The Objectivity Of The Mathematical World Viewed Through Systemic Functional Linguistics

Penrose (2004: 12-3):

Nevertheless, one might still take the alternative view that the mathematical world has no independent existence, and consists merely of certain ideas which have been distilled from our various minds and which have been found to be totally trustworthy and are agreed by all. Yet even this viewpoint seems to leave us far short of what is required. Do we mean ‘agreed by all’, for example, or ‘agreed by those who are in their right minds’, or ‘agreed by all those who have a Ph.D. in mathematics’ (not much use in Plato’s day) and who have a right to venture an ‘authoritative’ opinion? There seems to be a danger of circularity here; for to judge whether or not someone is ‘in his or her right mind’ requires some external standard. So also does the meaning of ‘authoritative’, unless some standard of an unscientific nature such as ‘majority opinion’ were to be adopted (and it should be made clear that majority opinion, no matter how important it may be for democratic government, should in no way be used as the criterion for scientific acceptability). Mathematics itself indeed seems to have a robustness that goes far beyond what any individual mathematician is capable of perceiving. Those who work in this subject, whether they are actively engaged in mathematical research or just using results that have been obtained by others, usually feel that they are merely explorers in a world that lies far beyond themselves — a world which possesses an objectivity that transcends mere opinion, be that opinion their own or the surmise of others, no matter how expert those others might be.

Blogger Comments:

From the perspective of Systemic Functional Linguistic Theory, 'agreed by all' means that the validity of mathematical propositions is intersubjectively defined in terms of the internal relations of mathematical semiotic systems, regardless of the identity or personal opinion of any individual mathematician. It is because of the strict internal logicality of mathematical relations that mathematical systems can be intersubjectively agreed to 'possess an objectivity that transcends mere opinion'.

By the same token, the exploration by mathematicians of 'a world that lies far beyond themselves' is the instantiation of hitherto uninstantiated mathematical potential.