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November 10, 2006

The Tasks of Philosophy

Posted by David Corfield

Kenny Easwaran and I had a brief but interesting exchange, starting here, concerning the detail of mathematical practice into which a philosopher of mathematics should enter. The attitude he reports is quite typical, and there lies the problem which has made my academic career so difficult. The dominant Anglo-American way is to analyse statements from different walks of life, such as:
  • Murder is wrong.
  • Copper conducts electricity.
  • All even numbers greater than 2 are the sum of two primes.
  • Liberal democracy is the best form of government.
  • Nothing beyond the artwork is needed to appreciate it aesthetically.
As regards mathematics, an example such as the third statement will do to represent the whole subject. Now, we need to try to understand what it means for the statement to be true, to include an account of what numbers are, and to understand what it would be to know such a proposition. I, on the other hand, share with Alasdair MacIntyre a conception of philosophy which makes us delve more deeply into different practices. Here is his description of the necessary tasks:

By 1953 I had acquired not only from my Marxist teachers, both in and outside the Communist Party, but also from the writings of R. G. Collingwood, a conception of philosophy as a form of social practice embedded in and reflective upon other forms of social practice. What I did not then fully understand was that philosophy needs to be conceived as having at least a fourfold subject matter and a fourfold task. There is first of all that which has to be learned empirically: the rules and standards, concepts, judgments, and modes of argumentative justification, actually embodied in or presupposed by the modes of activity which constitute the life of the social order in which one is participating. Secondly, there are the dominant ways of understanding or misunderstanding those activities and the relevant rules and standards, concepts, judgments, and modes of argumentative justification. Thirdly, there is the relationship between these two in respect of how far the second is an adequate, and how far an inadequate and distorting representation of the first. And finally there is that of which a philosopher must give an account, if she or he is to vindicate the claim to have been able to transcend whatever limitations may have been imposed by her or his historical and social circumstances, at least to a sufficient extent to represent truly the first three and so to show not just how things appear to be from this or that historical and social point of view, but how things are.

Philosophy thus understood includes, but also extends a good deal beyond, what is taken to be philosophy on a conventional academic view of the disciplines. It is crucial to the whole philosophical enterprise, on any view of it, that its enquiries should be designed to yield a rationally justifiable set of theses concerning such familiar and central philosophical topics as perception and identity, essence and existence, the nature of goods, what is involved in rule-following and the like. But, from the standpoint towards which Marx and Collingwood had directed me, the discovery of such theses was valuable not only for its own sake, but also because it enables us to understand about particular forms of social life what it is that, in some cases, enables those who participate in them to understand their own activities, so that the goods which they pursue are genuine goods, and, in others, generates systematic types of misunderstanding, so that those who participate in them by and large misconceive their good and are frustrated in its achievement. (‘Three perspectives on Marxism: 1953, 1968, 1995’, Ethics and Politics, CUP 2006)

So when I try to understand what is involved in a dispute over whether someone interested in symmetry ought to extend their tools from groups to groupoids, I am attempting to fulfil this conception of philosophy by looking at an allusive example of late twentieth century mathematical argumentative justification. Those in the dominant Anglo-American mainstream can only see it as an unnecessary departure from the study of the “really philosophically interesting” topics of logic and arithmetic.
Posted at November 10, 2006 11:37 AM UTC

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Re: The Tasks of Philosophy

I’m not a philosopher (except to the extent that we are all philosophers, at least temporarily, whenever we consider philosophical questions). But the strange thing for me about this dispute (if it can be called that) between David and Kenny (not that it is just them) is that I find both David’s interests and Kenny’s interests to be philosophically interesting.

Kenny’s interests (if I understand them correctly) are metaphysical (ontological or epistemological), and he’s concerned with how mathematical truth or knowledge fits in the general framework of truth and knowledge. These are interesting questions, and I’m inclined to agree (with Kenny) that they are all already present within elementary number theory (although one would want to keep an eye out for the possibility that some aren’t).

In contrast, David’s interests (again if I understand them correctly) are social as well as epistemological, and he’s concerned with how mathematics works as a scientific discipline, including not only how we come to know mathematics but also how we come to know this mathematics rather than that and how mathematicians organise that knowledge. These are also interesting questions, and I’m inclined to agree (with David) we must go beyond elementary number theory (and in particular, must look at category theory) to understand them.

