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Applications are open for the 6th Heidelberg Laureate Forum

HLF LogoYou may have noticed Aperiodical team members Paul and I were blogging from the Heidelberg Laureate Forum back in September. The HLF is an opportunity for young researchers (PhD, MSc and post-doc) to meet the winners of prestigious prizes in maths and computer science, including the Abel Prize, Fields Medal, ACM AM Turing Prize and Nevanlinna prize.

The next HLF will take place in September 2018, and applications open today for Young Researchers who want to participate. If you’re a maths or computer science researcher and want to be invited on a trip to Germany with lots of interesting talks, delicious food and good company, you can apply on the HLF website from today.

Taking Maths Further podcast launch

The Further Maths Support Programme (FMSP), which provides support for students and teachers in the UK doing Maths and Further Maths at A-level, has commissioned a series of podcasts, called Taking Maths Further, showcasing different people who work using maths as part of their job, and the mathematical tools they use.

Surds: what are they good for?

Here is a question I was asked:

Why is rearranging equations containing square roots on the curriculum for GCSE? What might it be useful for in later life?

This is a two-part question, one part of which is dynamite. When I put the question to Twitter, Paul Taylor @aPaulTaylor was the first to take the bait:

Is usefulness in later life a necessary condition for inclusion on the GCSE curriculum?

Let’s set that aside for now. Whether usefulness is necessary or not, asking what a topic might be useful for in later life is a perfectly valid question for a fourteen year old who is being asked to study that topic.

Surds is one of those confusing areas that I vaguely remember but have to look at a definition to recall properly. The BBC GCSE Bitesize website has “a square root which cannot be reduced to a whole number” and says “you need to be able to simplify expressions involving surds”. Rearranging surds, then, is the business of noticing that the square root of 12 multiplied by the square root of 3 can be combined to give the square root of 36, which is 6.

Surds, then, are a part of general algebraic fluency. I expected, therefore, that one answer would be that this is the kind of manipulation that helps generally with higher mathematics; though I wonder when such neat numbers arise in reality. I also expected to hear that surds were useful in very efficient computation. I remember once speaking to someone who was programming computers to go on board aeroplanes. These had very limited computing power and needed to work in real time; the programming involved all sorts of mental arithmetic tricks to minimise the complexity of calculations.

For the latter, I am not sure how relevant this is to modern engineering or programming. For the former, it might be that we are including this for every student at GCSE simply as part of the algebraic fluency that we hope of from incoming mathematics students at university. When I put the question to Twitter, two responses reflected my cynicism on this point. When are surds useful in later life?

Other, less cynical responses, were available. Early responses:

  • Ian Robinson ‏@IanRobinson said: “it allows you to work with precise fractional values without rounding errors in calculations. Useful for engineering etc.”
    Later, Colin Beveridge @icecolbeveridge suggested something similar: “in computing, roots are expensive — if you can consolidate them, you save computing time.”
    This rings true for me but it was a mathematically-inclined structural engineer who asked the original question. Is this really used in engineering?
  • Christian Perfect @christianp said “anything involving making rectangles” thinking particularly of “carpentry and landscaping“.

I put these suggestions – rounding errors and rectangles – to Twitter.

John Read ‏@johndavidread said (tweet 1; tweet 2):

I think it’s unlikely anyone doing practical work would need the accuracy. Feels more pure Maths than Applied. But is it used? For engineers, landscape, carpentry etc expansion to a few decimal places so you can measure to reasonable accuracy is fine.

Carol Randall ‏@Caro_lann said: “engineering isn’t just measurement! There’s lots of heavy maths involved in getting a B.Eng (and beyond).”

John Read ‏@johndavidread asked: “where in Maths do equations with square roots come up that you’d want to simplify without calculating numerical value?”

To this, Daniel Colquitt ‏@danielcolquitt wrote what on Twitter must be considered an essay, a four tweet message (1, 2, 3, 4):

Very simple examples: Computing the eigenfrequencies of beams, or reciprocal lattice vectors & hence in various Fourier transforms. In this case, exact form is required, decimal expansion will not do. For the beam example, a numerical value can be computed for a given set of parameters, but if you want to know that frequency for *any* set of parameters, you need to know how to hand surds.

