There is a profound issue that is being touched upon here. In the development of mathematical ideas, one important initial driving force has always been to find mathematical structures that accurately mirror the behaviour of the physical world. But it is normally not possible to examine the physical world itself in such precise detail that appropriately clear-cut mathematical notions can be abstracted directly from it. Instead, progress is made because mathematical notions tend to have a ‘momentum’ of their own that appears to spring almost entirely from within the subject itself.
Mathematical ideas develop, and various kinds of problem seem to arise naturally. Some of these (as was the case with the problem of finding the length of the diagonal of a square) can lead to an essential extension of the original mathematical concepts in terms of which the problem had been formulated. Such extensions may seem to be forced upon us, or they may arise in ways that appear to be matters of convenience, consistency, or mathematical elegance.
Accordingly, the development of mathematics may seem to diverge from what it had been set up to achieve, namely simply to reflect physical behaviour. Yet, in many instances, this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turn out to mirror the physical world in a much deeper and more broad-ranging way than those that we started with. It is as though Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought.
Blogger Comments:
From the perspective of Systemic Functional Linguistic Theory, mathematics is the metaphenomenal reconstrual of quantitative phenomena.
The development of mathematical ideas is the building of mathematical potential through instances that establish new systems of relations that are interpersonally assessed as mathematically valid. This is the sense in which mathematical ideas 'have a momentum of their own'
From this perspective, the 'divergence' of mathematics from the physical world is the distinction between mathematics as second-order meaning and the physical world as first-order meaning, and the reason why mathematics models the physical world 'in a much deeper and broad-ranging way' is that it is a reconstrual of the quantitative meanings of the physical world in terms of strict logical relations.
It is not that 'Nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought', but that Nature is first-order meaning and mathematical thought is the reconstrual of Nature as second-order meaning.
Importantly, the use of the words 'mirror' and 'reflect' here misrepresent mathematics and the physical world as being of the same level of abstraction (order of experience).
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