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December 17, 2007

Is Mathematics Special?

Posted by David Corfield

There’s to be a conference held in Vienna next May which asks Is Mathematics Special?

A bringing together of researchers from different fields interested in the question what makes mathematics special (if anything). Even if mathematics presents itself or is presented as a (quasi-)empirical matter, the status of an epistemic exception that mathematics forms among the sciences asks for explanations.

So, is mathematics special? Well, of course, any discipline is special. No other discipline than particle physics builds 27 km. long pieces of experimental apparatus. But one very noticeable feature of the Anglo-American philosophical treatment of mathematics is that it takes it to be a discipline very unlike the natural sciences. This divergence has developed out of the Russellian conviction that mathematics is a (dressed-up) form of logic, and so true in all possible worlds, while the sciences deal with particular empirical events in our universe.

There’s a quotation on the conference website from Felix Klein which suggests some will be looking to play down differences:

Quite often you may hear non-mathematicians, especially philosophers, say that mathematics need only draw conclusions from clearly given premisses and that it is irrelevant whether those premisses are true or false – provided they don’t contradict themselves. Anybody who works productively in mathematics, however, will talk in a completely different manner, In fact, those people base their judgements on the crystallized form in which mathematical theories are presented once they’ve been worked out. The research scientist, like any other scientist, does not work in a strictly deductive way but essentially makes use of his imagination and moves forward inductively with the help of heuristic aids.

Personally, given where we are today in philosophy, I should prefer people to side with Klein and attend to the similarities with other disciplines. In particular, we need to focus on what has been terribly neglected – mathematics’ higher-level aspirations. In its pursuit of phantoms, dreams, or vistas, my sense is that it is not so very different from other forms of intellectual enquiry.

John recommended a short article by Minhyong Kim – Mathematical Vistas – in which after a passage by Grothendieck the author says

It is hoped … that Grothendieck’s candid expressions might convey even to the casual reader that curious sense of the unknown attached to any process of deep learning and thinking, and the urgent conviction at the core of worthwhile endeavor.

If mathematics is special it’s because its practitioners have been deeply learning and thinking for such a very long time.

Posted at December 17, 2007 11:42 AM UTC

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Re: Is Mathematics Special?

It seems that the distincion between

empirical natural science

and

pure logic

is indeed a subtle one, or at least one that deserves further attention.

Math is only the extreme case of various sorts of “intellectual enquiries” that are more or less rooted in phenomenology and more or less in pure logic.

To start on the far side, there is exobiology: people are reasoning about which kind of organisms are thinkable in principle, which might exist somewhere, or, if not existing already somewhere or having existed at some point, might be forced into existence by suitable experimental setups.

Similar thoughts are being thought in chemistry, clearly. I guess here it can already happen that reputable practitioners spend the better part on their career investigating properties of compounds, that possibly nobody has ever, so far, created in the lab. Or maybe at least not probed in detail in the lab.

Then of course physics. I don’t have to remind anyone here about the huge discussion about to what extent it is healthy for theoretical physics to move into territory completely detached from experiment.

Viewed from this perspective, math seems to be not singular but just extreme. I think.

Posted by: Urs Schreiber on December 17, 2007 1:01 PM | Permalink | Reply to this

Experimental Mathematics; Re: Is Mathematics Special?

“Experimental Mathematics publishes formal results and conjectures inspired by experimentation, algorithms and software for mathematical exploration, and related surveys.”

Welcome to Experimental Mathematics, a journal devoted to experimental aspects of mathematics research. It publishes formal results inspired by experimentation, conjectures suggested by experiments, descriptions of algorithms and software for mathematical exploration, surveys of areas of mathematics from the experimental point of view, and general articles of interest to the community.

Experimental Mathematics, refereed in the traditional manner, is led by a first-rate editorial board. A high standard of exposition is maintained, in order to reach as many readers as possible.

Posted by: Jonathan Vos Post on December 17, 2007 4:05 PM | Permalink | Reply to this

Re: Experimental Mathematics; Re: Is Mathematics Special?

Mathematics is a very wide subject, some of which is strongly “foundational/high-concept” and some is more “constructional”. As a practical example, on Friday afternoon I scribbled on graph paper all discrete kernels on an 16x16 grid which satisfied certain conditions and symmetries that I could think of, then kicked off a random computer search over the weekend to find other kernels with this property. (They all have to be mutually orthogonal so doing as many as finding the obvious ones by hand reduces the search space.) It seems there aren’t any more, but if it had I’d have stared at the set to see if I could discern a group structure and “explained” it. If I’d found one and I’d written it up (as part of an application using them, so in an engineering rather than a math forum), I probably wouldn’t have mentioned the experimental search component in the work.

