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February 16, 2009

Dialectical Realism in Utrecht

Posted by David Corfield

Last Friday saw me in Utrecht to deliver a couple of talks for the From Plato to Predicativity seminar. I spent a very pleasant morning in a typical Dutch café chatting with Klaas Landsman about mathematics, physics and philosophy.

At the seminar in the afternoon I spoke first on Lakatos and then on Lautman, uniting them under the banner of ‘dialectical realists’. One obvious difference between the two is how contemporary were Lautman’s case studies. In the 1930s he’s talking about class field theory, where Lakatos’s main work from the late 1950s and early 1960s concerned early to mid-nineteenth century mathematics. His aesthetic antenna was so finely tuned to detect a certain kind of structure similarity that, had his life and work not been curtailed by the 1939-45 war, I wonder whether Lautman might have prompted his Bourbaki friends to take up category theory more rapidly.

The talk I gave on Lautman was similar to an earlier version, which includes the glorious example of the commonality between Galois theory and deck transformations. Jean Dieudonné said of his work on this:

La ‘montée vers l’absolu’ qu’il y discerne, et où il voit une tendance générale, a pris en effet, grâce au langage des catégories, une forme applicable à toutes les parties des mathematiqures: c’est la notion de ‘foncteur représentable’ qui joue aujourd’hui un rôle considérable, tant dans le découverte que dans la structuration d’une théorie.

I presented the algebraic number theoretic and topological manifestations side by side, with the ‘imperfect’ rationals \mathbb{Q} finding their perfected form in the algebraic closure, and the ‘imperfect’ circle finding its perfected form in the universal covering space, the real line sitting as a helix above the circle. Klaas wondered aloud why in the algebraic number theory case the object appears to become more complicated as we move up from \mathbb{Q} to its algebraic closure, while the real line seems simpler than the circle.

What should one say to that? That the covering space is not just the line, but the line equipped with a map to the circle, so that it possesses symmetries unseen by the circle?

Posted at February 16, 2009 9:56 AM UTC

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Re: Dialectical Realism in Utrecht

Well, the analogy between universal covering spaces and algebraic (or say separable) closures isn’t perfect. Basically, a separable closure is just a way to package all the finite separable extensions together, so that a closer topological analog would (morally) be a suitable inverse limit over finite covering spaces – and this will be a pretty complicated space, even when you’re starting with the circle.

The passage from the universal cover to this more complicated guy amounts to only remembering the profinite completion of the fundamental group – which is all the algebra can see in general (c.f. etale homotopy theory).

Also worth noting is that, from this perspective, the circle is much closer to a finite field than the rationals: the profinite fundamental group is, in both cases, Z hat.

While I’m making a post, I’ll go off-topic a bit and say that I really appreciate your philosophical contributions to this blog, David, even if I have nothing to say about them.

Posted by: anon on February 17, 2009 1:02 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Thanks for your final comment. I must say what a great privilege it is for me to be able to have discussions with mathematicians here, as much, I imagine, as it was for Lautman to have Herbrand and Chevalley to talk with.

And thanks for the other information. I guess I ought to knuckle down to everything in this thread.

Regarding a “suitable inverse limit over finite covering spaces”, would I be right in thinking that topologists don’t study them?

Posted by: David Corfield on February 17, 2009 11:53 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Well, I’m not sure if topologists really study it, but it has a name: the solenoid. It’s an LCA group, in fact Pontryagin dual to Q (because this guy is the _direct_ limit over all multiplication by n maps from Z to itself, and Z is dual to the circle). It’s also isomorphic to the adeles of Q modulo Q, a fact crucially used by Tate in his thesis.

But it doesn’t behave very nicely from the perspective of algebraic topology; there one would work with profinite spaces (a homotopical construction) rather than doing this more naive inverse limit in spaces.

Posted by: anon on February 17, 2009 4:34 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Can I chip in on this? The point at stake here is that the pro-space is better to handle than its limit. The solenoids can be studied (that is where I came in to algebraic topology!) but the method using shape theory is really to pass back out to the pro-space. It is instructive to look at SGA1 and how Grothendieck thought of this. He has a prorepresentable functor which is the `fibre’ functor corresponding to the system of all the fibres of the various covering spaces over a given point. He then takes the automorphisms of that functor. Many years ago I tried to push this further and to build an algebraic 2-stack type object from the etale homotopy type. This was before `Pursuing stacks’ suggested that (i) it was more difficult than I imagined and (ii) more important than I had dreamed of. That is another story and I still have not finished it!
Posted by: Tim Porter on February 17, 2009 5:39 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

From Plato to predicativity
“The relationship between mathematics and physical
reality, and that between provability and truth have
been studied from Plato till the present day.”

