Is topology used outside of cosmology in philosophy?

Question

It seems like topology is used to model spacetime, but outside of cosmology, it seems like topology has absolutely no use in philosophy. Is topology used to create models that relate to abstract and non-physical concepts in philosophy? I was looking for evidence of this, but couldn't find anything.

Answer 1

For an unfortunate preliminary example, Christopher Langan's infamous theory of everything uses the concept of topology in a way similar to how Alessio Moretti uses the concept of geometry. My take on Langan's theory is not that it is so bad that it is pseudoscientific (or pseudophilosophical), but instead that it occupies the grey zone of informed-but-still-weak attempts by mathematical/logic-minded folk to apply their special and important concepts to metaphysical questions. Other examples include Tegmark's "ultimate ensemble" (that he apparently had to backtrack to countable worlds) or Stephen Wolfram's "ruliad."

So, anyway, depending on where you situate Langan in the social history of philosophy, he either is or isn't an example of a philosopher who applies topology to metaphysics. But so in fact, shadowing the nod to Moretti, we should say that what has historically been a "geometrization" of metaphysical systems could be, or become, a "topologicalization" of the same, given the slightly obscure distinction between geometry and topology.

Actually, we in fact already see this, from a certain angle, in the question of topos theory. The Wikipedia article says:

Since the early 20th century, the predominant axiomatic foundation of mathematics has been set theory, in which all mathematical objects are ultimately represented by sets (including functions, which map between sets). More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set-theoretic mathematics. But one could instead choose to work with many alternative topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets.

It is also possible to encode an algebraic theory, such as the theory of groups, as a topos, in the form of a classifying topos. The individual models of the theory, i.e. the groups in our example, then correspond to functors from the encoding topos to the category of sets that respect the topos structure.

The first paragraph especially is reminiscent of plenitudinous platonism and "the" set-theoretic multiverse. So although "particular" results in mainline topology might not evidently be relevant (or, attempts at relevance have not been well made) to metaphysical problems as interpreted in philosophy (historically-speaking), yet the nature of topology is arguably relevant to the nature of all mathematics whatsoever, in the sense of the search for "foundations" of mathematics.

Answer 2

Stone Spaces

If you consider formal logic to be part of philosophy then topology is relevant. The basic result is about Boolean Algebras. A Boolean algebra is something like a bunch of sets where you can take unions and intersections and complements. Another example of a Boolean algebra is a bunch of formal statements and you can take disjunctions and conjunctions and negations.

For any such algebra there is an associated topological space called the Stone Space. In fact there is a one-to-one correspondence between boolean algebras and Stone spaces. That means all the logical properties of the algebra are encoded somewhere in the topology of the space and vice-versa. Depending on the context, it can be more useful to look at the Stone space than the algebra itself.

Stone spaces are fairly exotic. The most well-known is the Cantor set, which is what you get by continuing the process depicted below ad infinitum:

You will notice the Cantor set contains no intervals. It is a bunch of totally disconnected dust sprinkled on the number line. The manner of arrangement of the dust is what makes the Cantor set the Cantor set and not, for example, the set of irrational numbers. Stone spaces in general have this property.

There are many generalizations of Stone's theorem for things that are not quite Boolean Algebras. For example you might not want the law of excluded middle to hold for your collection of statements. In that case we get something more general than a Stone space.

The upshot is that if you are interested in studying formal systems in general, rather than a particular one, it might help to know some point-set topology.

Answer 3

I will elaborate a bit more on the connection between topology and the law of excluded middle (LEM), which Daron and Kristian have already mentioned.

In logic, the basic objects of study are formulas (or sentences), and studying these amounts to studying the syntax of a theory T. The Lindenbaum-Tarski algebra of T is a type of reification of T, which offers semantics of the theory.

Boolean algebras are somewhat familiar in that they are the Lindenbaum-Tarski algebras of classical propositional theories (and so reflect the LEM). By the Stone representation theorem, Boolean algebras arise from sets, and so one does not need topology to generate these.

The Lindenbaum-Tarski algebra of a propositional intuitionistic theory is a Heyting algebra. It turns out that the lattice of open sets of a topological space is an example of a Heyting algebra, hence giving rise to a semantics for propositional intuitionistic calculus. In fact, there is a kind of duality: Every Heyting algebra H is naturally isomorphic to a bounded sublattice L of open sets of a topological space X.

