Yesterday I gave a talk to the Nottingham Trent University Maths Society, ‘A brief history of mathematics: 5,000 years from Egypt to Nottingham Trent’. I had a slide in this where I said something about what the Greek style of proof means for mathematics. It has helped me put my finger on something of why mathematics isn’t like science, and I thought I would share it here so I can look it up when I’ve forgotten again.

Up to a point, it might seem reasonable to explore an issue by finding a bunch of examples and extrapolating a general rule that your examples seem to obey. I realise there’s a little more to it than that, but this is basically what science does. This process is called inductive reasoning, because a general theory is ‘induced’ from the ground up.

Mathematics, on the other hand, follows a deductive process. A set of basic ideas are assumed (we call these axioms), and a series of propositions are ‘deduced’ from these via proof. Of course, in reality there are mathematicians on the applied side who are effectively doing science, but at its heart, mathematics is a process of deductive reasoning.

So science induces from evidence, while mathematics deduces from assumed truths. This is why a mathematical truth (a true statement within a constrained system) remains true throughout time, while scientific truth (an idea based on a lot of evidence) can be overturned by new observations.

My officemate, a forensic scientist, is currently reading some Sherlock Holmes stories. I’ve never read these, and am only aware of the material through various screen incarnations, but I’m aware that what Holmes does is sometimes referred to as “deduction“. Actually, Holmes is proceeding from close observation and works to establish the facts of a case from these — a process of inductive reasoning.

We approached the case, you remember, with an absolutely blank mind, which is always an advantage. We had formed no theories. We were simply there to observe and to draw inferences from our observations.

(The Adventure of the Cardboard Box)

It is interesting to me that Moriarty, nemesis of Holmes, is set up as a Professor of Mathematics. I wonder if this is deliberate, to establish Moriarty as Holmes’ opposite, who reasons completely differently to Holmes (but is almost his match), and whether this distinction features overtly in the stories. This ability to reason appears crucial to Moriarty’s evil nature; he is not just any villain, because his “criminal strain” is “increased and rendered infinitely more dangerous by his extraordinary mental powers” (The Final Problem).

Of course, this makes science the hero and maths its diabolical nemesis. So be aware, dear reader, that we may be on the wrong side of good.

Nice article. Do you think it’s fair to say that Science is subjective (working from observation) and Maths is objective (working from a fixed foundation)? It could be the lack of subjectivity that makes Maths seem less human and therefore evil…

I’d say maths is subjective – you get to choose your axioms!

I would have said “yes, perhaps”, but then I read Christian’s reply, which I am fairly convinced by!

It’s not always true that science finds a bunch of examples and extrapolates. The positron was predicted from Dirac’s equations way before it was found. Higg’s boson was postulated long before it was found.

In a way, weren’t neutrinos and Higgs bosons sophisticated extrapolations of the currently available evidence? For example, Dirac’s equations first seem to govern some observable evidence, and an extrapolation of their applicability predicts the neutrino, no?

Thumbs up from me for starting your talk in Egypt and talking about the Greek ‘style’ of proof.

I’d agree with you that being a mathematician can be used as a short hand for being cold and emotionless – literally calculating. What I find more troubling though is films like Good Will Hunting that perpetuate the myth that either you are a mathematician with innate abilities or you are not. Huge problem for all of us that are trying to do maths outreach.

Scientific problems require mathematical thought. I didn’t know that Newton needed to understand differentiation to deduce F=ma. At school I was taught that he observed apples falling! You are right though. I often hear the words mathematical model to describe relationships. Science doesn’t rely on experimental evidence as it’s made out to be.

” A set of basic ideas are assumed (we call these axioms), and a series of propositions are ‘deduced’ from these via proof. ” This seems a bowdlerised view of mathematics. Have you never read “Mathematics and the imagination”, by Kasner and Newman?

Can you imagine a research method which I called : “An idea for a proof in search of a theorem”?

Of course constructing a proof is part of the craft of mathematics, but that is not what maths is “about”.

Have a look at articles on my Popularisation and Teaching page, http://pages.bangor.ac.uk/~mas010/publar.html

e.g. on The Mathoetic Mode

“`The poet’s eye, in a fine frenzy rolling,

Doth glance from heaven to earth, and earth to heaven,

And as imagination bodies forth the forms of things unknown

The poet’s pen turns them to shapes, and gives to airy nothing

A local habitation and a name.’ ”

I particularly like the third and last lines, as echoing what mathematicians do. The notion of “name” is very subtle, and is a way knowledge advances.