Thus, I conclude that we should all get along and respect each other’s interests. So perhaps the only problem appears in this fragment that I quote from David:

[…] which has made my academic career so difficult […]

If Kenny’s partisans control academia, then David’s are going to have trouble! I don’t know the facts of the case, whether it’s true that academic intolerance of David’s questions exists (although I can say that the one course that I took in college on philosophy of mathematics was concerned solely with Kenny’s questions). But if it is true, then I’ll side with David in his quest for academic respect!

Posted by: Toby Bartels on November 10, 2006 4:47 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

It’s not so much a dispute between me and Kenny. You’ll note he wants to be able dig more deeply into mathematical practice as a naturalist. (This term means many things. I’ll leave him to explain his meaning.) On the other hand, he does admit that this extra work is not something other philosophers need directly care about. This I disagree with for two reasons:

1) Many of the most important moments in the history of Western philosophy owe something to encounters with the frontiers of mathematical research: Aristotle and the founding of geometry; Descartes and analytic geometry; Leibniz and the calculus; Frege’s encounter with Riemann’s mathematics, to give a few obvious examples. If this trend is to continue I see no reason that we should expect it to be due to future encounters with ‘foundational’ work which fails to make serious contact with the forefront of mainstream mathematics. If, say, as all of us on this blog hope, n-categories transform mathematics, physics and computer science, wouldn’t this be enough to justify philosophical attention? Already it’s challenging our notions of ‘identity’ and ‘sameness’.

2) You mention my interests as ‘social and epistemological’. Either I’d want to see them as inseparable, or I’d go with MacIntyre and drop the term ‘epistemology’. Either way, I’m with him whole-heartedly in believing a theory of knowledge, call it what you will, must be thoroughly social. Just as his moral philosophy was transformed by reading the engaged historically-oriented philosophy of science of the 1970s, I would hope for an extremely useful cross-fertilisation between those studying the fascinating practice that is mathematics, and those studying other practices, scientific, artistic, political, or whatever else.

Kenny is still at the PhD stage, and his research may pan out in any number of directions, so one shouldn’t call them his partisans. It is quite clear that a strong grip is maintained by people whose agenda is not mine, as you can see if you follow up some of the reviews of my book, e.g., the Bays, Melia and Page reviews. Philosophers of science tend to be much more sympathetic.

Posted by: David Corfield on November 10, 2006 5:28 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Either way, I’m with him whole-heartedly in believing a theory of knowledge, call it what you will, must be thoroughly social.

Just to be clear, is your position that all truth is socially constructed, as, for example, many/most Marxists and deconstructionists would claim?

If that is so, would there be any point in studying the actual content of mathematical theories? If truth is irreducibly social, to understand its nature would only require understanding the process by which mathematicians convince each other their theories make sense and collectively arrive at “truth”;
there couldn’t be any objective value of the theories themselves independent of such a context.

This approach might lead to very interesting philosophical insight, however, I personally would find it incomplete.

Posted by: Marc Hamann on November 10, 2006 7:57 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Just to be clear, is your position that all truth is socially constructed, as, for example, many/most Marxists and deconstructionists would claim?

I’m surprised you include Marxists here. Although I’m not all that fluent in Marxist, I’ve never heard anyone self-identified as one advocate the idea that truth is a social construct. In fact, by claiming that Marx’s theories form an accurate picture of objectively existing economic and social reality, they seem to be directly contradicting that view.

That aside, and I’ll leave David to describe his own position, I can imagine many ways of reading his statement that don’t imply your suggested interpretation. To talk about a theory of knowledge - as opposed to, say, a theory of mathematics, or of evolutionary biology - involves bringing human cognition into the picture. A theory of knowledge ought to account for things like learning, and the development of new knowledge and ideas.

The processes of learning, at least in cases where we learn from other people, are going to involve social interactions that aren’t exclusively limited to the content of the knowledge being learned, and that affects how we acquire knowledge. The development of ideas has a dynamic that involves personal rivalries, the existence of widely held assumptions that are hard to challenge, etc. etc. Some ideas about matters of fact (whatever that means - which is what such a theory has to account for) have politically controversial overtones - evolution, climate dynamics, and others. You can probably name other examples. A theory of knowledge ought to account for how different people can look at the same data, experiments, observations, etc. and reach different conclusions, in part depending on their predispositions, or social pressures to believe one thing or another.

None of which is to suggest that the truth of the matter is socially constructed, or a subjective-idealist stance that the external world doesn’t exist, or that the content of the knowledge isn’t relevant… only that it’s not the exclusively relevant fact to studying how we come to know things, and even what that means. Presumably, whatever we conclude about the meaning of “truth”, it ought to be the case that some claim being true will help its social acceptance (at least under some conditions), but to say how requires some grappling with how social dynamics work as well as considering the content of the idea, and how its truth is evident (is it empirically testable? supportable by logical argument? what are the prevailing attitudes to those kinds of evidence?)

