Author: charmingnumbers.com

Criteria for Mathematical Beauty

Quite naively, when I originally decided to to analyse mathematical beauty, my goal was to develop a set of criteria to describe it as objectively-as-possible. Of course I knew then that aesthetic responses are within the eyes of the beholder, but I still believed that it would be possible to at least provide common aesthetic features of mathematical ‘objects’ and concepts. Research into this highlighted common criteria quite quickly (I’d already found a decent set of criteria within 30 minutes), so then my goal changed to dig deeper into the common features of mathematical beauty to find more meaning within each feature, but also between features. I’m going to highlight the common criteria/features/dimensions here using a very helpful ATM article written by Paul Ernest at the University of Exeter called “Mathematics and Beauty”. This will provide the foundations to help build many of the following posts.

Ernest (2015) reminds us initially how beauty in mathematics is perceived. To quote:

“the beauty of mathematics 
is not a response to something perceived through
 our sense organs, as with paintings, music or even landscapes… it must be experienced cognitively, through reason, the intellect, intuition, and affect (feelings).”

This is an incredibly important aspect of mathematical beauty which we will come back to in a future post, given that a direct consequence of this results in an amendment to the common phrase mentioned above. The phrase “beauty is in the eyes of the beholder,” transforms into “beauty is in the mind of the beholder,” from a mathematical standpoint. Experiencing mathematical beauty therefore necessitates some level of knowledge and understanding… (to be continued in an upcoming post with a discussion on basic and performative aesthetic responses).

After this, he provides a set of seven dimensions of mathematical beauty, noting that are ‘significant differences’ between the opinions of mathematicians. That implies that a mathematical object may only relate to one of these dimensions and be considered more beautiful than another object which relates to three of them, for example. Individual judgement and taste matters immensely, just as it would if you asked a group of people what their favourite food is.

1. Economy, simplicity, brevity, succinctness, elegance

The compression of a formula or a theorem of wide generality or an argument (proof) into a few short signs in mathematics is valued and admired.

Note: In relation to simplicity, many mathematicians would argue that the right balance of simplicity and complexity is key to promote a positive aesthetic response. If something is too simple, it is deemed as obvious and unremarkable. 

2. Generality, abstraction, power


The breadth and scope of a generality or a proof is one of the key characteristics of mathematics and evokes appreciation.

3. Surprise, ingenuity, cleverness

Unexpectedness, like wit, is appreciated and valued when it reveals a new knowledge connection, method or short cut in solving a problem.

Note: Unexpected simplicity in complexity, unexpected connections and unexpected applications all excite positive aesthetic responses. It is has thus become my belief over time that unexpectedness is one of the most important elements of mathematical beauty for the mathematician.

4. Pattern, structure, symmetry, regularity, visual design 


The discernment of pattern in its various and abstracted forms is the closest the values
of mathematics come to those of art and general aesthetics in the visual eld, although in mathematics these properties are largely abstract. Nevertheless, mathematics is the science par excellence for elucidating the meaning of structure and pattern.

Note: See my previous posts on pattern – specifically posts 2, 3 and 4. Seeking and analysing pattern cold be considered the major pursuit of mathematics, and hence mathematics as a domain could be considered inherently beautiful.

5. Logicality, rigour, tight reasoning and deduction, pure thought 


The development of logical reasoning to its ultimate forms of rigour and purity of thought is a valued part of mathematics and the steps in a well constructed mathematical proof evoke admiration like a gold necklace with well forged links.

Note: Look no further than Euclid’s Elements for verification of this dimension. See Michael Enciso’s book, “The True Beauty of Math” for more on logic , reason and mathematical beauty.

6. Interconnectedness, links, unification


The evidence of connections between different concepts and theories within mathematics is intellectually exciting and attractive. It combines economy, generality, ingenuity and structure and so it could be argued that it is reducible to these first four dimensions of beauty. Or it can be seen as sufficiently valuable in its own right so as
to deserve independent listing, as I have done here.

Note: Gian Carlo Rota famously wrote about mathematical beauty being misunderstood in his popular paper on “The Phenomenology of Mathematical Beauty”. He referred to mathematical beauty as being just a moment of enlightenment, in which the mathematics suddenly becomes well placed within the topic and the mathematical domain generally, and also more deeply understood. There have been a number of rebuttals to this work, but connections/unification/deepened understanding – i.e. enlightenment – nevertheless stands as an important aspect of mathematical beauty.

