“A Mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” (Hardy, 2012, p. 84)
What is mathematics?
One way to convince someone that mathematics is beautiful – although I don’t think it is the best way (that will come later in the tour) – is to dive deeper into the pursuit of mathematics. That’s not a simple task, and it’d take much longer than a short blog post to fully develop, but nevertheless it’s important that we peer into this discussion. Let’s try to start with a definition…
What would your students say if you asked them to define what mathematics is? How would you define it? It’d be lovely to read some attempts in the comments section below. I’ll hazard a guess that most would get close to that given in the Oxford English Dictionary. In summary, it refers to number, shape and space. It also mentions that mathematics is either abstract or applied to other real-world disciplines. Right, great! We’ve defined it and can clearly move on (sarcastic remark number one).
If I defined art – and I mean art in the traditional sense of the word – as a discipline which is about colours and paint which can either be abstract or about the real world, people may agree on parts of the definition but most would doubt its suitability. The problem is that it doesn’t give anyone a deep sense of what art is and why people bother to create or analyse it. Like art, mathematics is too varied a discipline to define in a few short sentences – and mathematicians from different fields are likely to disagree about the exact definition. It also evolves, so any strict definition is time-dependent. Indeed, Pythagoras may well have defined mathematics differently to the developers of a recent computer-assisted proof, for example.
Of course, mathematicians find some irony in this. A famous former mathematics lecturer, Paul Lockhart, acknowledged that mathematics as a pursuit is very difficult to ‘pin down’, but the definitions within mathematics must be extremely precise in order to logically deduce new mathematical theorems for the subject to develop rigorously (Lockhart, 2009). Thus there’s no formal agreed upon definition that spans the entire mathematical community (arrgghhhhh!!!). Even though the mathematician in me struggles with this, I do tend to agree with Ernst Gombrich, the popular art historian:
“Luckily it is a mistake to think that what cannot be defined cannot be discussed. If that were so we could talk neither about life nor art.” (In Schiralli, 2007, p. 117)
Instead of defining mathematics then, I’d like to provide a few ‘informal’ ideas about the pursuit of mathematics in order to give a better sense of what mathematics really is. I’ve compiled a list of musings by mathematicians about our field:
- “To some extent, the whole object of mathematics is to create order where previously chaos seemed to reign, to extract structure and invariance from the midst of disarray and turmoil.” (Davis & Hersh, 1981, p. 172)
- “Mathematics is the catalogue of all possible patterns – this explains its utility and ubiquity.” (Barrow, 2015, p. 2)
- “The primary drive for the mathematician’s existence is to find patterns, to discover and explain the rules underlying Nature, to predict what will happen next.” (Du Sautoy, 2004, p. 24)
- “A Mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” (Hardy, 2012, p. 84)
- Mathematics is the science of patterns. (Steen, 1988; Devlin, 2012)
It doesn’t take an expert mathematician to notice the connection between these five assertions, but well done if you did. They all refer to mathematics as a fundamental search for, and analysis of, pattern and order. As Devlin (2012) rightly points out, assertions such as these wouldn’t come as a huge surprise to a mathematician, but they certainly aren’t specific enough for someone who really wants to understand more about the nature of pattern within mathematics. If you’re reading this as a non-mathematician, I might then point you towards Devlin’s book, “Mathematics: The Science of Patterns” to develop a more attuned understanding of what pattern means in mathematics. I’d also encourage you to read “A Mathematician’s Apology” by the famous number theorist G. H. Hardy. If you haven’t the time for that then not to worry, hopefully the posts in this site will be clear enough without the need to explore extra material.
It might be of interest, and could possibly be confusing to secondary school students, that six prominent mathematicians would describe mathematics in this way. Some might question why these mathematicians haven’t mentioned equations and symbols, for example. To be blunt mathematicians don’t bother to refer to symbols and ‘mathematical objects’ when discussing the pursuit of mathematics, but rather to the form and structures that these objects represent when they are combined and related to one another (Resnik, 1999). This is a well-known property of aesthetics in the abstract arts. As Anjan Chatterjee describes in his brilliant book, the Aesthetic Brain:
“Clive Bell introduced the idea of “significant form,” which refers to particular combinations of lines and colours that excite aesthetic emotions… [thus] the aesthetic response is to the forms and relations to forms themselves.” (Chatterjee, 2014, p. 118)
Just as musician does not describe the act or pursuit of music in terms of the symbols written down on the sheet in front of them, mathematicians would not describe mathematics in this way either.
The mathematician Walter Sawyer said that ‘where there is pattern, there is significance.’ [Sawyer defined mathematics as the classification and study of all possible patterns, which provides further fuel to our fire]. This is the important aspect of pattern when you think about it. The fact that a pattern exists is fundamental to open the door to analysis, but it is in the understanding of why the pattern occurs that is the ultimate pursuit. The meaning in mathematics always matters! That’s not to say that when learning mathematics, one must always develop a deep understanding of meaning. Understanding often requires a higher knowledge base than doing, and sometimes it’s beneficial for the confidence and motivation of learners to do first, and understand later.
One might argue that truth was the ultimate pursuit of mathematics, and there’s no doubt that truth is an important goal, although truth cannot be the ultimate pursuit. As Gauss put it, if this were so we’d struggle tremendously to find a reason why so many mathematicians continue to search for the most beautiful and simplest proof after determining the truth of the matter (The 4-colour theorem is of course a prime example of this).
