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April 3, 2009

A River and a Trickle

Posted by David Corfield

I was sifting through The Oxford Handbook of Philosophy of Mathematics and Logic to see whether it included any trace of my philosophical heroes. Lautman was always going to be an outside bet, but I reckoned on finding Lakatos there.

Conspicuously absent in the index, I resorted to Google Books to help me search, and there, along with some mentions of Lakatos as editor of a book that Kreisel had contributed to, I finally found in the introduction:

Sometimes developments within mathematics lead to unclarity concerning what a certain concept is. The example developed in Lakatos [1976] is a case in point. A series of “proofs and refutations” left interesting and important questions over what a polyhedron is. For another example, work leading to the foundations of analysis led mathematicians to focus on just what a function is, ultimately yielding the modern notion of function as arbitrary correspondence. The questions are at least partly ontological.

This group of issues underscores the interpretative nature of philosophy of mathematics. We need to figure out what a given mathematical concept is, and what a stretch of mathematical discourse says. The Lakatos study, for example, begins with a “proof” consisting of a thought experiment in which one removes a face of a given polyhedron, stretches the remainder out on a flat surface, and then draws lines, cuts, and removes the various parts–keeping certain tallies along the way. It is not clear a priori how this blatantly dynamic discourse is to be understood. What is the logical form of the discourse and what is its logic? What is its ontology? Much of the subsequent mathematical/philosophical work addresses just these questions. (Shapiro 2005, p. 10)

What on earth does that final sentence mean? Why bother dealing with Lakatos so briefly in the introduction to a 800 page handbook if he is not to be considered at greater length later?

Where we represent the state of philosophy of mathematics as two streams, it seems as though those working in one of them take theirs to be the Ganges and ours an irrigation ditch.

But the ditch continues to issue forth good, if expensive, work. Jean-Pierre Marquis has recently published From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory.

The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective.

I also have my hands on Ladislav Kvasz’s Patterns of Change, one of which,

…called relativization, is illustrated by the development of synthetic geometry, connecting Euclid’s geometry, projective geometry, non-Euclidean geometry, and Klein’s Erlanger Programm up to Hilbert’s Grundlagen der Geometrie. In this development the notions of space and geometric object underwent deep and radical changes culminating in the liberation of objects from the supremacy of space and so bringing to existence geometric objects which space would never tolerate.

One book I don’t have yet is an explicit attempt to bridge philosophy’s two cultures. It is called The Philosophy of Mathematical Practice.

Posted at April 3, 2009 4:04 PM UTC

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Re: A River and a Trickle

`left interesting and important questions over what a polyhedron is/

reading current math literature, I think the term is not yet uniquely defined!

polytope seems doing a bit better

Posted by: jim stasheff on April 3, 2009 5:50 PM | Permalink | Reply to this

Re: A River and a Trickle

David Corfield wrote:
Where we represent the state of philosophy of mathematics as two streams, it seems as though those working in one of them take theirs to be the Ganges and ours an irrigation ditch. ———-

I thought this was a very witty remark. Accordingly, I chose a few choice quotes about Lakatos from “The Philosophy of Mathematical Practice” edited by Mancosu.

“In addition to Lakatos’ work, the philosophical opposition
took shape in three books: Kitcher’s The Nature of Mathematical Knowledge (1984), Aspray and Kitcher’s History and Philosophy of Modern Mathematics (1988) and Tymoczko’s New Directions in the Philosophy of Mathematics (Tymoczko, 1985) (but see also Davis and Hersh (1980) and Kline (1980) for similar perspectives coming from mathematicians and historians). …

Evaluating the analytic philosophy of mathematics that had emerged from the foundational programs, Aspray and Kitcher (1988) put it this way:

Philosophy of mathematics appears to become a microcosm for the most general and central issues in philosophy—issues in epistemology, metaphysics, and philosophy of language—and the study of those parts of mathematics to which philosophers of mathematics most often attend (logic, set theory, arithmetic) seems designed to test the merits of large philosophical views about the existence of abstract entities or the tenability of a certain picture of human knowledge. There is surely nothing wrong with the pursuit of such investigations, irrelevant though they may be to the concerns of mathematicians and historians of mathematics. …

It is enough to think that Lakatos’ Proofs and Refutations rests on the interplay between the ‘rational reconstruction’given in the main text and the ‘historical development’ provided in the notes. The relation between these two aspects is very problematic and remains one of the central issues for Lakatos scholars and for the formulation of a dialectical philosophy of mathematics (see Larvor (1998)). Moreover, in addition to providing an empiricist philosophy of mathematics, Kitcher proposed a theory of mathematical change that was based on a rather idealized model (see Kitcher 1984, Chapters 7–10).”

