Academe

So. Initially when I came to the University of Ottawa, I was interested in developing some of the stuff I had worked on at Concordia. Basically I wanted to expand some loose remarks I had made about the historical portion of my work on emotional rationality into something serious. I thought I would rely heavily on stuff in Nietzsche and maybe also in Michel Foucault's later work, but none of that ended up panning out. I took a course on Max Weber and I was very impressed with him, and started thinking more seriously about sociology and history, and also about education. So I drafted up a plan for my dissertation which was basically designed as a critique of the concept of multiculturalism as it is employed within secondary education in Canada. My idea was to actually use policy documents and things like that for a change instead of staying within the sphere of academia. But this program, too, fell by the wayside. I took my first course with my now-supervisor. The course was about the Austrian logician/mathematician Gottlob Frege and it opened my eyes to a lot of things that I had been vaguely interested in for a while but never pursued. And now I am mainly interested in problems in the philosophy of logic and the philosophy of mathematics, though in a certain respect my interests remain the same.

I say they remain the same because the reason why I was interested in the things I was previously was partially 'ontological' and super general in nature. And now that I focus on mathematics more than politics, I find it is easier (for me at least) to draw out this really general aspect of philosophy and concepts that I've always been interested in anyway.

But I still have a lot of learning to do about math and logic and so on, I'm really a novice at it and I have to be more competent, but I think I'm halfway decent working with some of the more general concepts even if the specific ways of manipulating them in mathematical contexts often eludes me, and that is a nice enough feeling in itself.


I've been trying to formulate the proposal for my thesis for a while and I haven't been working all that hard at it, but I'd like to get it done soon so I'm trying to clarify exactly what it is that I care about in all this stuff.


I've been thinking about my work as an attempt to clarify the concept of 'structure' within the philosophy of mathematics. That's the general aim, but there are obviously lots of ways to go about doing that. Because I'm always interested in conceptual history, I've been pursuing it partially in a historical way and partially in a conceptual analysis sort of way.

So in the first chapter, I'm planning on doing a miniature history of the development of non-Euclidean geometries, leading up to the publication of David Hilbert's 'The Foundations of Geometry' in 1899. That particular book, for me, is probably the clearest starting point for a view of geometry (and then the rest of mathematics) as a science of abstract structures, rather than a science of 'space' or a science of 'numbers' (whatever numbers may be).

After establishing all that in a loose kind of history, I plan on analyzing the correspondence between Hilbert and Frege. Frege himself had a kind of old-fashioned view of geometry as the science of space, but I will argue that his criticisms of Hilbert's methodology have very little to do with this outdated view of geometry.


(Actually, in a pretty interesting short paper from the 90s my supervisor (Paul Rusnock) argued that Frege's own views about axioms and the nature of logic contained the elements of sophisticated rejection of his own outdated view of geometry, though he never pursued them for whatever reason).


Anyway, Hilbert's views on geometry are really commonplace in mathematics now, but if you don't know much about math today (and I didn't until very recently, and I still don't really), his views seem really weird.

Basically, in the 'traditional' Euclidean/Fregean view of geometry, all the 'axioms' of Euclid's famous book The Elements are taken to be intuitive and self evident truths about the actual 3-dimensional physical space we inhabit. So, like, the claim that a straight line can be drawn between any two points, or the claim that given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center...these kinds of claims were seen as intuitively evident truths about the actual world. And geometry was the science which collected together all the basic truths about space, more or less.

But, like I said, people started developing so-called 'non-Euclidean geometries'. The first serious non-Euclidean geometries were developed between 1829 and 1832 by a Russian mathematician named Lobachevsky and, separately, by a Hungarian mathematician named Bolyai. Now today there are millions of different non-Euclidean geometries of all different kinds, but Bolyai and Lobachevsky both developed what can more specifically be called 'hyperbolic' geometries.


In Lobachevsky's version, Euclid's 'parallel postulate' was denied. This postulate goes something like this:

'If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. '

There are several different versions of the parallel postulate, and it had always been troublesome for geometry, but the basic idea is that if you have a straight line and a point NOT on that line, then there exists one and only one line which intersects the point and never intersects the first line. That sounds kind've complicated but it really isn't. All it says is that for any line and any point not on that line, there is one line parallel to the first line which intersects the point. If you picture it it DOES seem kind of obvious, too. But Lobachevsky denied this postulate, and developed a geometry from the resulting system.

