Mathematics[1] is a semiotic field in which quantities — as descriptions of categorisables — can be related to each other quantitatively. Equations are symbolic re-presentations of identifying clauses in which quantities function as mutually-identifying participants, each an elaboration of the other. The process of deriving a solution involves the sequential elaboration of equivalent clauses until the value of the unknown token is identified.
As a lens for physics, mathematics provides a univariate means (quantity) of testing ideational consistency between and within models — the “hardest” of tests, providing the greatest certainty. As a lens, it also provides symmetry[2], as embodied in the equation, as an organising principle of description, even to the extent that the modelled and the model are conceived as if two sides of an equation. Importantly, the two sides of an equation — token and value — are of different orders of experience, token lower, value higher. In this case, the categorisable form that is modelled, ‘being’, is the (lower order) token, and the functional model, ‘describing’, is the (higher order) value.
Arising from this experiential asymmetry, an ideational inconsistency in the interpretation of the epistemological status of mathematics has had an influence on interpretations in physics. Many mathematicians reify their semiotic system as a transcendent ‘reality’, as an abstract thing existing in a Platonic realm. For example, Ruelle (1993: 161;165) writes:
But mathematicians have no doubt that there is a mathematical reality beyond our puny existence. We discover mathematical truth, we do not create it…But mathematicians and physicists know they deal with a reality that has laws of its own…
That is, the phylogenetic expandability of the semiotic system of mathematical potential, in ideationally consistent ways by mathematicians, is equated with the “discovery” of “real” properties that “exist” independently. But it is not ideationally consistent to construe mathematical discoveries as objects with eternal existence; mathematical discoveries are system-internal consistencies in a resource for quantitatively modelling the categorisable.
The view, widely-held in the mathematical community, that ‘mathematics would exist even if mathematicians did not’ (Coveney & Highfield 1995: 24) reflects this reification, and makes the category error of confusing ‘potential existence’ with ‘existence’.[3]
Footnotes:
[1] Mathematics is usefully distinguished from metamathematics: in which mathematics is used to model mathematics itself, in the same way that linguistics is metalanguage: language is used to model language itself.
[2] All fields have their own aesthetic values (appreciation probabilities), and symmetry is of very high value in the field of mathematics.
[3] The ‘many universes’ hypothesis might be seen as an example of the same category error, of confusing ‘potential existence’ with ‘existence’; in this case, other possible trajectories of this universe are construed as other existing universes.