I’ve just come back from a few weeks in Thailand, where I gave talks at Chulalongkorn University, Thammasat University and Chiang Mai University. I also attended a conference on Geometry and Graphics in Bangkok, and met my mathematician friend Paolo Bertozzini, who is always a pleasure to talk to, full of insights and anecdotes from his long experience in the Land of Smiles. By the way, Thailand is a fascinating and wonderful place, visit if you can!
Travelling gives me time to muse; I can’t always be at the computer (even though on this particular trip I did spend a lot of time working on two papers with my graduate students) and waiting for airplanes or sitting in them gives one time to speculate. A topic that I have been pondering is: to what extent are mathematicians scientists? Or are we actually something else?
Probably I am steering towards the something else. Sure, mathematics and science have a lot in common. Science uses lots of maths first of all in setting up its theories. This used to be much more true of the physical sciences, but increasingly the biological sciences are also becoming more mathematical–or at least some aspects of them are. And some applied mathematicians are pretty close to being physicists, but not really experimental ones. I believe a strong case could be made that most mathematicians do research like scientists: we observe patterns, try to formulate theories to explain them, and then subject those theories to experiments–in our case calculations–to discover that they are probably wrong and need to be modified.
But in my experience this somewhat standard view-point misses an important distinction that needs a historical perspective to appreciate. Mathematics has been around a lot longer than science. The Greeks were doing mathematics at a very high level more than two thousand years ago. Mathematicians have a long sense of history, humbled and awed by the great minds which have preceeded us, of accomplishments in centuries gone by which we can no longer hope to surpass, or even equal.
The scientist thinks, and feels, quite differently. Science really only kicked off about 500 years ago in Europe, when people slowly started thinking thoughts like: how do we really know when something is true? Can belief and truth be separated? Does our desire that the world be a certain way prevent us from seeing it as it really is?
I remind you of the answers to these kinds of questions that people came slowly to appreciate: that the source of all true knowledge is observation: careful, unbiased and thorough. From the observations we make, we formulate theories to explain them. The simplest and most powerful theories take precedence. Finally we examine the implications and predictions of our theories and see if these are born out. If so, we strengthen our faith in our theories but do not become dogmatic about them. We are prepared to be wrong, and to change our minds when confronted with new evidence and explanations.
How much deep knowledge and power resides in the understanding which I have crudely summarized in the previous paragraph! A whole brave new way of thinking, of seeing, of understanding the world. Brave, because we are prepared to face the music, however it may sound. No longer must the heavens dance to a tune of our liking. Maybe we are small, and insignificant, and weak. But we will have the courage to admit it, and to carry on none-the-less in understanding the world, unconcerned if we are no longer at the center of God’s great plan, should such a One exist. And we are not upset if our theories overturn and disprove the thoughts of a previous generation–in fact we welcome such, and strive towards the breakthrough that upturns the applecart.
I believe that modern mathematics has lost its way logically, and that a new and far more interesting mathematics awaits us. I have a fair amount of evidence to support this point of view. Rational trigonometry gives a much simpler and more powerful approach to trigonometry and geometry, making computations easier—but the kicker is that it actually makes logical sense, as opposed to classical trigonometry, the development of which is a logical basketcase! And Universal hyperbolic geometry is likewise a complete logical overhaul of hyperbolic geometry, again replacing pictures and wishful thinkings with simpler and much more careful reasonings. Both of these new developments result in many novel and beautiful theorems.
It has been interesting, and I will admit somewhat (but not overly) disappointing, to see how uninterested my fellow pure mathematicians are in contemplating really new directions of thinking, and how unsure they are in applying their own critical analysis to weigh the evidence, rather than rely on authority and precedence.
The force of habit in people’s thinking weighs heavily on them, the mark of a heavy and bloated subject. How can I inject more scientific thinking amongst my fellow pure mathematicians? How can I make the subject lighter? These are the kinds of thoughts I have been thinking in Thailand.