce the obsolete ban.

Therefore my answer to both questions is negative.

Answer 2

You probably have been thinking sometimes about everything I am going to say here but I will try to express myself in such a way that I will try to stay as much as possible inside the boundaries of the question, in a way in which I interpret your question.

First, it seems that question in itself includes a need for as precise as possible (with respect to the current state of this branch of science) the definition of "mathematics" itself.

Because, how to talk about the boundaries of mathematics if there is no clear enough definition of mathematics.

The definition of mathematics could be time-dependent and could be that it will have to change as the subject evolves (or avolves) in time and with such processes it could be that the boundaries of the subject will also "expand" to include more and more inside the subject.

Even if we do not have clear enough definition of mathematics, we could try to define the concept of "theorem" and of "proof (valid or invalid)".

When armed with some definitions of "theorem" and "proof" we can then argue what is and what is not a theorem, and what is and what is not a valid or invalid proof, but only with respect to the definition we use (or definitions).

This could lead to different "theorem theories" and different "proof theories", and, more generally, "theory theories", but all definition-dependent.

So, all of this is very subjective and opinion-based, at least if you ask me.

I do not know what else to say and I am extremely tired so I will stop here, aware that I explained almost nothing.

Answer 3

TL;DR your question hits at the Nominalism vs Realism debate. If stuff have actual mathematical properties in real-life (without people), then model-making does belong in mathematics since mathematics is essential within the universe. I think people give the mathematical properties to objects, and that model-making begins in the mind, not in reality nor mathematics proper.

As user Petr Naryshkin pointed out in the comments, your question is asking whether model creation belongs as a part of mathematics proper. I do know that the process of making models cannot belong in a formal proof, and I'm somewhat certain that it does not belong in mathematics proper.

If you ever try to make a formal proof of those aforementioned propositions, or really anything which applies the tools of mathematics to the real world, you'll find yourself importing axioms. That's your base model.

Where do these axioms come from? Are these axioms true?

You can't answer these within the proof theory. Maybe you can use proof theory to develop more basic axioms, and figure out whether the underlying axioms are consistent with each other (or whether it's possible to know consistency), but it makes no assumptions otherwise about the axioms.

To answer those two questions, you need to reach outside. Euclid never made any assumptions about what points, lines, and the "lying upon" relation were, only that these entities had a certain way of interacting with each other. And that process of meaning-making is purely social and unrigorous. It's a psychological process of developing analogies with previously-experienced objects and phenomena, in exactly the same way that language is developed. Pushing worldly notions into ideal forms of mathematics, and subsequently pulling out the consequence, is a psychological process. That a space shuttle correctly orbits and docks with immense precision, or that particles decay while conserving numerical quantities, simply reflects an effective, socially established common mapping of objects to concepts; the modelers "correctly" mapped their objects to their corresponding mathematical notions.

I don't know what name this philosophical view has, maybe Nominalism? I think it might be at-odds with the metaphysics of scientific realism, which says that "The entities described by the scientific theory exist objectively and mind-independently."