June 23, 2006

Power of speculative thinking

Quantum Theory, Elementary Particles: Limits to Divisibility
In the physical sciences, certainly, mathematics is an essential and powerful language and mode of thinking, which of course gives an abstract quality to the basic theories of physics. One can get a feeling for this abstractness by appreciating that,
  • firstly, the equations of physics are more fundamental than the individual phenomena which they describe. This of course is to be expected.
  • But secondly, going beyond this, one finds that the symmetries of the equations are more basic than the equations themselves!

In Dirac’s words, "both relativity and quantum theory seem to show that transformations are of more fundamental importance than equations". So the ‘stuff’ out of which one imagines nature to be made, and the principles underlying our understanding, get more and more refined...

However, all these comparisons and discovery of parallels between ancient thinking and modern physics must be understood properly. As Heisenberg carefully clarifies, it is the appeal to controlled experiment that gives to modern science a really solid and serious foundation and meaning. The statements of quantum theory are in a very specific context, and are not largely poetic imagery and speculation, however elevating these may be. For one ancient ‘insight’ corroborated today, may be there are many others not so corroborated. But there is no surprise in all this. It shows the power that speculative thinking can bring in.
At the end of his discussion, Heisenberg concludes by saying: "All the same, some statements of ancient philosophy are rather near to those of modern science. This simply shows how far one can get by combining the ordinary experience of nature that we have without doing experiments with the untiring effort to get some logical order into this experience to understand it from general principles".
Coming back to modern physics, it is the reliance on mathematics that provides guidance and rigour in thinking, and helps avoid internal contradictions and inconsistencies. But here it is a quite startling fact to realise that the true impact of careful and rigorous thinking within mathematics itself is a matter of very recent origin. Namely it is no older than the later half of the last century, through the work of mathematicians like Weierstrass, Dedekind and others. Even in this century we have had the striking results of Hilbert and Godel — the former trying to prove the internal consistency and completeness of an axiomatic framework for mathematics, the latter then showing that these can never be achieved!

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