CAUSALITY: Models, Reasoning, and Inference by Judea Pearl
[A related dialectic?]
“Ten years ago, when I began writing “Probabilistic
Reasoning in Intelligent Systems (1988), I was working
within the empiricist tradition. In this tradition,
probabilistic relationships constitute the foundations
of human knowledge, whereas causality simply provides
useful ways of abbreviating and organizing intricate
patterns of probabilistic relationships. Today my view
is quite different. I now take causal relationships to
be the fundamental building blocks both of physical and
of human understanding of that reality, and I regard
probabilistic relationships as but the surface phenomena
of the causal machinery that underlies and propels our
understanding of the world. Accordingly, I see no greater
impediment to scientific progress than the prevailing
practice of focusing all of our mathematical resources on
probabilistic and statistical inferences while leaving causal
considerations to the mercy of intuition and good judgment.”

GERALD EDELMAN | “The most important thing to understand is
that the brain is “context bound.” It is not a logical system
like a computer that processes only programmed information;
it does not produce preordained outcomes like a clock. Rather
it is a selectional system that, through pattern recognition,
puts things together in always novel ways.”
Nonetheless, mathematics relies on cognitive mechanisms that
were empirically acquired during phylogenetic evolution, and
this explains their “unreasonable effectiveness”.

SH: I think that pattern recognition is the evidence our minds
process to establish causality as the underlying shaping metaphor,
of which cultural evolution is a more recent abstract development.
I think the enigma occurs before noticing “unreasonable effectiveness”.

Posted by: Stephen Harris on February 17, 2009 2:17 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

I found Edelman’s stance and Mercier’s point of view very appealing. Thank you for posting the link.

Posted by: Eugene Lerman on February 17, 2009 4:02 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

My point of view is just that sometimes you define the base space first and the universal cover second, and other times you do it the other way around. Typically if you need object AA to define object BB, then BB is rightfully seen as more complex than AA.

We need the field QQ to define its algebraic closure, but we need the space RR to define the circle. In each case, the logically earlier concept seems more simple.

We can come up with an example similar to the Galois one in topology. Let XX be two circles glued together at a point. Then I think most people would say that its universal cover is more complicated than XX is.

We can do it the other way around, too. It’s possible to start with an easy algebraically closed field LL and carve out a subfield KK over which LL is algebraic. Then the simpler object is LL and the more complicated one is KK

Posted by: james on February 17, 2009 9:33 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

David’s question was “What should one say to that?”

Here is one possible answer: consider the equation x^n + y ^n = 1, with n > 2. If you choose any value of x in the algebraic closure Q-bar of Q, one can obviously find a corresponding y in Q-bar that solves this equation. On the other hand, if we choose x in Q, it is a difficult theorem that we can’t then solve for y in Q (unless we are in a trivial case where x or y equals zero).

This brings out the sense in which an algebraically closed field is simple, while a non-algebraically closed field such as Q is not (despite initial appearances).

This example might seem a little flippant, and I wouldn’t entirely disagree with such a sentiment, but I do think it brings out something important. Algebraic closure is a kind of “simpleness” (in a non-technical sense; I just mean in the everyday English use of the word simple, as in David’s post) for fields, of a very specific nature, just as simple connectedness is a very particular kind of “simpleness” that a topological space can have. But since topology is normally easier to visualize than algebra (especially for non-experts), the “simplicity” of an algebraically closed field such as Q-bar can be hard to appreciate at first (say in comparison to
the evident simplicity of the real line, when contrasted with the more intricate shape of the circle).

Giving the example of a famously difficult Diophantine equation is then a genuinely useful way to illustrate this “simplicity” in the algebraic setting.

Posted by: Matthew Emerton on February 18, 2009 3:57 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

A continuation of my previous comment:

One need not resort to Fermat’s Last Theorem either, if one doesn’t want to.

Especially for the philosophically inclined, it might be enough to give the example of the equation x^2 = 2. This is tautologically solvable in Q-bar, but the fact that it can’t be solved in Q is not obvious, and was (perhaps) once the source of considerable philosophical controversy. This already illustrates that Q is less “simple” then it appears at first (and that Q-bar is more “simple” than it appears at first).

Posted by: Matthew Emerton on February 18, 2009 4:02 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

This is just a repetition of Matthew’s explanation, but in the context of this cafe, it might be worth spelling out the sense of simplicity involved.

