Monday, January 15, 2007

Transgressing the Boundaries: Toward a Non-Hegemonic Feminist Mathematical Theory

The title is an homage to Alan Sokal's witty demonstration of the vapidity of the whole postmodernism industry. I've always considered myself rather fortunate, as a mathematician, to engage in research safely outside the reach of the PC commissars. Yet every now and again the hydra thrusts one of its many heads into our pristine art; this time, under the heading of "social justice". I am not going to attempt to define justice here (I am not even sure that I can at all), but if there's one thing I'm sure of, it's that mathematics has nothing to do whatsoever with justice, or for that matter, with any aspect of the physical world. Sure -- physicists and engineers have used mathematical tools with great success to build scientific theories and neat gadgets. But math lives in its own separate Platonic world and we mere mortals can only hope for an occasional peek inside (much as no one "invented" electricity, there are no mathematical "inventions", only discoveries).

Which brings us to this pearl of wisdom (via Alexandre Borovik):

The content of mathematics is typically thought of as neutral. That is, to most, mathematics is considered a domain that is devoid of ethical-moral implications. One can use mathematics for whatever purposes one wishes, but the mathematics itself is not good or bad, it just is. If I am right and mathematics classrooms frequently teach powerlessness, then the notion that mathematics is essentially neutral needs to be revisited.

This reminds me of a Russian joke. A teacher in elementary school is explaining long division on the board. Suddenly, a KGB officer walks into the classroom and listens in for a while. During break, he walks over and exchanges a few quiet words with the teacher. When class resumes, the teacher announces, "Class, we've received directives from above: from now on, division will be done like this..."

I've been maintaining for years -- I believe it's my original observation -- that some of humanity's greatest atrocities were committed when people mixed the rational and the ethical spheres of reasoning. The rational/scientific sphere deals with measurable quantities and exact facts. It is a tool, completely devoid of ethics and morality: it can help you cure cancer or build a bomb -- your choice. The ethical/moral/religious sphere deals with questions such as, "What is the right thing to do?" When religion encroaches on matters of science, we get the Inquisition. When rationality overrides morality, we get Nazi experiments on humans.

It is a common philosophical mistake to endow inanimate objects with moral characteristics. A gun is not inherently good or evil -- it can be used to commit murder or save lives. It's a tool, like a hammer or a ruler. Only conscious sentient beings can be judged on a moral scale and labeled as good or evil.

The author of that piece -- Kurt Stemhagen -- makes two fundamental mistakes. First, he conflates mathematical constructions with their real-world applications. Second, he attempts to impose a moral rubric on an inherently a-moral realm of reasoning. (Teichmüller was a Nazi mathematician who actively persecuted Jews. Should we boycott the study of Teichmüller spaces?)

19 comments:

Anonymous said...

The first flaw in your argument is to imagine that it is somehow possible to separate ethical and rational discussions. Even to claim, as you do, that they are distinct, is to adopt a particular ethical stance. This stance, although common among scientists, is not uncontested -- much of XXth-century philosophy and sociology is devoted to demolishing it. For the consequences for our society of such a stance over the last 3 centuries, see the book "Cosmopolis" by Stephen Toulmin (Chicago UP, 1990). Your argument would have more weight with me were you to address the issues Toulmin raises in his book.

A second flaw is to imagine that there are such things as "measurable quantities and exact facts" which are independent of those doing the measuring or the fact-checking. Let us start by asking which quantities and which facts will be measured and assessed, and why? As soon as this question is asked social, historical, cultural, idiosyncratic and other contingent factors enter the picture, and once again, ethical issues are intertwined with the so-called rational.

If you don't believe this applies to pure mathematics, you should read the history of non-Euclidean geometry in the XIXth century. This history is one long battle between competing views of what geometry is or could be, views informed by different ethical positions. Ditto, for the history of constructivist mathematics in the XXth century; for the history of uncertainty formalisms (probability theory and its competitors) over the last 30 years; for the arguments over category theory versus set theory as a foundation for mathematics in the same period.

Modern pure mathematics is pervaded with ethical issues, just as is every other human activity. This is something to celebrate, not to deplore.

Leo said...

I'm glad we've got a debate going. I won't be able to respond for a couple of days due to giving talks and traveling. But I'll address this as soon as I can, promise! In the mean time, Aaron, Steve -- wanna jump in?

Anonymous said...

"Mere purposive rationality unaided by such phenomena as art, religion, dream and the like, is necessarily pathogenic and destructive of life; and \ldots its
virulence springs specifically from the circumstance that life depends upon interlocking
circuits of contingency, while consciousness can see only such
short arcs of such circuits as human purpose may direct."