Whatever “facts” may be, expressing them in words or concepts, if only to communicate them, has to use constructs like words, which evolve through some partly social process. If the “truth” of the concepts behind those words were the only relevant idea, we wouldn’t have any difficulty in translating ideas between languages, right? But that implies that the relation between “facts” and representations of facts in language, or even concepts, is at best indirect. It seems to me David is saying that a theory of knowledge which abstracts all this away is thereby an incomplete theory, because it implies a simplistic view of how direct is the relation between external “facts” and knowledge of them. If so, I agree with that.

Posted by: Jeff Morton on November 11, 2006 6:14 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

Excellent, Jeff. You’re not thinking of moving into philosophy, are you?

Marc, I hold a non-relativised truth to be the aim of every intellectual enquiry. Our only path to this goal is through overcoming challenges made by others about the partiality of our viewpoint. In mathematics this goal is certainly not the accumulation of all true mathematical propositions, but rather relates to the relationship between the mind and its objects. That a mathematician has a better understanding of an area than another does not reduce to the true statements she or he knows, but includes also knowing how to use them, and more importantly knowing in which directions to proceed to improve understanding.

You’ll see in the virtual reality discussion, I seem to be more of a realist than the mathematicians, certainly d and perhaps also John.

Posted by: David Corfield on November 11, 2006 9:36 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

I hold a non-relativised truth to be the aim of every intellectual enquiry.

To be honest, having perused your publication list, etc. I was inclined to believe this was so. Nothing like an exteme misintrepetation to flush clarity out of the bushes though… ;-)

As I said in my response to Jeff above, given that we accept that knowledge is socially-mediated, and some truths socially-constructed, how do we winnow out the objective truths so that we can identify and evaluate the objective claims about those truths?

Without such an approach, I don’t see that much headway could be made finding such a non-relativised truth.

Posted by: Marc Hamann on November 11, 2006 6:20 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

I’d rather speak about truth rather than truths with its suggestion that we’re only after true statements.

How do we make headway towards non-relativised truth? Don’t we think this has happened in science and mathematics? Don’t we think our understanding of, say, electrons or 3-manifolds is better today than it was a century ago? And not just better from our perspective, but better from the viewpoint of any more complete understanding. We may be wrong. The loss of understanding occurs, as Rota once said when looking back at 19th century mathematical work. But any claim to the effect that earlier understanding was better than ours is meant in a non-relativised way. It too could be wrong, but non-relativised truth is assumed.

How we can encourage more rapid headway is to engage with other existing viewpoints, both those of our contemporaries and those of our predecessors, hence the importance of history. Learning to see your own position from the perspective of another in order to overcome your own partialities is not at all easy. I describe this process beginning at p.14 of this.

Posted by: David Corfield on November 12, 2006 9:40 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

In fact, by claiming that Marx’s theories form an accurate picture of objectively existing economic and social reality, they seem to be directly contradicting that view.

I am in fact inserting my own analysis there: the way they accomplish this “objective” truth is by declaring one socially-defined point of view, that of the communist proletariat, to be the “correct” one. All the “bad” points of view are merely socially contstructed to serve “bad” interests. Once you stop privileging one “right” social construct, you end up with modern deconstructionist relativism.

It seems to me David is saying that a theory of knowledge which abstracts all this away is thereby an incomplete theory, because it implies a simplistic view of how direct is the relation between external “facts” and knowledge of them. If so, I agree with that.

I would also agree with this position, but I think in order to proceed with this line of inquiry, we need to be able to tease apart these different modes of knowledge perception.

For example, in some of the other threads here at n-Category Café, the notion has been discussed that Category Theory provides a better foundation to mathematics than Set Theory (and its super-structures). There are two immediate ways to conceive of this question: one is as a purely socially-constructed claim that one community of mathematicians has a better language with which to discuss mathematical ideas and, by extention, has a better view of the truths of mathematics as a result.

(This reminds me of discussions I’ve heard at various times about French, German, English, Classical Chinese, Sanskrit, etc., being a more “logical” language than others, and thereby being “better” for discussing logical, philosophical, mathematical, etc ideas)

The other immediate approach is to try to assess the objective merits of each theory. The challenge here is to find some neutral third “language” or model that describes the range of “real” mathematical structures and then to compare the power of each foundational school to describe that range.