I suspect that there are lots more mathematicians than just Ramanujan and Gauss (who are both famously reported to have shocked people with their books full of calculations) who are inspired by “empirical” (in the sense of “concrete calculations”) results but who don’t mention them because they don’t add any insight into the “technical story”.

Posted by: dave tweed on December 17, 2007 6:58 PM | Permalink | Reply to this

Re: Experimental Mathematics; Re: Is Mathematics Special?

Apart from the kind of “experimental math” mentioned here, there is the following “phenomenological” aspect which commonly heavily influences math as a whole:

of course math is all about “hand me those axioms, whatever, and I crank out their consequences”. But there are many possible sets of axioms. Do we really not care if there actually “is” anything satisfying these axioms?

Maybe in single cases we don’t. But on the large we do: we tend to be more interested in axiom systems (like Riemannian geometry, say) that we do know to be relevant for things that exist, than in axiom systems (like non-associative geometry) which one could equally well study, but where it is less clear what the point would be.

And on top of that, it seems that, all in all, those axioms sets actually relevant for some part of nature turned out to also be more interesting.

Just rememer how much trouble nature had to finally convince us that the axioms of Euclidean space, while looking nice, were actually not nearly as interesting as those of non-Euclidean geometry.

Posted by: Urs Schreiber on December 17, 2007 7:38 PM | Permalink | Reply to this

Re: Is Mathematics Special?

In some deep way mathematics may be close to the natural sciences, but the practice is certainly different. A story:

Back in high school I took a course entitled “Mathematics Research”, which along with “Technology Research” had recently been split off of the long-standing “Science Research” course. I coupled this with two sections of study release to enroll in UMBC’s MATH399 (“Independent Research”) under the guidance of Sam Lomonaco, who tossed me a problem on getting Maple to work with braids and braid groups. The upshot was that I could leave school at noon every day to go off and do my own studying on this problem.

Anyhow, all of us in any of the three branches of the class were working on our own stuff for most of the time, but we were expected to come back together at times to present our findings to-date to the rest of the class. The class as a whole, having been split into parts from the science-based class, was run by a science teacher.

She knew plenty about research in the natural sciences, but evidently very little about the actual practice of mathematics. Every time my turn came up, I made a presentation like I would at an AMS special session: some background motivation, leading into what I’d recently gotten to work. And every time she complained. There was no data, she said. There were no experiments. I clearly wasn’t doing research without data and experiments. She begrudgingly granted me a D- each quarter rather than fail me outright. She just couldn’t envision that “research” could take any different form than that in the natural sciences.

Incidentally, at the end of the year I took first place in the Baltimore Science Fair with my mathematical work (not the grand prize, but one below that). It was the first time that a math project had ever gotten near that level in the BSF. The people in charge of the judging clearly understood that mathematics research just doesn’t behave like physics or biology research.

Posted by: John Armstrong on December 17, 2007 5:34 PM | Permalink | Reply to this

Re: Is Mathematics Special?

As a followup to John Armstrong’s illustration of how math research is different from research in experimental sciences, I’ll contribute a somewhat similar story (although not involving any cool prizes ;-) which however will hopefully illustrate the point I tried to make above that the dichotomy we see here, (pure logic vs. observation of the existing world) is, while very real and very pronounced when we compare math with the rest of the scientific world, not really about math versus science, but rather a dichotomy which one can find within the sciences themselves:

something like half-way through my work on my PhD in theoretical physics, I was visited in my office by a former student colleague who I knew from our freshman times. She did biology and rememebered me when she needed some advice on a computation concerning some biochemical reaction.

So, she came to my office and we chatted a bit before getting to work. What kind of experiments I am doing, she wanted to know. None, I said. She grinned and asked me not to kid her. She was going to tell me about her experiment, so I should tell her about mine. But there was none. She literally had trouble believing that. She didn’t seem to find it reassuring when I pointed out that not only I was not doing any experiments (not since the experimental guys down on the base floor forced us to do them for credit points, anyway), but nobody else on the entire floor (though all of them physicists) was either.

Back then I thought that one of us was being mighty naïve, and that it wasn’t me, but possibly I was simply overestimating the common knowledge of what “theoretical physics” is. Certainly school didn’t prepare us for what either math or theoretical physics is really like.

Incidentally, some other time I was visitited by some other former student colleague of mine, this time somebody from experimental physics. She came around a little corner of my office, which was hidden from the door, and saw in the shelf I had there the rather huge collection of empty bottles which I had lined up there.