Gregory Bateson [1972]: "Style, Grace and Information in Primitive Art." p. 146 in: Steps to an Ecology of Mind. New York: Ballentine Books.

Osame Kinouchi said...

Sorry, Leo, but you also must remember that the platonic view of mathematics is very much questioned and even minoritary in the philosophy of mathematics. Notice that I am sympatetic with the platonic stance, but I do not have strong arguments to support it.

Noam said...

I don't see how the status of mathematical platonism has any role in determining the ethical content of mathematics -- but maybe I don't understand your version of platonism? Can you explain your statement, "mathematics has nothing to do [...] with any aspect of the physical world"? I think most platonists would acknowledge that most mathematical constructions at least model real-world phenomena. And I don't mean necessarily physical phenomena -- the fact that not only physicists, chemists, and biologists, but also economists, sociologists, philosophers, musicians, writers, etc., can all apply mathematics seems to me to be clear evidence that mathematics is very much about the real world. (Correspondingly, I think the relevance of mathematics to real-world "ethical phenomena" is also clear.)

SJMiller said...

There are many things in mathematics that are easily arguable: whether or not to accept a certain axiomatic system (and there are some very heated arguments here), whether or not certain types of mathematics can or should be done, as well as whether or not mathematics corresponds to the real world (whatever that is). Once you make your decisions on those points, however, I basically agree with Leo on the 'purity' of mathematical argument.

In regards to what Sokal's argument demonstrates: I think there are many people who could be classified as Unclothed Emperors, both in terms of the content of their work and their lack of understanding of their status. Of course, different people will value the same work very differently. I know I've proved theorems which I like far more than my colleagues. Fortunately for my career it seems I've also proved a few results that they like as well!

Anonymous said...

Noam said: "And I don't mean necessarily physical phenomena -- the fact that not only physicists, chemists, and biologists, but also economists, sociologists, philosophers, musicians, writers, etc., can all apply mathematics seems to me to be clear evidence that mathematics is very much about the real world."

I think we should be very careful about terminology here. Rather than applying pure mathematics, much of physics, et al, is using pure mathematics to describe and reason about some real phenomena. In that case, it is very hard to see how pure mathematics could be other than useful. In other words, if pure mathematics was not useful for string theory (say), how could we tell, since our only means of apprehending the phenomenon (that which string theory is about) is through the language (mathematics) we use to describe, model and reason about it.

Thus, the use of mathematics in physics, et al, is not necessarily support for the statement "that mathematics is very much about the real world." It is merely support for the statement that human beings have a prediliction for using mathematics to study the world.

Anonymous said...

"mathematics has nothing to do [...] with any aspect of the physical world"


I think for pure mathematicians, this has been the case for over a century. Jeremy Gray, in his history of non-Euclidean geometry, wrote this about the work of Mario Pieri, the first axiomatic treatment of geometry (published 1895, which predated Hilbert's Foundations of Geometry by four years):

"A third Italian, Mario Pieri, took the decisive step of breaking with Pasch, and abandoned completely any intention of
formalizing intuitions based on experience. Instead, as he wrote
in his "Sui principi che reggiono la geometria di posizione"
of 1895, projective geometry is treated "in a purely deductive
and abstract manner . . . independent of any physical
interpretation of the premises," and primitive terms, such as
line segments, "can be given any significance whatever, provided
they are in harmony with the postulates which will be
successively introduced." In his presentation of plane
projective geometry of 1898, the "Principii della Geometria
di Posizione", he put forward nineteen axioms (typically: any
two lines meet). The idea that geometry should be studied
entirely rigorously and with no appeal to intuition, which had
become something at best redundant and at worst dangerous, was now abroad."

-- Page 114 of:

@BOOK{gray:book04,
author = "Jeremy Gray",
title = "Janos Bolyai: Non-Euclidean Geometry and the Nature of Space",
publisher = "Burndy Library",
year = "2004",
address = "Cambridge, MA, USA"}

Leo said...

I am dead tired and taking a quick breather, so will not attempt to address all the issues.

(1) To the one or more anonymous commenters -- could you at least pick a consistent nickname for this thread, for ease of reference?

(2) Anon #1 -- could you summarize the main points of Toulmin's book?

(3) Being that I'm barely coherent even to myself, let me sidestep the various subtleties and be as blunt as possible. Pythagoras's theorem is a statement about objects that have no width, mass, or time duration. It is not a statement about depressions in sand, sticks, or strings. If this theorem happens to help a physicist model some aspect of nature, so much better for the physicist/nature. If it happens to be of use for easing the human condition, so much better for the human condition. If it is used to commit evil, so much worse for the perpetrator/victims. The fact that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse was true long before Pythagoras or even planet Earth was around; that it was discovered by some humans (long before Pythagoras, actually) has no bearing on its validity. What does ethics have to do with any of this?