Using my own (admittedly idiosyncratic) internal faculty of “raw mathematical ideas” as such a “neutral” language, my own conclusion (already expressed) is that sets and categories are both describing the same underlying ideas with different languages and different emphases. Though for some particular practical end, one may be more handy or more suitable, it is absurd to claim that one is “better” overall to the point of obviating the other.

If we are going to be able to discuss and investigate such notions, we will need to have some methodology by which we can tease apart the competing modes of knowledge so that we can assess what is actually meant by a particular claim, and whether such a claim is even falsifiable within its own social context.

Posted by: Marc Hamann on November 11, 2006 6:04 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

I’d rather not march into those great Serbonian bogs of discussing Marxism, comparing the merits of “foundations”, or claims about languages. I do want to say that I’m not too clear what the problem is with concepts of truth being “relativized”, or what level of expectation anyone here has of finding a “non-relative” truth. Part of the point of Einsteinian relativity is that one doesn’t do away with the external world by saying that measurements of it are to be understood as relative to a particular observer. When there are discoverable relationships between the observations of different observers, one can distinguish between “relative” and “subjective”.

Carlo Rovelli has some papers about the “relational interpretation” of Quantum Mechanics where he gives nice exposition of the idea that “facts” should be understood as residing in the relation between two systems, one of which we customarily call the “observer”. (At least, this is for all facts about the physical world - and pending evidence the nature of which I’m unable to imagine, I’m agnostic about the existence of another). To me, this makes sense, and seems to resolve all sorts of problems - but the philosophical attitude behind it does seem to undermine the expectation of being able to find, given two different observations, or descriptions, or whatever, a really “neutral” third language.

I’d expect to be able to find a third viewpoint which can get information from, and find relations between, both of two apparently conflicting viewpoints. But a fourth viewpoint might also be able to do this. Then one has to compare the third and fourth - and so on, ad nauseam. I don’t mind speculating that this process might have some limit that deserves the name “absolute”, but I wouldn’t care to assume it. None of this is implying that the “facts” or conceptual representations each viewpoint embodies is purely subjective.

I guess I just feel like there’s a lot of conflation of the duality of subjective/objective and that of relational/absolute, or at least I don’t understand which one is being talked about at all times. Is there some reason to conflate them I’m not thinking of, or am I just misinterpreting here?

Posted by: Jeff Morton on November 12, 2006 1:53 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

I guess I just feel like there’s a lot of conflation of the duality of subjective/objective and that of relational/absolute, or at least I don’t understand which one is being talked about at all times. Is there some reason to conflate them I’m not thinking of, or am I just misinterpreting here?

Basically, this is just the slippery world of epistemology, where how you know and the thing that is known are notoriously difficult to tease apart.

But for the sake of our current discussion ( and our sanity ;-) ) there really are only three types of knowledge/truth we need to worry about:

  • purely subjective preferences, e.g. I like studying Category Theory
  • objective facts, e.g. if you assume the axioms of CT, the Yoneda Lemma follows
  • strong beliefs held by particular communities of mathematicians, e.g. Category Theory is the cutting edge, offering original, new insights into mathematics

The problem is that these three categories aren’t as nicely separated in real life as we’d like: what category you put each fact in is itself conditioned by other facts in each category. One way to describe the enterprise of philosophy is the setting up of principles and frameworks of thought that can bring order to this spaghetti.

Posted by: Marc Hamann on November 12, 2006 4:34 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

It’s a fascinating subject the extent to which our physical theories describe our, or any possible, capacity to know or effect the world, rather than describe the world as such. E.g., Caves and Fuchs see the quantum state as a state of possible information, rather than an independently existing thing. Still, I think one must be careful not to conflate the relativity of an Einstein or relationalism of Rovelli with relativism. Einstein didn’t propose the general theory of relativity in a relativistic spirit. It is quite possible that an improved understanding of physics, improved in a non-relativist sense, will involve taking Rovelli’s step.

Posted by: David Corfield on November 12, 2006 10:15 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

I guess thinking hard about the distinction between a theory purportedly describing any possible knowledge of the world, and one describing the world as such - assuming there is such a distinction - doesn’t hold my interest. I don’t understand what a fact is if it’s intrinsically, as opposed to pragmatically, unknowable.

Apparently I also don’t understand this use of the term “relativism”, if it doesn’t denote the view that attributes of the world have meaning in relation to particular observers.

Posted by: Jeff Morton on November 12, 2006 10:30 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Hasn’t Einstein taught us to be able to compare statements such as “My observation of the length of that rod is x” with other people’s observations? You don’t say “Well that’s just the way you measure things. From my perspective it has length y.”

