# On Something Like an Operational Virtuality

## Abstract

**:**

## 1. (Re-)Naturalizing the Virtual

## 2. Exteriorization and Proof

… may I not […] be deceived every time I add two and three or count the sides of a square, or perform an even simpler operation, if that can be imagined?

...even where all the rules of thought are applied with formal correctness, there always remains a possibility that the contents of thought, instead of being repeated in identical distinctness, may change unbeknown to us. As we know, Descartes saw no epistemological but only a metaphysical way out of this labyrinth: his invocation of “God’s veracity” does not appease or resolve the doubt but simply strangles it. Yet here precisely lies the point of departure for Leibniz’ development of the technique and methodology of mathematical proof. It can be shown historically that Descartes’ skepticism about the certainty of the deductive method was the force that impelled Leibniz to his theory of proof. If a mathematical proof is to be truly stringent, if it is to embody real force of conviction, it must be detached from the sphere of mere anemic certainty and raised above it. The succession of steps of thought must be replaced by a pure simultaneity of synopsis.

It is therefore infinitely more reasonable and more worthy of God to suppose that, from the beginning, he created the machinery of the world in such a way that, without at every moment violating the two great laws of nature, namely, those of force and direction, but rather, by following them exactly (except in the case of miracles)……we can easily judge that this hypothesis is the most probable, being the simplest, the most beautiful, and most intelligible, at once avoiding all difficulties…

## 3. Rules and Demonstration

## 4. Artificial Equivalences

Perception is not a process of reflection or reproduction at all. It is a process of objectification, the characteristic nature and tendency of which finds expression in the formation of invariants.

## 5. Functorial Intuitions

In ordinary life we have all sorts of criteria for equality. ...equal weight, equal color, equal number, etc. Aren’t there very different criteria for equality in all these cases?

“Let’s look at the very contemporary example of the virtualization of a company. The conventional organization gathers its employees in one building or a group of buildings. Each employee occupies a precisely defined physical position, and his schedule indicates the hours he will work. A virtual corporation, on the other hand, makes extensive use of telecommuting. In place of the physical presence of its employees in a single location, it substitutes their participation in an electronic communications network and the use of software resources that promote cooperation.”

Rodin argues that this overcoming of set theory, which the Bourbaki authors had hoped to achieve through the structuralist program, was actually only achieved through a non-structural modification of our intuitions. He stresses that it was the taking of equalities for isomorphisms, rather than the taking of isomorphisms for equalities, that really allowed for Lawvere’s big leap into the top-down view offered by the category of categories. For it allows us to make sense of some of the “similarity” we informally observe between different domains of mathematics, and indeed between mathematics and the empirical world or the psychological domain. Indeed, with Cassirer, Rodin wants to conceive of mathematics as a “part of physics”, somewhere on the spectrum between the purely ideal and the empirical. Categories are not structures, he claims, they do not deal with invariances. Functors and their “natural equivalences” (transformations between functors that have a “dual”, in the reverse direction) are not “invariants” in the old sense. The tendency to view functoriality as a generalization of invariance, for Rodin, is symptomatic of a “conceptual inertia” possibly preventing us from doing full justice to the discovery of functoriality, and its adjustment of intuition. The new epistemic criterion introduced in functorial thinking does not in his view reduce to the Platonic or structuralist criterion according to which only invariant features are epistemically significant, while all the variable features are accidental and irrelevant. Rather than just tracking invariances, the functor tracks ways in which things can be taken for other things, it maps out different modalities of the as-if, it develops a diagrammatical logic for modeling ways of selecting, indicating, and picking out, or grouping, fusing and gluing things together, and thus of “moving” from one universe of discourse or thought or practice to another. The “hollowing out of substances” is replaced with an intuition for something like a cartography of worlding, a mapping out of virtual transitions between contexts, between worlds, between subjective experiences or objective constructions. It seems to detrivialize the opposition between Leibniz’s wishful realism and Kant’s claustrophobic finitude: it provides conceptual, formal, and geometric tools that allow a finer analysis, as it were, of what is really going on in the transition between theory and practice, or between perception and cognition, than ever could the comparably much blunter metaphysical tools of process, time, and becoming. The functors are not trivial tansitions from one context to the other: they divide into covariant and contravariant, such that their adjunctions do not recover what we had hitherto come to conceive as strict identities. Passing from the left to the right and back again, does not necessarily ensure that we have recoverend the original entity, as is the case with the one-to-one correspondence. Rather, as in a game of chinese whisperers, each passage through the circuit changes the message. It suggests, furthermore, that there are no ultimate invariants at the bottom of the real, but rather axiomatizes the inherent incompletion and relativity of both substance and structure. It is, in this way, more “honest” about cognition: it takes as a given that whatever is right for me is right, that identities and equalities are always pragmatic articulations rather that pure ontological entities. Thus, Rodin argues that category theory’s real intuitive leap beyond set theory actually makes it more concrete. Far from being “abstract nonsense”, as it sometimes is accused, we might more accurately say that category theory is concrete nonsense: it makes the “nonsense” between regimes of intelligibility concrete. It is not that we can always treat isomorphisms as equalities, but rather that all equalities are always already only equivalent “up to isomorphism”.hypothetical category CAT of all categories as an intended model of [elementary theory] ET and then add to ET new axioms which distinguish CAT between other categories; then pick up from CAT an arbitrary object A (i.e., an arbitrary category) and finally specify A as a category by internal means of CAT (stipulating additional properties of CAT when needed).

