The Invention Of Complex Numbers Viewed Through Systemic Functional Linguistics

Penrose (2004: 72-3):

Presumably this suspicion [of complex numbers] arose because people could not ‘see’ the complex numbers as being presented to them in any obvious way by the physical world. In the case of the real numbers, it had seemed that distances, times, and other physical quantities were providing the reality that such numbers required; yet the complex numbers had appeared to be merely invented entities, called forth from the imaginations of mathematicians who desired numbers with a greater scope than the ones that they had known before. 

But we should recall from §3.3 that the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be. We cannot directly see the minute details of a Dedekind cut, nor is it clear that arbitrarily great or arbitrarily tiny times or lengths actually exist in nature. One could say that the so-called ‘real numbers’ are as much a product of mathematicians’ imaginations as are the complex numbers. Yet we shall find that complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. 

It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the invention of complex numbers by mathematicians is the logical expansion of the metaphenomenal system that realises the field of mathematics. This is achieved through the conscious processing of mathematicians: the verbal projection of texts that are instances of the system.

From this perspective, the 'unity' of complex numbers with Nature derives from the meanings of (metaphenomenal) mathematics being reconstruals of the meanings of (phenomenal) Nature.

To be clear, it is mathematicians, as the part of Nature that is conscious of the complex-number system, who are impressed by its scope and consistency.