She was puzzled at that sight for a second. Then her face cleared up, she smiled at me and exclaimed: “Ah, you are doing an experiment here.”

But I had to disappoint her. It was just a bad case of laziness: all these bottles, which I had bought at the cafeteria, were refundable. But I never remembered to return them.

Interestingly, when I later moved to Hamburg, where I got Jens Fjelstad’s former office, I opened the cupboard there and saw — tons of empty bottles! :-)

(Jens, are you reading this? Hopefully you don’t mind me mentioning this…)

Posted by: Urs Schreiber on December 17, 2007 6:22 PM | Permalink | Reply to this

Re: Is Mathematics Special?

That’s very funny to read! I’ve had a few conversations like this myself, sometimes even with other physicists, who just can’t believe that I don’t do any experiments!

I suppose many academics, even in sciences, often don’t enounter any theoreticians until well into their PhDs. We should advertise…

Posted by: Jamie Vicary on December 17, 2007 8:43 PM | Permalink | Reply to this

Re: Is Mathematics Special?

I’ve had a few conversations like this myself, sometimes even with other physicists, who just can’t believe that I don’t do any experiments!

And a physical chemist will be able to tell similar stories. As might a theoretical biologist.

So, i think, the difference between math and the other natural sciences is ultimately really the restriction one imposes on which sets of axioms one considers worthwhile of investigation and to which degree on is interested in finding in the space of all axiom systems extrema of certain functions that measure the “fitness” of these axioms with respect to something. In math one tends to have very flat such fitness functions, that don’t very much prefer one set of axioms over another.

But just because these fitness functions are rather sharply peaked in the natural sciences, that doesn’t mean that other natural sciences aren’t interested in pure deduction. Science only begins after some kind of axioms have been picked and some pure reasoning is possible.

Posted by: Urs Schreiber on December 17, 2007 9:37 PM | Permalink | Reply to this

Re: Is Mathematics Special?

The claim about flat fitness functions just doesn’t seem to be true about mathematics. Maybe there are a broader range of sets of axioms that are considered to be of interest, but a truly random collection of axioms is generally of no interest whatsoever. I’m pretty sure there are even subsets of ZFC that haven’t been studied in any detail, even though this is probably the most investigated set of axioms, including many of its subsets.

Also, while theoreticians in other fields will be interested in sets of axioms, I think that experimentalists (which in practice seems to include most chemists, biologists, and medical researchers, and thus the vast majority of scientists) aren’t really that interested in axioms at all. They sometimes discuss laws, but not in anything like the explicit form that a mathematician or theoretical physicist would like.

It’s true that there’s some amount of deductive reasoning going on in all the sciences of course, just as it’s true that there’s some amount of inductive reasoning going on in the practice (if not the published version) of mathematics, but there is a very sharp different here, I would think.

Posted by: Kenny Easwaran on December 18, 2007 7:15 PM | Permalink | Reply to this

Re: Is Mathematics Special?

Am I right in thinking you take this “very sharp difference” to differentiate mathematics from all of the natural sciences? And that the sharpness has something to do with the preponderance of deductive or inductive reasoning in a discipline?

The trouble with this kind of portrayal is that one rarely makes plain what is meant by ‘deductive’ and ‘inductive’ reasoning.

It’s easy to think we know cases of each when we see them, perhaps the reasoning in Euclidean geometry (as improved by Hilbert) and that which takes us from white swan observations to a general statement. But how much of any discipline is composed of these forms of reasoning?

If you hang out in the Café for a even a brief time you come to realise just how much effort is devoted to working out, either in discussion or perhaps by yourself, how to get your hands on better concepts or how to view existing material more clearly, in sum how to tell a better story of the domain of study. Part of this process is to find out what it is to tell a better story.

Calling it ‘deductive’ or ‘inductive’, or even a mixture of the two, does no justice to it.

Posted by: David Corfield on December 18, 2007 7:43 PM | Permalink | Reply to this

Math, Science/Fiction, and Gonzo Journalism; Re: Is Mathematics Special?

Calling books and magazine articles “Fiction” or “Nonfiction” – or even a mixture of the two, does no justice to it. That is at the heart of the genre wars in (nontechnical) Literature, and the allegedly “special” roles of “gonzo journalism”, Hunter Thompson, Truman Capote, Norman Mailer, Tom Wolfe, and the endless debates about the boundaries between Science Fiction, Fantasy, and the like.