7. Applicability, modelling power, empirical generality 


Like metaphors in poetry the capture of empirical situations in mathematical models and more generally in applied theories and concepts is something appreciated both within and outside mathematics as a demonstration of its power and ‘unreasonable effectiveness’ in the physical world (Wigner 1960), as opposed to the world of pure mathematics.

Note: Paul Ernest rightly points out that Hardy did not include applicability in his famous set of criteria on the beauty of a mathematical proof, mostly because of his bias towards pure mathematics. I will refer to Cedric Villaini’s thoughts on this in a future post. 

So there you have it – a relatively strong set of criteria for mathematical beauty. There are many others who have attempted to develop criteria, such as Doris Schattschneider’s beauty criteria for mathematical proofs, but I am happy that Paul Ernest’s dimensions are foundational enough to help us springboard into deeper and more complex discussions on mathematical beauty.

The key for public understanding is to somehow explain all of this with appropriate examples to non-mathematicians. That will come soon enough…

Why no one can argue that mathematics is not inherently beautiful, even if they themselves do not experience that beauty.

If you’re reading this post it’s likely that you already fully comprehend that beauty by it’s very nature is subjective (i.e. beauty is in the eye of the beholder). A consequence of that is there are many individuals who will not be able to experience the true beauty of mathematics, just as I may never be able to experience the true beauty of poetry (although I do believe everyone is able to have basic aesthetic responses). Every aesthetic response is uniquely individual, and of course dependent on the knowledge, understanding, experience and motivation of each person. Having said that, what we can do is use evidence to convince anyone that mathematics is inherently beautiful, even if all people can not experience all of that beauty laid bare.

Given that beauty is subjective, you could easily argue that anything has inherent beauty, and it depends on the individual as to whether the aesthetic response is negative or positive. I do not dispute that point. I simply enjoyed finding evidence on why mathematics is inherently beautiful from a number of different perspectives.

This is based on an argument I made in a previous post about the nature and pursuit of mathematics as finding, analysing and understanding pattern (Link to that post).

Mathematical Beauty Pre-Pillar:

The study of mathematics, which fundamentally seeks to find, analyse and understand patterns, provides aesthetic experiences of beauty from the perspectives of neuroscience, psychology and evolutionary biology.

Psychology: According to Dietrich Dorner’s PSI-theory, beauty is based on a need for reducing uncertainty. Human beings crave explanations of our surroundings, and when we are able to discern order from something that may initially appear disorderly, we satisfy a basic need for reducing chaos. Once uncertainty diminishes we open the door to feeling pleasure, and this is precisely the point at which we access aesthetic appreciation (cited in Delle-Donne, 2010). Hence, given that a fundamental aim of mathematics is to reduce uncertainty by seeking, analysing and understanding pattern, there becomes little to dispute from a psychological perspective that mathematics is inherently beautiful.

Evolutionary Biology: Anjan Chatterjee, Professor of Neurology at the University of Pennsylvania, wrote a supremely detailed account of aesthetics in his book, ‘The Aesthetic Brain’, in which there are numerous references to the integral connection between pleasure and beauty – he understandably referred to the work of Edmund Burke who may have popularised this idea. Chatterjee provides an evolutionary biological interpretation of why human beings have a need to reduce chaos. He draws on research which shows that people prefer landscapes and scenes which predict safety and nourishment due to their relative uniformity. Human beings were more likely to survive if they could see patterns in landscapes amenable to survival. This idea is elaborated on in Denis Dutton’s TED Talk, “A Darwinian Theory of Beauty,” and also in the idea of patternicity put forth by Michael Shermer,  which is the phenomenon in which human beings have a predisposition through the process of evolution to search for patterns in anything, even in meaningless noise (Shermer, 2008).

Of course, one might say that the leap between drawing on visual patterns amendable to survival – to abstract pattern recognition in mathematics – is a leap too far. However, as Mattson (2014) argues, the increased size of our cerebral cortex dramatically increased our superior pattern processing (SPP), which he argues is the “fundamental basis of, if not all, unique features of the human brain including intelligence, language, imagination and invention.”