The algebraic closure Q¯\bar{Q} is simple in that it is easy to have maps

Spec(Q¯)Y,Spec(\bar{Q}) \rightarrow Y,

while maps

Spec(Q)Y\Spec(Q) \rightarrow Y

are quite difficult. This is the same for universal covering spaces in topology. In many concrete situations, the awkward notion of a ‘multi-valued map’ from XX to YY comes up naturally, which is then clarified by unwinding to a map

X˜Y.\tilde{X} \rightarrow Y.

The problem of getting to an actual map

XYX \rightarrow Y

is then one of these problems of descent, in either situation.

Perhaps the kind of complexity that one usually associates with Q¯\bar{Q} is actually the price of this simplicity: The automorphism group of Q¯\bar{Q} is quite large and complicated. This phenomenon seems to be common in usual geometry as well. At least if one includes the complex structure, then the automorphism group of a compact hyperbolic Riemann surface is finite, while its universal covering space is acted on by PSL 2(R)PSL_2(R). [I would *not* venture at the moment to argue that this is necessarily more complicated than a finite group, but it is unquestionably larger.]

There is another perspective, similar to that expressed by James, which is a bit hard to make precise: the simplicity of R is deceptive. If we view it in the same ‘bottom-up’ fashion as Q¯\bar{Q}, it’s horribly complicated.

Posted by: Minhyong Kim on February 19, 2009 12:20 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

I’ll mention just one other example, hopefully relevant. An injective module is simple in a certain sense from a category-theoretical viewpoint. But if you ‘look inside,’ e.g., at the construction, you get another picture.

Posted by: Minhyong Kim on February 19, 2009 12:37 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Furthering Minhyong’s remarks, I might go so far as to say that all mathematical objects of interest are simple from one point of view and complicated from another. If the object is one of primary interest, such as the ring of rational numbers, then it is probably simple to come to terms with, otherwise we wouldn’t care about it; but probably certain aspects of it are very complex, otherwise people would have moved on long ago.

On the other hand, objects of secondary interest, by which I mean tools for understanding those of primary interest, such as the ring of algebraic numbers, are typically simple in the ways that the primary objects are complex, otherwise they wouldn’t be very useful as tools; but for this reason, any explicit construction of such a tool is likely to require some work, and so understanding them in a straightforward way is often not easy.

As I said, you can view almost any object as an example of one or the other. For example, the category of RR-modules versus the derived category of RR-modules. Also, maybe I should add that the distinction here between primary and secondary is a bit simplistic. Almost any mathematical object is more primary than some and less than others.

Posted by: James on February 19, 2009 2:22 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Thanks everyone for these great comments.

Regarding the ‘imperfections’ of \mathbb{Q}, Ittay Weiss was wondering after the talk why it was that polynomial splitting was the only imperfection to be considered. Why not a trigonometric function?

To the example of sinx=0sin x = 0, we can say something. Yves André discusses here the Galois extension brought about by adding π\pi. Its conjugates could be thought of as all the non-zero rational multiples of π\pi.

But the list of conceivable imperfections is endless. Why are some so much more interesting than others?

Posted by: David Corfield on February 20, 2009 1:22 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Let me first add a bit to my previous remarks. I think it should be emphasized that the complexity we see in Q¯\bar{Q} as manifested in its automorphism group really is hidden complexity of QQ itself. This is true at least from the view of Aut(Q¯)Aut(\bar{Q}) as a fundamental group of Spec(Q)Spec(Q), reflecting its enormous topological complexity. Spec(Q¯)Spec(\bar{Q}), on the other hand, is simply-connected. That is to say, even if Q¯\bar{Q} might look more complex than, say, RR, we should consider it as simpler than QQ. Making this viewpoint really convincing is a good task for category theory: complexity in the category of covering spaces truly is complexity of a space, even an arithmetic one.

I’m not sure what you intend with the terminology of ‘imperfections.’ If the question concerns, for example, why the field of algebraic numbers should be considered especially interesting, my own feeling is that it shouldn’t. For example, one can consider towers

QQ¯FCQ\subset \bar{Q}\subset F\subset C

where FF is algebraically closed of transcendence degree one. Such an FF will be a union of fields of rational functions on all algebraic curves over Q¯\bar{Q}. An interesting problem, that is addressed in part by Andre, Kontsevich, and others, is to coherently distinguish different sub-fields FF of CC with this nature, say the one that contains π\pi from the one with ee.

It does seem plausible that at some point, certain natural ‘exhaustive’ towers

QQ¯F 1F 2CQ\subset \bar{Q}\subset F_1\subset F_2\subset \cdots \subset C

that provide a fine structure to CC will become of interest. In usual topology, there are various categories of fiber bundles with non-discrete fibers.

