In this post I attempt to dispel a few misconceptions about mathematics, and reveal what it is actually like to study higher level maths. Or at least; reveal what my taste of higher level maths has indicated to me. Below are seven reasons why I think that maths is awesome, compiled in such as way as to accentuate the inaccuracies of the layman's understanding of what maths entails. Enjoy!
1 - Maths is not just about numbers
"It is not the job of mathematicians to do correct arithmetical operations. It is the job of bank accountants." - Samuil Shchatunovski
It saddens me slightly to know that a lot of people think that a degree in maths is "like school maths, just more". Like we spend our degrees memorising our 147 times tables, dividing 10 digit numbers in our heads, or honing our calculator speed. I wonder what they think the tens of thousands of full time research mathematicians are paid for! Perhaps tax payers fund them to "look for bigger and bigger numbers."
Of course, maths does include some work on numbers. And in fact number theory contains many of the most beautiful results that I have encountered in mathematics. But they are not treating numbers in the way that is taught in high-school. The results of calculations of the sort that crop up in school are considered tedious, unnecessary and mostly trivial in content in higher mathematics.
Often, when ugly expressions such as 16!*2^100 crop up in work, rather than calculating it's actual value, the mathematician will either leave it as expressed to save calculation, or do a very quick mental calculation to establish that it is less than some more more manageable number - here one might write < 10^50.
A mathematicians job is to look at numbers, yes. But also to look at space and time, and structures within this realm. They create a system, observe it, spot various patterns and conjecture about precisely what rule underlies the pattern they see. They subsequently try to prove (or disprove) their conjectures. In this way they learn about and get a feel for various logical systems. One such system - which, granted, has some interesting properties - is "the whole numbers" under usual rules of addition and multiplication. But this is by no means the only (or even the main) system under inspection!
2 - Maths is simple
"The essence of mathematics is not to make simple things complicated, but to make complicated things simple." - S. Gudder
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is" - John Louis von Neumann
"That's all very well", you may say, "but when I look at the work of mathematicians it's full of nonsensical symbols. They just complicate everything when there's no need."
Remember that mathematicians have chosen to use the notation that they use. If they could make their lives easier, they would. It is, therefore, sensible to suppose that the formal language of mathematics is necessary to be able to express abstract and convoluted statements simply, precisely and unambiguously. This is indeed true. Furthermore, the language required in order to make such progress at understanding the abstract, as is seen in mathematics, even possible.
To see how the language of mathematics has evolved, let us first look at the solution to the quadratic equation used in the 7th century A.D:
"To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."
In contrast, here is how modern mathematicians express the solution:
3 - Maths is creative
“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” - G.K. Chesterton
"Mathematics is, in its way, the poetry of logical ideas" - Albert Einstein
This is perhaps the biggest misconception of them all. Again, this thought finds its origins in the lack of creativity required in high school maths. But the very opposite is true! Mathematicians are constantly searching for methods that make things as simple and elegant as possible. We hate unnecessary clutter, and to find neat, easy proofs and results requires a lot of creativity.
In the 19th century, a maths teacher told his seven year old students to work out what the sum of the first 100 numbers was, and started making his way out of the classroom. The kids started scribbling on their blackboards; 1+2=3, 3+3=6, 3+4=10, ... They had a long way to go. But, according to legend, before the teacher had even reached the door, a certain seven year old by the name of Carl Gauss stuck his hand in the air. He had finished; on his blackboard was written "5050", and he was correct.
The other kids had maybe calculated 1+2+3+4+5+6+7=28 by that time. Those that most people would consider good mathematicians had maybe already summed the first 15 to reach 105. How did he do the calculation so fast?
The answer? He cheated. That is to say, he did some actual maths.
To calculate the sum by hand is doable. But it's also laborious, boring and slow. And what if they finished and the teacher said "Now sum up to 10,000"? They would have no chance!
What Gauss had realised was that instead of adding like everyone else: 1+2+3+4+...+98+99+100, he could rearrange the terms and add like this instead: (1+100)+(2+99)+(3+98)+...+(50+51). Each of these pairs of numbers summed to 101, and there were 50 pairs of them. So the answer was simply 50*101=5050. What he had done was lazy and creative; far closer to mathematics than what everyone else was doing.
The real genius with this inspired approach is that there is nothing special about the number 100 here. We can just as easily sum up to 10,000; (1+10,000)+(2+9,999)+...+(5000+5001)=10,001*5000, so we reduce the problem to a simple multiplication. In fact, summing up to any even number, n, we will pair off the numbers into n/2 pairs, each summing to n+1 and so the sum is n(n+1)/2. This also works for odd numbers, though a slight adjustment to the method is needed. Perhaps you can figure it out?
