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February 26, 2009

Lakatos as Dialectical Realist

Posted by David Corfield

Reading again Proofs and Refutations for my first Utrecht talk, I came across this part of the dialogue:

Omega: But I want to discover the secret of Eulerianness!

Zeta: I understand your resistance. You have fallen in love with the problem of finding out where God drew the boundary dividing Eulerian from non-Eulerian polyhedra. But there is no reason to believe that the term ‘Eulerian’ occurred in God’s blueprint of the universe at all. What if Eulerianness is merely an accidental property of some polyhedra? In this case it would be uninteresting or even impossible to find out the zig-zags of the demarcation line between Eulerian and non-Eulerian polyhedra. Such an admission however would leave rationalism unsullied, for Eulerianness is then not part of the rational design of the universe. So let us forget about it. One of the main points about critical rationalism is that one is always prepared to abandon one’s original problem in the course of the solution and replace it by another one. (pp. 67-68)

The students have come to the point of defining Eulerianness for a polyhedron as possessing the property that VE+F=2V - E + F = 2, for vertices, edges and faces. They were driven to devise this term having believed all polyhedra had this property, when counterexamples, such as the picture frame, emerged.

As it turned out, Eulerianness did pick out something worth demarcating – those polyhedra homeomorphic to a sphere. But, as Zeta remarks, the transformation of which problems and which properties are taken as central is no cause for concern for the rationalist. Indeed, if these changes are brought about for good reason, we should celebrate such changes.

My main problem with Lakatos is that he takes the vehicle for such transformation to be his method of proofs and refutations:

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common is their name. But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification, the balance changes in favour of the realist. (p. 92n)

The method takes the form:

(1) Primitive conjecture.

(2) Proof (a rough thought-experiment or argument, decomposing the primitive conjecture into subconjectures or lemmas).

(3) ‘Global’ counterexamples (counterexamples to the primitive conjecture) emerge.

(4) Proof re-examined: the ‘guilty lemma’ to which the global counterexample is a ‘local’ counterexample is spotted. This guilty lemma may have previously remained ‘hidden’ or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem–the improved conjecture–supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.

(5) Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at the crossroads of different proofs, and thus emerge as of basic importance.

(6) The hitherto accepted consequences of the original and now refuted conjecture are checked.

(7) Counterexamples are turned into new examples - new fields of inquiry open up. (pp. 127-8)

Lakatos was right to point us in the direction of a dialogue, but it would do no justice at all to the discussions we have had on, say, generalized smooth spaces to interpret them as applications of the method of proofs and refutations.

He also usefully points us in the direction of natural kinds in mathematics.

Posted at February 26, 2009 12:37 PM UTC

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61 Comments & 2 Trackbacks

Re: Lakatos as Dialectical Realist

Is the method of proofs and refutations supposed to be prescriptive or descriptive? Could you remind us?

Posted by: Eugene Lerman on February 26, 2009 3:33 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

The method is described as “a simple pattern of mathematical discovery - or of the growth of informal mathematical theories”. Lakatos noted that

…the actual historical pattern may deviate slightly from this heuristic pattern. (p. 127)

He credits Seidel with the discovery of the methodology in the 1840s when he proof-generated the concept of uniform convergence from Cauchy’s ‘proof’ that the limit of continuous functions is continuous.

He blames the prevailing ‘Euclidean methodology’ for preventing the previous generation finding the flaw, where truth is taken to flow deductively from clear first principles.

As I wrote in my book, this is rather an odd picture. What are we to make of the totality of pre-1840s mathematics? And what of that which has left the informal stage? Lakatos doesn’t give any clue as to how conceptual change can take place with axiomatised theories, i.e., how a new informal may re-emerge after formalisation. (‘Formalisation’ and ‘axiomatisation’ tend to get used interchangeably.)

Yuri Manin describes this latter process beautifully in Von Zahlen und Figuren:

…the most fascinating thing about algebra and geometry is the way they struggle to help each other to emerge from the chaos of non-being, from those dark depths of subconscious where all roots of intellectual creativity reside. What one “sees” geometrically must be conveyed to others in words and symbols. If the resulting text can never be a perfect vehicle for the private and personal vision, the vision itself can never achieve maturity without being subject to the test of written speech. The latter is, after all, the basis of the social existence of mathematics.

A skillful use of the interpretative algebraic language possesses also a definite therapeutic quality. It allows one to fight the obsession which often accompanies contemplation of enigmatic Rorschach’s blots of one’s imagination.

When a significant new unit of meaning (technically, a mathematical definition or a mathematical fact) emerges from such a struggle, the mathematical community spends some time elaborating all conceivable implications of this discovery. (As an example, imagine the development of the idea of a continuous function, or a Riemannian metric, or a structure sheaf.) Interiorized, these implications prepare new firm ground for further flights of imagination, and more often than not reveal the limitations of the initial formalization of the geometric intuition. Gradually the discrepancy between the limited scope of this unit of meaning and our newly educated and enhanced geometric vision becomes glaring, and the cycle repeats itself.

Posted by: David Corfield on February 26, 2009 4:15 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

I agree with you that Lakatos’s method of proofs and refutations is inadequate for describing what mathematicians actually do. So what would you add? (I am sorry if I am asking you to retell your book for the n-th time).

Posted by: Eugene Lerman on February 26, 2009 6:57 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

The short (and somewhat evasive) answer is to say that there’s going to be no simple repertoire of techniques which will account for what mathematicians do. We can describe allusive examples of mathematics done well through the centuries, but there’s little reason to believe that there is a recipe characterising these. We should expect mathematicians to learn how to learn, and for novel ways of thinking about fundamental concepts to emerge.

We might still worry that the speed of such changes is reduced by the lack of a certain kind of dialogue. About ten years ago, I would have said that there was a very insufficient amount of informal chat about basic ideas. Now, here we are in the middle of a blog, characters in a huge Lakatosian dialogue, discussing logic, geometry, and the rest.

I wonder though sometimes whether things wouldn’t work better if there were greater pressure to respond to others’ views. Let us say you work on category theoretic formulations of universal algebra and you devote your time to monads, shouldn’t you be prepared to respond to Hyland and Power’s position that in many respects Lawvere theories are better suited? I’m a great fan of bringing together slightly oblique viewpoints.

Posted by: David Corfield on February 27, 2009 3:51 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

I wonder if one can use mathematics to model the propagation of ideas in mathematics. Think of ideas as mems and think of their propagation as a dynamical system on a graph. More specifically use the approach of Erez Lieberman, Christoph Hauert and Martin A. Nowak (Nature 433, 312-316 (20 January 2005) | doi:10.1038/nature03204).

Here is the abstract:

Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations1. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process3, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.

Posted by: Eugene Lerman on February 27, 2009 9:26 PM | Permalink | Reply to this

Are you smarter than an 8th grader?; Re: Lakatos as Dialectical Realist

Tomorrow, Friday 27 Feb 2009, I shall be videotaped (not for public) a Geometry lesson at Stern Science and Math School, to predominantly Latino kids from East L.A., a lesson on Archimedean Solids. They’ll build them from paper and glue, and I’ll see if they can self-discover Euler’s Polyhedral Theorem (for Genus 0) by my having them count vertices, edges, faces on each, and write that on the whiteboard as they build and examine. It took quite a while to have my lesson plan approved, find that the AP Calculus teacher did not have room in the curricululm until May, get Assistant Principal to approve, then notify Principal. I consider such self-discovery exciting to watch (will they get it?) and philosophically fascinating as well.