So to get an idea of what failure of the Law of Excluded Middle looks like, form the Heyting algebra of open subsets of any non-discrete topological space X. Then there will be an open set U whose is complement not open (this will correspond to a proposition P), and so the interior of the complement, V (corresponding to the negation of P), will be strictly smaller than the complement, and hence the union of U and V does not comprise all of X. There will be some nonempty boundary Z in between U and V, and this Z is the No Man's Land in between the proposition P and its negation not P.

To see where I initially learned of these ideas, see the answer here, the first comment, and the attendant link: http://mathoverflow.net/a/120737

The topos theory also mentioned elsewhere on this page is a staggering, grand study of this circle of ideas as well as utterly mind-blowing generalizations.

Answer 4

The average person is more familiar with space-time as a concept then mathematical theories, so the other domain in which topology matters, and may not be noticed, is the philosophy of mathematics (SEP).

More specifically, there are two domains of discourse where topology would be relevant in discussion. One of them is mathematical foundations which has come to include category theory. Category theory deals with representations that are graph-theoretic, and therefore they are topological in nature. Are topological structures better for reducing other forms of mathematics including logic? Why would a set, which extends the notion of containment or extension in space not be sufficient? The other is in defense of structuralism (SEP) as a mathematical metatheory. Some mathematicians believe that mathematics is the study of how objects are constructed and how they and their structures relate to each other. Topology is an interesting case because like a rubber band, one can hold the structure of points invariant, but change the distance between them.

Besides that? The only other thing that comes to mind is measurement theory (SEP) which can make use of the invariants that come along with topological spaces. Engineers and scientists, for instance, may have an interest in measurements of things that deform and translate in space.

Answer 5

There are some works on Analytic Philosophy. One important work on this area is called mereotopology. In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts. I recommend this paper to read something about topology in Analytic Philosophy and Mereotopology: http://philsci-archive.pitt.edu/20495/1/ITAPv3-1.pdf

Answer 6

Topology is, in fact, very intrinsically linked with more or less entirety of mathematics and all of its derivatives. It also has a very big role in theoretical computer science. Let me highlight some of the connections for you:

Homotopy theory is very closely linked with category theory, a framework which is used in basically all of modern algebra since Grothendieck popularised algebraic geometry. In fact every infinity category can be viewed as a topological space, with higher order morphisms being viewed as homotopies.

Topology is very closely linked to logic. Someone mentioned Stone Spaces here before, but the connection goes even deeper. Curry-Hkward correspondence shows that there is an isomorphism between logical propositions and types, showing a connection between logical systems and computations. And a lot of modern intuitionistic type theories have topological interpretations. So basically logic, topology and models of computation are all closely linked.

I am pretty sure due to it's connection to both logic and category theory, you can derive the entirety of mathematics from topology. And the fact that topology describes the very ways we talk about maths (logic and category theory) makes it useful not to just prove things, but prove statements about how we prove things.

Also take a look at Homotopy Type Theory, my dissertation topic. It provides univalent foundations for mathematics and uses topological spaces as a model.

Hope this helps. You can do more research on your own with the details I've given you I'm pretty sure.

Tl:Dr - The entirety of logic and computation theory more or less boils down to topology in our modern understanding, and those are very basic mathematical fields that the entirety of maths is built upon. Moreover, these fields are also very important in philosophy.

Answer 7

I'm assuming that formal logic also counts too. Topology is a crucial component of Dana Scott's denotational semantics / model theory of lambda calculus. The fixed-point combinator, which is what you need in order to do recursion, is defined as the limit of a sequence in a topological space. See for example Scott's papers Lambda Calculus: Some Models, Some Philosophy or proposition 3.14 in Continuous Lattices, or Park's The Y-Combinator in Scott's Lambda Calculus Models.

Answer 8

I guess it would depend on what the geometric representations of ideas are and what constitutes a deformation.

An argument (linear thinking) consists of inferential links (vector lines) connecting the premises (nodes) to the conclusion (another node). Refuting an argument is to break these inferential links (the vector lines). What's the topological relevance? The line is torn. In other words, philosophy is an enterprise dedicated to finding toplogical ideas aka sound arguments where the logical/inferential vector lines are unbreakable.

Also, in re graph theory, is it applicable to the ideaverse consisting of a network of ideas? Is it possible to traverse the entire ideaverse such that each inferential connexion is crossed once and only once?