Posted by: Stephen Harris on April 4, 2009 9:55 AM | Permalink | Reply to this

Face it!; Holey Mackeral; Re: A River and a Trickle

“The Lakatos study, for example, begins with a ‘proof’ consisting of a thought experiment in which one removes a face of a given polyhedron, stretches the remainder out on a flat surface, and then draws lines, cuts, and removes the various parts–keeping certain tallies along the way. It is not clear a priori how this blatantly dynamic discourse is to be understood….”

Was this an insufficiently rigorous derivation of Euler’s Polyedral Formula? I’ve experimented with self-discovery of this in High School Geometry, and made a mini-DVD of 20 minutes of the collaborative learning. I’ll be trying again in 2 weeks in another high school with 2 days of students building Platonic and Archimedean solids, and tabulating vertices, edges, faces. On a 3rd day, I’d give them a genus-1 solid (say, a cube with a rectangular parallelopiped hole cut from the Center of one face through to the center of the opposite face). I’ve had a small stellated icosahedron visible to them (made from a Zome kit) and carefully NOT pointed out its connection to the Euler material.

A formula relating the number of polyhedron vertices V, faces F, and polyhedron edges E of a simply connected (i.e., genus 0) polyhedron (or polygon). It was discovered independently by Euler (1752) and Descartes, so it is also known as the Descartes-Euler polyhedral formula. The formula also holds for some, but not all, non-convex polyhedra.

The polyhedral formula states
V+F-E=2, where V=N_0 is the number of polyhedron vertices, E=N_1 is the number of polyhedron edges, and F=N_2 is the number of faces. For a proof, see Courant and Robbins (1978, pp. 239-240).

The formula was generalized to n-dimensional polytopes by Schläfli (Coxeter 1973, p. 233), For genus g surfaces, the formula can be generalized to the Poincaré formula
chi = V-E+F =chi(g), where
chi(g) = 2-2g, is the Euler characteristic, sometimes also known as the Euler-Poincaré characteristic. The polyhedral formula corresponds to the special case g=0.

There exist polytopes which do not satisfy the polyhedral formula, the most prominent of which are the great dodecahedron {5,5/2} and small stellated dodecahedron {5/2,5}, which no less than Schläfli himself refused to recognize (Schläfli 1901, p. 134)…

Weisstein, Eric W. “Polyhedral Formula.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PolyhedralFormula.html

Posted by: Jonathan Vos Post on April 4, 2009 9:45 PM | Permalink | Reply to this

Re: Face it!; Holey Mackeral; Re: A River and a Trickle

uh oh, what is the difference
polytope versus polyhedron?

Posted by: jim stasheff on April 5, 2009 2:19 AM | Permalink | Reply to this

Polyhedra

Yeah, good question. The Mathworld article Jonathan is quoting from is rather confusing, especially when one tries to track down definitions according to other Mathworld articles, in an attempt to find out how the author is using his/her terms.

From the Mathworld article on polyhedra:

The word polyhedron has slightly different meanings in geometry and algebraic geometry [sic]. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is “polyhedra” (or sometimes “polyhedrons”).

The term “polyhedron” is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such “building blocks” as line segments, triangles, tetrahedra, and their higher dimensional analogs by “gluing them together” along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space of a simplicial complex (with the additional constraint sometimes imposed that the complex be finite; Munkres 1993, p. 9). In the usual definition, a polyhedron can be viewed as an intersection of half-spaces, while a polytope is a bounded polyhedron.

The first paragraph is maddeningly vague; the only attempt at precision comes a little later in the article where convex polyhedra are defined (compact set in 3\mathbb{R}^3 given as the intersection of finitely many half-spaces), which of course rules out the stellated things.

(The second paragraph just makes me think, “Well, which is it? Simplicial complex, or convex hull of finitely many points? Well, whatever, the generalized Euler formula holds in either case.”)

