“What is Mathematics?”

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“What is Mathematics?”

February 1, 2012 Leave a comment Go to comments

It might be a little annoying that a description of mathematical activity that satisfies everyone has not yet been found. People have been doing, teaching and learning mathematics for centuries, in schools, colleges and universities throughout the world. Math is something there for all to see. So, what is being asked for? Moreover, what could be learnt from any such description, given that it is of something already known? There are several reasons for this situation.

Usually, when such a kind of question in relation to some activity (medicine, for example) is asked, there is behind it some concern with a potential or actual source of professional intrusion, that is, a concern with some distortion undermining honest professional practice. I think that this is, in part, the motivation of some of Frank Quinn’s writings (links to a couple of them can be found at my previous post). Quinn worries that “a century of difficult adaptation to precision might be eroded by a new wave of heuristic work”. According to this author, heuristics may play their role in “mathematical science”, but they should not be used to justify results in “core mathematics”, because results so obtained cannot, and in fact must not, be considered reliable. In contrast, another author, Carlo Cellucci, defends that mathematics cannot be understood without the ‘analytic method’, which makes essential use of heuristics, or rules of discovery (see the Suggested Readings, below). So there is a demarcation problem here. “What is mathematics?” stands for “Which means are to be considered mathematical and which not? Should some open-endedness be allowed so that as new means are discovered they can be added to the collection of those already known? How would one then decide whether candidate means deserve the right to join the previously accepted ones?”, and so on.

But it is also common that what is sought with this kind of question is to put the discipline in a broader context than the most immediate to its practice, to look at it from a different perspective, so to speak, and there are a host of possible choices of perspective (historical, sociological, philosophical, educational, … and any combinations and ramifications of these). Actually, there could be so many different expectations behind a single formulation of the question that it would be impossible that one answer could satisfy all of them at once. Gian-Carlo Rota described several senses of the question “What is mathematics?” under various circumstances in a chapter of the Lezioni Napoletane, ed. by F. Palombi, La Città del Sole, Naples 1999 (an English version of this chapter, entitled What “Is” Mathematics?, can be found here. I thank Cellucci for the reference). Rota concludes that the feeling of wonder expressed by this question is the start of a philosophical journey that will eventually disclose the ‘conditions of possibility’ of mathematics. Maybe Quinn would make the point that this ought to be a scientific journey, rather than a philosophical one. It might seem strange, however, that to understand or to explain what is mathematics one needs to offer an explanation in terms of ‘conditions of possibility’ (just imagine one had to give an account of carpentry or tailoring in such terms). This approach to mathematics springs from Kant’s transcendental idealism in relation to science and the general problem of knowledge as a response to Descartes’ methodological skepticism. But there is also a modern motivation for it. In connection with Gödel’s realism (I borrow here terms from Penelope Maddy) ‘common features of [mathematics] are taken to derive their justification’ from ‘Robust Realism’ for which ‘the justification — or lack of justification — for mathematical methods is based on a metaphysical account of its subject matter’ (Maddy [2010], p. 87). By calling for metaphysical or epistemologycal accounts, one attempts to leverage the exploration of the frontiers of incompleteness, trying to answer exactly the same questions as above but now with an entirely different purpose in mind.

It is, hence, natural that different answers exist to the question “What is Mathematics?” each of them in relation with a different purpose or throwing light on a different aspect of mathematical activity.

Conditions of Possibility and Objectivity

If one had to look for what makes it possible that human beings do mathematics probably a good place to start would be our ability to establish precise rules and accurately follow them, checking whether each rule has been perfectly observed, and to make plans and carry them through, controlling that they produce the desired results; all of this together with our ability to disregard any rules and change any plans at convenience.

Creating rules is for our minds like creating the images or the sounds of our perception. It happens naturally and possibly it is creating rules what underpins the meaningfulness of our experiences. When we think, both rules learnt in the past and new rules emerging from current experience come into play. Ideas, concepts, etc. are ultimately systems of rules that are frequently involved in our thinking processes. Moreover, what we call reality could be understood as just one of those systems. All of them are systems of rules with an instrumental value. Mathematical ideas, in this sense, are no exception: they are an aid for thinking and understanding. Mathematics could then be seen as a style of reasoning and looking at things. But this, all alone, does not seem to explain its objectivity.