Neuroscience: Scientists in the UK recently conducted an experiment in which fifteen mathematicians were asked to rate sixty equations on an aesthetic scale of -5 to 5 (ugly to beautiful). MRI scans showed that the region of the brain which connects sensory experience, emotions and decision making, was highly active when the mathematicians saw equations that they considered to be beautiful. It happens that the same region of the brain is similarly active when we look at art of listen to music that is perceived as beautiful. So, if anyone is to claim that particular pieces of music or art are beautiful, neuroscientists would also claim that particular parts of mathematics are perceived in a similar way (Newman, 2014). In addition, one of the reasons we enjoy music is based precisely on pattern recognition and prediction. As Salimpoor et al (2013) found, when we listen to an unfamiliar piece of music, our brains predict how the music is likely to develop, and we get a sense of reward from that. You might be forgiven for thinking that reward is purely sensory, but the researchers found that this reward is a direct intellectual one also, which is what we could reasonably expect from pattern searching and recognition in mathematics.

References:

Chaterjee, A. (2014). The Aesthetic Brain: How we Evolved to Desire Beauty and Enjoy Art. UK: Oxford University Press.

Delle-Donne, V. (2010). How can we explain Beauty? A Psychological Answer to a Philosophical Question. Proceedings of the European Society for Aesthetics, vol 2.

Mattson, M. (2014). Superior pattern processing is the essence of the evolved human brain. Retrieved from: https://www.frontiersin.org/articles/10.3389/fnins.2014.00265/full

Newman, S. (2014). Beauty in Math and Art Activate the Same Brain Area. Retreived from https://www.scientificamerican.com/article/beauty-in-math-and-art-activate-same-brain-area/

Samimpoor, V. N., Van den Bosch, I., Kovacevic, N., McIntosh, A. R., Dagher, A., & Zatorre R.J. (2013). Interactions between the nucleus accumbens and auditory cortices predict music reward value. Science, 320(6129), p. 216-219.

Shermer, M. (2008). Patternicity: Finding Meaningful Patterns in Meaningless Noise. Scientific American. Retreived from https://www.scientificamerican.com/article/patternicity-finding-meaningful-patterns/

Patterns, Intuition and Enlightenment: The Towers of Hanoi

Let’s start where the last post ended:

“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…”

So, I thought before moving on that I’d pick one of an infinite number of possible examples to seek and analyse pattern, thus reducing uncertainty. It wasn’t easy picking a piece of mathematics to do this – I wish I could go deep into 50 patterns – but I promise that I will do my best with this example.

Let’s take a lovely little puzzle called the Towers of Hanoi puzzle (or Towers of Brahma), in which the aim is to move all of the disks on the first pillar to the third pillar in the least number of moves. Only one disk can be moved at a time and it is not permitted to put a larger disk on top of a smaller disk.

In the video above, we’re left with 10 disks on the first pillar – how many moves do you think it will take to move all of the disks to the third pillar?

It might be that your intuition is pulling a curtain over the depth of the problem (as intuition often does) – i.e. you think it’s a fairly simple game and the number of moves can’t be much more than one hundred, for example. Alternatively if you’re more attuned to mathematical thinking you may have already began to gain a sense of some complexity in the puzzle. Whatever the case there’s little point moving forward without doing what mathematicians do best, i.e. seek and analyse patterns. So let’s develop an understanding by building up from simpler cases. As you saw, the three disk puzzle takes 7 moves, and the four disk puzzle requires 15 moves. See the table below to find out how much you were fooled by your basic intuition.

Number of Disks, d

 Number of Moves, M

1

 1
2

3
3

7
4

15
5

31
6

63
7

127

Looking at the pattern in the right column of the table, you may have noticed that we can take the previous number of moves, multiply it by two and add one, and we’ll get the answer for the next row. Hence, there are 255 moves for the eight disk puzzle, 511 moves for the nine disk puzzle and 1023 moves for the ten disk puzzle. That might be something of a surprise, especially if you haven’t had a go at the game or analysed it in any detail. But it is lovely that we’ve used patterns to understand something about the minimum number of moves required.

If you were intrigued to know the minimum number of moves for a fifty disk puzzle it’d be cumbersome to have to multiply by two and add one continuously to get the answer. This is why we look for a relationship between the number of disks and the number of moves. The pattern almost doubles each time which makes it exponential (or geometric), and it is simple by inspection in this case to see the relationship between the number of disks, d, and the number of moves required, M.

M = 2d – 1

The simplicity of this formula from a game that seems partially complex will hopefully leave you with a sense of satisfaction that uncertainty has been reduced and at least a slight feeling of aesthetic pleasure!