## 6. Isomorphism and Computability

This history of ideas has been launched into new territories by the late Vladimir Voevodsky, and his influential introduction of univalent foundations. Voevodsky’s project imports a category theoretic intuition, an influence he gained in his early reading of Grothendieck’s (1997) Esquisse d’un programme… (which was written in 1984 and circulated in the mathematical community long before its publication), but also combines it with a very Leibnizian quest for the mechanical verifiability of mathematical proofs.[For Duns Scotus] the understanding … objectively apprehends actually distinct forms which yet, as such, together make up a single identical subject. ... Formal distinction is definitely a real distinction, expressing as it does the different layers of reality that form or constitute a being. … Real and yet not numerical, such is the status of formal distinction.

## 7. Operational Virtuality

Mathematics has been historically kind of static. If something has been proved it has been proved forever. One can speculate about the possibility of a kind of “dynamic” mathematics in that sense. It is very hard to imagine at this point…

This natural mathematics is only the rigid unconscious skeleton beneath our conscious supple habit of linking the same causes to the same effects; and the usual object of this habit is to guide actions inspired by intentions, or, what comes to the same, to direct movements combined with a view to reproducing a pattern.

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1 | His argument also applies to more recent structuralisms like James Ladyman’s Ontic Structural Realism, which denies the existence of objects, and develops a rather compelling realist theory of structure, where there are only relations, no relata (Ladyman, Ross, Spurrett, Collier 2009). Be that as it may, I do believe OSR is compatible with the operational notion of virtuality. |

2 | Voevodsky supposes that any foundation for mathematics should have the following three components: first, a formal deduction system; second, an informal aspect that provides a natural language equivalent that is “intuitively comprehensible to humans”; and third, in the reverse direction, a “structure that enables humans to encode mathematical ideas” into this framework (Voevodsky 2014). In ZFC, the first component is built on top of predicate logic, supplemented with a layer of patchwork axiomatics. Its second component, according to Voevodsky, is an implicit assumption that humans have the “ability to intuitively comprehend hierarchies”. Its third component, its main advantage, is that it provides an intuitive way of encoding mathematical objects as sets. |

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On Something Like an Operational Virtuality. *Humanities* **2021**, *10*, 29.
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Wilson A.
On Something Like an Operational Virtuality. *Humanities*. 2021; 10(1):29.
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Wilson, Alexander.
2021. "On Something Like an Operational Virtuality" *Humanities* 10, no. 1: 29.
https://doi.org/10.3390/h10010029