What is mathematics, really?• mathematics • teaching • Permalink
Greg Ashman recently published two provocative posts (the first and the second) in response to Dan Meyer’s post, claiming that “A lot of people don’t seem to understand what mathematics is”. Dan Meyer’s statement:
“Math is only objective, inarguable, and abstract for questions defined so narrowly they’re almost useless to students, teachers, and the world itself.”
formed the starting point of this. Greg’s central thesis is that mathematics uses deductive reasoning, as opposed to other subjects which use inductive reasoning. From this follows the argument that calculating things like p-values is mathematics, whereas evaluating their meaning or usefulness is science. (I encourage you to read the full posts; I have just picked a couple of points out.)
But this seems to be quite a strange argument. Let us consider a metaphor: what is Art? A painting or a sculpture would be examples of Art (though I would not like to begin defining “Art” itself). We might admire them, study them, and so forth, but that would not be “doing Art”. An artist may well do this to gain inspiration, as well as studying art techniques so that they can produce their own novel art.
Mathematics is similar. Let us focus first on pure mathematics.
Doing “pure” mathematics
A beautifully presented piece of deductive reasoning is a piece of mathematics. Mathematicians will study existing such arguments to learn ideas and techniques, but one of their main focuses is to generate novel such arguments, showing that some result is true. This is always a creative process, and sometimes hugely so. The process is messy, exploratory, and full of inductive reasoning. It often takes the form of “I’ve noticed that in all these cases I’ve tried, such-and-such seems to happen. Maybe I can prove that, and that will help me to show that the main result is true.” The deductive argument which results is a product of doing mathematics, just as the painting is the result of the artist’s messy exploration, trial and error, and so on.
And even having a deductive argument is not the end of the story. When presented with a deductive argument, how does one check that it is correct? Perhaps one could do so just by following the given argument and checking every step? Unfortunately, the challenge of justifying any but the simplest theorems using automated theorem provers shows that this is generally far more involved and complicated than it may superficially seem. The skill of seeking counterexamples to arguments, of finding errors in proofs and ambiguities in terminology is a highly creative one, not a logical-deductive one. So even in the seemingly purely logical world of deductive mathematical arguments lies a huge amount of exploration, insight and perhaps even induction.
And those questions which have an objectively correct answer at school level are generally calculations. And even there, we can start asking interesting (and valuable) non-objective questions such as: Is this the best approach? How do these two different methods compare? When would you use one rather than the other? When would we want/need to perform such a calculation? These questions are as important or - given the ability of calculators and computers to do the grunt-work - arguably more important than the calculation themselves.
Another vital aspect of (pure) mathematicians’ work is the generation of conjectures. This is what spurs others on to try to find arguments (proofs) for these conjectures being true, or to show that they are false. At the extreme lie the Millennium Problems, which are very important conjectures which had defied all attempts at solving them. (One has since been solved.) There are also many smaller (but still significant) conjectures, which are some of what continues to drive mathematics. But how did mathematicians come up with them? By doing experiments, and lots of … you guessed it … inductive reasoning.
I am not questioning the central importance of teaching and learning calculation techniques (in the very broadest sense), for it is impossible to do mathematics without them, and I am not arguing for or against any particular pedagogical approach. I likewise strongly agree with Greg that we must teach deductive reasoning in mathematics, because it is so central to the subject and one of the unique qualities of school-level mathematics in comparison to other subjects. (Indeed, I am in the middle of writing a book about this.) But I do claim that just performing calculations, such as calculating p-values or whatever, is not “doing mathematics” - it is simply doing calculations. And likewise, reading other people’s deductive arguments is not “doing mathematics” either (though it is a prerequisite to creating one’s own arguments). “Doing mathematics” must involve some form of exploration, generation of conjectures, and justification (to paraphrase John Mason), and that must take place in the classroom, just as school art lessons are not only about learning specific techniques.
So much for pure mathematics being deductive; where does that leave us when we consider applied mathematics? We must first recognise that the boundary between Mathematics and Non-Mathematics is quite fuzzy; a cosmologist or fluid dynamicist might equally find themselves in a Physics Department or an Applied Mathematics department, depending upon the institution. (Likewise, a pure mathematician might end up in a Philosophy Department or Computer Science department if they are studying logic or category theory.)
But returning to the statistical examples with which we began, how do we handle those? I would contend that calculating a p-value is just performing a calculation; without thinking about its meaning or significance, that cannot be considered “doing mathematics”. Is considering the meaning of such calculations actually part of a different discipline, such as Statistics or Physics? That is returning to the question of where one draws the dividing lines, but what seems clear is that we should not be teaching the calculation of something without also teaching the meaning of the calculation. Whether this takes place in a maths lesson or a physics lesson or a statistics lesson is immaterial.
So how does applied mathematics fits into the mathematical family, and should we consider statistics as part of mathematics? That itself is a long discussion, and I am sure that disagreements abound. So a brief thought will have to suffice here: applied mathematicians use their mathematical reasoning powers to apply mathematical tools to interesting and novel problems in the “real world”. And that is, in some sense, very similar to pure mathematicians using their mathematical reasoning powers to apply mathematical tools to interesting and novel problems in the “mathematical world”. So perhaps they are not so different, after all.
Finally, a key educational question
Perhaps the argument between Dan Meyer and Greg Ashman comes down to the question of what we want to teach in our classrooms. Do we want to just teach pure calculation? Or do we want to teach the creativity and messiness of mathematics, alongside the deductive aspects?
Some students are very well-served by having a subject in which they can be objectively right or wrong: it gives them a stability and safety which is lacking in many other subjects. But for other students, who could end up becoming excellent and creative mathematicians, this objectivity is stifling and off-putting, and they will never see mathematics for what it truly is. In the world today, we don’t need people who can just perform calculations: computers are so much better than humans at that. We rather need people who can think and ask good questions, and as a small part of that, who can decide what calculations need to be performed.
The debate about how we educate our students is hugely important. There is no “right” answer. It is, instead, a question of what we believe a mathematics education is aiming to achieve, and there it seems that Dan and Greg fundamentally disagree.
A relatively readable and very interesting book on the nature of proof and deductive mathematics is Imre Lakatos, Proofs and Refutations, and several of the ideas in this post have been inspired by it. It is quite old, but it highlights how complex the Philosophy of Mathematics actually is.