Start of Interlude: The Dangers of Intuition in Mathematics

Time-out! True mathematicians would be dismayed to let this stand on such shaky premises. As Marcus du Sautoy remarks in his popular book, “The Music of the Primes,”

“The mathematician is obsessed with proof and will not be satisfied simply with experimental evidence for a mathematical guess…[indeed] Goldbach’s conjecture has been checked for all numbers up to 400, 000, 000, 000, 000 but has not been acknowledged as a theorem. Most other scientific disciplines would be happy to accept this overwhelming numerical data as a convincing argument, and move on to other things.” (p. 31)

The point here is that mathematicians do not trust their intuition. We have no idea whether that pattern will continue forever, or break down at some unknown point – right now we’re relying on a very small sample of data alongside an intuition that pattern will continue. The funny thing about intuition is that it reminds me of that friend we all have who somehow manages to convince us to do something we’re not 100% sure about. At times, it leads us to great experiences in which we wondered why we were ever worried in the first place, but every so often we’ll find ourselves in a situation we wished we’d never been a part of. Intuition is absolutely that same friend to a mathematician – sometimes it leads us to fruitful pathways, but if left unchecked, it can cause us huge problems down the road.

Probably the most famous example of intuition leading a mathematician astray presented itself in number theory, in which Ramanujan thought that his function approximating the number of primes below a given number would always predict less than the actual number of primes. John Littlewood proved that there existed a number in which Ramanujan’s function would begin to predict more than the actual number of primes (actually, that it would alternate between less and more forever). Then, some years later in 1933, Stanley Skewes found such a number which is so big you can’t compare it to anything in the observable universe, it’s 10101034. Given that estimates of the number of atoms in the observable universe lie around 1070, which is minuscule in comparison with Skewes’ number, a non-mathematician might forgive Ramanujan for relying on his intuition. If you’d like to know more about this I’d advise watching the wonderful Numberphile video which explains it with simplicity and clarity (Grime, 2015).

I can’t now move on without peering into a few more examples…they’re just too interesting not to point out.

  • Moser’s circle problem (14-16 years example) – this involves splitting a circle into discrete regions by creating points on the circumference and joining them with chords (with no three chords being allowed to pass through the same point).  If you do so, you get the following progression:

Picture1.png

The first 5 iterations follow a simple exponential pattern for the number of regions, very similar to that of the Towers of Hanoi puzzle: 1, 2, 4, 8, 16.

The sixth iteration is more intriguing; instead of giving 32 regions as intuition would suggest, it gives 31 regions. Defining n as the number of vertices, and R as the number of regions, we obtain a formula which is much more complex than our intuition might have suggested:

R = 1⁄24(n4 – 6n3 + 23n2 – 18n + 24)

On the surface this formula looks down right atrocious –  how dare you be so complicated given the simplicity of the patterns in the first 5 iterations! It certainly doesn’t enlighten my understanding – which if you go back to the previous post – is a major element of mathematical aesthetics and one of the goals of mathematics. If you’d like a deeper understanding of this formula – and you have some background knowledge of Pascal’s triangle – check out this lovely video by 3Blue1Brown.

  • The Powers of 11 and Pascal’s Triangle (16-18 years Example) – Have you ever noticed the connection between the rows of Pascal’s Triangle, and the powers of 11? It behaves beautifully, up until the 6th row…

Screen Shot 2019-02-19 at 10.41.35

This is horrible at first site, but then once you look at the binomial expansion of (1 + 10)1, (1 + 10)2, (1 + 10)3, …, (1 + 10)5, then you quickly become enlightened as to the reasoning behind the breakdown.

  • The Borwein Integral (post-18 years example) – this churns out π⁄2 in the first seven iterations, but then ‘mysteriously’ churns out a number ever-so-slightly less on the eight iteration.

Picture1

Image: Wikipedia (https://en.wikipedia.org/wiki/Borwein_integral)

End of Interlude: The Dangers of Intuition in Mathematics

I hope you found that to be an interesting tangential path on our walk through the landscape of mathematics, but let’s make our way back to the previous path. Intuition is essential, but never sufficient, and that is why proof is more important to a mathematician than butter is to bread. I’ll continue to come back to the notion of proof, understanding and aesthetics later in later posts. As I said in the previous post, the truth of the matter is sometimes only the first stage in the game of mathematics. If the proof doesn’t enlighten the mathematician on the meaning of why something is true, then we love to get inside the engine of the car to see how everything works. That is if we have the tools to do so of course…

To give some insight on enlightenment with the Towers of Hanoi, see the pictures below with a five disk puzzle. A very important part of this puzzle is to free up the bottom largest disk so as to be able to move to the 3rd pillar. In order to free this up, we need move all of the disks above it to the 2nd pillar. In this case that means we need to make 15 moves to move all 4 disks to the 2nd pillar.

