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Friday, July 14, 2006

Conceptual essentialism

John Baez recently added a comment to this post, which is too old now for comments to appear in 'recent comments'. I had remarked that something he had said earlier sounded like it came straight from the Jaffe-Quinn debate. For those of you who don't remember it, these two mathematical physicists launched a passionate attack on slipping standards in mathematics, brought about by an imitation of the sloppier ways of physicists. Many very interesting responses were made, not least William Thurston's wonderful On proof and progress in mathematics.

Anyway, John replied:

I hope it's clear that I'm not complaining about the lack of rigor. I'm complaining about a swarm of people writing hundreds of short papers on the same subject in a short time, each referring to many of the previous ones, nobody taking the time to distill the matter to its essence. Even if all the papers contained nothing but rigorous theorems, I would still find this annoying. It's fine if you wish to devote yourself to one specialized subject, rapidly master the literature, and compete with the crowd to extract some big nuggets before this vein of ore looks exhausted and it's time to move on. I'm sure this is fun for people with a competitive streak. But there are other people who like to slowly mull over one topic and nurse it to perfection - or like me, mull over lots of topics and gradually form a web of connections until something interesting emerges. And, you know, it's just possible that some of the people in that Jaffe-Quinn dispute were secretly annoyed about the fast-paced "swarming" style of theoretical physics more than any lack of rigor. I forget if any of them came out and said this.
I think the phrase 'nobody taking the time to distill the matter to its essence' is the key one here. Remember, two posts ago we had Borovik saying "The work of three generations of mathematicians confirmed that matroids, indeed, capture the essence of linear dependence" (my emphasis).

I've done my damnedest to get the idea of mathematical activity at its highest level aiming to extract the essence of a situation to be the principal topic of philosophy of mathematics, but with little success. It's not that I'm the only philosopher thinking about such things. For instance, Kenny Easwaran posted Do Mathematical Concepts Have Essences? on his blog, where you can follow up the reference to a paper I wrote on the subject. But it never stays on the agenda for long.

I think what is needed is a name. Essentialism is overused. Conceptualism is also already taken. It concerns the kind of problem faced when wondering what the tallness is shared by, say, a 2 metre man, a 30 metre tree, and a 300 metre building. As this paper explains:

Conceptualism, along with nominalism and realism, is one of three traditional families of views about universals. There are many species of each family, but the story line goes like this. Realists hold that there are universal properties and that these solve the problems of universals. Conceptualists deny this, arguing that concepts can do most of the work realists invoke properties to do. And nominalists, at least traditional ones, spurn both universals and concepts, arguing that words alone can do all the legitimate aspects of this work.
Blending the two, conceptual essentialism has been used in philosophy of science to designate a similar position. But is it snappy enough? How much of Kuhn's success was down to his choice of the word revolution?


Clark Goble said...

Isn't one big issue the issue of community inquiry and refinement versus individual inquiry and refinement. Yes physics has its Einsteins. But the more common way it develops is Quantum Mechanics. It sounds like mathematicans prefer the Einsteins to the community in QM. I understand this. Math has that odd connection to what one might almost call poetry. Probably a remnant of all that Platonism among mathematician of the past. (grin)

Honestly though it seems hard to put one above the other. But I can certainly understand why someone with an aesthetic preference for one above the other would be upset. But I do think the "hive mind" so common among physicists can reach essences. Although perhaps the current situation in theoretical physics doesn't inspire confidence.

July 15, 2006 6:41 AM  
david said...

It's not the communal vs individual dimension that John's worrying about. He himself has collaborated on many papers, frequently works out ideas with James Dolan, and has played a major role in promoting the higher-dimensional algebra program to the rest of the world. Indeed, he's a living embodiment of the kind of mathematician I present as Aristotelian-Thomist in my paper as one who recognises that mathematics is a tradition of enquiry, after Alasdair MacIntyre.

What's bothering him is the kind of work that doesn't care to get to the bottom of things. That a certain effect of fashionability is to encourage this more superficial kind of work does not mean that communal co-operation necessarily leads one away from depth. On the other hand, working alone is unlikely to promote clarity, whereas a willingness to share ideas is.

July 15, 2006 1:20 PM  
david said...

I should add, this getting to the bottom of things is not at all the same as the production of rigorous proofs. What tends to happen, though, is that when you do get to the bottom of things matters have become simple enough that proofs become a whole lot easier to produce.

July 15, 2006 1:29 PM  
John Baez said...

I'm not sure philosophy needs another "-ism". The great thing about Kuhn's idea of a "revolution", as far as salesmanship goes, is that it's not a full-fledged "-ism" complete with tenets, doctrines and splinter movements. It's just a catchy concept!

Personally I think "the philosophy of real mathematics" is also a catchy concept.