Usually you think of relativism as dealing with conceptual frameworks. E.g., a moral relativist might say that “Killing babies is wrong” is not a universal truth, but only relative to the moral code of particular communities. To be a full blown relativist with regard to Einstein’s GTR, one might say that it’s no more correct for Western scientists to hold GTR to be true, then some pre-industrial society’s holding a theory of the cyclical nature of time.

You may find amusing Sokal’s criticism (quarter of the way down) of Latour’s use of GTR for his own relativist purposes.

Posted by: David Corfield on November 13, 2006 8:26 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

Okay - I guess I was right that I’m not up on the terminology here. I would have used a word like “subjectivism” for a position that says different observers have their own facts and they can’t be related in any coherent way. Actually, I find it difficult to understand what that position even is. If the hypothesis is that they can’t compare and correlate their perceptions, I fail to imagine the two observers exchanging signals to discover the disagreement in the first place. Or is it just a semantic quibble about the strength of the meaning of “true” or something?

Moral relativism seems very different, because it involves normative, rather than empirical, statements. The issue seems to arise exactly when you turn one kind into another. Replacing “normative” with “imperative” to make the analogy, I imagine this as something like the transition from “GO!” and “STOP!” to “The traffic light says GO, but the pedestrian signal says STOP.” Is this issue even much similar to, I guess I should say, “relational” issues for empirical statements?

Posted by: Jeff Morton on November 13, 2006 11:02 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

Incommensurability in science is an interesting topic. One extreme position would have us say that the Newtonian and Einstein each uses the term ‘mass’ in such a different way that communication between them is impossible. This is surely going too far. Both are in the game of trying to work out how things move. On the other hand, I can happily accept that two scientists may have sufficiently diverse beliefs that even if they can agree on some basic statements, e.g, what instruments display, this doesn’t extend to an agreement of what is being observed theoretically. Where the Galilean and Aristotelian each saw a heavy mass hanging by a chain from the ceiling, one saw it as a pendulum, the other as a body prevented from reaching its natural resting place. Similarly, today astronomical observations may be interpreted differently.

Fortunately science generally has the resources to resolves such differences eventually, which certainly doesn’t involve deciding from a third neutral standpoint, but rather depends on how each theory does in explaining the rival’s problems, whether it branches out into unexpected new areas, etc.

Moral matters are a different issue. We seem not to have the resources to resolve them. MacIntyre has an interesting twist on this. The thing to note, for him, is that the irresolvability of moral questions in the modern liberal state coincides with what are the most flexible languages we have ever seen. Given enough time, modern English, say, seems perfectly adequate to represent the belief systems of very diverse societies, as anthropological reports suggest. But this modern English cannot play an arbitrational role for this very reason. So there’s a spectrum running from a very rigid language which can make no sense of other systems to a modern language flexible enough to understand everything. For MacIntyre, neither of these extremes is conducive to progress.

The further idea is that some languages have been situated closer to the middle of the range, in some sense more like the languages of the sciences, and that progress was possible then.

For a brief sketch of this position, see here, where I play with the idea that set theory is like the modern language in being able to represent everything, without being able to say what’s important.

Posted by: David Corfield on November 13, 2006 11:50 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

I follow MacIntyre in holding that there is no third position, no neutral standpoint. One solution commonly adopted is to ignore the other position and ‘let history decide’. But I prefer the path of open engagement. If, after perhaps many years, any resolution is possible, and its occurrence is by no means certain, it may involve what you would largely describe as the overcoming of one tradition by another (Galileo over Aristotle), or the formation of a composite integrating aspects of both (for MacIntyre, Aquinas integrating Augustinianism with Aristotelianism).

What is worse is for one movement to overtake another without proper engagement, and important insights of the overtaken thereby be overlooked. Rota was keen to flag up examples of this phenomenon.

As I said elsewhere, taking on the role of the external critic of categorification got me started on this Klein 2-geometry business. If we fail to turn anything up here, where of all places it ought to work, my confidence will be severely dented.

Posted by: David Corfield on November 12, 2006 10:32 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

It was once said:

Klein geometry: good. Categorification: good. Groups naturally categorify to 2-groups, with lots of nice examples: good. Klein geometry is about differing degrees of structure, and categorifying structure gives stuff: good. So how could we possibly not do well by categorifying Klein geometry? It would be like failing to find sand in the Sahara!

So if we fail to find anything, either we’re extremely incompetent, or we’ve been sidetracked by a lush oasis, or the sand was just a mirage. The latter would be the most troubling.