Picture1

Fig 1: www.softschools.com, Fun Games, Logic Games

Once this has been achieved, we need one move to transfer the largest bottom disk to the 3rd pillar. After that, we then have to move all four disks which are currently residing on the 2nd pillar to the 3rd pillar and this requires another 15 moves. Hence, in this case, we have 15 moves + 1 move + 15 moves = 31 moves for the five disk puzzle.

Picture2

Fig 2: www.softschools.com, Fun Games, Logic Games

If you think about it, this reasoning applies to any number of disks. We always follow the same steps:

  1. Move all of the disks except the bottom one to the second pillar.
  2. Move the bottom disk to the third pillar.
  3. Move all of the disks from the second pillar to the third pillar.

Using some symbolism, if we say that the number of moves for the four disk puzzle is denoted by M4 and the number of moves needed for the five disk puzzle is M5, then:

M5 = M4 + 1 + M4

M5 = 2M4 + 1 

This agrees with our initial observation of the pattern in the table, i.e. multiply by two and add one. With any number of disks, n, we have a recurrence relation which tells us that if we know the previous number of moves, then we can find the number of moves for when we add another disk.

Mn+1 = 2Mn + 1

At this point, we have reasoning which gives us confidence that this pattern continues in the same way forever, no matter how many disks we have on the first pillar – no more intuition required. We can then use basic recurrence relation mathematics to generate the formula given previously, M = 2d – 1, but this case was simple enough to see the relationship via inspection.

Going back to the previous post, we now see why mathematicians would neglect to discuss the symbolism and equations of mathematics; because they are merely the tools we use to reduce uncertainty, analyse pattern, develop relationships, and crucially, to understand these relationships. Every part of the process can be aesthetically pleasing for a mathematician, as you’ll continue to find out!

References:

  1. du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. HarperCollins. ISBN 0-066-21070-4.

Patterns as Pathways: The Pursuit of Mathematics

“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:

  1. “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)
  2. “Mathematics is the catalogue of all possible patterns – this explains its utility and ubiquity.” (Barrow, 2015, p. 2)
  3. “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)
  4. “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)
  5. 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…

References:

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)[1940]. 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.

 

Fantastic Beasts: Imaginary and Complex Numbers

Before starting our tour of charming mathematics I’d better introduce you to some of the most fantastic beasts in our world – the imaginary and complex numbers. In fact, whilst introducing them I’d like to convince you that extending our number system beyond what we can see in the natural world is a relatively common affair in mathematics, and that the imaginary and complex numbers are therefore less imaginary than some people think. If you’re already something of an expert, I’m hopeful you’ll still find the historical elements of this insightful (and please do leave a comment if you’ve noticed any inconsistencies with sources you’ve come across).

Section 1: The Trials and Tribulations of Accepting Negatives and Irrationals

Let’s set the scene by beginning the story with none other than the fabulous Pythagorean theorem. I’d like you to close your eyes and imagine yourself back in Ancient Greece (approx 500 BC) at the time when the Pythagoreans were having a blast defining, inventing and discovering new things about mathematics. Given the time period, your understanding of number is wildly different, extremely tedious to work with, and in many ways quite basic when compared to what a normal primary age student understands about number today.

They used their alphabet to represent different numbers (e.g. α = 1 and π = 80, so that πα = 81) and they put a ‘diacritical’ mark before a letter to denote it as a unit fraction (e.g. ‘πα= 1/81) – although it’s important to note that they didn’t think of fractions as numbers lying on the number line, but rather proportions of positive integers . Anyhow, I’m sure – once you’d developed fluency with this system – that it wasn’t too difficult to work with, but this notation does instil me with a sense of gratitude for what we have now. The Greeks did have a way to represent fractions with a numerator greater than one but they didn’t have the equivalent decimal form that we’re now accustomed to, nor did they have a way to represent negative numbers; not to mention the fact that a number line was centuries away from being conceptualised.

