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.

LMS Education Day

This Tuesday I attended the LMS Education Day, a yearly event focussing on how we teach mathematics to our undergraduates. The topic this year was curriculum development: are our curricula ready for the 21st century?

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Science as Sport

What is science as sport? In sports we measure achievement by winning and so for science it means measuring success in research using metrics that measure esteem: How many citations do I have? What is my H-index? How prestigious are the journals in which I publish? How many grants have I obtained? How soon have I been promoted?

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Machine Learning – My First MOOC

I have been engaged with higher education for the past 15 years, starting first as an undergraduate and now as lecturer teaching undergraduates myself. The rise of MOOCs, from 2012 onwards happened after I had finished my graduate studies and from then on I learned new material mostly by reading textbooks, research papers and spoke directly to colleagues. Although I was aware of their popularity, MOOCs never seemed important enough to warrant a closer look. Until now that is.

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Christmas Lecture

For the Christmas lecture of my multivariable calculus module I tried find something entertaining to present. This turned out to be quite difficult, but in the process I across this multivariable calculus themed comic.
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What is the Point?

Today the Guardian reported that a Polish man who went to the police to report an assault on his wife was questioned about his immigration status and handed over to immigration officials who detained him in Colnbrook Immigration Removal Centre. He has been in detention for the past two months.

There are genuine arguments for controlled immigration and “taking back cotrol of the border”. There is also merit in finding and deporting those who are in the country illegally. But what can possibly be the point of making life hell for those who have done nothing wrong? What purpose is served by casting the net of suspicion so wide that it causes collateral damage?
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Writing a Bachelor Thesis

One of my duties as lecturer is supervising final year projects. Seeing students engage with a piece of mathematics on their own is an interesting and rewarding experience. As part of the process of guiding and advising, of observing students and with the benefit of hindsight one distills little nuggets of advice. Advice that, one hopes, might help students engage deeper with the mathematics and benefit more from the process.
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Adventures with Android

A geocaching adventure gone wrong saw my phone disappear into the depths of a Latvian river and so I was in need of a replacement phone. My parents were kind enough to let me use one of their old phones and after unlocking it and acquiring a new sim card—both tasks ending up being more time-consuming than they should have been—nothing stood between me and the enjoyment of a new phone.

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What is Mathematics?

A couple of weeks ago I was asked to help out with the upcoming induction week for our mathematics undergraduate students. I am given one hour to give 70 students some idea of what they have come to study. After thinking about the question, “What is mathematics?”, I have found the following four-fold answer.

Mathematics helps us to…

  • … find precise answers to precise questions.
  • … find approximate answers to vague questions.
  • … interpret answers that seem precise but are not.
  • … figure out the right questions to ask.

Below I have tried to develop these points and to supplement them with examples. The text below is addressed at first year students of mathematics at Brunel University.

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Effect of Student Loans

Student loans are an everyday reality for thousands of students in the UK. The average debt for a graduate in England is £32,220, but it can be more than £50,000 for students from poorer families on a four-year course.

One can look at these and other numbers and make mathematical statements: What is the average time it will take a graduate to repay the loan? How to model the average value of a university degree compared to the hopefully higher salary of a graduate with the loan owed to the Student Loans Company. We ask our first-year students at Brunel to create a simplified model of their projected income and loan repayment. It is usually an educational experience for them.

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