Posted by: Minhyong Kim on February 20, 2009 11:31 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Lest we get too grandiose, we should of course remind ourselves that at present, towers at the zeroth-level

QE 1E 2Q¯Q\subset E_1\subset E_2\subset \cdots \subset \bar{Q}

are already difficult enough to command the undivided attention of most arithmeticians.

Posted by: Minhyong Kim on February 20, 2009 11:37 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

I’m not sure what you intend with the terminology of ‘imperfections.’

I’m just wondering how far we can take Lautman’s thesis. We’ve already discussed the idea that it is more accurate to speak of ‘ambiguities’ rather than ‘imperfections’.

One could say that not very much hangs on these informal terms. What we want is just the best mathematical account of Galoisian situations. We might also consider it to be very unlikely that such deep mathematics will manifest itself anywhere other than in physics.

Could you imagine any other walk of life with a structural resemblance to Galois theory?

Posted by: David Corfield on February 23, 2009 10:50 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

I guess I should admit I hadn’t been following that earlier discussion except very briefly, hence my confusion. Regarding imperfection vs. ambiguity, it seems to me that they do refer to quite distinct things, albeit closely related. If I understood you/Lautman correctly, a typical imperfection is a non-contractible loop. A typical ambiguity, on the other hand, is the plurality of inverse images of a point in some covering space. That the two are related is of course a very interesting fact.

I’m sure you’re aware of this, but there is a mathematically precise version of ‘constructing the perfection from imperfections’ in the case of the universal covering space of a space XX. This is when we choose a base-point bXb\in X and simply define it as

X˜:= xXπ 1(X;b,x),\tilde{X}:=\cup_{x\in X} \pi_1(X;b,x),

endowed with a natural topology.

Do you see this in Descartes somewhere? :-)

Incidentally, this seems an off-beat and interesting foundational question in Galois theory: To give a convincing and direct analogue of the construction above for Spec(Q¯)Spec(\bar{Q}).

Posted by: Minhyong Kim on February 23, 2009 2:08 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Is there any mental imagery you use for Spec(¯)Spec(\bar{\mathbb{Q}})? And for its map to Spec()Spec(\mathbb{Q})?

Posted by: David Corfield on February 24, 2009 5:45 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Does it make sense to compare Descartes intuition to the notion of a universal object? Given a category, one can speak of an initial object, for example, whose definition is prefigured in the category itself, consisting perhaps of various imperfect objects. Of course it’s important that one knows not just the objects ‘in and of themselves’ but the relations between them, e.g., morphism.

It occurred to me in any case as amusing that while the *definition* of a universal object is prefigured in the category, its existence is not (cf. some standard refutation of the ontological argument).

I realize this is somewhat diluting your original inquiry, by boiling down the essentials of the Galoisian to just the notion of a universal object, as far as the Cartesian argument is concerned. If we wish to make Galois theory in particular the relevant metaphor, it seems to me we need more detail on the nature of the perfection.

Posted by: Minhyong Kim on February 23, 2009 2:38 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

I’ll read some Descartes and report back.

Posted by: David Corfield on February 23, 2009 5:31 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Well, last night I finally looked into your Lautman lecture and had a vague glimpse of the things you have in mind. I saw at least three notions at play that need to be carefully distinguished:

(1) A universal concept, such as a cat, a polygon, or a group. Eventually, we bundle these into a *category*. That is, a universal concept seems to just correspond to a category. (For the moment, I’ll ignore higher-dimensional extensions.)

(2) Given a category, a universal object in it, say, an initial object or a final object. Usually, the definition is straightforward, but the existence is not (e.g., God, or a universal covering space *in the category of algebraic varieties*).

(3) A dualizing object in a category, as a special case of a universal object.

Duality is the most tricky of these concepts, and it’s not at all clear that the notion of a dualizing object captures all the instances where we wish to refer to some duality.

Posted by: Minhyong Kim on February 24, 2009 11:24 AM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

Hmm. I knew Descartes was not straightforward to read, but now I see how steeped in the history of philosophy you have to be to understand him properly. One thinks of him, perhaps along with Bacon, as marking a radical turn away from Scholasticism, but you have to be careful about this.

If you try to read the Meditations, for example part III where the existence of God is argued, you must understand ‘natural light’, ‘formal reality’ and ‘objective reality’.

