63. Physics: Mathematics As Lens

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.

ChatGPT revised:

The Reification of Mathematics

Mathematics[1] is a semiotic field in which quantities — as descriptions of categorisables — are related to one another quantitatively. Equations symbolically re-present identifying clauses, in which quantities function as mutually identifying participants: each an elaboration of the other. The process of deriving a solution involves the stepwise elaboration of equivalent identifying clauses until the value of the unknown token is specified.

As a lens for physics, mathematics offers a univariate means — quantity — of testing ideational consistency both within and between models. It is, in this sense, the “hardest” of tests: the one that confers the greatest epistemological certainty. It also brings with it a powerful organising principle — symmetry[2] — as embodied in the structure of the equation. In fact, the modelled (token) and the model (value) are often conceived as if they are two sides of an equation. Crucially, however, these two sides are of different orders of experience: the token belongs to a lower order, the value to a higher. That is, the categorisable form being modelled — being — is the lower-order token, while the functional model — describing — is the higher-order value.

From this experiential asymmetry arises a persistent ideational inconsistency — one that has deeply shaped epistemological interpretations in physics. Many mathematicians reify their semiotic system as a transcendent reality — an abstract realm of ideal entities existing independently of the human observer. For instance, 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…

Here, the phylogenetic expandability of the mathematical system — its capacity to be extended consistently by users — is conflated with the “discovery” of intrinsic properties that pre-exist that activity. But it is ideationally inconsistent to construe such “discoveries” as existing objects. They are not eternal entities; they are system-internal consistencies — relations within a semiotic resource for modelling the categorisable.

The widespread view that “mathematics would exist even if mathematicians did not” (Coveney & Highfield 1995: 24) exemplifies this reification. It conflates potential existence (as meaning potential) with actual existence (as instantiated meaning).[3]

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Notes

1. Mathematics is usefully distinguished from metamathematics, just as language is from linguistics: in metamathematics, mathematics is used to model mathematics itself; in linguistics, language is used to model language itself.

2. Every semiotic field values certain kinds of patterning. In mathematics, symmetry has particularly high appreciation value.

3. The “many universes” hypothesis may be seen as a similar category error — confusing potential with instance. In that case, alternative trajectories of this universe are construed as parallel universes already instantiated.