Now all kinds of crazy things happen when you deny the parallel postulate. One of the results, for instance, is that the sum of the angles of a triangle is always LESS than 180 degrees in hyperbolic geometry. But the most basic result is that the 'meaning' of the term parallel shifts significantly, and in a way so does the meaning of the basic terms 'point' and 'line'. Because in a way, which you could make more precise, the very meaning of Euclid's 'lines', etc. is bound up with the basic fact of the parallel postulate.

Anyway, all this is just to say that people became more interested in geometry in a way that had very little to do with 'intuitive truths' or even with 3-dimensional space. Perhaps the final death knell of the traditional view of geometry as the science of our actual space was signaled by the German mathematician Bernhard Riemann, who developed a whole slew of new concepts and theories. In 1854 he gave a lecture which generalized some aspects of geometry well beyond 3-dimensional space, he even talked about n-dimensional spaces or manifolds, which might seem a bit strange but which are so common in physics and even in engineering now that it is hard for physicists (probably impossible) to work with old-style geometries. Anyway, the interesting part about all this history stuff for MY concerns is that the philosophical apparatus for understanding what was going on with all these new geometries was not there at all, people really didn't have any idea what geometry (or the rest of mathematics, really) WAS anymore.

That's what is interesting about Hilbert, he provided a kind of picture of what mathematics was up to that many other mathematicians immediately accepted and which has since become developed and extended well beyond geometry into almost every avenue of mathematics, logic, and formal semantics (among other places). Hilbert's basic idea was to remove the reference to a single, concrete space in his axioms and treat the terms 'line', 'point', 'plane', etc. as general, abstract concepts which could then be applied to specific concrete instances via interpretations. An 'interpretation' in this sense is the assignment of actual referents to each of the abstract concepts of the theory, and an interpretation in which all the axioms still remain true is what we would today call a 'model'.

(Basically, Hilbert invented the modern notion of model theory, more or less.)

This codified in a certain respect the way that mathematicians had been working for 50 or more years, it gave a bit of sense to what they were doing. They were no longer studying concrete systems like the properties of physical space. Instead, they were studying abstract structures which could then be mapped onto particular concrete instances via determinate and rigorous procedures.

That's the basic way that most mathematics works today, too, and that's why a lot of philosophers interested in mathematics call it the 'science of structure'. So when I first started learning about all this stuff, I always wondered what these structures were. My first inclination was to think about them along the same lines as abstract objects. Like, I don't know if you know much about Plato, but he basically thought that there was a whole other realm of reality (actually for him it was the 'most real' realm) where abstract objects, like 'redness' and stuff like that were to be found. A lot of people think that math is kind of like the study of objects like this, like numbers are abstract objects or whatever.

So then I really started thinking about the nature of OBJECTS themselves, which sounds super abstract and almost impossible even to really think about, and it is in some ways, but a few people have really spent a lot of time thinking about it already so it is nice to see their work and go on from there. Frege, for instance, had a really rigorous distinction between objects (abstract or otherwise) and concepts (which for him are a specific kind of function). And another very interesting philosopher, an Austrian guy named Alexius Meinong, also developed an entire approach to the subject which he called his Gegenstandstheorie, or theory of objects. Basically what Meinong did was try to see how we can understand everything in the world as a kind of object, so he had a whole crazy hierarchy of different objects from impossible objects to abstract objects to contradictory objects and all kinds of things in between.

Anyway, the more I thought about it, the less reasonable it seemed to think that we could understand what mathematical structures were by using the category of 'objects' as our basic ontological category. For one thing, if they are objects they will be quite strange ones, and we will require the whole extended hierarchy of objects that Meinong proposes. For me this is still a semi-viable option, but it seems really elaborate and ornate, and I am interested in seeing if we can explain things a little better without having to go the Meinong route, though it is very interesting.

So I started reading this strange fellow named Gustav Bergmann. Bergmann wrote an interesting book criticizing Meinong's view because he saw it as a failed attempt to create a NEW (non-object) ontological category. He says the same thing about the earlier Austrian psychologist and philosopher Franz Brentano as well.

Bergmann himself was a German, I believe, who had moved to America because of the Second World War and ended up in Iowa writing books that very few people have read, even today for whatever reason. Anyway, Bergmann's own work tries to develop an ontology in which there is a non-object like category, which he variously calls a 'nexus' or a 'fundamental tie'. I find his work interesting but it is very difficult to read and it is extremely dense. Anyway, reading it made me think more about structures as a unique ontological kind of category, so I started investigating views in the philosophy of mathematics which try to establish 'structures' on an ontological footing.