Posted by: David Corfield on November 12, 2006 1:45 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

David wrote:

So if we fail to find anything, either we’re extremely incompetent, or we’ve been sidetracked by a lush oasis, or the sand was just a mirage.

Or, we didn’t get there yet! I’ve been very remiss on this Klein 2-geometry project, but when I last checked, we hadn’t even properly categorified the theory of incidence relations, much less seen what it gives in a bunch of examples.

Math is not quite so easy as you seem to think. We’ve only been at this since May, and since the summer I haven’t done much work at all on this project. Throughout the whole course of the project, I’ve spent very few solid days doing calculations.

By comparison, I started working on higher gauge theory around 2000. I worked on it very hard, I had two students who wrote their theses on it, I had three more collaborators who helped me write papers on it, but only in 2004 did results come in that really confirmed it was a good idea - the fake flatness condition for 2-holonomies, and the construction of the String group from a 2-group.

So, I’d be sort of shocked if we’d gotten anywhere interesting on Klein 2-geometry this quick, with so little work.

One problem, of course, is that both of us are involved in lots of other projects. If we were locked in a cell and only released when we showed the jailer nice theorems about Klein 2-geometry, we could probably get out in a few months.

Wanna try it?

Posted by: John Baez on November 13, 2006 12:54 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

My “If we fail to find anything” was written implicitly in the future perfect tense “If we shall have failed”. No time limit was specified, so we have the ambiguity of what I take to be a reasonable point at which to declare failure. We’re certainly nowhere near that point!

Posted by: David Corfield on November 13, 2006 8:46 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

Okay, so I won’t call the Riverside jail and reserve a room for two just yet.

I misread you because I’m feeling guilty for not working harder on this Klein 2-geometry project. I’m just way too busy right now.

Posted by: John Baez on November 13, 2006 7:03 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

One solution commonly adopted is to ignore the other position and ‘let history decide’. But I prefer the path of open engagement.

History certainly provides a “third language”, but it may or may not be neutral.;-)

I enthusiastically agree that respectful engagement is really the only path to getting at shared truth; “third languages” necessarily take the form of shared dialogues.

I guess that is why I’m so wary of CT triumphalism (or any other kind of triumphalism): it’s an impediment to the process of truth.

Posted by: Marc Hamann on November 12, 2006 5:43 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

I am in fact inserting my own analysis there: the way they accomplish this “objective” truth is by declaring one socially-defined point of view, that of the communist proletariat, to be the “correct” one. All the “bad” points of view are merely socially contstructed to serve “bad” interests. Once you stop privileging one “right” social construct, you end up with modern deconstructionist relativism.

Wouldn’t this apply to adherents of any theory that thought they had it right, but sought to explain away all the other stances out there?

If we’re seeking the truth, once we have it, shouldn’t the false theories actually be explained as socially constructed truths?

Posted by: Aaron Denney on November 13, 2006 8:10 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

We should certainly want to explain why we take rival theories to be partial. Ideally, we’d be able to explain how they did as well as they did, and why they met with problems, and how they were not going to be able overcome these problems. If you’re implying by “socially constructed”, that we’ve also got to give an account of how adherents to the other theory were irrational, driven perhaps by social interests to avoid taking our step to the truth, then I don’t think this is necessary. Maybe they just backed the wrong horse. Maybe our own successful choice owed something to political interests. Marxists found in Darwin’s writings British Imperialism written into his picture of nature. Even if he had relied on Imperialist imagery, it wouldn’t stop his theories being a step towards the truth.

Posted by: David Corfield on November 13, 2006 8:37 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

If we’re seeking the truth, once we have it, shouldn’t the false theories actually be explained as socially constructed truths?

How do you know you have the truth and not just a new socially constructed version that you just happen to be comfortable with?

The other possibility is also that the other theory wasn’t “false” per se, but had a different partial truth, or had completely different goals than the ones of the new “true” one.

The danger of assuming that whenever another group disagrees with you that they are victims of a socially contructed truth (by implication not a “real” one) is that you become blind to your own socially constructed truths, or the socially constructed applications of more “objective” truths, e.g. taking political positions based on scientific theories.

The safe course is to have a respectful awareness that there are various “truths” serving various purposes, but without losing sight of the pursuit of foundational philosophical or scientific truths.

Posted by: Marc Hamann on November 13, 2006 3:09 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Some important discussions in mathematics seem to involve epistemological issues directly, e.g., discussion of the use of computation, of diagrams, and of very long proofs.

Dennis

Posted by: Dennis Lomas on November 11, 2006 5:40 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Thank you for the discussion, and the clarification of some potential misconceptions in other comments!

