The Independence Of Natural Numbers From The Physical World Viewed Through Systemic Functional Linguistics

Penrose (2004: 63-4):

We can, however, raise the question of whether the natural numbers themselves have a meaning or indeed existence independent of the actual nature of the physical world. Perhaps our notion of natural numbers depends upon there being, in our universe, reasonably well-defined discrete objects that persist in time. Natural numbers initially arise when we wish to count things, after all. But this seems to depend upon there actually being persistent distinguishable ‘things’ in the universe which are available to be ‘counted’. Suppose, on the other hand, our universe were such that numbers of objects had a tendency to keep changing. Would natural numbers actually be ‘natural’ concepts in such a universe? Moreover, perhaps the universe actually contains only a finite number of ‘things’, in which case the ‘natural’ numbers might themselves come to an end at some point! We can even envisage a universe which consists only of an amorphous featureless substance, for which the very notion of numerical quantification might seem intrinsically inappropriate. Would the notion of ‘natural number’ be at all relevant for the description of universes of this kind?

 

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, it is not that natural numbers 'have a meaning or existence independent of the actual nature of the physical world' but that the natural numbers of mathematics and the physical world constitute different orders of experience. The physical world is first-order meaning (phenomenal), construed of experience, whereas the natural numbers are second-order meaning (metaphenomenal), construed of first-order meaning (quantities).

To be clear, natural numbers arose in a universe in which experience came to be construed as countable things, through the linguistic systems that evolved in humans.

To be clear, in a universe where the numbers of objects keep changing, there are numbers — since it is these that are changing — and so the natural numbers of mathematics could be construed from them.

To be clear, in a universe that contains a finite number of things, it is the things that 'come to an end at some point', not the numbers used to count them.