Posted by: Jonathan Vos Post on February 26, 2009 4:50 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

David: this is the second post where you mention ‘dialectical realism’ without saying what that means. Is this something we’re all supposed to know? I know about lots of ‘isms’, but not this one. How far behind the times am I?

Posted by: John Baez on February 27, 2009 5:05 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Heh. Let me anticipate the teacher and take a wild guess: This is the tenet that mathematical objects are real, but not as immutable entities. Rather they transform through the dialectics of mathematicians who work with them.

David, I’ll answer your question about images of Spec(Q)Spec(Q) at some point. It’s a hard question.

Posted by: Minhyong Kim on February 27, 2009 2:01 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Would this analysis of spec([x])spec(\mathbb{Z}[x]) help?

Posted by: David Corfield on March 1, 2009 6:07 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

It’s a good picture, but perhaps not the most useful one. That’s why I’d prefer to struggle with the answer a bit. By the way, one aspect is that Spec(Z)Spec(Z) does somehow make sense as a line or a two-manifold, but Spec(Q)Spec(Q) is usually just some cloudy point. One does eventually need to do better than that…

Posted by: Minhyong Kim on March 2, 2009 1:29 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Here’s my own (and probably lots of other people’s) way of picturing these things.

I like to picture Spec QQ as something like a 2-manifold which has had all its points deleted. The extra complication is that what we think of as the points are actually very small circles. So it’s really a three manifold with all of the loops inside it deleted.

To do this we use the original Rosetta Stone, at least in mathematics. :)

For example, let’s look first at function fields. Spec C[z]C[z] is just the complex line CC. As we start inverting elements of C[z]C[z], as we must do to make C(z)C(z), the effect on the spectrum is to remove bigger and bigger finite sets of points. The limit is where we remove all the points and we’re just left with some kind of mesh.

If we had started with a Riemann surface of genus gg, then we’d be left with a mesh of genus gg, a surface sewn out of the cloth from which fly screens for windows are made. If we want to recover the original surface from the surface mesh, we just put it out back in the shed for a while and let the mesh fill up with dirt. This is just the familiar fact that a (smooth compact, say) Riemann surface can be recovered from the field of meromorphic functions on it.

If we replace CC by a finite field FF, then everything is the same but what we thought of as the point is now a very small circle, and so our original surface reveals itself to be a 3-manifold fibered over a very small circle when we zoom in. And when we delete points, we’re really deleting not just single-valued sections of this fibration but also multivalued sections. So Spec F(z)F(z) is some kind of 3-manifold fibered over the circle with all the loops over the base circle deleted.

For the passage from ZZ to QQ, I don’t have anything better to say than that it’s sort of the same but there’s no base circle. We’re just removing lots of loops from a 3-manifold. Maybe some should be seen as bigger than others, corresponding to the fact that there are prime numbers of different magnitudes.

Posted by: James on March 2, 2009 3:09 AM | Permalink | Reply to this

Voodoo spectra

James wrote:

I like to picture Spec Q as something like a 2-manifold which has had all its points deleted.

You may or may not know Jim Dolan’s eloquent term for this idea: he calls it a ‘voodoo spectrum’.

This is supposed to remind you first of the blatant nonsense involved — like ‘voodoo economics’ — but then, more deeply, of a voodoo doll poked full of holes. ‘Death by a thousand pinpricks’.

As you note, the analogy to the field of meromorphic functions on a Riemann surface is a reassuring sign that this idea is not crazy. There are very few holomorphic functions on a compact Riemann surface, but the more holes you poke out, the more holomorphic functions you get on the complement — and in some sense, if you poke a hole everywhere, you get all the meromorphic functions.

For what it’s worth, I believe we’re engaging in the process ‘dialectical realists’ embrace: using dialogue to find, or perhaps create, mathematical reality.

Posted by: John Baez on March 6, 2009 7:57 PM | Permalink | Reply to this

Re: Voodoo spectra

and in some sense, if you poke a hole everywhere, you get all the meromorphic functions

I want to make sure I have this right.

Given any finite subset KK of a Riemann surface SS, we get a ring Hol(K˜)\Hol(\tilde{K}) of homolomorphic functions on the complement of KK. This is actually a (directed-colimit-preserving) functor from the poset Sub fin(S)\Sub_\fin(S) of finite subsets of SS to the category of rings. And the field Mer(S)\Mer(S) of meromorphic functions on SS is the (directed) colimit of this functor?

Certainly the (directed) colimit of the inclusion functor from Sub fin(S)\Sub_\fin(S) into the poset Sub(S)\Sub(S) of all subsets of SS is SS itself, so we pretend that Hol(˜)\Hol(\tilde{-}) is a directed-colimit-preserving functor on all of Sub(S)\Sub(S) so that Mer(S)=Hol(S˜)\Mer(S) = \Hol(\tilde{S}). Which is not true (since S˜\tilde{S} is the empty set \varnothing and Hol()\Hol(\varnothing) is the trivial ring), but boy isn't it cute to think so?

And in fact we could extend Hol(˜)\Hol(\tilde{-}) from Sub fin(S)\Sub_\fin(S) to all of Sub(S)\Sub(S) by requiring it to preserve directed colimits, although it would no longer mean literally the ring of holomorphic functions on the complement of the argument. And then Hol(S˜)\Hol(\tilde{S}) really would be Mer(S)\Mer(S)?

If I've got this holomorphic stuff right, then I can probably understand this stuff about spectra too.

Posted by: Toby Bartels on March 6, 2009 9:19 PM | Permalink | Reply to this

Re: Voodoo spectra

This sounds about right. I like James’ mesh image as well. However, some people seem to think that we should just replace the spec of a field or a ring of integers by a suitable category of sheaves, and try to view schemes like Spec(Q)Spec(Q) using a geometry of categories. One hopes thereby to find natural varying families that contain spectra of number fields. Of course this is far-fetched, but curiously compelling. I suppose we’re vaguely aware of a sense in which (n+1)-categories of (n-)stacks on a space are fancy incarnations of triangulations. The challenge is to make this really compelling to the geometric intuition, and to find characterizing (n-)categories for spaces like Spec(Q)Spec(Q). I’m sure many people here have a better sense of this than I do.

Posted by: Minhyong Kim on March 6, 2009 11:50 PM | Permalink | Reply to this

Re: Voodoo spectra

John: “Voodoo spectrum” is very nice and sums up the idea perfectly, like all of Jim’s appellations.

Toby: I agree with Minhyong that that “looks about right”. Let me add that it can be made precise and is not just a cute story. It’s simply that Spec C(z)C(z) is the limit in the category of affine schemes (hence all schemes) of the collection of open subschemes of the affine line which are complements of finite sets of points (or, better, “complex-valued” points). But, as you suggest, the point is not the mathematical content but the interpretation of that content. (And the point of the interpretation, as always, is so that the less formally gifted of us can use the formalism without effort.)