On algebraic fluency, Christine Corbett ‏@corbett_inc suggested “the umbrella of ‘simplifying equations'”.

To this, John Read ‏@johndavidread asked: “but then why not teach it as ‘simplifying equations’? No kid had heard of a surd in the 1980’s”.

Daniel Colquitt ‏@danielcolquitt replied: “For GCSE & roots of reals >0, I would tend to agree with you. Complex roots are somewhat different”.

But we’ve swayed back rather close to the dynamite, haven’t we? I’ll stop there.

My sense is that I haven’t had a satisfactory answer really. This sort of rearrangement is good for building up the background knowledge of the undergraduate mathematics student or perhaps engineering student, but no one seems to be claiming they are an engineer who uses this outside of the classroom. No one seems to have claimed this topic develops mathematical thinking in an interesting way, or that engineers who don’t think they are using it really are relying on it in the black box of software, or that the topic somehow contributes to an appreciation of the beauty of mathematics in the teenagers who are learning it. (This may be due to my experiences and the experiences of those who have replied, or the way I have misinterpreted their words.) It may be that there’s a bunch of stuff on the GCSE syllabus just for those who go onto A-level or degree-level mathematics, and perhaps that’s fine, but it would be nice to have a more satisfying answer to give. So, dear reader, are you satisfied with these answers? Do you have a better answer?

"I’m not a mathematician, the maths I’m doing is really just basic modelling"

Last week I attended the first Institute of Mathematics and its Applications Employers’ Forum. The theme was ‘Employability of Mathematics Graduates’. This was an interesting event with many useful views and viewpoints on display.

One speaker, talking about how mathematics student applicants to the graduate training scheme fare, mentioned that during the technical interview some such applicants seem to expect that they will be asked detailed questions about their final year modules. In fact, the questions asked are more like A-level mechanics and this trips up many students. This chimes with a problem I’ve thought about previously about attitudes to mathematics from mathematicians.

I have noticed that many graduate mathematicians who work in mathematical jobs will tell me “I’m not a mathematician, the maths I’m doing is really just basic modelling”. Students and graduates (including, if I think back, me when I graduated) seem to think that if the mathematics they are doing after graduation isn’t at least as hard as final year undergraduate mathematics, then it can’t be ‘real mathematics’ and they can’t be a ‘real mathematician’. As they haven’t moved onto a higher degree to do more advanced mathematics, they must have failed as mathematicians.

I came across this problem somewhat when I worked for the IMA because someone who doesn’t consider themselves a mathematician might ask: since I’m no longer a mathematician, why would I join the mathematicians’ professional body?

I think it is terribly sad when graduates think this. I must be careful here: of course there is more advanced applied mathematics but many graduates find themselves applying fairly basic mathematics to problems and therefore think that they have regressed to an earlier stage of their mathematical development. This rigorously hierarchical view of mathematics – particularly from people who are using mathematics to make a substantial contribution – seems to me to be a real shame. In fact, final year undergraduate mathematics is pretty far up the tree – so far, if we continue the analogy, that it can’t support very many people – but it’s hard to appreciate this when, to overuse the analogy, you’re only looking at the few academic researchers balancing on higher branches.

“If I apply for a job using mathematics, they must want to quiz me about what I learned at the culmination of my degree. And since they’re asking me questions about forces and moments using techniques from A-level, then this can’t be real mathematics and I can’t be a real mathematician.” It’s a real problem.

This is part of where I think the value lies in the IMA series of conferences for the ‘Early Career Mathematician’. Since many mathematicians in industry think of themselves as ‘someone who used to do mathematics’ and may well be the only mathematics graduate in their team/department/company, it can be a very powerful experience to come together and meet others in similar positions. If you’ll excuse a small plug, I am chairing the next of these conferences, the IMA Early Career Mathematicians’ Autumn Conference 2012, in Greenwich in November. Registration is now open. Come along!