4 - Maths is beautiful
"Mathematics is the music of reason" - James Sylvester
"Mathematics has beauty and romance" - Marcus du Sautoy
"It’s like asking why is Ludwig van Beethoven’s Ninth Symphony beautiful. If you don't see why, someone can't tell you. If [numbers] aren't beautiful, nothing is.” - Paul Erdos
Perceived beauty is largely due to a combination of symmetry and chaos, of patterns and of fluidity. But symmetry and patterns are inherently mathematical properties, as are chaos and fluidity. Visually beautiful things often contain lengths with the golden ratio, a particular number with many curious properties. (See http://www.instant-analysis.com/Principles/straightline.htm)
For example, music is beautiful primarily due pitch and rhythm. But rhythm is, of course, all to do with patterns in time. Similarly different notes sound pleasant together because they are chosen according to mathematical formulae that describe how well their different wave forms interact. Beauty is also present, of course, when musicians transcend the rulebook and mess around with the timing, or play notes that shouldn't work.
But this more creative beauty necessarily presupposes the existence of basic mathematical structures; if there was no "correct time" then what would it mean to "mess with the timing"? And what would a "blues note" be without the standard scales? Furthermore, this form of creativity is a sign of more subtle rules at play, as opposed to the true absence of rules, and as such is underpinned by maths in the more direct sense, too. Similarly with any other beautiful thing. Mathematics not only provides a basis for beauty, but actually underpins and explains the human concept of beauty.
But, of course, this is not enough. Biology, one may argue, can explain the social and survival value of a concept of beauty. It does not mean that the biological systems themselves are themselves beautiful.
Indeed, it does not. It is conceivable that a very inelegant and ugly biological system could birth conditions in which beauty is desirable. But with mathematics, we are not talking about something that exists contingently as mere product of circumstance, in the same way our biosphere and evolutionary history does. We are talking about something that, as far as we can tell, exists necessarily . Something, certainly, that is intrinsic to our universe.
And thus, the intelligibility and simplicity of mathematics, combined with its enormous explanatory power gives us reason to think that this simplicity and symmetry - to which, and indeed by which, we are affected to attach elegance and beauty - is a property (and thus, in some sense, desirable) of our existence.
This is a somewhat weak argument for the idea that beauty is a mathematical property. But of course the much weaker idea that maths can be beautiful is easier to provide a convincing case for. Namely, the vast majority of people who have ever understood higher level maths have found it elegant and beautiful, And so, in the weaker subjective sense, maths can be safely assumed to be beautiful.
If you are keen to actually see some of the beauty in mathematics, you can get a taste of it by you-tubing "Mandelbrot set" and exploring from there. If you want a full dosage, I'd recommend sticking with formally taught maths long enough for it to get going properly!
5 - Maths is practical
"But, in my opinion, all things in nature occur mathematically" - Rene Descartes
"The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve." - Eugene Wigner
This is perhaps the single biggest misconception about mathematics; that it is void of practical application. It is one thing to think that a degree in mathematics is useless in the real world, but quite another to suggest that mathematics is itself valueless to humanity. Granted, they are both factually incorrect statements. But the former I feel somewhat sheepish arguing against. The latter view, however, is simply unacceptable. Everything requires mathematics, at least indirectly.
For centuries maths has been invaluable to the Physics and Astronomy, and often - unbeknownst to the viewer - to the visual arts. But nowadays it really has become ubiquitous. Biologists model population growth with differential equations. Philosophers borrow propositional logic and set theory. Sociologists use hypothesis tests and statistics. Lawyers *should* use Bayesian probability. (see http://understandinguncertainty.org/court-appeal-bans-bayesian-probability-and-sherlock-holmes)
We are not talking just about arithmetic, but proper, high-level maths. Particularly in the physical sciences and in computer science, the maths required is on a similar level to that being discovered at the very frontiers of maths research! The need for maths crops up literally everywhere. The unreasonable effectiveness of mathematics in the natural world is one of the most mysterious and curious observations we have made about our universe.
Of course, such is the delight of mathematics itself that a lot of mathematicians care little for its practical application. If it didn't happen to be a solid career choice with prospects of a decent salary then I'd be doing it in the evenings as a hobby. The intrinsic beauty of maths has enough appeal without having to relate to the physical world. But it would not be an understatement to say that without mathematics being incredibly useful, we would be living in a much more primitive world.
6 - Maths is personal
"God used beautiful mathematics when he created the world" - Paul Dirac
"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere." - W.S. Anglin
In school, maths didn't seem to be personal in the slightest. Everyone learnt the same method(s) to solve a problem, and tended to work individually. There was not collaboration in the same way that there was in more discussion based subjects like English and RE, where we would bounce ideas off each other. However, this changes at degree level - at least in my experience!