Minyhong: OK, now I see what was behind your mysterious suggestion. My comment was a bit besides the point, then. But I still think it’s worth mentioning that the generic point of an algebraic variety is more than just a smudge, as the textbooks would have it. It’s a very geometric thing. In some sense, it’s just a variety with a finite number of hypersurfaces deleted, but where your adversary is allowed to keep changing which hypersurfaces.

Posted by: James on March 7, 2009 2:59 AM | Permalink | Reply to this

Re: Voodoo spectra

Toby wrote:

I want to make sure I have this right.

Given any finite subset SS of a Riemann surface S, we get a ring Hol(S˜)Hol(\tilde{S}) of homolomorphic functions on the complement of KK. This is actually a (directed-colimit-preserving) functor from the poset Sub fin(S)Sub_fin(S) of finite subsets of SS to the category of rings. And the field Mer(S)Mer(S) of meromorphic functions on SS is the (directed) colimit of this functor?

Something like that. We’ve got 1) the contravariant functor sending a finite subset XSX \subset S to the complement XSX - S, and 2) the contravariant functor sending a Riemann surface XSX - S to the commutative ring of meromorphic functions on this surface. Composing them, we get a covariant functor, and this ‘does the obvious good thing’ with respect to unions of finite sets.

But is ‘the obvious good thing’ the same as ‘preserving all directed colimits’? Hmm.

Does the empty diagram count as a directed colimit? The initial finite subset of SS is the empty set. This gets sent to the ring of all holomorphic functions on SS. If SS is compact and connected, that’s just \mathbb{C}. That’s not the initial commutative ring — but it’s the initial commutative \mathbb{C}-algebra.

So, maybe we should say this functor goes from finite subsets of SS to commutative \mathbb{C}-algebras. And then, in the case where SS is compact and connected, I believe this functor preserves all finite colimits.

And I guess it also preserves all directed colimits that actually exist in Sub fin(S)Sub_fin(S).

If I’ve got this holomorphic stuff right, then I can probably understand this stuff about spectra too.

Right, they work the same way.

James wrote:

And the point of the interpretation, as always, is so that the less formally gifted of us can use the formalism without effort.

While the point of the formalism, as always, is so that the less intuitively gifted of us can get the right answer by following the rules instead of just seeing the truth.

Posted by: John Baez on March 7, 2009 4:06 PM | Permalink | Reply to this

Re: Voodoo spectra

Does the empty diagram count as a directed colimit?

No.

Posted by: Toby Bartels on March 7, 2009 11:14 PM | Permalink | Reply to this

Re: Voodoo spectra

John wrote:

Does the empty diagram count as a directed colimit?

Laconically, Toby replied:

No.

Okay. I may have seen a directed set defined as a poset such that given elements xx and yy, there is an element zz with zxz \ge x and zyz \ge y. This makes the empty set a directed set. Your definition does not. I imagine yours is better.

In particular, your definition makes your comment on meromorphic functions true, without needing the Riemann surface to be connected.

Wikipedia currently defines a directed set as a nonempty poset such that given elements xx and yy, there is an element zz with zxz \ge x and zyz \ge y.

Posted by: John Baez on March 8, 2009 11:50 PM | Permalink | Reply to this

Re: Voodoo spectra

One rule of thumb I like is that for finitary structure/properties, almost always the most important definition is one that’s equivalent to one starting “for any finite set, …”. And, of course, the empty set is finite. Thus, a category with finite limits (“every finite diagram has a limit”) has a terminal object. A join-semilattice (“every finite subset has a least upper bound”) has a bottom element. An associative monoid (“every finite string has a unique specified product”) has an identity. And a directed poset (“every finite subset has an upper bound”) is nonempty.

Posted by: Mike Shulman on March 9, 2009 1:47 AM | Permalink | Reply to this

Re: Voodoo spectra

Of course, probably you know that, and the question is more about how people use the word. But I’ve (almost?) always seen “directed” used in the more reasonable sense including nonemptiness.

If you allow empty things to be called directed, then all sorts of things go wrong, like what you observed above but even worse. For example, if XX is a finite set then Set(X,)Set(X,-) preserves directed colimits, but not the empty colimit unless XX is empty.

Posted by: Mike Shulman on March 9, 2009 3:48 AM | Permalink | Reply to this

Re: Voodoo spectra

John wrote at last:

Wikipedia currently defines a directed set as a nonempty poset such that given elements x and y, there is an element z with zx and zy.

Are you upset because Wikipedia used the word ‘nonempty’ instead of the more constructively friendly term ‘inhabited’? I wouldn't worry about that; constructivists know how to interpret ‘nonempty’ when reading classical mathematics, and some of them even use it that way themselves, since the literal meaning is almost never wanted.

Or are you upset because you misquoted Wikipedia by saying ‘poset’ instead of ‘proset’ (or ‘set equipped with a preorder’)? I wouldn't worry about that either; you can always edit your comment, even now since it's your blog.

Otherwise, I don't see what would upset you about that!

Posted by: Toby Bartels on March 9, 2009 2:26 AM | Permalink | Reply to this

Re: Voodoo spectra

I forgot to say that I'm pretty sure that there's no disagreement about this in the literature. I suppose that somebody might forget to say that a directed set is inhabited, but I'd be very surprised if they didn't mean that!

Posted by: Toby Bartels on March 9, 2009 3:24 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

This is the tenet that mathematical objects are real, but not as immutable entities.

That’s interesting. But if they’re mutable, what does it mean to say that they’re ‘real’?

Posted by: Jamie Vicary on March 1, 2009 8:25 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

I am, but I am mutable, sadly.

In any case, I was just giving a smart-aleck student answer!

Posted by: Minhyong Kim on March 2, 2009 1:01 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

If we could put a question to Lakatos and Lautman, an interesting one might be:

You describe the autonomous development of mathematics in some supra-human realm, but why not say that it’s not the mathematical entities or concepts themselves that are changing but rather the human understanding of these entities or concepts, so that the movement is all on the side of the human knower? We can be interested in following the history of physics, but we don’t need to see the development of physical understanding as a projection of some dialectical activity in the laws of physics.

I’m not sure how they’d answer. Lakatos’s worry is that we’ll mistake a particular presentation of mathematics as definitive, and so stop questioning basic assumptions. About Euclideanism (theories as deductive consequences of clear and true axioms) and theories of a ‘new quasi-truth like probability’, Lakatos says:

The main danger of both delusions lies in their methodological effect: both trade the challenge and adventure of working in the atmosphere of permanent criticism of quasi-empiricial theories for the torpor and sloth of a Euclidean or inductivist theory, where axioms are more or less established, where criticism and rival theories are discouraged. (p. 42)

Euclideanism makes the axioms seem to have been always there. Inductivism takes truth or probability to trickle up from facts to theories, when we know that only falsity flows in that direction.

I don’t see why to escape ‘torpor and sloth’ we are forced to postulate an autonomous movement of ideas. Why not say that we believe we are becoming clearer about the ends of mathematics, while expecting that many profound changes of viewpoint are still to be made. We may have lost some wothwhile understanding of our predecessors, but by and large we have more powerful resources, and can understand their successes and difficulties. And we expect our descendants to understand that our understanding was superior to that of the ancestors, even if they see it as partial in many respects.