There is a mapping from such matters in nontechnical Literature to the arguments in technical (Science/Math) literature as characterized by the previously written line: “Calling it ‘deductive’ or ‘inductive’, or even a mixture of the two, does no justice to it.”

David Corfield is right, I think, to emphasize “narrative.” As editor Teresa Nielsen Hayden says (I may be slightly paraphrasing from memory): “Plot is a convention, story is a force of nature.”

This may be related to why John Baez and I especially enjoy the uncategorizable writings of Greg Egan.

Posted by: Jonathan Vos Post on December 18, 2007 8:12 PM | Permalink | Reply to this

Re: Is Mathematics Special?

David writes:

If you hang out in the Café for a even a brief time you come to realise just how much effort is devoted to working out, either in discussion or perhaps by yourself, how to get your hands on better concepts or how to view existing material more clearly, in sum how to tell a better story of the domain of study.

Yes, ‘deductive reasoning’ is only one of many tools we mathematicians use… and to me, perhaps the most dull. It’s important, but I’m much more interested in things like ‘changing my perspective so patterns snap into focus’, or ‘sensing the presence of a hidden mystery’.

Posted by: John Baez on December 18, 2007 8:03 PM | Permalink | Reply to this

Re: Is Mathematics Special?

Maybe there are a broader range of sets of axioms that are considered to be of interest, but a truly random collection of axioms is generally of no interest whatsoever.

Notice that this was actually one of my points, more explicitly stated further above:

of course math is all about “hand me those axioms, whatever, and I crank out their consequences”. But there are many possible sets of axioms. Do we really not care if there actually “is” anything satisfying these axioms?

Maybe in single cases we don’t. But on the large we do: we tend to be more interested in axiom systems (like Riemannian geometry, say) that we do know to be relevant for things that exist, than in axiom systems (like non-associative geometry) which one could equally well study, but where it is less clear what the point would be.

Still, the “fitness functions” are in general much, much flatter than in the rest of the natural sciences. That makes the difference.

experimentalists (which in practice seems to include most chemists, biologists, and medical researchers, and thus the vast majority of scientists) aren’t really that interested in axioms at all.

They don’t handle them like mathematicians do. They don’t even call them axiom systems. They call them theories.

But a theory in the natural sciences is nothing but an axiom system.

Posted by: Urs Schreiber on December 18, 2007 9:03 PM | Permalink | Reply to this

Re: Is Mathematics Special?

Although I have little to contribute to this discussion, when I read the comment

…a theory in the natural sciences is nothing but an axiom system

I couldn’t resist adding my hearty agreement. This was the subject of a thread long ago in sci.physics.research, in the days when I kept better track of these discussions. The concern at that time was the relation between a model in the sense of model theory (in logic) and a `mathematical model’ as used by physicists or applied mathematicians. There was, and still is, a fashion among model-theorists to claim that the two notions are completely different. It seems obvious to me, and I guess to Urs Schreiber, that the usage of both `theory’ and `model’ is essentially the same in physics and logic.

MK

Posted by: minhyong kim on December 18, 2007 10:35 PM | Permalink | Reply to this

Re: Is Mathematics Special?

the usage of both ‘theory’ and ‘model’ is essentially the same in physics and logic.

Yes!

Here at the nn-Café we enjoy dreaming about how to remove the word “essentially” in this statement, such that it becomes plain

the usage of both ‘theory’ and ‘model’ is the same in physics and logic.

Just recently we had again a big discussion about this, triggered by the statement of The Principle of General Tovariance, which we proposed to refine to an “Enriched Principle of Context and Concept”.

As John explicitly pointed out here (somewhere in the middle) and here (towards the end) this amounts pretty much to saying:

A physical theory is (or should be after we think about it hard enough) an nn-category AA. A model for this theory is an nn-functor f:AT. f : A \to T \,.

As John pointed out, this is nothing but Lawvere’s notion of theory and model.

On top of that we might find it useful to demand everything to work in the world of topoi. Quoting John’s comment here this means:

Clearly AA is the ‘concept’ here — a bunch of statements about the system we’re trying to describe — while TT is the ‘context’. But, we can put this idea into the framework I was describing if we form the ‘classifying topos’ Th(A)\mathrm{Th}(A) of the axiom system AA. Then a model of AA in the topos TT is the same as a geometric morphism f:Th(A)T. f : \mathrm{Th}(A) \to T \,.

Physicists often say system instead of model. For some reason the term “model” is popular mostly in particle physics and high energy theory, where model building is an industry all in itself.

One should beware, though, that in order to confuse themselves and everybody else, they also sometimes say “theory” when they mean “model”, which may account for the

fashion among model-theorists to claim that the two notions [in physics and logic] are completely different

which you observed.