Now we all know what the Pythagoreans main contribution to mathematics was. But you may not be aware that the same connection between areas of squares on the sides of right angled triangles was also known to the Babylonians. Why then, you might ask, are the Pythagoreans credited with the theorem? Simply put, it wouldn’t be a theorem if the Greeks hadn’t mounted their flag on it by proving it to be true in every possible case. The credit in mathematics almost always goes to the person or people who lay the matter to rest and provide ever-lasting certainty in the form of proof. (Note: whilst Pythagoras himself did exist, much of what is attributed to him is likely based on the work of his followers [1].  Furthermore, “most of what we know about him is conjecture, with a great many anecdotal fables thrown in.” [2]).

As an aside, I love how you can stick any old shape on the side of a right angled triangle, and the same theorem holds as long as you stick similar shapes on the other sides. Alex Bellos represented this perfectly in his fabulous book, “Alex’s Adventures in Numberland” by plonking a Mona Lisa shape on each side of a right-angled triangle and showing that the area of the two smaller Mona Lisa’s sum up to the area of the larger one.

Anyhow, after revelling in their success of proving the theorem it must have come as quite a shock to the Greeks to realise that using it resulted in numbers that didn’t exist (at least in their notion of number). When attempting to find the length of the hypotenuse of different right-angled triangles they didn’t understand why the answers weren’t nicely defined integers or fractions. There comes a huge problem; if we know our theorem is correct, but it churns out answers that don’t make any sense to us, what do we do? Maybe we put our heads under our pillows and sing ourselves to sleep without ever thinking about these un-Godly, inexpressible numbers ever again. Indeed, it’s said that they were named “unspeakables” by the Pythagoreans. Maybe we could go one step further and throw whomever discovered them overboard [3], or maybe, instead of hiding from these terrible monsters, we could conceptualise things in a new way in the hope of understanding them.

Without digging more deeply into the history of irrationals, the problem rested on the way Greeks thought of quantities as ‘discrete objects’, made up of a finite number of parts. When you think in these terms it’s darn near impossible to accept something which can’t be split up, and can’t be expressed as a well-behaved proportion. Fortunately for us, a Greek called Zeno, and then Eudoxus after him, did a good deal to lead the revolution on this, and fought to ensure that quantities be conceptualised as continuous entities rather than as discrete packets [4].

Irrationals weren’t the only stop-gap for the Greeks; they also struggled immensely with the concept of negative numbers. Indeed, if geometry is your fundamental mathematical toolkit then what would a line of negative length even mean??? (the number zero was unknown for the a similar reason). Diophantus referred to equations with negative solutions as absurd, and to be honest I don’t blame him. With no way to conceptualise what a negative number could mean I’m sure I’d say the same thing. This same lack of conceptualisation prevented negatives from being accepted in Europe for longer than we’d like to admit.

Even as late as the early 18th century, a few mathematicians continued to oppose negatives which is quite difficult to believe given how naturally we think of them today as part of the ‘real’ number line. But then we probably take our current view of the real number line for granted. It was only in the late 16th century that Bombelli and Descartes popularised the idea of numbers lying on a line spanning infinitely in both directions. This lies in stark contrast to the Chinese viewpoint on number given that around the time of Diophantus in Ancient Greece, the Chinese found negatives relative easy to accept due to the importance placed on duality in Chinese philosophy [5].

When you sit down and think for a moment, it’s not at all obvious that numbers should be represented on an infinitely spanning line. Given how we experience number in the real world in terms of discrete, countable objects, or even measurable objects which can be approximated as discrete quantities, representing numbers on a continuous line spanning in two directions is not intuitive. The number line is an abstract representation of numbers that does not match with our observations in the real world, and this is especially the case with negative numbers. Thus, extending the number line to span in the opposite direction to account for numbers that we do not directly see in the world is an unintuitive, abstract leap, but it is a necessary one in order to progress our understanding.

