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
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?
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.
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.