Posted by: David Corfield on March 2, 2009 9:18 AM | Permalink | Reply to this

Idea Stuff; Re: Lakatos as Dialectical Realist

I agree with David Corfield here. I deny that we are forced to postulate an autonomous movement of ideas.

It may be isomorphic to say that there is an autonomous movement of ideas, and that the representations of ideas propagate with mutation, recombination, and Natural Selection in the social network of Mathematicians.

I wouldn’t emphasize putting one’s favored spin on the means of production and distribution of ideas, but on finding (if ideas are the STUFF) what are the STRUCTURES of the mutable network of ideas, and what are the PROPERTIES of that structure.

Posted by: Jonathan Vos Post on March 2, 2009 7:44 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

My take on

“why not say that it’s not the mathematical entities or concepts themselves that are changing but rather the human understanding of these entities or concepts, so that the movement is all on the side of the human knower? “

is this it how it *feels* to someone doing mathematics. And it is this feeling that’s responsible for the appeal of the notion that Ideas are Real. But I suspect that this feeling is a cognitive illusion.

Posted by: Eugene Lerman on March 2, 2009 10:36 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

A realist might say that the path humans take doesn’t just reflect mathematical reality in terms of the destination, but that the course of the path itself also reflects something of the intricate structure of that reality. Just as we might say that to understand a building we should follow its construction.

I think the interesting question with regard to reality is how constrained you are as you develop concepts and theories, and where those constraints originate. E.g., could you imagine a similarly sophisticated mathematics to today’s which had bypassed Lie groups? If not, why?

Posted by: David Corfield on March 3, 2009 10:10 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

For myself, I’ve tended to assume the reality of a *space of ideas,* rather than a specific idea. Humanity moves across it’s landscape. This is somewhat dual to the propagation of ideas through humanity that Eugene was proposing to study, so maybe it’s better to think of a correspondence HZIH \leftarrow Z \rightarrow I [Sorry, in this post, I can’t seem to do a correspondence diagram with ZZ at a higher level.] between humanity and the space of ideas.

The question of what ideas are stable, what results or structures are robust under a change of ideas, and so on, become rather natural in terms of the geometry and dynamics of II (or perhaps the correspondence). Needless to say, I’m not proposing that one can do anything definite with this imagery. But perhaps it can aid in the discussion of what is or is not changing.

Posted by: Minhyong Kim on March 3, 2009 12:01 PM | Permalink | Reply to this

Stability, fruitfullness, in the Ideocosm; Re: Lakatos as Dialectical Realist

Mathematical objects are, to begin with, stable within the mind of the individual mathematician, and then robust in representation, and then stable on transmission of that representation to other mathematicians, which map back into mathematical structures stable in their minds.

The fruitfulness of the mathematical ideas are the next step beyond stability, where under perturbations (I had said mutations and natural selection previously) there are other mathematical ideas that survive and spread in the network of mathematicians.

In other threads I’ve discussed the topology of the Ideocosm (Zwicky’s term for the space of possible ideas).

Gregory Chaitin addresses the robustness of Mathematical thoughts in:

VII. Mathematics in the Third Millennium?
[Based on my talk on “Mathematics in the third millennium” at Tor Nørretrander’s fabulous Mindship institute, Copenhagen, summer of 1996. Also based on the interview with me conducted by Guillermo Martínez and published June 1998 in the Buenos Aires newspaper Página/12.]

That’s the fact that physicists know that no equation is exact—they’re merely good approximations in which one ignores lower-order effects, in which one ignores perturbations that operate on smaller scales. As Jacob Schwartz so beautifully put it in an essay in M. Kac, G.-C. Rota, and J.T. Schwartz’s anthology Discrete Thoughts—Essays on Mathematics, Science, and Philosophy, physicists know that all equations are approximate, so they prefer short, robust, unrigorous proofs that are stable under perturbations, to long, fragile, rigorous proofs that are not stable under perturbations (but that are perfectly okay in pure mathematics)… I also strongly recommend Gian-Carlo Rota’s anthology Indiscrete Thoughts. Among his other fascinating observations on doing mathematics, Rota makes the point that some mathematicians are mental athletes who like finding new proofs and settling old problems, while others are dreamers who prefer to find new definitions and create new theories.

Posted by: Jonathan Vos Post on March 3, 2009 4:05 PM | Permalink | Reply to this

If I may ask, Re: Stability, fruitfullness, in the Ideocosm

Mathematical objects are, to begin with, stable within the mind of the individual mathematician, and then robust in representation, and then stable on transmission of that representation to other mathematicians, which map back into mathematical structures stable in their minds.

Fine but…
Does anyone know (or care about?) how mathematical objects become stable?
Also, whenever “map[ing] back into mathematical structures stable in their minds”, doesn’t this happen via some discrete and finite text (the publication) of “reasonable length”, how does this works?
Would it be possible to model this communication bottleneck in spite of it being somehow “informal”?

Posted by: J-L Delatre on March 6, 2009 5:22 PM | Permalink | Reply to this

Re: If I may ask, Re: Stability, fruitfullness, in the Ideocosm

I think you are in effect asking how it is possible for humans to have stable concepts and to communicate the concepts. A possibly glib answer is that those humans that couldn’t didn’t leave many descendants. A less glib answer is: ask a cognitive psychologist.

In other words, I think your question is interesting and it must have a scientific answer.

Posted by: Eugene Lerman on March 6, 2009 5:55 PM | Permalink | Reply to this

Chaotic Attractors in the Ideocosm; Re: If I may ask, Re: Stability, fruitfullness, in the Ideocosm

“how it is possible for humans to have stable concepts and to communicate the concepts”

(1) How is it possible for humans to have stable concepts?

Cognitive Science establishes that we do not passively store memories and retrieve them. Rather, there is a dynamic reconstruction of a distributed system of sensory, abstract, emotional components into something close to the originally remembered concept.

“T.S Eliot in his essay ‘Hamlet and His Problems’ (1919)” introduced the metaconcept of Objective Correlative: “the only way of expressing emotion in the form or art is by finding an ‘objective correlative’; in other words, a set of objects, a situation, a chain of events which shall be the formula for that particular emotion; such that when the external facts, which must terminate in a sensory experience, are given, the emotion is immediately evoked.” (Selected Essays, [London: Faber and Faber, 1951], pp. 144-5).

I suggest that what we call a “concept” is akin to an attractor in the space of all possible concepts (ideocosm) that many trajectories of changing representations of concepts almost always stay close to.

(2) How is it possible for humans to have stable concepts and to communicate the concepts? The witten representation of a mathematical or literary concept (or musical expression of a melody, or kinaesthetic expression of a choreographic concept) is of much lower dimensionality than the subspace of the ideocosm from which it emerges, precisely because the attractor is of lower dimension than the space in which it is embedded.

To make this speculation crisper, we would need to establish the topology of the ideocosm.

In other threads of this blog we had some consensus that there must be a narrative, even in a Mth paper, to be effective, because our brains evolved to be good at interpreting, remebersing, reconstructing and telling, and extrapolating from stories.

