Friday, April 3, 2009

The Continuum Assumption

The engineering mathematics course at Swinburne is typical of most engineering mathematics courses around the world, in that, there is a heavy emphasis on the use of functions in analysing the physical world. In particular, there is underlying assumption (often unstated) that we can deal with physical data as a continuum (e.g. analysing radioactivity measurements using exponential functions). It is this assumption that underpins the "classical" paradign of engineering mathematics, which I would describe as:

A. analyse the physical relationships of the system being studied (e.g. force balance of a particle),

B. form equations that reflect these relationships, making appropriate simplifciations and assumptions (e.g. particle is spherical),

C. solve these equations for a given set of boundary conditions or limitations using either analytical or numerical techniques, and

D. analyse the solutions obtained against physical data, returning to first two steps if the solutions obtained are inaccurate or not credible.

This approach, and many subtle variations, has proved to be very powerful in analysing engineering problems, though complex systems where subtle changes in geometry and boundary conditions can produce large variations in behaviour (e.g. turbulence in fluids, movement of fine particles and "chaotic systems" in general) have proved difficult to model using this approach. Stephen Wolfram, in his book "A New Kind of Science" (2002) (see http://www.wolframscience.com/) argued that the classical approach was fundamentally flawed and need replacing with a new approach called "cellular automata". At the heart of Wolfram's claims was this central observation:

All of our measurements of the world are made discretely, that is, we obtain discrete numbers from our instruments (e.g. the temperature measurement from a thermometer) including our senses, and artificially impose continuous relationships upon the world by forming equations around fundamentally discrete phenomena. We could more easily, and naturally, use discrete mathematical models to describe the physical world and dispense with the classical approach.

Quite a claim !! As you imagine this book caused much debate, some of it polite and in some cases, quite inpolite ! You might find the overheads of a lecture I gave on the book interesting - see http://www.swin.edu.au/feis/mathematics/staff/gbrooks_pres.html - and there are literally hundreds of sites on the web discussing this book. You may also interested to read a much earlier (and more modest) version of the same idea by Konrad Zuse (1910-1995) who published "Computing Space" in 1967. An English translation of this pioneering work on "digital physics" is available at http://www.idsia.ch/~juergen/digitalphysics.html. Zuse was also an early pioneer in the development of the computer and was, clearly, a highly imaginative and interesting thinker.

I think the claims, details and repercussions of Wolfram's claim are a bit detailed to discuss here but I do think the first part of his central claim is uncontroversial, that it, the measurements we make of the world are discrete and the equations we impose on this discrete data reflect our intellectual choices not an underlying physical connection between equations and nature (i.e. cannon balls do not have a parabolic equation written into their structure, it is "us" that chooses a parabola to model the motion of the ball). I think this is underlying assumption to appreciate as we continue along our path of differentiating/integrating/ etc. continuous functions to describe the physical world.