                                            ## Posts Tagged ‘mathematics’

### The Role of Mathematics and Rational Arguments in Science

Friday, April 20th, 2012

Mathematics is a tool used by scientists to help them construct models of how the universe works and make precise predictions that can be tested against observation. That is really all there is to it, but I had better add some more or this will be a really short essay.

For an activity to be science, it is neither necessary, nor sufficient, for it to involve math. Astrology uses very precise mathematics to calculate the planetary positions, but that does not make it science any more than using a hammer makes one a carpenter (Ouch, my finger!). Similarly, not using math does not necessarily mean one is not doing science any more than not using a hammer means one is not a carpenter. Carl Linnaeus’s (1707 – 1778) classification of living things and Charles Darwin’s (1809 – 1882) work on evolution are prime examples of science being done with minimal mathematics (and yes, they are science). The ancient Greek philosophers, either Plato or Aristotle, would have considered the use of math in describing observations as strange and perhaps even pathological. Following their lead, Galileo was criticized for using math to describe motion. Yet since his time, the development of physics, in particular, has been joined at the hip to mathematics.

The foundation of mathematics itself is a whole different can of worms. Is it simply a tautology, with symbols manipulated according to well defined rules? Or is it synthetic a priori information? Is 2+2=4 a profound statement about the universe or simply the definition of 4? Bertrand Russell (1872 – 1970) argued the latter and then showed 3+1=4. Are the mathematical theorems invented or discovered? There are ongoing arguments on the topic, but who knows? I certainly don’t. Fortunately, it does not matter for our purposes. All we need to know about mathematics, from the point of view of science, is that it helps us make more precise predictions. It works, so we use it. That’s all.

I could end this essay here, but it is still quite short. Luckily, there is more. Mathematics is so entwined with parts of science that is has become its de facto language. That is certainly true of physics where the mathematics is an integral part of our thinking. When two physicists discuss, the equations fly. This is still using mathematics as a tool, but a tool that is fully integrated in to the process of science. This has a serious downside. People who do not have a strong background in mathematics are to some extent alienated from science. They can have, at best, a superficial understanding of it from studying the translation of the mathematics into common language. Something is always lost in a translation. In translating topics like quantum mechanics—or indeed most of modern particle physics—that loss is large; hence nonsense like the “God Particle”. There is no “God Particle” in the mathematics, only some elegant equations and, really, considering their importance, quite simple equations.  One hears question like: How do you really understand quantum mechanics? The answer is clear, study the mathematics. That is where the real meat of the topic and where the understanding is—not in some dreamed up metaphysics-like the many worlds interpretation.

Closely related to mathematics are logical and rational arguments. Logic may or may not give rise to mathematics, but for science, all we require from logic is that it be useful. Rational arguments are a different story. Like mathematics, they are useful only to the extent they help us make better predictions. But that is where the resemblance stops. Rational arguments masquerade as logic, but often become rationalizations: seductive, but specious.  Unlike mathematics, rational arguments are not sufficiently constrained by their rules to be 100% reliable. Indeed, one can say that the prime problem with much of philosophy is the unreliability of seemingly rational arguments. Philosophers using supposedly rational arguments come to wildly different conclusions: compare Plato, Descartes, Hume, and Kant. This is perhaps the main difference between science and philosophy: philosophers trust rational arguments, while scientists insist they be very tightly constrained by observation; hence the success of science.

In science, we start with an idea and develop it using rational arguments and mathematics. We check it with our colleagues and convince ourselves using entirely rational arguments that it must be correct, absolutely, 100%. Then the experiment is performed. Damn—another beautiful theory slain by an ugly fact. Philosophy is like science, but without the experiment. Perhaps the real definition of a rational argument, as compared to a rationalization, is one that produces results that agree with observations. Mathematics, logic, and rational arguments are just a means to an end, producing models that allow us to make precise predictions. And in the end, it is only the success of the predictions that count.

Additional posts in this series will appear most Friday afternoons at 3:30 pm Vancouver time. To receive a reminder follow me on Twitter: @musquod.

 I believe this observation comes from one of the Huxelys but I cannot find the reference.

### Physics and Mathematics

Tuesday, April 7th, 2009 Airy pattern

When I was an undergraduate I enjoyed both my mathematics and physics subjects. Pure maths was an exercise in precision, when in proving a simple enough looking theorem, you should be concerned about the minutest detail. Physics I enjoyed because it was mysterious, it was about the world and it involved maths. That seemed like a compelling combination – that a whole class of physical phenomena could be encapsulated in a single mathematical expression. However, there is always the question lurking in the background, what exactly is the relationship of physics to maths?

There are of course many different attitudes and indeed deeply philosophical attitudes. A mathematician might think something like this:

In mathematics, the pure notions of numbers and other structures do not need physics to exist or explain or even justify them. But the surprising thing is that often some newly discovered abstract formulation in mathematics turns out, years later, to describe physical phenomena which we hadn’t known about earlier. The only conclusion I can bring myself to is that mathematics is not just a tool of physics; it must be much, much more.

Conversely, mathematics alone is not enough to determine a physical system. For instance in studying magnetism historically, the English physicist Michael Faraday invisioned lines of forces in an invisible medium stretching between, say, your fridge magnet and your fridge as you bring the former towards the latter. Continental physicsts like Laplace and Poisson envisaged centres of force acting over a distance across empty space. James Maxwell showed that the two different visions were identical mathematically. However physically they were completely different systems giving rise to long debates and experiments about the existence of a universal aether which may transmit Faraday’s lines of force George Airy

One concept which occurs often in physics – and which gives rise to interesting mathematical expressions is that of symmetry. For instance the image above is caused by focusing light onto a circular hole and resulting in a centrally symmetric diffraction pattern. The mathematical function which describes how the brightness of the pattern varies is called after its creator – the Airy function.

These Airy functions (written Ai(z)) occur whenever we try to describe any physical system with the same type of symmetry. And indeed I have an ulterior purpose in making such a long-winded introduction – I study a particular physical system with such a symmetry – in fact an interaction between particles embedded in a strong centrally symmetric field. So naturally in my study I obtain Airy’s functions – and not just one or two, but an awkward combination of Airy’s and other functions.

To be frank this “awkward combination” has been driving me nuts for quite sometime – with me wishing that my undergraduate maths lectures hadn’t occurred so long ago. You didn’t think I was going to spare you the gory details did you? Maths conundrum

There are two ways of simplifying this little bit of maths. Firstly, the squiggly, almost vertical line on the left – the integration – can be done analytically. That is we can perhaps find an algebraic expression exactly equivalent to the above, but without the integration. This is usually the preferred result – to be able to see a physical system described in assimple mathematics as possible is not only asthetically pleasing, but leads to deep insights about the physical system in question. For instance the formula describing the entropy S (or amount of disorder) of a gas, developed by physicist Lugwig Boltzmann, was considered so important that it is engraved on his grave. Boltzmann's Tomb

In the second method of simplification, my problem integration could be done numerically by using a computer to plot the functions to the right of the integration sign and then calculating the area under the plot. This is the “brute force” method and not very satisfactory if you expect a physical system to be written simply in the language of mathematics. On the other hand, since all simple functions like the Airy function are themselves written in terms of integrations over other functions, then it may be the case that I’m dealing with a new type of function that deserves to be “fundamental” in some sense.

This goes to a deeper question – what is more important, the abstract formulae that describe a physical system, or a the real numbers that arise from calculating such formulae and which are compared with real experiments on the system in question? I suspect the answer is that both are equally important and it is the interplay between the numbers and the formulae – the experiment and the theory – that leads to a deeper understanding.