What is mathematics?

This morning on Twitter Tony Mann asked the question: “This morning’s class is “What is Mathematics?” Answers in a tweet please.” Answers were collected via the #MATH1103 hashtag.

Some of the answers were what you might expect: patterns, abstraction, order.

Stuart Ravn sent a series of tweets giving his views:

Math is everything you can do with the abilities to count and deduce. There’s literally no end to the fun you can have. No joke.
Math is the only thing which is truly universal; it underlies and makes it possible to understand and communicate with everything.
Everything, immediately or ultimately, is mathematical and arises from mathematics.
Ask yourself what isn’t mathematics, and try to prove yourself right.
Noam Chomsky said of love, “I can’t tell you what it is, but life’s empty without it.” The same is viscerally true of mathematics.
I don’t just have enthusiasm for maths. I love it. It’s the closest thing to my heart after my family. I’m emotional about it.

Noel-Ann Bradshaw noted that “What is mathematics?” is the name of “an excellent book by Courant & Robbins revised by Stewart“. I have the tenth printing from 1960. Although this has a lot to say on the subject, it opens a discussion of historical development with:

Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasise different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.

I was interested in a related question: When does mathematics become something else? At some point some topic is clearly applied maths and at some point it is physics, astronomy, engineering, economics, computer science, biology, and so on.

We struggle with this a little on the Math/Maths Podcast, where we try to report news from mathematics and its applications. On Twitter I said that I think I tend to stray a little further from that which is unambiguously mathematics than does Samuel Hansen. We both report applications but I think mine are often more tangential than Samuel’s. This was quite noticeable on episode 80 this week when Samuel picked me up on an astrophysics story I was defending as involving statistical models. He said:

This is ‘Math/Maths’, not ‘Stat/Stats’, and it’s definitely not ‘Astronomy/Astronomies’. I’m assuming you put ‘s’ at the end of every single science – you have ‘Chemistries’ and ‘Physicses’, right?

and later

Now you’ve turned us into ‘Geology/Geologies’.

Answering my question on Twitter, Samuel said: “if an application has been around long enough to have own name, Physics astronomy or thermodynamics. It’s not math“. I don’t fundamentally disagree with this and some disciplines, notably computer science, were born this way. However, Sharon Evans made a very practical (if teasing) counter-point: “so it’s only maths if it hasn’t got a name? You’re not leaving much to report on in [the podcast]“.

Clarissa Wornack replied to say “Well, when you start writing code; it’s IT/software eng/comp sci; if you create something that is a material object; it’s eng” and Charles Brain said “Applied maths becomes Engineering when it hits the real world and money becomes part of the equation!” I don’t particularly agree with these. I know people who use high end computing to do mathematics and just because they are using computing as a tool (and writing bespoke code) this doesn’t mean they are doing computer science research. I also don’t agree that it stops being applied maths when it creates a material object. Defining mathematics as that which doesn’t involve the real world or money seems very self-defeating.

Felipe Pait offered this definition: “Applied math interests mathematicians and non mathematicians. Otherwise it’s pure math, or pure engineering. Math stops being applied math and becomes pure physics when it doesn’t interest mathematicians. An operational definition.

There’s something in this. When I think about “what is mathematics?” I am really thinking “what can mathematicians do?” I am particularly interested in what university mathematics graduates might become and would like this to be a broad as possible. I meet a lot of mathematics researchers working in different application areas. For example, back when I was doing the Travels in a Mathematical World podcast for the IMA I spoke to Paul Shepherd. I am much more naturally inclined to consider Paul a mathematician working in architecture than an architecture researcher who once did a maths degree. By extension, I am happy to include Paul’s use of geometry in architecture as part of mathematics than to exclude it.

Daniel Colquitt suggested “a lot depends on the user” and “in many cases, the distinction is arbitrary“. I think this may be the wisest view on the subject I have heard. From my point of view I am biased towards including topics on the edge within mathematics rather than excluding them, and maybe even collecting a little of the host subject along with them. I would rather cast the arbitrary net as wide as possible.