Emulating Hilbert
Dennis Lomas has pointed out to me that various translations of Paul Bernays' writings are available on-line. Go to The Bernays Project and click on translations. Bernays is perhaps best known for his collaboration with David Hilbert in their studies of the foundations of mathematics. The paper Die Bedeutung Hilberts für die Philosophie der Mathematik (1922) gives an interesting snapshot of Bernays' views on the significance of Hilbert's work as philosophy, long before the shadow of Gödel fell over the programme. Not only is Hilbert's axiomatic method praised for its importance to mathematics, but at the end of the piece it is promoted as important to physics too, providing the simplest presentation of relativity theory, and pointing Hilbert to a way to unify this theory with electrodynamics, carried further by Weyl.
At the beginning of the article Bernays expresses his delight that mathematical thought had at least regained influence over philosophical speculation. I wonder what he would make of the current situation. The really curious thing is that so few of those philosophers who would want to emulate Hilbert have turned to category theory. Not only is it evidently important for the axiomatic formulation of mathematics, but it is looking very likely that it will play a critical role in whichever reconciliation of general relativity and quantum field theory wins out.
For some of the latest research in category theory, you can take a look at the slides from the CT2006 conference, including one by Makkai, whose radical idea is to remove equality from mathematics (further papers here).
At the beginning of the article Bernays expresses his delight that mathematical thought had at least regained influence over philosophical speculation. I wonder what he would make of the current situation. The really curious thing is that so few of those philosophers who would want to emulate Hilbert have turned to category theory. Not only is it evidently important for the axiomatic formulation of mathematics, but it is looking very likely that it will play a critical role in whichever reconciliation of general relativity and quantum field theory wins out.
For some of the latest research in category theory, you can take a look at the slides from the CT2006 conference, including one by Makkai, whose radical idea is to remove equality from mathematics (further papers here).
8 Comments:
I noticed one of the papers did first order logic via categories, and John Baez has emphasized the replacement of set-oriention with category-orientation in deriving stuff. Which raises in my mind the question, can Goedel be avoided along this route? As I understand it, if you can do without the second order quantifiers, and mathematical induction, then Goedel has no grip on you.
What do you call the elimination of equality a radical idea? Isn't it the very point of category theory and homotopy theory? I agree that it might have been have been radical at one time, but nowadays how could you argue otherwise?
One problem I've had with all approaches n-categories I've seen is that the n-categories are built with lots of structure that you're then not supposed to use because you're not allowed to speak of equality, except perhaps at the top level. For example, in a usual category, you can speak about equality of objects -- it's just that if you do it, people will tell you you're not allowed to. The solution is that you shouldn't even be given that oppotunity of making that mistake.
I can't think of any way of doing this other than throwing away equality from the foundations.
Oops. Pressed publish instead of preview. Please forgive the typos above.
I would be curious to know what other people think of Makkai's approach. I'm not very good at actual work in foundations.
Voevodsky's been going around talking about the "homotopy lambda-calculus"; it sounds similar. Mathematicians already need, or will need soon, a computerized method of checking proofs. ZF is not viable syntax for such a computer program. Voevodsky seems to think homotopy lambda-calculus is.
Let me first say that I'm a big fan of your weblog, just because people on the Internet only express their negative thoughts, not their positive ones.
That said, I think this is as wrong as could be. I think the idea of a set as a list of its members is the conceptual breakthrough that makes all of modern mathematics possible.
dick
Sorry, can't say much about your question. In passing I note that Manin says in Georg Cantor and his heritage:
Baffling discoveries such as Godel’s incompleteness of arithmetics lose some of their mystery once one comes to understand their content as a statement that a certain algebraic structure simply is not finitely generated with respect to the allowed composition laws.
By the way, in this piece he speaks of higher categories as the next stage in the quest for a foundational language:
The following view of mathematical objects is encoded in this hierarchy: there is no equality of mathematical objects, only equivalences. And since an equivalence is also a mathematical object, there is no equality between them, only the next order equivalence etc., ad infinitim. This vision, due initially to Grothendieck, extends the boundaries of classical mathematics, especially algebraic geometry, and exactly in those developments where it interacts with modern theoretical physics.