There are a few different ways of doing this, one offered up by a guy named Stewart Shapiro suggests that structures are just like universals. Universals, if you've never encountered that term are things like 'redness'. So, while there are lots of red things in the world, beyond all these red things, there is also 'redness'. Mathematical structures are like this, sort of: there are a number of things which exhibit certain properties of the natural numbers, but beyond this there is also the natural number structure, for instance. That's what structures are for Shapiro. Now he tries a bunch of different ways of explaining how this works but I still find that his work tends to treat structures as objects; he hasn't really 'solved' the real ontological problem that I found pointed out in Bergmann, he's really just shifted it into a new and perhaps more familiar vocabulary. (The problem of universals is a famous old philosophical problem that goes back at least as far as Plato and Aristotle).

There are other views too, like Geoffrey Hellman's 'modal eliminative' structuralism. His view is called the structuralism without structures view, too, and he kind've treats structures as useful fictions or more like 'possibilities', but there are a lot of different problems for me with his conception of possibility, besides which I feel a strong inclination to defend the 'reality' of abstract structures and abstract objects, so I'm not heavily inclined to eliminate them or think of them as merely ways of speaking.

One view of structures which I'm beginning to really like stems from mathematics itself, as opposed to the philosophy of mathematics, and has developed out of a branch of mathematics called 'category theory'. I'm probably going to talk a lot about this in my dissertation, so maybe it will help if I explain it against the back drop of a (historically) better understood like set theory.


Basically, since around the time of Hilbert, maybe a bit after (the 1910s and 1920s) 'set theory' became a popular way of grounding all mathematics inside of one particular structure. Set theory deals with sets, obviously, and sets are just collections of entities. So, there can be a set of the coins on my desk. And then that set itself is an entity, so you can make another set of the set of coins on my desk and combine it, abstractly of course, with the set of all the hairs on my head, or whatever. Abstract set theory generalizes this and just talks about sets without really caring about what they are 'sets of'.

So when set theory became sophisticated enough, people figured out how to 'reduce' all the various branches of mathematics to set-theoretic terms. So there is a way, for instance, to reduce all the natural numbers and the operations on them (like root extraction, addition, multiplication, etc, etc) to sets and functions on sets. And, even more interesting, we can figure out ways to characterize the functions on sets in terms of sets themselves, so really all we have is a whole mess of sets at the basis of mathematics. This idea was very appealing, for epistemological and other kinds of reasons that I won't really get into, but it also develops its own problems, which is why, partially at least, I still think Frege's criticisms of Hilbert are so valuable. Anyway.

Category theory is a very different way of looking at things. Whereas set theory is almost entirely about 'objects' (namely sets and functions construed as sets), category theory [which developed out of things like group theory and topology beginning around about 1945 but mostly throughtout the 60s and 70s until today] is really about 'morphisms' or 'maps'. Generally, the morphisms are thought to hold between objects, so that there are objects within category theory as well. But, really, the interesting part is that we can always construe the objects of category theory as morphisms themselves: there is no real need for 'objects' in the final, absolute, rigid set theoretic sense. Thus, category theory is the kind of radically abstract, structural theory that Hilbert was sort of fumbling around trying to create with his approach to axiomatics. The difference, I think, is that category theory doesn't have restrictive absolute quality that Hilbert's view still latched onto, and that Frege rightly criticised him for. Hilbert's view was too 'concrete' insofar as it only moved the level of abstraction up one step; category theory, rightly understood, moves the abstraction not just a 'level' up but untethers it from the whole absolute hierarchy of levels entirely, allowing for what I think is a very interesting ontological freedom.

Despite what I think is the super relational, or relative, aspect of category theory, my interest is still in linking its concepts to a more general ontological foundation. The difference between my desire to do so and, say, the set-theoretic or Fregean desire to do so, is that I am interested in providing an ontology that can actually help to explain why the 'map'-centered picture of category theory allows for an ACTUALLY relational account of mathematics, and also why the veiled referential picture of set theory or, even, of Stewart Shapiro, does not allow for this. In other words, I want to develop a broad ontological picture in which the difference between objects and structures is taken seriously from the beginning.


Hm. I guess that's really what I want to do, but I'm really not at all sure that I'm capable of doing so, or really how to go about all the specifics. In any case, I think it is an interesting topic and there is a lot to learn about it. I don't know if you'll bother reading through all of what I wrote, but if you do maybe you can ask me questions about the unclear parts and I can explain them in a better way, because I think it is really worthwhile being able to explain like super abstract mathematical stuff in a way that a non-specialist (which I still am) can understand and appreciate. And I think it is possible to do that too, even if I can't yet do it that well.