Posted by: Jonathan Vos Post on March 6, 2009 11:11 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Conc.: “Humanity moves across it’s landscape.”:
Lars Gustafsson once described the human mind as symbiosis of biology and language in analogy to lichens being a symbiosis of fungus and photobiont. Perhaps one should extend that analogy of the human mind, as symbiosis of biology, language and ideas? In contrast to the former two constituents, the “ideas”-part would have not streamlined and roughly equally distributed by evolutionary pressure.

Posted by: Thomas on March 6, 2009 3:29 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

What troubles me about this suggestion is that biology, language and ideas are (implicitly?) put on equal footing. But try to imagine language without biology! And, no, I won’t accept programming “languages” as counterexamples. And there is no evidence that anyone besides humans can have ideas.

Posted by: Eugene Lerman on March 6, 2009 5:47 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

David wrote:

E.g., could you imagine a similarly sophisticated mathematics to today’s which had bypassed Lie groups?

I can imagine imagining that. There’s lots of math to think about; we could keep ourselves quite happily entertained without Lie groups, though we’d have trouble doing physics.

What I can’t imagine is a version of mathematics where they decided to classify simple Lie groups and didn’t come across E 8E_8 — unless, of course, they were just stupid.

So, I say E 8E_8 exists. If you look in the right direction, and look hard enough, you’ll see it… just like any star or planet.

I find it strange when people think it’s obvious that physical entities exist before we look for them, while mathematical entities don’t. It could be true, but it sure isn’t obvious to me… unless for some reason you’ve decided ahead of time build this asymmetry into your definition of ‘exist’.

Posted by: John Baez on March 7, 2009 4:15 PM | Permalink | Reply to this

Rules of Thumb to giant amoeba and Egan-Diasporae; Re: Lakatos as Dialectical Realist

Greg Egan, Greg Benford, Robert Forward and other have provided compelling narratives of how different Physical or Mathematical objects are “obvious” to different species of organisms in different ambient spaces.

The issue becomes, to what extent can we leverage our humanity to do good Physics and Math, without being so stupidly antropocentric as to miss what most civilizations find obvious?

We’ve agreed on the human evolution for Narrative here. We’ve agreed about how many dimensions we observe by unmediated sensory and sensory-motor hardwiring.

I still want to have someone do the experiment of designing a really good 4-D geometry computer game, and see how little kids master the play and then find Minkowski space “obvious.”

My friends able to do this – such as Michael Zyda who has 200+ students in the program at the USC GamePipe Laboratory, Research money flowing in, and is advisor to 6 startups plus trustee to 1 non-profit – well these people are VERY busy and have their own network of agendae.

Posted by: Jonathan Vos Post on March 7, 2009 7:52 PM | Permalink | Reply to this

agenda

Nitpick: the Latin word agenda is already a plural form (sing. agendum, thing to be done). Although it is commonly used as a singular form in English, the plural is definitely not “agendae”. I would say “agendas”.

Posted by: Todd Trimble on March 7, 2009 11:45 PM | Permalink | Reply to this

Re: agenda

This is one of my favourite words. Not only did it change from a foreign plural count noun to a vernacular singular mass noun (which is fairly typical); it went one step further and became a singular count noun. (We now use the back formation ‘agenda item’ in place of the original singular count noun, which was never well naturalised.)

Posted by: Toby Bartels on March 8, 2009 12:18 AM | Permalink | Reply to this

Re: Rules of Thumb to giant amoeba and Egan-Diasporae; Re: Lakatos as Dialectical Realist

“4-D geometry computer game, and see how little kids master..”
Toy cubes suffice.

Posted by: Thomas on March 8, 2009 1:11 PM | Permalink | Reply to this

Alicia Boole Stott and the Mimzies; Re: Rules of Thumb to giant amoeba and Egan-Diasporae; Re: Lakatos as Dialectical Realist

Yes, I was quite influenced in this conjecture by Alicia Boole Stott, about whom I’d read as a child. I worked hard at 4-D visualization when my brain was still plastic. My son rebukes me for not giving him the same advantage.

Alicia Boole Stott rediscovered the breakthroughs of the regular polytopes that were discovered before 1852 by the Swiss mathematician Ludwig Schläfli. He was not just a visionary Mathematician, but an expert linguist speaking many languages including Sanskritt and Rigveda. His complete treatise – rejected by the Imperial Academy of Science – “is an attempt to found and to develop a new branch of analysis that would, as it were, be a geometry of n dimensions, containing the geometry of the plane and space as special cases for n = 2, 3. I call this the ‘theory of multiple continuity’ in the same sense in which one can call the geometry of space that of three-fold continuity,” was published posthumously in 1901, and only then did its importance become fully appreciated.

Posted by: Jonathan Vos Post on March 8, 2009 7:04 PM | Permalink | Reply to this

Rigveda

… speaking many languages including Sanskritt and Rigveda.

Rigveda is not a language, but rather part the Vedas, which are ancient sacred texts of Hinduism, written in Sanskrit of course.

Not that this has anything to do with Lakatos of course. I just thought a correction was in order.

Posted by: Todd Trimble on March 8, 2009 7:29 PM | Permalink | Reply to this

Re: Rigveda

I stand corrected. Thank you, Todd.

Posted by: Jonathan Vos Post on March 9, 2009 4:14 AM | Permalink | Reply to this

Is God a Mathematician? by Mario Livio; Re: Lakatos as Dialectical Realist

Is God a Mathematician?
by Mario Livio
The Structure of Everything
A review by Marc Kaufman

Did you know that 365 – the number of days in a year – is equal to 10 times 10, plus 11 times 11, plus 12 times 12?

Or that the sum of any successive odd numbers always equals a square number – as in 1 + 3 = 4 (2 squared), while 1 + 3 + 5 = 9 (3 squared), and 1 + 3 + 5 + 7 = 16 (4 squared)?

Those are just the start of a remarkable number of magical patterns, coincidences and constants in mathematics. No wonder philosophers and mathematicians have been arguing for centuries over whether math is a system that humans invented or a cosmic – possibly divine – order that we simply discovered. That’s the fundamental question Mario Livio probes in his engrossing book Is God a Mathematician?

[JVP: we discussed whether math has “coincidences” in another thread]

Posted by: Jonathan Vos Post on March 7, 2009 9:03 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

What I can’t imagine is a version of mathematics where they decided to classify simple Lie groups and didn’t come across E 8E_8 — unless, of course, they were just stupid.

So, I say E 8E_8 exists. If you look in the right direction, and look hard enough, you’ll see it… just like any star or planet.

I consider myself a mathematical fictionalist, not a mathematical realist. But I agree that they would have to come across E 8E_8, because that's the only way that the story makes sense. Even in a story intended merely as entertainment, there's a difference between a narrative that hangs together and one that's internally inconsistent, but that doesn't make the features of the former narrative real. In mathematics, things are more clear cut (perfectly clear cut at the level of formal proof), so we don't get people arguing like movie critics about whether E 8E_8 should be taken out (although importantly, they may still argue about its significance).

Of course, I'm a reality fictionalist too; reality is a narrative that ties together my direct perceptions (including thoughts and memories). So I might say that nothing is real, except that when one goes through this sort of solipsism all the way and comes out the other side, one can redefine ‘real’ to refer to any feature of this narrative. So I talk like a naïve realist normally and refer to stars and planets as ‘real’ … but not mathematical ideas.

