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April 20, 2024
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A Conversation with Jonathan Kenigson on Mathematical Language in Modern Physics

And would it have been worth it, after all,

After the cups, the marmalade, the tea,

Among the porcelain, among some talk of you and me,

Would it have been worth while,

To have bitten off the matter with a smile,

To have squeezed the universe into a ball

To roll it towards some overwhelming question,

To say: “I am Lazarus, come from the dead,

Come back to tell you all, I shall tell you all”—

If one, settling a pillow by her head

Should say: “That is not what I meant at all;

That is not it, at all.”

-Eliot

Mathematics tends to proceed historically via a process of mathesis – a postulation of paradigms aimed at the classification of the common features or pre-existing systems. Mathetic existence as entitival (e.g. pertaining to the existence of a particular mathematical object) is bound to fail by Quinean arguments. Existence causa sui is too strong a claim for any entity that is in-principle incapable of empirical observation or incapable of expression without reference to some language L. The existence of formal languages sufficiently rich to permit quantification and qualification is a prerequisite for the definitive attribution of properties that mathematical entities possess. Mathematical thought possesses an inherent trend toward generalization.

One such prominent trend is the encapsulation of geometric symmetries in algebraic objects, and in the case of ST (String Theory) and related Conformal Field Theory (CFT), a mathesis from Lie theory is used for this purpose, as described in Schottenloher (2008). Roughly speaking, the first goal of CFT is to explore the energy- momentum tensor T and associated charge conservation principles. Conformal invariance guarantees the traceleness of T and produces local (infinitesimal) symmetry transformations that form a Lie algebra under standard commutation conditions. T is also shown to be preserved under the action of complex Moebius transformations, wherein the complex special linear group SL(2) acts on it as a tensor. Virasoro algebras are invented to describe invariances preserved under Lie actions. This approach allows one to provide a model for asymmetric quantum effects and symmetry breaking that occur due to violations of so-called local conformal invariance. Corollaries can be applied to black hole dynamics and cosmology to explain quantum phenomena that are unaccounted for by the standard model. One example is the resolution of the black hole (BH) information paradox and the introduction of tight string cores (so-called fuzzballs) instead of BH singularities. Indeed, String Theory makes many novel predictions about the compositions of BH singularities and many other broad domains of physical inquiry, and ST can be considered one of the best scientific theories available for the exploration of diverse events in cosmology. 

It is not sensible to inquire if mathematical entities exist prior to language if our putative definition is to exclude from existential discourse those entities that either cannot be empirically quantified or must be defined rigorously via linguistic means. The existence of mathematical objects cannot be interpreted without reference to the totality of the paradigms they inhabit, and such paradigms cannot be granted unequivocal existential status apart from the formal languages in which they are defined. Generalization is a trend in philosophical dialectic as well as mathematical dialectic. Language is a dialectic toward the general. In GR and ST, the mathematical entities demanded by the theory – Lie algebras, conformally invariant maps, Moebius transformations, special linear and orthogonal groups, Virasoro algebras, and the like – cannot be said to exist independently of humans and their cultural practices because all these structures and their relations cannot exist prima-facie apart from their definitions. Each mathematical structure reflects underlying symmetries in other pre-existing structures or reflects symmetries inherent in physical reality. The existence of each entity is inherently a linguistic postulation of existence. To state of some object X that “X is a Lie algebra” or “X is conformal under a transformation T” is to say that X satisfies certain properties defined in some formal language L. If L is chosen to reflect an underlying physical reality, then the object X can be conceived as a linguistically situated model for some at least potential state of affairs in the physical world. A prime example from ST is the worldsheet itself. Worldsheets, manifolds, complex functions, and Fourier series are indubitably indispensable to ST, but can be said to exist only contingently as models for proposed aspects of physical reality. One cannot conclude that woldsheets exist abstractly considering ST’s novel predictions. 

It may be impossible to say one way or the other whether mathematics could exist apart from language, then. Because it is impossible to conclude if a given set S exists independently of any language L, the same relativizing conclusion applies to other mathematical entities derived from sets using formal language. For the philosopher of mathematics, it is sufficient to appreciate these predictions considering their methodological and epistemological implications. Unlike GR, quantum mechanics is founded upon the state spaces of functionals (waveforms) acting on fields (Schottenloher 2008). A challenge of ST is to define these operators in the vocabulary of functional actions to produce a unification. Because both theories can be seen to obey wave mechanics, Fourier analysis is a natural candidate for unification. Given a fixed worldsheet W, one can define, for each point x in W, a complex exponential parameter that is an eigenstate of an initially unknown momentum state P(x) defined at x. Because P(x) is an operator whose states are complex exponentials that can be expanded as infinite operator series, the local behavior of each state is well-known. Initial conditions on the states (the so-called ground states of quantum mechanics) can be fixed from local string behavior. These give rise under suitable transformations to worldsheet correlation functions that have calculable amplitudes and are invariant under so-called Moebius Transformations, which are very useful symmetry operators whose actions are well-known from the study of special linear groups (informally, collections of matrices that perform rigid actions like rotation and inversion) acting on the complex plane through traditional matrix operations. Again, the relevant mathesis is a unification of ST with wave mechanics. The wave states of GR are species of affine transformations of waves that occur under integral operators. In ST, the integral species (most prominent of which are the Nambu- Goto and Polyakov actions) occur as expansions of complex waveforms on worldsheets. GR does not require worldsheets because – although its local geometry can be complex – point-particle propagation is intuitive without complex analysis. Conformal field theory (CFT) provides the next mathesis to understand propagation as a periodic deformation of an underlying field. The linguistic matrix of the resulting theories is one of translation, interdigitation, and missed connections that are rectified by appropriate academic and ideological work.

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