How is the concept of a topos in mathematics relevant to philosophy?

Question

https://en.wikipedia.org/wiki/Topos

Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology.

My understanding is that a topos is essentially a set associated with a location in a grid. How is the concept of a topos relevant to philosophy?

So I've been told that Topoi is used in philosophy, but it seems like it's a set with a location on a grid 2d or maybe 3d even, but I don't see how that's useful, especially in philosophy. Could you explain what are the various ways topoi are used in philosophy?

Answer 1

The word 'localization' here has a specific technical meaning; it's referring to the localization of a category:

[L]ocalization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms.

Answer 2

Could you explain what are the various ways topoi are used in philosophy?

Let's start with a quotation Robert Goldblatt's text on the matter:

The notion of topos has great unifying power. It encompasses Set as well as... sheaves, and so brings together the domains of set theory and algebraic geometry. But it also has ramifications for... logic, the study of the canons of deductive reasoning. The principles of classical logic are represented in Set by operations on a certain set - the two element Boolean algebra. Each topos has an analogue of this algebra and so one can say that each topos carries its own logical calculus. It turns out that this calculus may differ from classical logic, and in general the logical principles that hold in a topos are those of intuitionistic logic. Now Intuitionism is a constructivist philosophy about the nature of mathematical entities.

That's pretty heavy. The implication is simple. That the formalizations of classical logic, logic tied to the appellation Laws of Thought, are actually a special theoretical case of constructivist, intuitionistic logic as L.E.J. Brouwer advocated. Furthermore, classical logic, as Frege may have been familiar with, has structural isomporphisms with set-theoretic and topological formalisms. As I see it, this is just a further sign of how Frege's logicism is misguided, which is important in the pursuit of mathematical foundations. It is also circumstantial mathematical evidence that logical principles reduce to psychological principles. Mathematical intuitionism affirms Dedekind's notions that the formal sciences are a product of the mind. Math is neither discovered nor invented, but constructed.

Answer 3

I actually spent some time studying category theory this year so I feel I can provide a little more insight here than has been offered yet.

I have seen at least one research article in which it is succinctly expressed, “topos theory is basically modern logic”.

A topos is a specific, very technically defined mathematical object, construct, or structure (whatever you want to call it).

Having spent a few months this year studying category theory, I have come to feel pretty strongly, you cannot do mathematics justice through approximations. Many people have heard a layperson’s simplification of topics like Gödel’s incompleteness theorem, or quantum mechanics, or quantum computers, or blockchains, or string theory, and the issue is that if you do not study it authentically - having studied the actual mathematics in which the ideas are expressed - you pretty much never succeed in actually understanding it authentically either. The problem with this is that it leads to misunderstanding - inaccurate belief - which charades as comprehension. My entire life, I have felt like I kind of get the idea of Gödel’s incompleteness theorem, but as I have learned more logic slowly, I have often had moments of revelation where I realize, “I basically never had a clue what the theorem is even saying, let alone why anyone would think it is interesting, let alone very interesting”. (I am still in that process, which maybe hints that this still happens to people as they continue to study something more.)

Trying to explain category theory (which can be seen as the general field that topos theory comes from or lies within) in terms of loose generalizations simply cannot get you anywhere beyond vague-sounding truisms - “it’s a kind of mathematics of structure, which shows the structure of other mathematical objects, and their relationships to one another…” It doesn’t sound like a particularly illuminating idea.

I will be honest with you, through my exposure, I have come to feel that category theory may be the most extraordinary field of knowledge I have encountered in my life. It is hard to convey how amazing it is. And I know many people who get exposed to it feel the same way. It is profound.

Anyway. Maybe I can actually try to explain a topos in technical - yet clear - terms (I’ll come back and edit this answer). But a short answer to your question for how, is…:

Category theory is one of the most general fields of mathematics imaginable, because it starts with a very simple, basic structure (a category), and shows how many things in the world of math can fit into that structure. It then builds on itself by mind-blowing applications of the idea of “abstracting something into its structure” to itself: you can have a category of categories, for example. This is a category whose objects are also categories. Thus, you can study the relationship of different categories to each other, using a category. Also, there are many, many specific properties a given category can have, and this leads to acquiring a very large and esoteric-sounding vocabulary to express the relationship between all these properties - things like ‘functors’, ‘monads’, ‘natural equivalences’, ‘fibrations’, ‘quasicategories’, ‘dagger categories’, ‘infinity categories’, and so on.

Basically, a topos is one such special type of category, with certain special properties. Some people see it as having come to define the modern field of logic, because it provides a way to specify the unique properties of many, many different kinds of logic (yes, there are different kinds).

Basically, there is this finding that people find to be extremely deep - they call it “the computational trilogy”: https://ncatlab.org/nlab/show/computational+trilogy

I’ll leave you to read it; but the idea is that three very different ways of thinking about the mathematical world can be shown to be structurally identical, in a way: computation - logic - and structures.

I found category theory very hard to study. I would recommend you start with some simple books, for example by David Spivak, and join the chat group that some people from the nCatLab site run.

As I said, as you adapt to this mathematical way of thinking, one ceases to use “hands-wavy” verbal approximations or interpretations and becomes increasingly aware of the usefulness, compactness, economy, and necessary specificity of mathematical definitions.

Answer 4

Topos (space) is a bona fide subject of study in metaphysics, a branch of philosophy which also puts under the microscope, allied ideas such as time and causality and ontology and necessity/possibility and identity/change.