As part of my fundraising efforts for the Great Shave, I promised to write 500 words in praise of mathematics in return for some generations donations. I thought this would be the easiest of my promises to fulfil, but it’s actually pretty hard to write in praise of something that is (to me) so obviously wonderful. Anyway, here goes.
The most striking single thing about mathematics is that a collective endeavour, pursued for thousands of years primarily because of its beauty and pure intellectual interest should turn out, in our time, to be so amazingly useful. To take perhaps the most striking example, the amazing fact that
or better still, with five fundamental constants
is, or ought to be, adequate reward for all the effort that went into the discovery of calculus, trigonometry and complex number theory, and the effort each new generation puts into learning these things. But, it gives us, free of charge, the amazingly useful Fourier transform, the basis of all kinds of modern communications, and much, much more.
(BTW, I made a big effort to instal a plugin that would make pretty math, trying and failing with both LatexRender and itextoMathML. Both require lots of fiddly modifications, and any wrong step crashes the blog completely. If anyone has an easy way to implement this, please advise me).
Although they didn’t make much practical use of it, the ancient Greeks had already perceived the connection between mathematical reasoning and the workings of the universe, manifested in such observations as the golden ratio, and the relationship between mathematics and music. Mathematicians, then and now, are instinctive Platonists, thinking of themselves as discoverers of hidden truths rather than as inventors of new ideas.
As an aside, this kind of thought, extending into number mysticism, always used to be associated with Pythagoras. I was taught that, although Pythagoras himself was obscure, his school not only discovered the theorem that bears his name but followed it through to the scandalous discovery that irrational numbers have natural geometric representations (consider a right-angled triangle with two sides of length one. They square of the hypotenuse must be two, but it’s easy to show that the square root of two cannot be rational). The person who leaked this terrible secret was supposedly put to death by the vengeful Pythagoreans. It turns out, as I read in a recent London Review of Books, that this is all legend and that there is no good evidence that the Pythagoreans discovered anything, although they were mystics. As is the way nowadays, the Wikipedia entry on Pythagoras had been updated before my copy of LRB had arrived in the post, and you can read all about it there.
Coming down to our own time, as late as the early 20th century, pure mathematicians could propose the toast “To pure mathematics and may it never be of use to anyone�. GH Hardy is the archetype of this attitude, though I don’t know if he ever actually made this statement. But economists and others use Hardy’s work in inequalities all the time, and even his contributions to number theory are now hot topics in cryptography.
What Eugene Wigner has called ‘the unreasonable effectiveness of mathematics’ remains a mystery. Maybe the Universe was designed by a mathematician. Or maybe we fit the mathematics we have to the universe, because no other mode of reasoning is remotely as powerful. Either way, mathematics is one of the great achievements of our species.
John Quiggin 
Nicely put.
Following up on your Hardy reference, this from A Mathematician’s apology:
How times have changed. Your computer and cell phone depend critically on the mathematics of those “useless” folks.
Ok, then tell me why when I talk to the Taxation Department that it’s always more that my accounting?
Step 1. Open new MS word document. Then INSERT> OBJECT >MS EQUATION 3.0.
Step 2. Make beautiful formula.
Step 3. Take image of screen (on a PC press us the “Print Screen” key).
Step 4. Paste image into paintbrush and crop.
Step 5. Save image to webserver (as gif of jpg or png file format).
Step 6. Create relevant HREF link in your post.
Wolla! There it is.
I do think that e^(i.pi) + 1 = 0 is quite beautiful and very inclusive. However 1 + 1 = 2 has the merit of being easier to remember.
I like the idea of the ‘unreasonable effectiveness of mathematics’. R. Thom equivalently described mathematics as ‘successful magic’. Abstract daft notions such as a ‘straight line’ in the plane or a ‘point’ combine to provide the basis for a geometry that can be used to build bridges which often don’t fall down.