Regarding my thought of a Galoisian construal of Aquinas’s angelology, Descartes says in conversation with Burman:

D [although there is some dispute as to who is speaking]: As far as the idea of an angel goes, it is certain that we form it from the idea we have of our own mind: this is the sole source of our knowledge of it. And this is so true that we can think nothing in an angel qua angel that we cannot also notice in ourselves.

B: But on this view, an angel is going to be identical with our mind, since each is something that merely thinks.

D: It is true that both are thinking things. But this still does not prevent an angel from having many more perfections than our mind, or having perfections of a higher degree. Indeed, it is possible that they may even differ in kind. Thus St. Thomas wanted every angel to be of a different kind from every other, and he described each one in as much detail as if he had been right in their midst, which is how he got the honorific title of the ‘Angelic Doctor’. Yet although he spent more time on this question than on almost anything else, nowhere were his labours more pointless. For knowledge about angels is virtually out of our reach, when we do not derive such knowledge from our own minds, as I have said. We just do not know the answers to all the standard questions concerning angels, for example whether they can be united with a body, or what the bodies were like which they frequently took in the Old Testament, and so on. It is best to follow Scripture and believe they were young men, or appeared as such, and so forth.

Posted by: David Corfield on February 24, 2009 2:28 PM | Permalink | Reply to this

Descartes; Re: Dialectical Realism in Utrecht

Meeting Descartes and Klein somewhere in a noncommutative space

V.V. Kisil - Highlights of Mathematical Physics, 2002


… 1981 American Mathematical Society 0002-9904
… Now the physicists hesitated between
a corpuscular (Descartes and Newton, 17th century) and a wave (Huygens and …)

Tacit Knowledge in mathematical Theory

H Breger - The Space of Mathematics. Philosophical, Epistemological, 1992
… In: Descartes: il Me- todo ei Saggi, vol. 2, Rome, 1990, pp. 349-369. Page 105. … Transactions of the American Mathematical Society, 58, 1945, p. 231-294

Posted by: Jonathan Vos Post on February 24, 2009 5:34 PM | Permalink | Reply to this

Re: Dialectical Realism in Utrecht

As Minhyong wrote, studying algebraic extensions of Q is analogous to (in topology) studying covering spaces of some
base space X, which are the same as fibre bundles with discrete fibres. As he goes on to remark, lots of fibre bundles with non-discrete fibres are also of much interest.

Thus, one would expect there to be an interesting analogy back on the Q side of things, and indeed there is: if we consider a curve (e.g. y^2 = x^3 - x) with coefficients in Q, we can think of this as a fibre bundle of (in this case, genus 1) curves over Q.

(The analogy on the topological side would be to replace Q by some space, e.g. the line, with coordinate t, say, and then to consider a family of curves over the line, e.g. y^2 = x^3 - t x.)

If one focuses attention on a generic point of y^2 = x^3 - x, one rapidly discovers a field of transcendence degree 1 over Q (in this case, the field Q(x)[y]/(y^2 - x^3 + x) ), and indeed, studying such field extensions of Q is more-or-less equivalent to studying curves over Q.

But now one notices something: if we want to pursue the matter further, and make the curve y^2 = x^3 - x really look like a space, then we want to replace this equation (which is not in any concrete way a space – it is just a collection of symbols) with some actual set of points. Which set? Well, the set of rational solutions to the equation is not so good, in general, because there may not be many, or any. On the other hand, a theorem of algebra (Hilbert’s Nullstellensatz) says that the set of solutions in Q-bar is big enough to fill out the curve y^2 = x^3 - x, in any reasonable sense. And so one can use this set of solutions (equipped with its Zariski topology, and structure sheaf) as a geometric model for the curve; and now we have passed from fields and equations to actual spaces and sheaves (and so have begun to do algebraic geometry, at least in one if its many formal realizations).

Notice here that although we were studying something that had to do with a transcendence degree 1 extension of Q, the field Q-bar still played an important role. And this is one reason number theorists focus so much attention on Q-bar: because even when they want to study other, higher-dimensional, structures over Q, the set of Q-bar points is always there as a geometric model, and to understand it well requires one to understand the field Q-bar itself, and its automorphisms, arithmetic, and so on, as well as one possibly can.

Posted by: Matthew Emerton on February 21, 2009 2:47 AM | Permalink | Reply to this
Read the post Lakatos as Dialectical Realist
Weblog: The n-Category Café
Excerpt: Considering Lakatos's idea of mathematicians aiming at the real through dialogue
Tracked: February 26, 2009 12:51 PM
Read the post A River and a Trickle
Weblog: The n-Category Café
Excerpt: On the brief appearance of Lakatos in a Handbook of philosophy of mathematics
Tracked: April 3, 2009 4:44 PM

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