Posted by: Toby Bartels on March 7, 2009 11:57 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Maybe physical entities exist before we look for them, but I doubt that physical theories exist before somebody constructs them.

Posted by: Eugene Lerman on March 8, 2009 12:50 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

“Maybe physical entities exist before we look for them…”

To prove that seems suspiciously tricky…Some ideas coming from physical theories, like quantum cohomology, seem in contrast to have a more persuasive existence and even motivic incarnations.

Posted by: Thomas on March 10, 2009 12:15 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

“Maybe physical entities exist before we look for them…”

not

“Maybe physical entities exist before we look at them…”

In any case, I was not trying to suggest that one can separate perception and cognition. Perhaps I am making an obvious philosophical mistake, but it seems to me that physical entities don’t really have an independent existence of the theories they are embedded in. Some of these theories are acquired so early in our life we are not even aware of them, like object permanence. On the other hand I can imagine an alternative universe where humans don’t have something like the standard model. Do quarks exist for them?

Posted by: Eugene Lerman on March 10, 2009 4:25 PM | Permalink | Reply to this

When did atoms become real? Re: Lakatos as Dialectical Realist

Did atoms exist for us after Democritus, or not until Dalton, or not until Boltzmann, or not until Einstein explained Brownian motion, or not until Scanning Tunneling Microscopes? There were Big Name Scientists who considered atoms merely a mathematical fiction in 1900 A.D.

This makes it harder to relate epistemology and ontology of physical objects, if they are so linked to Physical Theory.

Did the luminiferous aether exist, and then vanish, to join its friends Phlogiston and Caloric? Or did it exist again after Einstein’s GR restored the Aether as space-time itself?

Cue Donovan: “First there is a mountain, then there is no mountain, then there is.”

Posted by: Jonathan Vos Post on March 10, 2009 6:33 PM | Permalink | Reply to this

Top 10 Beautiful Experiments; Re: When did atoms become real? Re: Lakatos as Dialectical Realist

Experimental Nonfiction

Science can’t solve the mystery of life, but it can make it a lot more fascinating.
By Jennifer Fisher Wilson

There are times in life when it seems that nothing ever changes — life goes on and on in the same frustrating old way, cliché after cliché. Decades go by. Then suddenly, or so it seems, important parts of life change instantly and forever: a word processing computer replaces my clumsy typewriter, a microwave oven defrosts my food in minutes, and a cell phone makes reaching me in Lisbon, London, or Milan as easy as reaching me at home.

Scientific notions change in much the same way: not at all and not at all, and then, boom, in a flash of inspiration, utterly completely. In The Ten Most Beautiful Experiments (released in paperback this month), noted science writer George Johnson describes how some of history’s most remarkable experiments overturned long-held theories about nature. At the denouement of each of these experiments, Johnson writes, “confusion and ambiguity are momentarily swept aside and something new about nature leaps into view.”…

Posted by: Jonathan Vos Post on March 10, 2009 9:36 PM | Permalink | Reply to this

Beautiful experiments

These comments are meandering far off topic, but with regard to

Scientific notions change in much the same way: not at all and not at all, and then, boom, in a flash of inspiration, utterly completely

that to me is a kind of romantic or heroic picture of science that’s pretty distorted. Mostly it’s not “not at all, not at all, then boom”; it’s mostly gradual incremental advances and patient assimilation, clarifying, and refinement, discovering the true ramifications and limitations of a theory, which may prepare the ground for the next boom. Not as splashy or fun to write about, but much more accurate.

Also, it would be great to use either quotation marks or the <blockquote> tags, so I can tell whether it’s you Jonathan or someone you’re quoting. By now I can usually tell the difference, but sometimes it gets confusing, and other people might get confused as well. Plus, it’s just good writerly form.

Posted by: Todd Trimble on March 11, 2009 7:07 PM | Permalink | Reply to this

Change theory, change objects? Re: Beautiful experiments

You’re right, Todd. The “Experimental Nonfiction” was entirely a quote (sorry I omitted punctuation) from the hotlink. I also reject that romantic notion; I was providing it to challenge Eugene Lerman’s suggestion that “physical entities don’t really have an independent existence of the theories they are embedded in.”

I was implicitly, maybe too cryptically asking: “When theory changes abruptly, did the physical objects change abruptly?”

Posted by: Jonathan Vos Post on March 12, 2009 7:20 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

On the other hand I can imagine an alternative universe where humans don’t have something like the standard model. Do quarks exist for them?

If by ‘quarks exist for them’ you mean that they believe in quarks, then of course no, since they don't know about quarks. So I'll assume that you mean that quarks exist in their universe (or even that they are made in part of quarks, like we are).

Now the answer depends on what you mean by ‘an alternative universe’ with ‘humans’. To begin with, suppose that this is simply somewhere else in the galaxy, with intelligent life. If they are in our universe, and particularly if we can interact with them (even if not so well that we can explain about quarks to them), then we must use quarks (at least implicitly) to understand them ourselves (even if our only understanding is in the context of this hypothetical discussion). So certainly I would say that they are made of quarks almost as much as I would say that we are made of quarks, with the ‘almost’ in there only for the very unlikely chance that the Standard Model is a merely local phenomenon.

At the other extreme, suppose that this is an entirely fictional universe which can never possibly affect ours (except in the entirely internal sense in which fiction in our universe affects reality in our universe by influencing people's actions). Then who knows whether quarks exist there; maybe they're made of magic; this is fiction! If we invent a fictional world, then we're quite capable of saying that they're made of quarks but don't know it, or that there are no quarks there, or even that they believe in quarks but are wrong.

Since you haven't specified this, then my answer is that I don't know.

Posted by: Toby Bartels on March 10, 2009 10:56 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

John wrote:

David wrote:

E.g., could you imagine a similarly sophisticated mathematics to today’s which had bypassed Lie groups?

I can imagine imagining that. There’s lots of math to think about; we could keep ourselves quite happily entertained without Lie groups, though we’d have trouble doing physics.

I’ve tried this question on other mathematicians who have been much more ‘inevitabilist’ than you. Of course, there’s a huge vagueness in the question as to what is meant by ‘similarly sophisticated’. Given that ‘we’d have trouble doing physics’, it could be argued that we wouldn’t be comparably sophisticated.

Anyway, this after dinner kind of question is raised more to point to the more general question of the kind and strength of the constraints under which mathematics is done. Let’s dig out that quotation from Weyl again:

Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.

(p. 136, ‘The Current Epistemogical Situation in Mathematics’ in Paolo Mancosu (ed.) From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s, Oxford University Press, 1998, pp. 123-142).

I’ve heard similar debates in philosophy of science about the way physics might have run, from extreme contingency, to the claim by Arthur Miller that there was only one way physics could have unfolded.