Clarke Goble mentioned the "hive mind" of physicists. Certainly this can and does reach essences, as the tremendous success of physics shows. Mathematicians also have a "hive mind", with even plodding drones able to produce proofs that add usefully to the store of knowledge. When I spoke of "swarming" behavior, I was referring to something less like the collective industry of the honey bee, and more like the "eat everything that's easy to get and then move on" strategy of the locust. It's this that I don't like.

July 16, 2006 5:48 AM  
Clark Goble said...

So your point is more about the quest for low hanging fruit?

July 16, 2006 6:05 AM  
Dennis said...

Perhaps the notion of meaningfulness is appropriate here.

Husserl lamented the distancing of science from meaning, in favour of formula manipulation.

Something like this may be what John Baez is getting at.
D. Lomas

July 16, 2006 5:50 PM  
david said...

Is this the Husserl of Die Krisis der europäischen Wissenschaften und die transzentale Phänomenologie? I should have thought that scientists of 1936 deserved to feel rather pleased with themselves. Or should we fault them for failing to give a satisfactory interpretation of quantum mechanics?

July 16, 2006 8:59 PM  
david said...

Personally I think "the philosophy of real mathematics" is also a catchy concept.

I rather think Kuhn's concept was doing better in 1965 than mine is in 2006.

I have been introduced before a talk as having written 'Towards a Real Philosophy of Mathematics', suggesting most philosophy of mathematics is unreal, rather than that what it's doing may be philosophy of mathematics but is not about the real thing. Perhaps the difference is not so great.

July 16, 2006 9:28 PM  
John Baez said...

Clark Goble wrote:

So your point is more about the quest for low hanging fruit?

If you read the discussion that led to my remark, you'll see I was talking about the way string theorists do mathematical physics, with large numbers of them writing about the same topic after Witten, Polchinski, or other leaders making progress on this topic.

If you know what happened after Witten wrote his paper on twistors for perturbative supersymmetric QCD, or on the geometric Langlands' program, you'll know what I mean. If you don't, you're out of it - out of the current scene in mathematical physics, that is. But don't worry: I forgive you. I'm out of it too.

To consider the former example, check out the 242 papers citing Witten's December 2003 paper Perturbative Gauge Theory As A String Theory In Twistor Space. Penrose has been working on twistors for decades (after having invented them), so he had mixed feelings about this sudden surge of interest. He described the situation from his viewpoint as follow: you're on a grassy plain watching huge herds of antelopes with your binoculars... they're running hither and thither... and then all of a sudden you see they're running towards you!

There are definite advantages to this approach to doing mathematical physics, which is why people do it... but I was pointing out some of the disadvantages.

July 17, 2006 2:27 AM  
Dennis said...

You asked:
Is this the Husserl of Die Krisis der europäischen Wissenschaften und die transzentale Phänomenologie?


I am relying on Tieszen’s account (*Phenomenology, Logic, and the Philosophy of Mathematics*, 2005. Cambridge, Cambridge University Press).

Husserl’s view has a broad historical sweep. Here is a paraphrase of Tieszen’s account of Husserl's “crisis of science”. It is contained in a review I wrote of Tieszen's book.

Failure fully to recognize and investigate the nature and implications of the conscious side of the scientific enterprise has led to a crisis in science (according to Husserl). Leading disciplines such as physics become wedded to a purely factual and objective outlook, excluding questions such as what science means for humanity. (Disciplines closer to human science attempt to follow suit.) In Galileo’s mathematization of science, nature is idealized, cast in terms of pure geometry. “Formula-meaning” – the phenomena captured in a formula – replaces true meaning. Even though this mathematization is tremendous achievement, a kind of superficialization results – a detachment from what formulas might be about. The technological and purely mathematical aspects of science produces a world unto itself. Tieszen remarks: “Modern science undergoes a far-reaching transformation and there is a covering over of its meaning” (p. 41). “A type of naiveté has developed. Galileo is both a discovering and a concealing agent” (p. 42). The alienation of science from original meaning, that is, from its historical emergence through the process of idealization from the real world should be redressed (according to Husserl). A historical disclosure of science needs to be undertaken by philosophers and phenomenologists. (MAA reviews, 2006)

I don’t think Husserl would have discounted tremendous advances of science. It has come at a price, however.

Whatever one may think of Husserl’s view, his “crisis of science” seems to capture something of the downside of today’s mad dashes toward publication.
D. Lomas

July 17, 2006 12:51 PM  
Dennis said...

Would the term "transparent" suit your purpose?
D. Lomas

July 17, 2006 1:04 PM  
david said...

Transparency is precisely what is to be aimed for, and yet a subject matter can only be transparent to someone who has traversed a certain path. We discussed this idea already in the context of Ian Hacking's false charge that Lakatos was a deflationist as regards mathematics.

