Maths in Minutes

Paul Glendinning Maths in MInutes: 200 Key Concepts Explained in an instant Quercus 2012

Both simple and accessible … illustrated by means of a straightforward picture and a maximum 200-word explanation.

Concepts span all of the key areas …

An excellent guide:

  • For anyone who wants a quick introduction to the topics.
  • Those who know about the topics, but struggle to explain them to others.
  • Those who want to see how mathematics being presented by a mathematician who leads in pedagogy and public engagement, and is well-respected within the appropriate institution.

But, at the risk of appearing unduly pedantic, I feel compelled to quibble.

Introduction

… In order to develop some of the ideas in more detail it seemed natural to focus on core mathematics. The many applications of these ideas are mentioned only in passing.

My quibble is that, at least by default, the book could be read as promulgating some misleading (and occasionally dangerous) myths: a lost opportunity that I would like to set right.

Natural number

Natural numbers are the simple counting numbers … . The skill of counting …

To me, mathematics as such is entirely logical, and therefore has no direct bearing on the real world. We may all ‘know’ that counting in the real world has a straightforward correspondence to arithmetic, and mathematicians might routinely make such a claim, but doing so goes beyond the domain of pure logic or mathematics as such. If it is true then it is a scientific truth, not a mathematical one.

Some argue that the above is common sense and ‘goes without saying’. I am not so sure. It seems to me that mathematics ‘proper’ can be taken as literally and relied upon as well as the logic that underpins it: any correspondence  between mathematical structures and reality cannot, – no matter how appelaing to ‘common sense’. Sometimes this seems to matter.

Introducing sets

A set is simply a collection of objects. The objects within a set are known as its elements. [In] many ways sets are the absolutely fundamental building blocks of mathematics – more basic even than numbers.

Actually, numbers can be constructed from sets. But sets can be constructed using category theory. So while sets can be used to construct most of the most familiar maths, it seems to be stretching a point to say that they are ‘the’ ‘absolutely’ fundamental building blocks.

… The elements within a set can be anything from numbers to people to planets … .

Again, if sets are mathematical then they can only include mathematical objects, not things like ‘counting numbers’. According to Wikipedia:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Are ‘people’ and ‘planets’ definite, distinct objects? If we have some definite test of whether an object of perception is a planet then our perceptions of planets might form a set that corresponds to real planets, but this is a scientific question, not mathematical. And the formal notion of a planet can change. To adapt Einstein:

As far as sets refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.

Gödel’s incompleteness theorems

Gödel’s results have profound implications for the philosophy of mathematics – but , in general, working mathematicians tended to carry on as though nothing has changed.

True: but the exceptions may be important.

Probability theory

Probability … is both an application of set theory, and an entirely new theory in itself.

[Mathematicians have developed] a set of axioms for probability, written in terms of the probability of sets and the operations defined on sets.

True. But how can such a theory be applied and interpreted? Are there any common mistakes to be avoided?

Introducing geometry

Geometry is the study of shape, size, position and space.

[New] axioms were introduced in the late 1800s to develop geometry within a strictly logical framework.

This whole section seems in need of qualification. To adapt Einstein again:

As far as Euclidean geometry refers to reality, it is wrong, and as far it is certain, it does not refer to reality.

Surely, the study of shape, size, position and space is a subject for science: all a mathematician can do is to model a scientific concept mathematically. The practical value of this is that in attempting to do so and failing, mathematics was used to show that the classical concepts were not scientifically credible, leading to huge advances in science and the modern world as we know it.

I would rather say:

I mathematics, geometry is the study of concepts of shape, size, position and space. (The realtionship of these concepts to any external reality is the subject of physics.)

Introducing calculus

Calculus is notoriously hard to understand, even for mathematicians. These difficulties largely disappear if we regard it as applying to mathematical models, not to ‘things in themselves’: that is, calculus is purely mathematical, not scientific.

Sensitivity analysis

The description provided could be interpret as being about within-model sensitivity, whereas the graphic is about between-model differences. The first is purely mathematical and may be captured by probability theory. The second is about the science, and is not simply mathematical. The difference matters. To adapt Einstein again:

As far as sensitivity analysis refers to reality, it is uncertain, and as far it is certain, it does not refer to  reality.

The Banach-Tarski paradox

[A] three-dimensional solid ball can be chopped up into a finite number of pieces that may then be re-arranged to make two balls identical to the first. …

The result relies on the existence of non-measurable sets … .

Quite so. But if – as they seem to – non-measurable sets exist, what (adapting an argument of Wittgenstein) would the probability of a point chosen at random from the original ball lying in such a set be?

Logic and theorems

‘Once you have eliminated the impossible then whatever remains, however improbable, is the answer.’ …

Ideally, mathematics starts with a set of objects – primitives – and axioms – properties of those primitives. …

From definitions and intuitions we create conjectures. …

But if you start with a collection of theories and eliminate all but one, that doesn’t mean that the remaining one is correct, or even that it would be a more pragmatic answer than the others.

Mathematics has to start somewhere, but perhaps not with conventional ‘objects’.

The dependence of conjectures on intuition is important: for example, applications of probability theory assume that what is intuitively impossible is actually impossible. But experience shows that our intuition is not so reliable, and sometimes it needs some help. Mathematics can help in creating seemingly impossible ‘virtual’ worlds. If the intuitively possible have been eliminated, we may need to consider such worlds (as in quantum mechanics).

Exhaustion and elimination

… The initial assumption, that we have a complete list of suspects, is often ignored, but it explains why isolated country houses feature in many detective stories.

Quite so. But an alternative to assuming that we have a complete list is to make the conclusion conditional on the true explanation having been one that has been recognized as a possibility. This is often doubtful, hence ‘radical uncertainty’. ‘Expect the unanticipated’.

My Conclusion

To adapt Einstein again:

As far as the book refers to applications, it is easy to misunderstand, and as far as it is reliable, it does not refer to reality.

It is tempting to say that we should take the book’s explanations seriously only in so far as they are narrowly mathematical, and leave other issues to physicists. Yet it seems to me that taking the issue of the proper scope of the book seriously has helped to illustrate to some important limitations on knowledge more generally, beyond which only ‘intuition’ can go. And moreover, mathematics can sometimes be an aid to such intuition, even if never conclusive.

Dave Marsay

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