When I click on the hyperlinked term polytope from that first paragraph, I am led to

The word polytope is used to mean a number of related, but slightly different mathematical objects. A convex polytope may be defined as the convex hull of a finite set of points (which are always bounded), or as a bounded intersection of a finite set of half-spaces. Coxeter (1973, p. 118) defines polytope as the general term of the sequence “point, line segment, polygon, polyhedron, …,” or more specifically as a finite region of n-dimensional space enclosed by a finite number of hyperplanes. The special name polychoron is sometimes given to a four-dimensional polytope. However, in algebraic topology, the underlying space of a simplicial complex is sometimes called a polytope (Munkres 1991, p. 8). The word “polytope” was introduced by Alicia Boole Stott, the somewhat colorful daughter of logician George Boole (MacHale 1985).

Neither of these senses of ‘polytope’ embraces the star polyhedra proposed as counterexamples to the Euler formula; really this is just repeating that second paragraph from the polyhedron article. I conclude there’s some internal inconsistency in these Mathworld articles.

In order to sort out the confusion, you’d probably have to turn to the Coxeter reference, which is the famous text

  • Regular Polytopes, Third Edition, Dover 1973.

But you have to pay very close attention to how Coxeter modifies his terms as he goes along; the attempt to lift stuff from his book and condense it as in the articles above leads to a mess! Can’t trust everything you read on the Web, can you?

(Apologies for continuing an off-topic thread.)

Posted by: Todd Trimble on April 5, 2009 3:59 PM | Permalink | Reply to this

Re: Polyhedra

(Apologies for continuing an off-topic thread.)

I think the thread is very indicative of what I found odd about with Shapiro (the editor of that handbook) wrote about Lakatos. Discussion of how best to define ‘polyhedron’ is a mathematician’s task. Lakatos’s interest in it was not for its own sake, but as a case study of the way concepts can change through a kind of questioning process (dialectic).

Posted by: David Corfield on April 5, 2009 5:43 PM | Permalink | Reply to this

Coxeter, Nash, and Jacob T. “Jack” Schwartz; Re: Polyhedra

You are right on all points, Todd.

As a child, I read Regular Polytopes, Third Edition, Dover 1973, over and over until it fell apart at the cheap binding. I stared at the photographic plates as if peering out of the 3-D prison cave to the Platonic multidimensional world outside.

At a number of international conferences, my coauthor Prof. Philip V. Fellman (U. Southern New Hampshire) and I ask the Mathematical Economists in the room to raise their hands if they’ve actually read the short and hard-to-grasp PhD dissertation by John Forbes Nash, Jr. – usually no hands go up, sometimes just one. It is really about polytopes!

It was the virtually unknown Jacob T. “Jack” Schwartz who came up with a proof for what is known as the Nash embedding theorem. According to Dave Shield, one of Jack’s colleagues said that when Nash learned that Jack was working on the same problem, he became furious at Jack.( Dave suggests that his fury made him work even harder to solve the problem, so Jack served as an unwitting assistant in helping Nash find a proof.)

He also said while Jack had come up with a proof, he did not appreciate the profound implications of the theorem once he had proved it.

Halmos wrote that the guiding idea of one of Jack’s books is the spectral theory of a single linear operator, and its varied applications. Algebras of operators, specifically B*-algebras, enter only in ancillary function as aids in the proof of the spectral theorem, or as convolution algebras. This is all to the good, in view of the emphasis on concrete applications and special linear operators. It is rumored that von Neumann algebras and group representations are to be treated in detail in a separate volume by J. T. Schwartz anyway.

Posted by: Jonathan Vos Post on April 5, 2009 8:38 PM | Permalink | Reply to this

Proper citation

Jonathan: would you please not paste in other people’s words without proper indication that they are not your own? Please apply either quotation marks or (what I’d prefer) blockquote tags. The author of the quotation should also be given.

I am sorry to have to say this so harshly, but this really offends me, and it happens frequently in your comments. I’ve brought this to your attention before.

I refer specifically to the last paragraph you wrote, which was taken from this blog post by Dave Shields (not Shield), who is quoting Gian-Carlo Rota. I had to track this down myself.

Posted by: Todd Trimble on April 5, 2009 11:13 PM | Permalink | Reply to this

Yes, and yes^2 and Sorry; Re: Proper citation

Sorry, Todd. You’re right.

I thought that I’d cut-and-pasted the Rota URL, but couldn’t very well link to the Facebook postings that I was excerpting and editing.

You’re also right that the N-Category Cafe form entry seems to strip out indenting, forcing the commenter to had re-code block quotes back in. Sorry again.

As someone who spent 15 years and over $10^5 in the longest running case in the world’s largest judicial district as a plaintiff in a severe plagiarism case, and as a 5-time elected officer in the L.A. chapter of (and 2-time elected delegate to the National policy-making entity of) the National Writers Union, I am acutely sensitive to this as well. As Mathematicians, Scientists, Authors, all that we won in our Intellectual Property and our Reputations.