I take this opportunity to point out a very nice proof of the Brouwer Fixed Point theorem stemming from the ideas of an economist H.Scarf. The proof is in Appendix B of “Economics for mathematicians” by J.W.S Cassells ( Cambridge University Press, 1981). Unfortunately, this is the only part of the book that I understood.
Er… it’s not easy to show that the square root of two is irrational, if all you have is geometry and arithmetic without algebra. Wikipedia has some stuff on the various proofs of its irrationality, including the one that is most likely the one the Greeks found. It relies on being able to construct a smaller right isosceles triangle from an initial one, using straight edge and compasses – which leads to an infinite regress, which is impossible if the ratio involved is rational.
Fair point, PML. I meant to indicate the argument for readers with algebra, not to say it was easy for the Greeks.
Terje: I can help with the installation of LatexRender. If you’d like to email me (address is in the readme_plugin.text file of the LatexRender for WordPress download) and let me know what went wrong or you couldn’t understand then I’ll try to help.
Fugato http://fugato.net/2007/01/20/latex-in-wordpress/ has an automated script which may also help.
For anybody who doesn’t want to bother with all this, at the expense of losing automation, then the Equation Editor at http://test.izyba.com/equationeditor/equationeditor.php will produce a mathematics image for downloading and uploading.
Whoops! Must read more carefully! It is John not Terje who would like to use mathematics!
Politics is a dread distraction from the cool eternal beauty of mathematics – thanks for this very nice piece John.
Speaking of Euler’s Identity, and unlike Gauss whose opinion was probably a little too hard as it was on most things, I’d say a mathematician is that person who on first acquaintance recognises
e^(i.pi) + 1 = 0
to be one of the most beautiful things they have ever seen, and suspects it’ll remain that way no matter how long they may live.
Euler’s Identity, along with beer, is the best evidence I know of for the existence of God.
I got around to looking at the wikipedia stuff again, and on following its links I found a good page with a lot of different proofs of the irrationality of the square root of two, here.
Some of the proofs don’t use algebra. I particularly like the Conway proof, although as presented it leaves out the lemma that the positive square root of two is greater than 0.5 (can you see why it’s needed?).
Proof 7 from PML’s link is my favourite. Pure geometry.
>Although they didn’t make much practical use of it, the ancient Greeks had already perceived the connection between mathematical reasoning and the workings of the universe,
not true, the great astronomical treaties of Ptolemy, known by its arabic name Almagest, is actually called Mathematical Treaties (Mathematike Syntaksys) in greek. Read the intro chapter, where Ptolemy explains that philosophy can be devided into three branches Theology, Physics, and Matematics. He says Theology is concerned with things that cannot be observed, so its conclusions cannot be totally trusted; whereas physics is concerned with the ever-changing state of materia and so lacks permanence. Mathematics derives its conclusions by logical proof and its conlusions are permanent and immutable, it is for this reason, Ptolemy says, it is best applicable to study astronomy.
True, mathematics was applied to astronomy, but they didn’t make much practical use of astronomy either, except in navigation. The separation between mathematics and physics is notable in this regard. Archimedes was the big exception to all this, but he seems to me to be an outlier.
I studied pure maths in the early 80s – we didn’t even use computers then (applied and stats did, but pure was just starting to). I remember the sheer beauty of so many of those proofs (and the magic of seeming to go around all over the place and the last line fitting perfectly – which always prompted those thoughts about maths and the universe).
I was at Monash that had an absolutely fantastic maths department. One of the lecturers even brought a picture of Euler to a lecture as he was very keen on him. The world has indeed gone the pure maths (and stats) way since then.
I am interested to know if maths (especially pure) has been decimated like so much of the arts since then in all the university cutbacks (I presume the 80s would probably now be regarded as a golden time).
I’m a public servant now – we use maths rather differently!!
If ever you read a maths textbook and it says something is obvious, and it doesn’t seem obvious at all, do not despair – it probably isn’t 😛