July 18, 2006 4:43 AM  
urs said...

John Baez wrote:

"I'm complaining about a swarm of people writing hundreds of short papers on the same subject in a short time, each referring to many of the previous ones, nobody taking the time to distill the matter to its essence."

To me, the cumbersome aspect of this behaviour is not so much the behaviour itself, but the fact that after the swarm moves on to another topic people rarely sit down to write coherent and comprehensive reviews of what has been accomplished - shallow or not, essential or not.

The result is that there are entire subfields in high energy physics about which you can learn only in two ways:

a) by chasing an immense list of scattered preprint literature, with no paper telling the full story from beginning to end, with no guarantee that any given paper of the bunch is actually worth it

b) by happening to being in close contact with people that are part of the swarm and benefitting from their private communication.

As a result, some topics are considerably more esoteric than they should be.

July 18, 2006 9:25 AM  
david said...

So what is the remedy? Surely it can only be administered by the powers-that-be in the relevant discipline by their declaration that the kind of work, the lack of which you describe, be recognised as being of the very greatest importance, and duly rewarded.

My small contribution to this change, as a philosopher, is to present a theory of enquiry which accuses disciplines which don't reward such activity of irrationality. We must recognise, however, that irrationality may be institutional. Even if many individuals were to agree with me, it takes more than this to bring about change. Perhaps more philosophers could lend their support.

July 18, 2006 9:44 AM  
urs said...

"So what is the remedy?"

In part the problem is self-healing, albeit only very slowly.

Namely if a subject attacked by the swarm is substantial enough to be of intrinsic interest, it eventually attracts the attention of people that are usually counted as "mathematical physicists".

As, if I recall correctly, Jürg Fröhlich once said half-jokingly, the job of mathematical physicists is to clean up behind the physicists.

To give an example that I was recently looking at more closely, there is an old physics idea called self-dual abelian p-form gauge theory.

I am not sure if there was ever a "swarm" of people working on that, but there is sure plenty of physics literature on this subject of the kind I mentioned: rich in details, but none of it tells the full story coherently and comprehensively.

Now, just recently, some high-powered mathematical physicists have taken another look at this issue and have managed to extract some deep principles - certainly conceptual essentials - facets of which had appeared before, but which had not (as far as I am aware) been spelled out like this before (I, II).

And they did not just do it out of benevolence to hapless swarm creatures. They did it because the swarm activity had revealed glimpses of a deeper structure of intrinsic interest.

This is just one example. There are more. You can tell how many there are by counting how often you hear string theorists complaining about mathematicians by saying "Pft, we knew that ten years ago, already, these guys are just reformulating material that we have put to the files a decade ago".

And partly that's true, partly not.

July 18, 2006 12:47 PM  
Dennis said...

david said...
Transparency is precisely what is to be aimed for, and yet a subject matter can only be transparent to someone who has traversed a certain path.

If transparency is precisely what is to be aimed for, then using “transparent” (and its cognates) would seem to be better than “conceptual essentialism” -- a term which might get mired in terminological nitpicking. Also, often for mathematicians a proof becomes transparent after initially appearing messy or difficult. This could be another benefit of using “transparent”. One could talk about “historically revealed transparency” or “historically engendered transparency” or “narratively revealed transparency”, etc.
D. Lomas

July 18, 2006 4:55 PM  
david said...


As you've heard me say, the position we're describing is very close to MacIntyre's, especially as presented in First Principles, Final Ends and Contemporary Philosophical Issues (details in 5th paragraph of this post).

July 18, 2006 7:52 PM  
Anonymous said...

My comments on the term "tranparent" were intended as a quick, off-the-cuff suggestion at the terminological level, nothing more.
D. Lomas

July 20, 2006 4:05 PM  
Pentcho Valev said...


The following two quotations are extremely important:
"Genius Among Geniuses" by Thomas Levenson
"And then, in June, Einstein completes special relativity, which adds a twist to the story: Einstein's March paper treated light as particles, but special relativity sees light as a continuous field of waves. Alice's Red Queen can accept many impossible things before breakfast, but it takes a supremely confident mind to do so. Einstein, age 26, sees light as wave and particle, picking the attribute he needs to confront each problem in turn. Now that's tough."

Einstein at the end of his career:
"I consider it quite possible that physics cannot be based on the field concept,i.e., on continuous structures. In that case, nothing remains of my entire castle in the air, gravitation theory included, [and of] the rest of modern physics."

So Einstein knew what had happened. Perhaps at that time (1954) the death of physics was still reversible. Now there is no hope. This civilization seems to be suicidal.

Pentcho Valev

September 20, 2006 12:07 PM  

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