As for the point about whether other philosophers should directly care about - I definitely agree that many important moments in the history of western philosophy owe something to contact with cutting-edge mathematical research (presumably as well as contact with cutting-edge research in psychology, linguistics, physics, and various other disciplines at other times). However, I don’t think that this means that this sort of contact with mathematics is something all philosophers should always pay attention to. Understanding of the notion of identity that one gets from category theory (or from quantum mechanics) may end up being very important for other philosophers to notice, but the same is also true about potential breakthroughs elsewhere, say in studies of context-sensitivity in semantics, or reliabilism in epistemology.

It’s clear why other philosophers should care about notions of logic and basic arithmetic, and the possibility of knowledge of abstract objects. Maybe there’s reason for them to care about higher category theory, but I don’t think this has been made clear yet.

Posted by: Kenny Easwaran on November 14, 2006 7:11 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Kenny wrote, to David, who had responded to me:

Thank you [David] for the discussion, and the clarification of some potential misconceptions in other [my] comments!

I apologise for treating you as the individual representative of a political force in academia that you may not entirely agree with.

Posted by: Toby Bartels on November 14, 2006 10:28 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Have we reached the stage where all philosophers of mathematics ought to care about it? If people like Shapiro tell us that mathematicians only care about entities up to isomorphism, and I can show that he’s wrong, have I done enough?

He’s wrong because different entities have different sameness conditions. So if you say that a mathematician only cares about a category up to isomorphism, this is plainly and simply wrong. A mathematician only cares about a category up to equivalence. Two categories will be considered the same if there are a pair of functors between them, such that when you compose them, either way around, the resulting functor is naturally equivalent to the identity functor. If isomorphism were at stake, we’d need the composites to be equal to identity functors.


You can see this by asking a mathematician a question such as “In your category of finite sets how many copies of the terminal object (i.e., a singleton) do you have?”. In our discussions we’ve been calling this kind of consideration ‘evil’. Anything you can do with a category of finite sets can be done with a category of sets with one object per cardinality.

Would this be enough for you, Kenny?

Posted by: David Corfield on November 15, 2006 9:02 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

If people like Shapiro tell us that mathematicians only care about entities up to isomorphism, and I can show that he’s wrong, have I done enough?

This is probably just a reference to something that I haven’t seen, but it strikes me as unfair. It’s an understandable historical anomaly that, as category theory first clarified the notion of isomorphism, that notion was applied incorrectly to category theory itself. Thus what were (and still are) called “isomorphisms” of categories are not the proper notion of equivalence of categories, leaving us to use the blander term “equivalence” for that. But if things had gone differently, we might well have used “isomorphism” for equivalence of categories all along. (Of course, if Shapiro themself misapplied the notion of isomorphism to category theory, then that would be a fair criticism.)

So Shapiro’s claim should properly be that mathematicians only care about entities up to equivalence, something that one might argue against, but not with a terminological quibble. The only problem with that formulation is that it doesn’t use the punchy buzzword “isomorphism”. I could forgive Shapiro for using the wrong word (either deliberately or out of ignorance), especially when speaking to nonmathematicians.

(We could have gone on to higher words like “2-equivalence”, “n-equivalence”, and “ω-equivalence”, which would make a nice buzzword. In fact, people do use the term “biequivalence” for equivalence of bicategories, but I think that this is only used by people that stop at bicategories; serious n-category theorists can use the simple term “equivalence” for all of these.)

Actually, saying that one cares about things only up to equivalence is almost tautological. The real insight of category theory (or at least groupoid theory) is that equivalence of equivalences is not trivial (and when applied recursively, this is the real insight of ω-groupoid theory = homotopy theory). If Shapiro misunderstood that point, then criticism is definitely warranted!

Posted by: Toby Bartels on November 16, 2006 10:29 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

I don’t have Shapiro’s books to hand, but I am morally certain that his definition of isomorphism is too strict to include the equivalence of categories, in that it requires cardinality of the collection of places in a structure to be preserved. Can anyone confirm this?

In any case, what you go on to say surely provides enough support for the proposition that higher category theory is of philosophical interest.

Posted by: David Corfield on November 17, 2006 8:52 AM | Permalink | Reply to this

Re: The Tasks of Philosophy

I don’t have Shapiro’s books to hand, but I am morally certain that his definition of isomorphism is too strict to include the equivalence of categories, in that it requires cardinality of the collection of places in a structure to be preserved. Can anyone confirm this?

Unfortunately you caught me right after I left New Haven for a week and change. My copy of his book is about 600 miles north right now .