There are certainly plausible and implausible historical unfoldings. In the sciences the implausibility of some alternative histories point us to truth. Here’s a passage from Alasdair MacIntyre’s Tasks of Philosophy:

What…worries Kuhn is…: “in some important respects, though by no means all, Einstein’s general theory of relativity is closer to Aristotle’s mechanics than either of them is to Newton’s.” He therefore concludes that the superiority of Einstein to Newton is in puzzle-solving and not in an approach to a true ontology. But what an Einsteinian ontology enables us to understand is why from the standpoint of an approach to truth Newtonian mechanics is superior to Aristotelian. For Aristotelian mechanics, as it lapsed into incoherence, could never have led us to the special theory; construe them how you will, the Aristotelian problems of time will not yield the questions to which special relativity is the answer. A history which moved from Aristotelianism directly to relativistic physics is not an imaginable history. (p. 21)

The threat to the realist about physics comes from the charge that physical theories swing about wildly through their course with no convergence as concerns fundamental entities.

The threat to the realist (in my sense) about mathematics is the charge that logical constraints are all there are, so that mathematics is merely the accumulation of logical truths.

I would suggest that the kind of rich history of problem situation, resolution and reformulation exemplified by the emergence of the Lie group concept, through, among others, the work of Gauss, Riemann, Jacobi, Klein and Lie, is a sign of considerable constraint.

Posted by: David Corfield on March 9, 2009 11:21 AM | Permalink | Reply to this

Weyl at the Edge of Chaos; Re: Lakatos as Dialectical Realist

Weyl’s: “we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself” is consistent with my handwaving about concepts being attractors in the ideocosm.

This is because “the essence of man” or to be politically correct “intersection of restraint and freedom that makes up the essence of humanity itself” could be interpreted as: “we formulate our Mathematical concepts AT THE EDGE OF CHAOS” in the evolutionary sense of the Santa Fe Institute folks.

Music, to please the human, needs to be on the edge between a note or chord seeming to be inevitable as it fits into the melody, and yet surprising enough the be interesting. Granted, these parameters of note-wise entropy secularly vary, but within a certain range.

“restraint” in that the attractor is of lower dimensiona than the ambient space; freedom in that there is a chaotic component, and “lines of thought” are sensitive to initial condition and diverge exponentially fast.

Posted by: Jonathan Vos Post on March 9, 2009 5:18 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

I am guessing that a dialectic realist is a dialectic materialist mugged by reality.

Posted by: Eugene Lerman on February 27, 2009 3:02 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

There are certainly differences between Lakatos and Laudan, but remarkable similarities too. At the very least, by following them you can at last as a philosopher talk about ‘real mathematics’.

In the case of Lakatos there’s a strong Hegelian influence. In the acknowledgement to his thesis he mentions Hegel, Popper and Polanyi as the three sources. By the time he publishes the work in 1963/4 in the Britisih Journal for the Philosophy of Science, Hegel has disappeared.

When a book based on the thesis was brought out in 1976, however, we see Hegel very much still there. Lakatos’s second case study was the development of the concept of uniform convergence.

As thesis we have Leibniz’s principle of continuity, that the limit function of any convergent sequence of continuous functions is continuous.

The antithesis is provided by Cauchy’s more precise definition. This ‘legalises’ Fourier’s counterexamples, and excludes the possible compromise that they be counted as examples by allowing vertical lines. (You know that Fourier sum which is a saw-toothed shape curve, with vertical lines joining extremities of angled lines.)

The ‘positive pole’ gets strengthened by Cauchy’s proof, which will be the proof-ancestor of uniform convergence. The ‘negative pole’ gets strengthened by more and more global counterexamples to the primitive conjecture.

So then to the synthesis:

The guilty lemma to which the global counterexamples are also local ones is spotted, the proof improved, the conjecture improved. The characteristic constituents of the synthesis emerge; the theorem and with it the proof-generated concept of uniform convergence.

So rather than objects, as such, it’s really concepts which form the subject matter of dialectical improvement. For a powerful piece of Hegelianism (one the editors of the posthumously published 1976 book played down) try this:

The Hegelian language, which I use here, would I think, generally be capable of describing the various developments in mathematics. (It has, however, its dangers as well as its attractions.) The Hegelian conception of heuristic which underlies the language is roughly this. Mathematical activity is human activity. Certain aspects of this activity - as of any human activity - can be studied by psychology, others by history. Heuristic is not primarily interested in these aspects. But mathematical activity produces mathematics. Mathematics, this product of human activity, ‘alienates itself’ from the human activity which has been producing it. It becomes a living, growing organism, that acquires a certain autonomy from the activity which has produced it; it develops its own autonomous laws of growth, its own dialectic. The genuine creative mathematician is just a personification, an incarnation of these laws which can only realise themselves in human action. Their incarnation, however, is rarely perfect. The activity of human mathematicians, as it appears in history, is only a fumbling realisation of the wonderful dialectic of mathematical ideas. But any mathematician, if he has talent, spark, genius, communicates with, feels the sweep of, and obeys this dialectic of ideas.

Now heuristic is concerned with the autonomous dialectic of mathematics and not with its history, though it can study its subject only through the study of history and through the rational reconstruction of history. (Lakatos 1976, 145-6)

Lautman I understand less well. Still the subject matter is concepts, or rather ‘ideas’. For example, one we’ve talked about before,

Any case of imperfection presupposes a corresponding perfection, and it is possible to understand the attributes of the associated perfection through the defects of the imperfect entity.

With Lakatos, dialectic concerns some sort of autonomous play of concepts, the shadow of which is projected into the history of mathematics. Humans can allow this to shine through if they ‘feel the spark’.

With Lautman, dialectical ideas involve a pair of coupled concepts imperfection/perfection, local/global, etc. Somehow there’s a movement generated by these oppositions which reveals itself many times in the course of the history of human mathematics.

Nous voudrions montrer, avant de conclure, comment cette conception d’une réalité idéale, supérieure aux mathématiques et pourtant si prête à s’incarner dans leur movement, vient s’intégrer dans les intérpretations les plus autorisées du platonisme. (Lautman, p. 230)

Now, I don’t really understand what he means by the last point. That there are other ways of taking ‘platonism’ than the rather bland belief in abstract objects is clear. I would imagine understanding Hegel’s views on Plato would shed some light.

Perhaps you might wonder whether we need all this rigmarole. Why not just take mathematics to be a long human conversation about number and space, which shows clear signs of improvement over the centuries? I have quite a sympathy for that position.

Posted by: David Corfield on February 27, 2009 3:26 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Looks like I’ll have a chance to find out more about Lautman the dialectician.

Posted by: David Corfield on February 27, 2009 4:27 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

About Lakatos not Lautman

‘It is widely recognised that all practice and theories of learning and teaching rest
on an epistemology, whether articulated or not. As Rene Thom puts it, for mathematics:

“In fact, whether one wishes it or not, all mathematical pedagogy, even if
scarcely coherent, rests on a philosophy of mathematics.” (Thom 1973, page 204)’

Philosophy is an informal science so the philosophy of mathematics is informal.
Philosophy draws its inspiration from observation of physical reality. If informal
natural English were to be formalized (as was Sanskrit) the range of expression is
reduced. So formalized mathematics is also constrained to its axiomatic foundations
which do include all of the basis of informal mathematics, physical reality.

The layman concept of *dialectical is “a method used in philosophy to try to discover
truth by considering ideas together with opposite ideas. I think many people would
see an analogical relation/relevance to ‘proof by contradiction’ and *dialectical.