Noam said...

Anonymous Three: that's a good distinction, though I'm not sure it's all that important to make it here. I spoke of models to appeal to Leo-the-Platonist. Whether mathematical abstractions are used as models or as actual descriptions in, say, physics, in any case they are relevant to the physical world. In the former instance they serve as approximations, in the latter as our best current understanding of the physical world.

Leo: your example of Pythagoras's theorem attacks two strawmen -- I will only address the second. No one is claiming that the theorem is an ethical judgment, nor that its truth is in any way dependent upon ethics. Are you reading this in Stemhagen's article? I thought the article was not all that exciting, as its claims are very modest. Basically (and I apologize for the pithiness) I read it as saying "mathematical education should teach critical thinking, since critical thinking is important for social justice." He's basically making the old criticism of teaching mathematics soley through the "Euclidean method" (cf. Imre Lakatos' Proofs and Refutations), though also combining it with an (I think more novel) pedagogical critique of constructivism.

Osame Kinouchi said...

What about the mathematics of games? Consider chess. I suppose that a lot of theorems could be found for chess, for example, if there is an always winning overture or not, the number of different plays, and other combinatorial issues.
But that does not means that chess existed before its invention in India... Or God plays chess?

About the ethical side, the chess example also could be relevant: chess is not only an abstract rule based game. It also embodies a worldview (an warrior society based in casts). So, by osmosis, we absorb this wordview (ok, this is not so serious, or is it?)

So, is it possible that pure mathematics cultivated at a given time also embodies some ideological emphasis of that time? For example, is Von Neuman game theory pure math or not? Is Nash equilibrium pure math or not? But there isnt some ideological wordviews that motivate these mathematical developments? Should we stay unaware of these subliminar influences?

Leo said...

Well, I'm back in Pittsburgh, though things will be hectic for a few more days.

I think we've touched upon some deep issues in this thread, and to address them all fully will require a new post.

In the meantime, let me concede several points. Yes, the boundary between "discovery" and "invention" is ineed blurry; I am not sure I can give a meaningful answer to whether chess was invented or discovered. I also concede to Noam's point (perhaps not explicit in the comments but certainly so in our conversation!) that rationality should definitely play a role in ethics; it was wrong of me to lump ratioinality on one side of the morals/science dichotomy.

There are a few points I am still prepared to fiercely defend, however.

1. Mathematical statements are only valid regarding abstract objects. Any mathematical inferences regarding the physical world are solely in the eyes of the model-builder, any false assumptions or erroneous conclusions are his responsibility.

2. The ultimate responsibility for the (mis)use of a tool lies with the user, not the tool-maker.

3. The phrase "social justice" has a particularly ominous connotation of intrusive, oppressive social engineering policies. To see this used in the context of mathematical education sends shivers down my spine.

Looking forward to elaborating on these next time I get a breather,
-L

Kevembuangga said...

The fact that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse was true long before Pythagoras or even planet Earth was around;

Bwahahaha!
A hard core platonist!
Suppose I am an ignorant alien in whose world I NEVER met anything like a "line", a "triangle", a "right angle", etc...
How would you convey to me an UNDERSTANDING of Pythagoras theorem?

P.S. I do agree about the disconnection of maths/logic from ethics.

Leo said...

Kevembuangga,

Your comment about the aliens raises some fascinating questions, far beyond the scope of what I can address in a humble comment of a lowly blog.

Before we can reason about coveying "lines" and "angles" to aliens, we need to figure out what possibilities there exist for life and intelligence radically unlike our own. I won't attempt to define either "life" or "intelligence" here, though meaningful definitions exist and I could probably produce some at gunpoint. But let me go out on a limb and say that any intelligence sufficiently complex to reason with at all, can grasp the notions of Euclidean geometry. This excludes, for example, chimpanzees -- who are highly intelligent, but lack the requisite power of abstraction.

Kevembuangga said...

Hi Leo,

Take it easy, I am not a religionist and you missed my point entirely.

Your comment about the aliens raises some fascinating questions, far beyond the scope of what I can address in a humble comment of a lowly blog.

My mention of aliens was not an attempt to introduce grand speculations about "life", "intelligence" and what have you, but meant only to have you look at UNHEEDED assumptions you might have about Platonic "entities".

Before we can reason about coveying "lines" and "angles" to aliens, we need to figure out what possibilities there exist for life and intelligence radically unlike our own. I won't attempt to define either "life" or "intelligence"

As per above, let's drop "life" but wouldn't it be of some interest to peer a bit into our ordinary human intelligence, yours and mine, no more no less.

But let me go out on a limb and say that any intelligence sufficiently complex to reason with at all, can grasp the notions of Euclidean geometry. This excludes, for example, chimpanzees -- who are highly intelligent, but lack the requisite power of abstraction.