Suddenly, as aforementioned, one is expected to solve problems creatively. Two mathematicians of similar competency will likely both have the tools at their disposal needed to solve the problem; the tricky part is coming up with an idea; a strategy or game-plan that will be employed to attack the problem.
This means that one person may not come up with a successful idea for hours, whereas another may see it instantly, even though they are of similar ability; the roles may well swap for the next problem! Thus collaboration becomes absolutely key. Maths, for the vast majority of practitioners, is inherently social.
As another corollary of this, it becomes important that mathematicians are not merely "machines for turning coffee into theorems" - here I beg to differ with the great Paul Erdos! - but, rather, are individual people with different traits and ways of seeing things. It is simply false that all mathematicians are similar in nature; there are common traits and eccentricities but mathematicians more diverse in nature and in outlook than one might suspect. It is conducive to healthy collaboration to have different perspectives and tool-kits at ones disposal when solving a problem.
Lastly, and on a related note, I am familiar with the notion that nearly all maths was done hundreds of years ago, and that there really isn't very much left to discover. This has an element of truth to it; all of "the basics" seem to have been covered, in that it is now quite rare for an amateur mathematician to solve a new problem with no knowledge of higher-level maths.
But the main motivation of the notion - namely, that new maths has all but dried up - is plainly false. The number of maths papers published per year went gone up from 3,000 in the 1940s to 160,000 in the 1990s, and shows no sign of retreating any time soon. While the collective human bank of mathematical results is incredibly vast, and while the easiest route to publishing original research now involves a significant number of years of preliminary studies, we appear to only have scratched the surface. Indeed, the vast majority of the most beautiful results in mathematics may still be waiting to be found!
7 - Maths is not (needlessly) pedantic
"I never did very well in math - I could never seem to persuade the teacher that I hadn't meant my answers literally". - Calvin Trillin
A lot of discoveries in mathematics rely heavily on intuition and creativity, as has been established. But once we use these skills to get to an answer that we think is correct, logic steps in. This is where mathematicians formalise their thoughts, and check to see whether their intuition holds up to scrutiny. Mathematics of this kind has a very rigorous layout; first we establish our assumptions (called axioms) and then we logically write down "in between steps" that lead us to our conclusion (often called a theorem), where each single step is deemed to be logically true.
Sometimes this means that mathematicians question and try to prove things that every person knows to be true already. For example, everyone knows that the shortest path between two points is a straight line, right?!
Well, why is that? Can you prove it? Your response will probably be something like:
"It is obvious. Take the straight line between the two points. Now change a small section of it so that it is no longer a straight line. That made the path longer. Therefore the straight line is the shortest possible path."
But that is not an acceptable standard of proof in mathematics. Two flaws are present in this argument. The first is that it assumes the result in the proof. Why is it true that a small deformation of the straight line results in a longer path? That only is necessarily true when the result you are trying to prove is true! Secondly, even if small deformations anywhere on the path made it longer, that means it is locally optimal but it does not follow that the path is globally optimal.
In other words, there could be some crazy path that you just haven't thought of yet which is shorter, that looks nothing like a straight line. Just because you've thought really hard and failed to think of it doesn't mean it doesn't exist! Consider standing on a rock out to sea, so that you are only knee deep in water. It doesn't matter which direction you go, the water gets deeper and deeper. However, it is incorrect to assume that you are on the highest surface in existence; infact, the mainland exists!
As it turns out, the shortest path between two points is, indeed, a straight line. But only once we've required a "path" to have a few tight conditions so that it is, in s strict sense, "well behaved". Indeed, there are conceivable definitions of "path" in which this doesn't hold. The two short edges of a piece of paper are, we suppose, at least a length of paper away from one another. Yet in three dimensional space we can simply bend the paper and make them touch; in a sense, the shortest path between them has zero length. (See http://www.instant-analysis.com/Principles/straightline.htm)
The reason that mathematics can seem pedantic is because human intuition is far from flawless. The Earth is not flat. Angles of a triangle adding up to 180 degrees only in a very special case. And in some sense 1+2+3+4+5+... = -1/12. Using well defined assumptions and consistent schemes of logic we can be sure that our results are true. Certainty of this kind really does not exist outside of mathematics.
Hopefully this post has enlightened readers who have not pursued maths beyond school level. It really is a fascinating field of study. While I do not hold that it is better than other subjects, I do believe that it offers something truly unique to what humans can achieve; namely discovery of absolute truth via pure reason. As such I wish that everyone was open to learning about the beautiful truths present in maths, in the same way that most are open to exploring the arts, social sciences and, more recently, the natural sciences. I would recommend you take a wander in the "strange wilderness" sometime, it may just be a very different experience to what you might expect.
Jeff
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