Richard Feynman, through Ralph Leighton, in “Surely You’re Joking, Mr. Feynman!”: Adventures of a Curious Character, Part 2 “The Princeton Years”, under the section “A Different Box of Tools”, shares with us some ideas I personally find very interesting.

“trivial” means “proved” […] mathematicians can prove only trivial theorems, because every theorem that’s proved is trivial […] there are never any surprises […] mathematicians only prove things that are obvious

I can find at least two senses in which what these words suggest could be regarded as true:

1. As soon as a proof of a theorem has been found, the logical connection between the premises and the conclusion, with the aid of the proof, becomes transparent. A proof can then be seen as a chain of platitudes.

2. Axioms fix beforehand which statements are theorems and which are not. What you get is what you put in the axioms. Therefore, there is no room for surprises.

I bet there isn’t a single theorem that you can tell me what the assumptions are — and what the theorem is in terms I can understand — where I can’t tell you right away whether it’s true or false

Feynman is claiming neither that all mathematical theories are decidable nor that he is an oracle. Also, by theorem here he refers to any statement known to be true or false after it, or its negation, has been proved from the assumptions (theorems are always true by definition). Even after these caveats, it is still a bold claim. Without the aid of the proof, in contrast to what has been said just above, most theorems are anything but obvious. He explains the method that justifies this exhibition of self confidence (emphases are mine):

I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) — disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”
If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample.
“Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.”

I adore this passage. I think it is a really good description of the process of understanding something. Moreover, together with the previous one, it is a perfect introduction to the issue of objectivity in mathematics. In a nutshell: If something is provable for some person, then it is provable for every person (me included!). Actually, knowing that a statement (or its negation) is a theorem, one could program a computer to produce one proof after another and simply wait until the proof of the statement (or its negation) is found. But I think this is certainly not what Feynman did have in mind. Moreover, he clearly manifests that if a counterexample can be found then either assumptions were wrongly understood or the statement is simply not true.

[…] although the mathematicians thought their topology theorems were counterintuitive, they weren’t really as difficult as they looked. You can get used to the funny properties of this ultra-fine cutting business and do a pretty good job of guessing how it will come out.

Again, emphasis is mine. This passage also points to something important: that some theorems might seem counterintuitive is probably only because the assumptions of the theory are not correctly understood.

The objectivity of mathematics would seem, at least in part, to be explained by the objectivity of logic. If $B$ follows from $A$ it is a waste of time to look for a counterexample.

Here go some more passages for reflection, now from QED: The Strange Theory of Light and Matter:

We physicists are always checking to see if there is something the matter with the theory. That’s the game, because if there’s something the matter, it’s interesting!

What could go wrong with a mathematical theory? As I have been saying, there are two possibilities:

1. The theory is inconsistent.

2. The theory is misunderstood.

you might think you do not understand what I am telling you [because] while I am describing to you how Nature works, you won’t understand why Nature works that way. But you see, nobody understands that.

I think that this message is important because in philosophy we make many questions, but science necessarily has to choose very well which battles to fight, its success depends critically on it. The same happens with mathematics. Mathematics has to make decisions as to which questions to answer, and that is only a part of mathematical activity.

It is not a question of whether a theory is philosophically delightful, or easy to understand, or perfectly reasonable from the point of view of common sense.

The question is that a theory makes good predictions. What could it be for a mathematical theory that it made good predictions? It could be revealing the strains of mathematical fruitfulness, as Penelope Maddy would possibly defend.

I’m going to explain to you what the physicists are doing when they are predicting how Nature will behave, but I’m not going to teach you any tricks so you can do it efficiently.

The price we have to pay for doing things efficiently is going technical, but what’s that exactly? This is another important issue to deal with because this is, again in part, what mathematics is about.

I wish I had more time to develop these ideas, and to compare and comment the different author’s opinions, but this is impossible for me right now. I hope to be able to do that soon.

And so on…