Section 2: The Development of Number 

So before moving on, let’s summarise the major points of that discussion. It took a long time for irrational numbers and negative numbers to be accepted because they do not appear in our perceived world view, and we had no immediate way to conceptualise them. Let me think? – are there any others number that fall into a similar category…ah yes, the imaginary numbers. Like negative numbers they initially revealed themselves as the nonsensical solutions to equations. Historically speaking this goes back to the 16th century in which the first people to accept the existence of imaginary numbers were the likes of Cardano and Bombelli in the solutions of cubic equations, but a simpler equation to begin thinking about imaginary numbers is the quadratic below:

                   x² + 1 = 0                        (1)

There is clearly no whole number, negative number, fraction or irrational which we can substitute in to solve this equation. Having said that, if we take a bold, abstract leap and assume that there is a solution – as we did with irrationals and negatives before-hand – then all we need to do is find a way to conceptualise that solution. Low and behold – just as we extended the number line in the opposite direction to accommodate for negative numbers – the likes of Casper Wessel in the late 18th century, and then Argand and Gauss in the early 19th century, decided to conceptualise imaginary numbers by extending the number line once again.

So whilst the development of the number line was far messier than this, let’s look at a simpler, logical progression in the context of extending our number system…

Step 1: Discrete natural numbers = {1, 2, 3, 4, 5, … }

Step 2: All Positive Numbers – Re-conceptualisation of numbers to accommodate firstly fractions, and then irrationals. i.e. transitioning from a discrete view of numbers to a continuous one.

Screen Shot 2018-12-21 at 21.52.40

Step 3: First extension of the number line in the opposite direction to accommodate negative numbers; all integers (including zero).

Step 4: Second extension of the number line to accommodate imaginary numbers. This second extension allows us to conceptualise solutions to equation (1):

Screen Shot 2018-12-21 at 22.08.03

Thus, there are two imaginary solutions to the quadratic equation,

Screen Shot 2018-12-22 at 21.12.34

Each co-ordinate on this grid, which is most often referred to as the complex plane, represents a complex number with both a real part and an imaginary part. For example, 1 + 2i can be represented as a co-ordinate or vector on the complex plane, and we can then begin to understand geometrically what it might mean to add, subtract, multiply and divide complex numbers.

To provide some small insight into the complex world, if we multiply 1 + 2i by i then we obtain:

Screen Shot 2018-12-22 at 21.17.38.png

If you then look to see what multiplying by i represents on the complex plane, then -2 + i is a rotation of 90 degrees anticlockwise from 1 + 2i. It is then trivial to show that multiplying any number by i – real, imaginary or complex – represents an anticlockwise rotation of 90 degrees in the complex plane – the geometric meaning of i is thus unveiled!

The moment geometric understanding is achieved is often the moment that community acceptance is catalysed.

I’d completely forgive you if this all still seems completely, well….imaginary, and unfortunately I can’t continue this post forever to show the applications of complex numbers to electrical circuits with impedance, for example, but I hope to have shown that irrational numbers and negative numbers went through a similar stage of scepticism until familiarity and geometric understanding begins to develop an acceptance, and real-life application further validates their use.

We now have a widened perspective of numbers which allows us to progress mathematics. Just because we do not immediately perceive negative numbers, irrational numbers and imaginary numbers in our worldview, it does not mean that we cannot conceptualise them in more abstract terms, and then develop our understanding to ultimately apply them to real world phenomena. It is unfortunate that these numbers were initially given names which make them appear more odd, foreign and strange then they should [6], and that is because they were named before they were understood and applied. Ultimately, the improvement of our concept of number from discrete to continuous, alongside the extension of the number line to incorporate negative numbers, shows that a third extension in the form of the Imaginary axis and the complex plane is not particularly shocking in the history of mathematical development.

References:

[1] Wilczek, F. (2016). A Beautiful Question: Finding Nature’s Deep Design. UK: Penguin Books: UK.

[2] Muir, J. (1963). Of Men and Numbers. New York: Dodd, Mead and Company.

[3] One legend claims that this very thing happened. Hippapus, Pythagoras’ student, is attributed with the proof of the existence of irrational numbers, and he died at sea some time later. Whether his death was accidental, or whether he was killed for exposing this fact to the public, is left to the imagination…

[4] See Zeno’s Paradox of Motion if you’re interested in digging deeper into this.

[5] Hodgkin, L (2005). A History of Mathematics: From Mesopotamia to Modernity. Oxford: Oxford University Press.

[6a] McAllister, J. (2005). Mathematical Beauty and the Evolution of the Standards of Mathematical Proof. In Emmer, M. (ed.), The Visual Mind II: MIT Press.

[6b] Sinclair, N. (2011). Aesthetic Considerations in Mathematics. Journal of Humanistic Mathematics, 1(1), 2-32. Retrieved from https://pdfs.semanticscholar.org/c01f/1b7cdbe2b9a649d09311f7b3e5e1bcb88310.pdf