Open Source complexifies the issues.

Perhaps this forum should have a Facebook and/or a LinkedIn presence?

Shields, on Linked-In has continued the dialogue on Jack Schwartz. The Schwartz proof of the embedding theorem has to be considered double hearsay until that’s resolved.

Posted by: Jonathan Vos Post on April 6, 2009 7:36 PM | Permalink | Reply to this

Re: Yes, and yes^2 and Sorry; Re: Proper citation

It would probably be best to continue this particular discussion offline. Just a few points: (1) I wouldn’t have come on so strong (or said anything at all) if this were an isolated or an infrequent incident; (2) it seems to me that since one is obliged to hit “preview” before posting a comment, I have to assume that you know and take responsibility for what your output looks like before posting; (3) you should know I am not charging any bad intent like ‘plagiarism’ here, nor could I possibly make that judgment, but in an open forum like this, we need to trust one another to talk straight. I want to know which words are from your mouth and which are from someone else’s. A few simple keystrokes can make all the difference; the software on this blog is more than sufficient for this purpose.

I will cease further public discussion of this unless future events cause me to reconsider.

On the separate matter of Jack Schwartz: I do not understand why you called him “virtually unknown”. I think that is wholly false, and a bit disparaging to his memory. Virtually unknown to whom? Wasn’t he just as well known at the time as John Nash?

Posted by: Todd Trimble on April 7, 2009 1:10 AM | Permalink | Reply to this

Re: Proper citation

If you choose a Markdown-based Text Filter from the options on the submission form, then you can put the blockquotes in by hand very easily:

A

>B

C

produces

A

B

C

Posted by: Toby Bartels on April 7, 2009 1:02 AM | Permalink | Reply to this

Re: Proper citation

Dialogue continues on Facebook about the Mathematical Geneology of Jack Schwartz, how his tree connects to those of others, and anecdotes about how Forsyth recruited Knuth away from Caltech. Knuth kept saying that he wanted to finish this little book first, for which he was “half done.” The Art of Computer programming.

Thanks for the tip on Markdown-based Text Filter. I’ll try it next time.

Have proofreading to do now on what I hope is the last PDF of a Springer paper on Quantum Cosmology. The QM equations and the text mess up in different ways. I’ve been warned that typesetting is more error-prone at Springer since they outsourced a lot to India. Anyone else know more?

Posted by: Jonathan Vos Post on April 7, 2009 7:51 PM | Permalink | Reply to this

Re: Coxeter, Nash, and Jacob T. “Jack” Schwartz; Re: Polyhedra

At the very least, a URI would be quite nice.

Posted by: Toby Bartels on April 6, 2009 2:07 AM | Permalink | Reply to this

Re: A River and a Trickle

David Corfield wrote: “One book I don’t have yet is an explicit attempt to bridge philosophy’s two cultures. It is called The Philosophy of Mathematical Practice.” —-

This post is intended to compare somewhat, Shapiro and Mancosu as editors.

I appreciated Mancosu’s understanding and
his way of relating to David Corfield’s book,
“Towards a Philosophy of Real Mathematics”
considerably more than the focus of Arana’s
review of Corfield’s book. I also like that
Corfield is interested in the relationship
between category theory and logical semantics
(model theory). Mancosu’s insight partnered
him with Maddy properly.

The Philosophy of Mathematical Practice
Edited by Paolo Mancosu [Introduction]

“Within the traditional background of analytic
philosophy of mathematics, and abstracting from
Kitcher’s case, the most important direction in
connection to mathematical practice is that
represented by Maddy’s naturalism. … (page 6)
We will see how Maddy combines both the
influence of Quine and Godel. -> [Section 3]
..
Developing a formal language, such as Frege
did, which aimed at capturing formally all
valid forms of reasoning occurring in
mathematics, required a keen understanding
of the reasoning patterns to be found in
mathematical practice.

My strategy for the rest of the introduction
will be to discuss in broad outline the
contributions of Corfield and Maddy, taken as
representative philosophers of mathematics
deeply engaged with mathematical practice, yet
who come from different sides of the
foundational/maverick divide. I will begin with
Corfield, who follows in the Lakatos lineage,
and then move to Maddy, taken as an exemplar of
certain developments in analytic philosophy. It
is within this background, and by contrast with
it, that I will present, in Section 4, the
contributions contained in this volume and
articulate, in Section 5, how they differ from,
and relate to, the traditions being currently
described. For Corfield, contemporary philosophy
of mathematics is guilty of not availing itself
of the rich trove of the history of the subject,
simply dismissed as ‘history’ (you have to say
that with the right disdainful tone!) in the
analytic literature, not to mention a first-
hand knowledge of its actual practice.”