Posted by: Urs Schreiber on December 19, 2007 11:15 AM | Permalink | Reply to this

Re: Is Mathematics Special?

Philosophers of science have categorised models into many kinds. As this article explains, there is an

…incredible proliferation of model-types in the philosophical literature. Probing models, phenomenological models, computational models, developmental models, explanatory models, impoverished models, testing models, idealized models, theoretical models, scale models, heuristic models, caricature models, didactic models, fantasy models, toy models, imaginary models, mathematical models, substitute models, iconic models, formal models, analogue models and instrumental models are but some of the notions that are used to categorize models.

But it goes on to say that some order can be brought to this vast range.

Posted by: David Corfield on December 19, 2007 12:21 PM | Permalink | Reply to this

Re: Is Mathematics Special?

Urs might have made his colleagues feel better by saying he was doing experiments…

… but that these experiments consisted of making marks with a pencil on a paper, following certain prearranged rules, and seeing which combinations of symbols actually showed up.

In other words, we could take V. I. Arnold seriously when he wrote:

Mathematics is the part of physics where experiments are cheap.

I’m not sure this is a good viewpoint, but it’s a consistent one.

Posted by: John Baez on December 18, 2007 6:27 AM | Permalink | Reply to this

Re: Is Mathematics Special?

More from Arnold (p. 18):

To facilitate the search of mathematical information, Russian mathematicians have tried to cover most of the present day mathematics in the more than one hundred volumes of the Encyclopaedia of mathematical sciences, several dozens of which have already been translated into English. The idea of this collection was to represent the living mathematics as an experimental science, as a part of physics rather than the systematic study of corollaries of the arbitrary sets of axioms, as Hilbert and Bourbaki proposed.

Posted by: David Corfield on December 28, 2007 11:37 AM | Permalink | Reply to this

Re: Is Mathematics Special?

One need only look at the accomplishments of applied mathematics, especially by engineers [applied physicists] to observe what a wonderful and special thing is mathematics.

Posted by: Doug on December 17, 2007 10:53 PM | Permalink | Reply to this

Re: Is Mathematics Special?

It seems to me that mathematics is special in that it is the only science that can claim to be about the structure of problems. In particular, math can be used to describe structure in all branches of science, including itself – viz. logic or, perhaps better, category theory can be said to capture the structure of mathematics.

Said another way: in the (poset) category of scientific inquiry, with morphisms being the “describes phenomena in” relation (as in “math describes phenomena in physics”), isn’t math the initial object?

(Those aren’t actually quite the morphisms I want, but I hope you get my point.)

And in response to the Russellian conviction mentioned above: sure, fine, maybe math is true in all possible worlds. But isn’t it more important that math can be used to describe all possible worlds?

Posted by: Josh on December 18, 2007 5:58 AM | Permalink | Reply to this

Imaginary Logic; Re: Is Mathematics Special?

“But isn’t it more important that math can be used to describe all possible worlds?”

That’s the central issue in N. A. Vasiliev and his (nonaristotelian) imaginary logic. How do beings in universe 1 using logic 1 describe beings in universe 2 using logic 2, and beings in universe 2 claiming to be using logic 3, where “universe” explicitly includes “fictional” universes and fictional characters. Lewis Carroll and Vladimir Nabokov also contributed to such analyses…

“Many Worlds” and the “Landscape” are limited by coherence and consistency. The deep question is deep indeed.

Posted by: Jonathan Vos Post on December 18, 2007 7:42 PM | Permalink | Reply to this

Re: Imaginary Logic; Re: Is Mathematics Special?

“But isn’t it more important that math can be used to describe all possible worlds?”

Yuri Manin’s just published essay collection may contain some interesting ideas related to this discussion (not yet in the library here, so I could not read it so far).

Posted by: Thomas Riepe on December 20, 2007 6:58 PM | Permalink | Reply to this

Re: Imaginary Logic; Re: Is Mathematics Special?

conc. Grothendieck:

” The first volume, Anarchie (in German) of a biography-in-progress of Alexander Grothedieck’s life entitled

Wer is Alexander Grothendieck? Anarchie, Mathematik, Spiritualitaet,

has now appeared in print. Anarchie, by Winfried Scharlau (scharla(at)math.uni-muenster.de) can be purchased for a price of 12.00 euros plus postage, directly from the author.”
http://www.math.jussieu.fr/~leila/grothendieckcircle/biographic.php

Posted by: Thomas Riepe on December 21, 2007 1:52 PM | Permalink | Reply to this

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