Assuming I (as an hypothetical alien/savage) am endowed with exactly the same potential "power of abstraction" as any human, do you mean that "the notions of Euclidean geometry" are already embodied in this potential, such that their "existence" is mandatory for any kind of intelligence of similar (or higher) "power of abstraction"?
Or is there "something else" which would be needed?
Be at rest that I do not mean by this any kind of "spiritual" or "vital" properties but some background experience held by the said potential intelligence.
In which case what would be the "nature" (not the right word but I don't have any other at hand for the moment) of this experience?

I am afraid you are probably not really interested in such "philosophical" arguments since they would distract you from the fun of doing "hard maths" but if you were to argue about Platonism (which I find a laughable position, sorry!) you could get some "flavor" of my opinions in this comment (not to needlessy repeat myself and clutter the comments space).

Leo said...

I actually find discussions of the nature of life/intelligence more satisfying, since they tend to lead to more insights.

What we're engaging in now is reminiscient of the old "if a tree falls in the forest does it make a sound" debate. Platonism is not a scientific theory; it's not even much of a mathematical philosophy. It has internal problems, as pointed out by Osame. While I feel comfortable tagging certain bits of knowledge as inventions or discoveries, others are not so obvious. Is chess an invention or a discovery? I'm not sure there's a meaningful answer...

Is the law of the excluded middle a product of our minds or a necessary fact of (any) reality? I would go with the latter, but that's intuition solely based on the experiences of someone who can't fathom a different kind of reality.

I am very much interested in philosophy of math -- though I tend to focus on other aspects, such as the process of mathematical discovery and proof techniques/validity. But you're right -- at the end of the day, I'm a mathematician, whose job is to produce math rather than reason about it.

Still, thanks for the pointer -- will read when I get a chance, and looking forward to more of your inputs.

Anonymous said...

First, I think it's patently unfair to use such an obscure essay -- whether it is good or bad -- to tar anything as broad as "postmodernism." A parallel that you might understand: imagine I dug up some crackpot's proof that pi is a rational number, eviscerated it, and then on that score attacked geometry as intellectually suspect.

Second, I don't think you have any understanding of the significance of the Sokal hoax; neither the tendencies that he was trying to lampoon, nor the nature of the journal that he was published in, nor the incoherencies of his critique.

Third, although the essay you linked to seems poor, I can't find anything objectionable about the overarching theme that justice is a good thing and that, in so far as possible, education should serve the ends of justice. Your intuitions about scientific rationality going poorly with ethics are humorous for two reasons:

(1) If there is anything called "postmodernism," its primary theme is to criticize attempts to join rationality and ethics, which seems to be your position exactly.
(2) Most reasonable people think this is a highly problematic position, and that most attempts to marry instrumental/scientific rationality with ethics have gone swimmingly.

Leo said...

Anonymous:

Your statement, "I don't think you have any understanding of the significance of the Sokal hoax" is an odd one to make, since you (presumably) don't have direct access to my mental states. Just how do you know what I do or don't understand? All the same, do feel free to enlighten us on the real significance of the Sokal affair.

To address your other points:
1. I was using Sokal's, not Stemhagen's piece to excoriate postmodernism. The latter is a sad and -- yes, I agree -- insignificant symptom of our PC culture.
2. If a crackpot proof of pi's rationality were published in, say, the Journal of Number Theory, that would certainly be bad news for the publication. Obscure or not, the Philosophy of Mathematics Education Journal is responsible for what appears in its pages.
3. Justice is indeed a good thing, but a rather elusive one, and I've amply argued that it has nothing to do with math.
4. Statements of the type "in so far as possible, education should serve the ends of justice" are naive at best and sinister at worst. What if you and I disgree vehemently about what's just? What if your idea of justice involves (say) a radical redistribution of wealth while mine involves minimal intrusion into people's lives and property? Should I be re-educated to adopt your more progressive view?
Education should impart to students the necessary knowledge and skills to become competent and productive members of the society. Rather than indoctrinating students with a particular ideologue's notion of justice, we should be teaching them math, science, history and literature.
5. On ethics and science: the two spheres of reasoning do have points of contact (cloning and stem-cell research are scientific problems with genuine ethical components). But any question you ask still cleanly falls into a scientific or ethical realm. Science deals with the limits of what humans, in principle, can do. Ethics deals with what they ought to do.

Kevembuangga said...

It happens that I am currently involved in discussions about scientism/platonism/epistemology/etc... at various places, if some people were interested in pursuing the arguments it would be nice to pick a single thread and stick to it.
I suggest using the GNXP thread because it is the latest and the comments facility is quite good compared to others.