The Era of Mathematics

Two weeks ago the report ‘The Era of Mathematics’ was presented in the House of Lords by Professor Philipp Bond. The report examines the wider impact of mathematics on the economy and makes recommendations how to expand and facilitate knowledge exchange between mathematicians and the industry.

Several parts of the report invite comments, but I will focus here on one particular aspect: the notion of ‘impactful’ mathematics. The report wants to overcome the traditional division of mathematics in ‘pure’ and ‘applied’ and so it creates a new category—impactful mathematics.

What is impactful mathematics? The report mentions several well-known examples intended to show that pure mathematics can be impactful. Graph theory is used to analyse social networks, harmonic analysis underlies much modern signal processing and number theory is the basis of modern encryption methods.

The problem with the label ‘impactful’ is that it can only be applied in retrospect. Sometimes decades pass between the mathematical discovery and its impact. Elliptic curves, for example, which are used in the encryption and signature algorithms underlying Bitcoin make their first appearance in the work of Diophantus and that they form an Abelian group was known at the time of Poincaré. The use of elliptic curves in cryptography was first proposed in 1985, independently by Neal Koblitz and Victor S. Miller, but only in the 2000s did their use become widespread. The situation is similar for graph theory and signal processing. Impact often takes time.

The report states that

We are often able to predict that a mathematical breakthrough will be important – but not always. G.H. Hardy, for example, famously boasted in his ‘A Mathematician’s Apology’ of the uselessness of his great love, number theory. Seventy years later, number theory lies at the heart of internet and e-commerce security, fundamental to the functioning of the world economy and of worldwide communications.

Two comments jump to mind. First, we may be able to predict the usefulness of a breakthrough once it has happened, but the research grant-oriented landscape we all live in require us to predict the usefulness of future breakthroughs. There our track record is much worse. Breakthroughs often happen serendipitously without much planning or anticipation and they certainly don’t come with a pre-written ‘Pathways to Impact’ statement as required by EPSRC.

Second, Hardy’s views on the usefulness of mathematics are often misrepresented. Hardy did not so much boast of the uselessness of number theory as take solace in it. Hardy was well aware that some mathematics is useful or impactful. (All following quotes are from ‘A Mathematician’s Aplogy’)

Now some mathematics is certainly useful in this way; the engineers could not do their job without a fair working knowledge of mathematics, and mathematics is beginning to find applications even in physiology. —Hardy §19

But then he drew the conscious decision that this is not the mathematics that he himself is interested in. For Hardy the pursuit of mathematics is an aesthetic pursuit, mathematics is to be judged by its beauty and depth. Interestingly, Hardy also anticipated the notion of impactful mathematics and that it differs from both pure and applied mathematics.

There is another misconception against which we must guard. It is quite natural to suppose that there is a great difference in utility between ‘pure’ and ‘applied’ mathematics. This is a delusion: there is a sharp distinction between the two kinds of mathematics, […], but it hardly affects their utility. —Hardy §22

While the Bond report gives examples of pure mathematics that has found impact, Hardy on the other hand gives examples of applied mathematics that—in his time at least—has no usefulness.

I count Maxwell and Einstein, Eddington and Dirac, among ‘real’ mathematicians. The great modern achievements of applied mathematics have been in relativity and quantum mechanics, and these subjects are, at present at any rate, almost as ‘useless’ as the theory of numbers. —Hardy §25

Hardy is also aware that his views might well be swept away by the tides of time.

It is the dull and elementary parts of applied mathematics, as it is the dull and elementary parts of pure mathematics, that work for good or ill. Time may change all this. No one foresaw the applications of matrices and groups and other purely mathematical theories to modern physics, and it may be that some of the ‘highbrow’ applied mathematics will become ‘useful’ in as unexpected a way; but the evidence so far points to the conclusion that, in one subject as in the other, it is what is commonplace and dull that counts for practical life. —Hardy §25

Hardy certainly did not boast about the ‘uselesness’ of number theory. In fact he wrote the exact opposite.

But here I must deal with a misconception. It is sometimes suggested that pure mathematicians glory in the uselessness of their work, and make it a boast that it has no practical applications. […] If the theory of numbers could be employed for any practical and obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering, as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to decry or regret such applications. —Hardy §21

And now we come to the difficult part: one can apply mathematics for good as well as for evil. Rockets that brought man to the moon also enable man to deliver a nuclear warhead anywhere in the world. The technology that enables Facebook to automatically tag people in photos also enables police to automatically identify people on CCTV. And so Hardy continues:

But science works for evil as well as for good (and particularly, of course, in time of war); and both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean. —Hardy §21

Today mathematics has found many applications and with the rise of artificial intelligence and machine learning there will certainly be many more. We are living in a time where mathematics can be used for both good and evil in our everyday life. Cathy O’Neil recently wrote a book, ‘Weapons of Math Destruction’ highlighting the potential of mathematics to cause harm if employed without care and reflection. Mathematics has certainly lost the innocence and harmlessness it still enjoyed in Hardy’s time.