The Platonic Existence Of The Mandelbrot Set Viewed Through Systemic Functional Linguistics

Penrose (2004: 15-7):

In Fig. 1.2, I have depicted various small portions of that famous mathematical entity known as the Mandelbrot set. The set has an extraordinarily elaborate structure, but it is not of any human design. Remarkably, this structure is defined by a mathematical rule of particular simplicity. … 

The point that I wish to make is that no one, not even Benoit Mandelbrot himself when he first caught sight of the incredible complications in the fine details of the set, had any real preconception of the set’s extraordinary richness. The Mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. 

If it has meaning to assign an actual existence to the Mandelbrot set, then that existence is not within our minds, for no one can fully comprehend the set’s endless variety and unlimited complication. 

Nor can its existence lie within the multitude of computer printouts that begin to capture some of its incredible sophistication and detail, for at best those printouts capture but a shadow of an approximation to the set itself. 

Yet it has a robustness that is beyond any doubt; for the same structure is revealed — in all its perceivable details, to greater and greater fineness the more closely it is examined — independently of the mathematician or computer that examines it. Its existence can only be within the Platonic world of mathematical forms.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, the sense in which the Mandelbrot set is 'objectively there in the mathematics itself' is as the meaning potential of mathematics. As meaning potential, its instances can be realised materially, and as material realisations, they can be perceived mentally as metaphenomena (representations of meaning). None of this requires a distinct 'Platonic world of mathematical forms'.

The 'endless variety', 'unlimited complication' and 'incredible sophistication and detail' of the Mandelbrot set are (nominalised) qualities attributed to the metaphenomenon, and as such, are not, in themselves, criterial to the location of the "existence" of the Mandelbrot set: mental, material or Platonic.