## Abstract

This paper considers the role that category theory can play in philosophy. Category theory is a source of problems, methods and inspiration when it comes to considering both some new and some longstanding philosophical issues. Among the former, the paper draws attention to the ontological interaction between categories and sets, as well as the quantificational criterion of being—to mention just two. Among the latter, it highlights the problem of cognitive access to mathematical objects, and that of the way in which such objects exist. In the context of the development and the ontology of mathematics, I argue in favour of the thesis that category theory is the most “platonic” theory in mathematics. I also point out that category theory impacts significantly upon many standard philosophical positions, providing many counter-examples to popular, often repeated, yet unjustified philosophical claims. The influence of category theory on the foundations and ontology of mathematics is also briefly explored here.

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## Notes

- 1.
However, some

*alter*-theories of sophisticated theories are trivial. - 2.
See the various works listed in the bibliography: for instance, [11].

- 3.
- 4.
- 5.
For Ingarden, mathematical objects are ideas with rigid essences. However, in the opinion of Ingarden, sets are only some sort of purely intentional object, without any such rigid essences. There is a difference between Husserl’s and Ingarden’s conceptions of intentionality. In the following, I prefer the concept of intentionality given in Husserl’s

*Logical Investigations*. - 6.

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## Appendix: The Commentary of an Anonymous Reviewer

### Appendix: The Commentary of an Anonymous Reviewer

The paper deals with very important and so far only partially explored topic of relations between category theory (CT) and philosophy. There are (as far as I know) not many works in this field. On the other side, many existing mathematical papers on CT contain strongly philosophical parts (e.g. Lawvere). Nevertheless, there is a shortage of systematic studies of CT and philosophy. The Author presents several examples, both in philosophy and mathematics, where such studies could be performed.

In particular I am aware of the Author’s work on alter theories which are mentioned in this paper. Let me comment on this point more carefully. In general an alter-theory is built as an associated theory to any existing axiomatic theory with recursive set of axioms, consistent and logically independent. Then one can negate one or more of these axioms which leads to the variety of possible alter theories. It is quite surprising that such set of alter-theories can be used as a tool for exploring the original theory. I think that this is also very nontrivial construction regarding existence in mathematics and possibly in philosophy. This also sheds some light on a would-be formal theory for ontology (at least in mathematics). To explain these points let me consider a model *M* of ZF(C). Then one can modify slightly the procedure of building an alter-theory: It relies rather on creating both the (model of a) theory and the (model of the) alter-theory. Historically such an approach has been realized by forcing extensions of *M*. There typically appear independent sentences on ZF(C) in the extensions, e.g. continuum hypothesis in one extension and its negation in the other. One extension can be seen as the model for a theory and the other for the alter-theory. One can ask the question: Is there anything fundamental for mathematics in the entire procedure of creating the pairs (Theory, Alter-theory) as above? Recent results of Joel Hamkins (e.g. [20]) show that the entire structure of the set of such forcing extensions (partial order) is a fundamental entity in mathematics, called the multiverse. The best recognized multiverse is the one where the initial model *M* is countable transitive and the forcing extensions are just Cohen extensions. The true challenge would be to recognize the multiverse starting from the constructible universe **L**. But the ultimate goal would be to investigate the multiverse corresponding to the universe *V* of sets of von Neumann. By now it remains among very ambitious, though beyond current possibilities, aims of the approach. In this way the procedure of passing to the pairs (Theory, Alter-theory) by means of forcing extensions appears as fundamental in foundations of mathematics. Finally, the categorification of the multiverse is possible and would lead to another interesting connections between CT, set theory and foundations of mathematics.

Regarding existence in mathematics let us make a simple observation: Given ZF(C) model *M* we can add extra axioms which leads to the more powerful capabilities for proving the existence of certain objects. The profound example is that of the large cardinal axioms added to ZF(C). Then one can prove the existence of things such as \(0^{\#}\). They correspond to real numbers which are not present in any forcing extension of *M* but exist in \(\mathbb {R}\) in *V* under suitable large cardinal hypothesis (LC). Without LC, \(0^{\#}\) does not exist. Another example is the constructible axiom \(V=\mathbf{L }\) added to ZFC. In such models, nonconstructible reals certainly do not exist and their existence relates again to LC when \(\mathbf{L }\subset V\) (under \(\mathbf{L }\ne V\)).

The question is whether there are any fundamental model *M* and ’the most fundamental’ LC axioms which would be ’the best’ for the entire mathematics. If they exist, this could shed light on a would-be formal theory for ontology in mathematics. As argued by Hugh Woodin, one candidate for such a formal theory could be certain extension of \(\mathbf{L}\) by all real numbers, i.e. \(\mathbf{L }(\mathbb {R})\); then, LC is given by a suitable set of Woodin cardinals (see e.g. [21]). The point is that \(\mathbf{L }(\mathbb {R})\) is stable under forcing extensions and it survives the modifications. This seems to be a very fundamental approach in mathematics, which leaves space also for categorical reformulation (cf. [19]). In all cases discussed above, an important thing was the behaviour of the models under forcing extensions. They can be seen alternatively as certain instances of alter-theories. However, building alter-theories seems to be a wider and simpler procedure than forcing extensions and consequently a less specific one. However, it is quite interesting that such simple point of view exists at all, referring directly to deep questions in mathematics.

Even though the above remarks are of rather purely mathematical nature, they carry also potential for philosophy. However, I am not competent enough to decide precisely this case. Similarly, I do not know whether these remarks should influence the current chapter or should be considered rather in a separate work. I will leave this issue to the Author’s judgement.

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Król, Z. (2019). Category Theory and Philosophy. In: Kuś, M., Skowron, B. (eds) Category Theory in Physics, Mathematics, and Philosophy. CTPMP 2017. Springer Proceedings in Physics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-030-30896-4_2

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