Anonymous,
If you a sufficiently far along the path to enlightenment, then maybe the elimination of equality no longer seems radical to you. But you are hardly in the majority. Makkai's trying to form a language in which you can't slip into mistakenly using equality. Most people would think that fairly radical.
Walt,
I think the idea of a set as a list of its members is the conceptual breakthrough that makes all of modern mathematics possible.
I don't have a problem with this, except I would probably say 'was' rather than 'is', and, like Manin, take it that mathematics moves on:
I will understand “foundations” neither as the para–philosophical preoccupation with nature, accessibility, and reliability of mathematical truth, nor as a set of normative prescriptions like those advocated by finitists or formalists.
I will use this word in a loose sense as a general term for the historically variable
conglomerate of rules and principles used to organize the already existing and always being created anew body of mathematical knowledge of the relevant epoch.
At times, it becomes codified in the form of an authoritative mathematical text as exemplified by Euclid’s Elements. In another epoch, it is better expressed by nervous self–questioning about the meaning of infinitesimals or the precise relationship between real numbers and points of the Euclidean line, or else, the nature of algorithms. In all cases, foundations in this wide sense is something which is of relevance to a working mathematician, which refers to some basic principles of his/her trade, but which does not constitute the essence of his/her work.
So what area of math do you work in? If you're following the Pro-Am event which is the categorification of Klein geometry, we keep butting into these problems the whole time. If you treat a category as a list of objects and arrows you will go wrong.
2-category theory is a hard, technical subject, one where you have to train yourself to think in a particular way. To make progress, you have to teach yourself new inhibitions. What I don't believe is that we need to make those inhibitions universal. Denying yourself the idea that sets have members, and that you can check two objects for equality will not make real analysis, say, easier. It will make it harder. For certain kinds of problems, that sacrifice might be worth it, but for other kinds of problems, it's a high price to pay.
Walt wrote:
What I don't believe is that we need to make those inhibitions universal. Denying yourself the idea that sets have members, and that you can check two objects for equality will not make real analysis, say, easier. It will make it harder.
Nobody is denying themselves the idea that sets have members, or that equality between elements of sets is a useful concept - not even Makkai.
In Makkai's foundations you can only talk about "equivalence" between objects, morphisms, 2-morphisms, etc. of an infinity-category. But, when your infinity-category happens to be a 0-category, or set, "equivalence" means "equality". When it happens to be a 1-category, or category, "equivalence" means "isomorphism", and so on.
So, we can say everything we'd ever want to say - and nothing we'd never want to say.
It's possible you're still influenced by an older myth, namely that doing topos theory is like studying "sets without elements". Lawvere, a founder of topos theory, has repeatedly pointed out the mistake in this idea - and he's right. An "element" of an object x in a topos is just a morphism
1 -> x
This is a special case of the more useful concept of a "figure" in x, namely
y -> x
This is why in algebraic geometry we define Spec using prime ideals, not just maximal ideals.
But the main point is: advocates of n-category theory aren't trying to take away useful tools from you. It's not a "puritan" movement like finitism or constructivism.
By the way, 2-category theory is no more hard nor technical than any other interesting mathematics. In every interesting branch of math, you have to teach yourself new inhibitions.
For example, as part of mastering integration theory, we have to learn it's against the rules to evaluate an L^1 function at a point! As part of mastering generalized functions (e.g. distributions or hyperfunctions), we have to learn even more scruples - for example, we need to check some conditions before multiplying two of these guys, and sometimes the answer is ambiguous (hence renormalization). To master operators on Hilbert space, we must carefully distinguish between the norm, strong, weak, ultrastrong and ultraweak topologies, and resist the temptation to say a sequence of operators converges to another operator without specifying the topology... people who haven't mastered this find it "technical" and "difficult".
Quite generally, learning how to do things also involves learning how to not do things. We shouldn't wiggle the steering wheel around wildly like a kid when driving a car down the highway. But this lets us do more, not less.
Post a Comment
<< Home