Now, I must admit that you’d be forgiven for critiquing my selection of five statements on the nature and pursuit of mathematics; they’re clearly biased towards patterns. If I wasn’t being so biased I could have chosen to include Gauss’ statement that “mathematics is concerned only with the enumeration and comparison of relations.” This does encompass a large part of what mathematics is about, and I doubt many would argue with one of the best mathematicians to have ever lived. However, all we have to do in this case is think briefly about what a ‘relation’ is.
- If a relationship exists between two things, then there must be order and pattern which allows this relationship to exist.
- We cannot relate two things when one or both of those things do not display pattern.
- Hence pattern is fundamental to Gauss’ statement and we can move on.
I could also forgive a critique on my lack of inclusion of logic and reasoning as integral aspects of mathematics. For example, Michael Enciso referred to mathematics in his delightful book, ‘The True Beauty of Math’ as: “the craft that unambiguously derives new incontrovertible truths from previously established incontrovertible truths, using a mode of reasoning that is itself incontrovertible.” This is undoubtedly also true of mathematics – it is the way that we progress our discipline. Who could argue with the aesthetic pleasure one feels in the logical construction of theorems from basic axioms and postulates as in Euclid’s Elements! However, if patterns did not exist, then relationships would not exist, and if we can cannot connect things to one another then we would not be able to use logical reasoning to develop new mathematical truths. I would argue then that the seeking of patterns – alongside the understanding of why they exist – is a more fundamental way to describe the pursuit of mathematics.
Now, I’ll hazard a guess that applied mathematicians and scientists reading this could well be boiling with rage at my lack of inclusion of mathematical utility, and they’re right to feel this way. I’ll be writing a post about applied mathematics and beauty later in the series so I won’t spoil that post by discussing that here. Although I will give you a very minor hint as to my line of thinking by pointing you back to the second quotation in the list of musings above.
I’ve now built up something of an argument to hopefully convince you that mathematics isn’t all about basic numeracy and solving equations; these are simply the tools which we use to meet a more fundamental goal. Unfortunately most would be forgiven for thinking that mathematics is wholly about these basic tools. To quote Peter Hilton, the Emeritus Professor of Mathematics at the State University of New York,
“The study of mathematics starts with the teaching of arithmetic, a horrible, wretched subject, far removed from real mathematics, but perceived to be useful. As a result, vast numbers of intelligent people become ‘mathematics avoiders’ even though they have never met mathematics…to those intelligent people, it must seem absurd to liken mathematics to music as an art to be savored and enjoyed even in one’s leisure time.”
Cited in Gullberg, 1997
Whilst I’m sure that Peter Hilton would agree with myself and many others, that a strong understanding of arithmetic is important for school children to grasp, we’d all also agree that basic arithmetic is far removed from what mathematicians do on a daily basis. A sense of satisfaction in completing a calculation or solving an equation that has no real meaning is a lovely feeling, often due to an increase in process fluency (saved for a later post), but this pales in comparison to the aesthetic experiences one can feel when the fog lifts on a difficult problem. All lovers of mathematics are familiar with this, but just in case you aren’t, a lovely example is in Andrew Wiles’ 7-year pursuit to prove Fermat’s Last Theorem. If you’ve never seen the clip, please do click the link to hear the short emotional account of his journey.
So there’s the start of our discussion on patterns…and this is only the start. In future posts we’ll look into examples of patterns, discuss how the pursuit of beauty has guided mathematicians over the centuries, and also whether seeking beautiful mathematics can lead us astray.
A major pursuit of mathematics is to seek and analyse patterns, and as we’ll see, the identification of patterns reduces uncertainty. This incites pleasure and unlocks the doorway to positive aesthetics responses…
Barrow, J. (2015). 100 Essential Things You Didn’t Know You Didn’t Know About Maths & The Arts. NY: W. W. Norton and Company
Chaterjee, A. (2014). The Aesthetic Brain: How we Evolved to Desire Beauty and Enjoy Art. UK: Oxford University Press.
Davis, P., J. & Hersh, R. (1981). The Mathematical Experience. US: Birkhauser Boston.
Devlin, K. (2012). Patterns? What patterns?. Retrieved from http://devlinsangle.blogspot.ch/2012/01/patterns-what-patterns.html
Du Sautoy, M. (2004). The Music of the Primes: Why an Unsolved Problem in Mathematics Matters. London: Harper Perennial.
Gullberg, J. (1997). Mathematics: From the Birth of Numbers. NY: W.W. Norton and Company.
Hardy, G.H. (2004). A Mathematician’s Apology. Cambridge: Cambridge University Press.
Lockhart (2009). A Mathematician’s Lament: How School cheats us out of out most fascinating and imaginative artform. NY: Bellevue Literary Press.
Resnik, M, D. (1999). Mathematics as a Science of Patterns. Retrieved from http://www.oxfordscholarship.com/view/10.1093/0198250142.001.0001/acprof-9780198250142
Schiralli M. (2007) The Meaning of Pattern. In: Sinclair N., Pimm D., Higginson W. (eds) Mathematics and the Aesthetic. CMS Books in Mathematics. Springer, New York, NY
Steen, L., A. (1988). The Science of Patterns. Science, 240(2852), p. 611-616.