Posted by: John Armstrong on November 17, 2006 1:09 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

I don’t have Shapiro’s books to hand, but I am morally certain that his definition of isomorphism is too strict to include the equivalence of categories, in that it requires cardinality of the collection of places in a structure to be preserved.

I don’t see how this even applies to categories in general! To speak of an object’s cardinality, you would need a concrete category, that is a category equipped with a functor to the category of sets. Actually, people usually require a concrete category to be equipped with a faithful functor to the category of sets; in that case, the statement is true (isomorphic objects will have equal cardinalities).

It sounds like Shapiro is considering only certain categories (in particular, only concrete ones in the strict sense). Then his statement would be OK, but it would be a mistake to apply it to categories themselves, because the 2category of categories in not concrete. (One sometimes sees a category of categories, or at least of small categories, defined so as to give the usual notion of ‘isomorphism’; this is concrete, but it is just the wrong context to put categories into. Shapiro would not be the first to make this mistake!)

Since I don’t have Shapiro to hand either (and, in all probability, have never read him in the first place), take my comment only as a defence of what Shapiro should have meant. Or better, just as a defence of the slogan ‘Mathematicians care about objects only up to isomorphism.’, including an explanation of what that slogan should mean (equivalence in the general ωcategorial sense).

Posted by: Toby Bartels on November 17, 2006 10:29 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

He’s wrong because different entities have different sameness conditions.

I think this hides some of the complexity behind the notion of “sameness” that CT has brought to light.

The notion of “same” is highly dependent on what kinds of structure (and structure preserving mappings) you are considering.

Consider oranges. If I’m hungry, and I want to eat an orange, and what I care about (and thus choose to model mathematically) is that I can find an orange that is good to eat, all isomorphic oranges are the “same” in the category of oranges with morphisms the “good to eat”-preserving mappings. No one would rationally argue about the identity of one or the other in that context.

On the other hand, if I take the same two oranges and send one into space to study the effects of cosmic rays on oranges, and keep the other on earth as a control, I’m now working in the category of “space-travel”-preserving mappings, where an otherwise identical orange that had been in space might well do just as well.

Most of the time when people insist on identity as a non-trivial property they are in fact smuggling an external mapping into a different system. From this point of view, it is perfectly reasonable to say that, when studying particular structural propreties, all you care about is isomorphically preserved instances of that structure: any more demanding notion of “identity” is either a trivial property of the system (e.g. identity arrows in a category) or an additional mapping in from some other system of interest that is also only good up to isomorphism.

Posted by: Marc Hamann on November 17, 2006 8:05 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Many of the most important moments in the history of Western philosophy owe something to encounters with the frontiers of mathematical research: […]. […] Already [research into higher category theory]’s challenging our notions of ‘identity’ and ‘sameness’.

This reminds me that there have been other reasons historically for philosophers to be interested in mathematics (or at least one other reason): applications of mathematics to philosophy. For example, Russell applied ideas from mathematical logic (including the theory of types in his foundation for mathematics) to clarify the use of logic in analytical philosophy. There are other examples, I’m sure. This too should not stop with hundred-year-old mathematics, as your example about the nature of identity shows.

Posted by: Toby Bartels on November 14, 2006 10:38 PM | Permalink | Reply to this

Re: The Tasks of Philosophy

Russell applied ideas from mathematical logic (including the theory of types in his foundation for mathematics) to clarify the use of logic in analytical philosophy.

In case readers take this to mean both enterprises were well established already, I’d like to flag up the fact that not only was Russell a founding father of analytical (now analytic) philosophy, he also contributed importantly to the development of mathematical logic. The two went hand in hand.

Anyone who thinks (higher) category theory could play a similar role might try to import its constructions into the ongoing analytic enterprise. But I’m with Michael Friedman in wanting something more. As I discussed here (and at greater length in the linked paper), one might also hope to alter the direction of philosophy more profoundly.

I could even argue that it must happen this way. Kenny said:

Maybe there’s reason for them to care about higher category theory, but I don’t think this has been made clear yet.

But you could run this line for Russell with regard to arithmetic and the new (at the time) mathematical logic. Imagine the idealists of the day saying about Russell:

Maybe there’s reason for us to care about arithmetic and Fregean logic, but I don’t think this has been made clear yet.

As far as their concerns went, Russell might well have seemed completely irrelevant. It’s easy to forget that Russell’s new style of philosophy wasn’t accepted overnight. Collingwood still thought it a mistake in the 1930s.

Posted by: David Corfield on November 15, 2006 8:31 AM | Permalink | Reply to this
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