We put cats and dogs in the same category, mammals. But at a sharper categorical
focus, we separate them by dogs having non-retractable claws and cats having
retractable claws (except one species). Thus I think the broader definition which
is applied to what the function of dialectic covers does not map that property
identically with the function of dialectic within a formal proof. This is probably
obvious, however, I think there is a subtler implied limitation; that our theories
are not going to absolutely realize the full depth of physical reality.

Lakatos supported the Marx dialectical position not Hegel’s. Marx:
“My dialectic method is not only different from the Hegelian, but is its direct
opposite. To Hegel, the life-process of the human brain, i.e., the process of
thinking, which, under the name of ‘the Idea,’ he even transforms into an
independent subject, is the demiurgos of the real world, and the real world is
only the external, phenomenal form of ‘the Idea.’ With me, on the contrary, the
ideal is nothing else than the material world reflected by the human mind, and
translated into forms of thought.”

This view forms the basis of the Lakatos position which is close to social
constructivism. And it eschews a vision of mathematical realism with an archetypal
virtual realm (Platonism) or mathematical objects with an independent existence,
which one might lump together as somehow external to human origin, “God-given”.

Paul Ernest wrote:
“Currently, there is a move in some quarters to reconceptualise mathematics and
the philosophy of mathematics in fallibilist, human-centred and even social terms
(Davis and Hersh, Kitcher, Lakatos, Tymoczko, Ernest 1991a).

This reconceptualisation represents a break from the traditional absolutist views
of mathematical knowledge which see it as monological in character. Monologicality
is a central assumption of Cartesian rationalism and the modernist outlook based
on it. Mathematical knowledge is presented as if it is God-given, not uttered by
human voice, let alone by a one of several voices (albeit a dominant one) in a
dialogue or conversation.

Instead, my argument is that mathematics is dialogical, and that conversation
permeates mathematics in deep and multiple ways.”

Lakatos disapproved of the extent of the authority invested in formal proofs.
I think people who are critical of Lakatos in this regard are going to tend to be
the same people who believe in the independent existence of mathematical objects,
which derive their ontological authority in some version of dualism, rather than
solely through the trial and error progress of human invention.

Descartes’ mind/matter dualism holds that the mind is a nonphysical substance,
which is not explained in terms of the body generating it. Lakatos/Marx thought
that the mind and any other nonphysical concept such as mathematics, originated
from the physical universe -> physical body, which generated the mind; there is
no reality which is independent of the physical basis (called Physicalism).

Fundamentally, the physical universe evolved intelligent life which produced
humans who had physically produced minds which further evolved abstract ideas
through an extended period of developing culture with its strong social aspect.

I’m interested in AI which means I’ve encountered Lakoff and his description
of mathematics in terms of embodiment. There is a Lakoff paper which includes
the Lakatos notion about and under Proofs and Refutations” in Section 5.1

http://homepages.inf.ed.ac.uk/apease/papers/note1647.pdf
Using Lakatos’s methods to refine metaphors
“The development of some of the conceptual metaphors in (Lakoff and Nunez) can
be described in Lakatosian style: i.e. metaphors which are refined via contact
with counterexamples. For instance, in the mathematical world we might ask the
question: “what happens when we subtract 7 from 7?”

Since we have the idea of closure, we might expect the answer to be another
number. However, there is no collection which corresponds to this number. In
order to accommodate the metaphor then, we have to conceptualise the absence
of a collection as a collection. The Arithmetic is Object Collection
will then map the empty collection onto a number: zero. This is called an
entity-creating metaphor. Zero, and the corresponding empty collection, are
created ad hoc. Further properties of numbers now follow as entailments of
the metaphor:…”

David Corfield wrote:
That there are other ways of taking ‘platonism’ than the rather bland belief
in abstract objects is clear. I would imagine understanding Hegel’s views on
Plato would shed some light.”

SH: Perhaps this is true of Lautman. But I don’t think it is true of Lakatos.
Hegel seems consistent with platonism, but Lakatos endorsed the Marx dialectic
which leaves no room (it is opposite) for the primacy of the the Idea(l). The often
repeated observation about the unreasonable effectiveness of mathematics:
Using a physical basis, it would be quite unexpected if mathematics were not
shaped to predict and deal with physical reality as a facet of survival.

The physical basis requires the Theory of Evolution, which actually is not a
complete theory. There are gaps (physical->fossil record) and conceptual gaps
which prompted Gould to formulate his theory of punctuated equilibrium. So
a purely physical basis can be disputed. I think there is a lot less marvel
in the unreasonable effectiveness of mathematics when the nonphysical aspects
of mathematics are removed as a generating (other than strictly abstractly)
possibility; this philosphical notion (of Idealism) is what produces an
artifact of supposed unreasonable effectiveness. It’s similar to the “mystery” artifact
which mind/body Dualism creates for explaining consciousness. However, you
may have been referring to thesis, antithesis, and synthesis, rather than
the focus of the Marx (anti-Hegelian) quote that I presented early on.

Posted by: Stephen Harris on February 28, 2009 10:45 PM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

This is getting rather tangled as we invite more and more great dead thinkers to the party. Nevertheless, I’d like to say a word for Hegel (I’m not a Hegelian–like all great dead thinkers, his work is an instructive failure).

Plato distinguished between the intelligible world of ideas and the sensible/empirical world. To cut a very long and subtle story short, Hegel says: it’s all intelligible. You can be a materialist so long as you don’t know any micro-physics. Once you learn some physics, ‘brute’ matter ceases to be brute for you. You see that it is mathematically ordered. Keep doing physics; you’ll find that matter is intelligible all the way down. Hegel was not a Platonist; for him, there is no ideal hereafter, no separate higher plane of pure ideas. We live in Plato’s heaven and always have done, but it has taken us a very long time to come to a position to see this.

Why am I not a Hegelian? Because I don’t think there is anything inevitable about our arrival at this point.

Posted by: Brendan Larvor on March 6, 2009 11:55 AM | Permalink | Reply to this

Re: Lakatos as Dialectical Realist

Stephen says:

This view forms the basis of the Lakatos position which is close to social constructivism. And it eschews a vision of mathematical realism with an archetypal virtual realm (Platonism) or mathematical objects with an independent existence, which one might lump together as somehow external to human origin, “God-given”.

Let me dialectically oppose this point of view by pointing to a quotation taken from Lakatos in an article written by Brendan:

I have very strong feelings against Popper’s linguistic conventionalist theory of mathematics and logic. I think with Kneale that logical necessity is a sort of natural necessity; I think that the bulk of logic and mathematics is God’s doing and not human convention. (Philosophical Papers vol. 2 p. 127)

The reason why we need to continue questioning presuppositions, concepts, definitions, etc. is because we may have them wrong.

As far as naïve classification is concerned, nominalists are close to the truth when claiming that the only thing that polyhedra have in common is their name. But after a few centuries of proofs and refutations, as the theory of polyhedra develops, and theoretical classification replaces naïve classification, the balance changes in favour of the realist. (Proofs and Refutations, p. 92n)

He really isn’t ‘close to social constructivism’.

Posted by: David Corfield on March 6, 2009 12:19 PM | Permalink | Reply to this
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 4:42 PM
Read the post Afternoon Fishing
Weblog: The n-Category Café
Excerpt: A question on a picture of Spec(Q).
Tracked: April 23, 2009 12:31 PM

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