Posted by: Stephen Harris on April 6, 2009 2:05 AM | Permalink | Reply to this

Re: A River and a Trickle

I know at least one mathematician (a category theorist, in fact) who is actively irked by the activities in the Ganges and would apparently like to see more mathematical philosophy from the irrigation ditch.

Off-topic: cafe regulars may find Michael Barr’s response to a related post on that blog provocative.

Posted by: Mark Meckes on April 7, 2009 4:34 PM | Permalink | Reply to this

Re: A River and a Trickle

Thanks. I see you (and Charles Wells) are from the home of my friend and fellow ditch-dweller Colin McLarty.

Posted by: David Corfield on April 7, 2009 4:47 PM | Permalink | Reply to this

superfluid craft of doing math; Re: A River and a Trickle

Wells writes: “I want to gain a scientific understanding of the craft of doing math. “

Yes! Yes in bold, italics, underlined!

That is also at the core of any proper Pedagogy of Mathematics. As a Math Teacher, I feel a keen responsibility to at least read such research, whether or not I actively contribute. Little flashes of this appear from time to time in the publications of the National Council of Teachers of Mathematics, and the ACM, and the AMS.

Most mathematicians pass through high school at some time, and how deeply their teachers understand “the craft of doing math” is at least as important as mere content knowledge of the State Standards for Mathematics, or other rigid framework.

I left as adjunct professor of Math (which I enjoyed) to teach in high schools because I so keenly feel the need to reach teenagers, engage their aesthetics, make Math fun, connect it to their lives, give them a taste of problems whose solutions we have not yet found, and show my joy for the field.

I am so grateful to Corfield, and Baez, and others here who inspire me even after hard days in the classroom (currently Algebra 1, Geometry, AP Statistics, and AP calculus). I am struggling to connect the philosophy discussed here, and new neuroscience, and other tributaries of the watershed.

Other streams may flow downhill, but it seems to me that Math is a fluid which can trickle uphill. Does that make it a superfluid?

Posted by: Jonathan Vos Post on April 7, 2009 8:05 PM | Permalink | Reply to this

Re: superfluid craft of doing math; Re: A River and a Trickle

JVP wrote:
how deeply their teachers understand the craft of doing math

I am forever grateful to my HS geom teacher
who knew that a valid proof did not have to be the one in the book

In those days, we had the statewide regents exam in NY which included `originals’ i.e. theorems of which we had not seen the prrof.

Posted by: jim stasheff on April 7, 2009 8:22 PM | Permalink | Reply to this

Regents; Re: superfluid craft of doing math; Re: A River and a Trickle

The Regents saved me time and time again. There’d be a class where I was doing indifferently, or even flunking, as the teacher thought, usually because I was bored. Then I’d get 100% on the Regents, and suddenly the teacher-student relationship would become warmer and more nurturing

For non-New Yorkers, Wikipedia’s entry begins [and I beta-test the Markdown-based Text Filter for blockquote]:

Regents High School examinations, or simply The Regents, are exams given to students seeking high school Regents credit through the New York State Education Department, designed and administered under the authority of the Board of Regents of the University of the State of New York. Regents exams are prepared by a conference of selected New York teachers of each test’s specific discipline who assemble a “test map” that highlights the skills and knowledge required from the specific discipline’s learning standards. The conferences meet and design the tests three years before the tests’ issuance which includes time for field testing and evaluating testing questions.

Posted by: Jonathan Vos Post on April 7, 2009 8:42 PM | Permalink | Reply to this

Re: Regents; Re: superfluid craft of doing math; Re: A River and a Trickle

In senior year, a very good friend and I had scheduling conflicts so he took Trig and I took Solid Geometry (or vice versa)
and then each `taught’ the other. The difference was clear - we each scored 100 on the class we had taken and only 98 on the subject we learned from each ohter.

Posted by: jim stasheff on May 15, 2009 3:31 PM | Permalink | Reply to this

Re: A River and a Trickle

No Cassirer in the Handbook either.

Posted by: David Corfield on May 15, 2009 8:11 AM | Permalink | Reply to this

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