## Math as Semiotic: Mediation, Mathemes, and Politics

by Christopher Vitale (crossposed at *Networkologies*)

It is time for philosophy and theory to re-engage with math. This has already begun via the hugely influential work of Alain Badiou, but the sort of math that Badiou has introduced into contemporary continental theory is, I think, part of the problem. That is, it continues to mystify, and uses that mystification to produce some of the slight of hand maneuvers that give it its force. The goal of a re-engagement with math needs be to de-mystify, not the other way around.

*Some Historico-Political Stakes: Math Education, Racism, and Structuralism*

Much of the reason for philosophy’s disengagement with math has to do with historico-social factors. Take the case of so-called ‘analytic’ philosophy in the Anglophone world. It is impossible to study ‘continental’ philosophy – that which is considered philosophy in the majority of the world, and has been in the Anglophone world as well for most of its history – in these departments. Much of the hegemony maintained by the analytics in the Anglophone world is maintained, I believe, by the use of mathematical and logical symbols. In a university system dominated by capitalist interests, anything that looks like science is likely to get more funding, because it is easier, I think, to persuade non-specialist university administrators to fund what looks ‘like’ science and math.

Math symbols, and math in general, is often used by those who posses certain types of knowledge to exclude others. The case of analytic philosophy is a minor version of this, but there is a much more consequential version of this. For anyone who has read my postings on secondary education, I firmly believe that math education is used to do one thing in the country at this point – make sure there is a permanent underclass who fail high-school. That is, I believe math education in this country as it is currently structured serves to perpetuate a racist status-quo, and I believe that the perpetuation of this racist status-quo is the largely (unconscious) reason why math is taught the way it is.

Here’s the quick version of this argument. The large majority of folks today never need more than first year algebra – if that – in their everyday lives. And in the age of computers, few of us will ever even do long division again! Why then teach this difficult subject? Firstly, you do need math to do science, but most of the math needed to do even college level science goes beyond that same first year of high-school algebra. That which does can require a single ‘math for advanced science’ prep-course in college. Beyond this, however, proponents of math say it produces logical and rigorous modes of thinking in situations in which multiple answers might be required. Students will then, supposedly, carry these skills into any area they pursue later in life. But why not teach the ability to deal with difficult situations in the real contexts they might face later in life? We’d do much better teaching high-school students how to multi-task and problem solve in situations they might encounter in real life, rather than in the abstracted versions thereof, divorced from any context whatsoever, that they encounter in math class. Why then do we teach math this way?

Much of this has to do with the history of math in the twentieth century. Math education went through a massive overhaul as a result of the desire for ‘rigor’ and the unification of math under the aegis of set-theory in the early twentieth century. We need to recognize that this movement is a thoroughly modernist phenomenon. Math, like so many other fields of study, from literature to painting to architecture, went through a foundations crisis. And math decided that its foundations were at the most abstract level, those most divorced from practice, and that students needed to be taught these fundamentals before applications. A similar situation occurred in art education – seemingly the most different field of study than math! But in art education, the influence of the Bauhaus, with its desire to systematize the fundamentals of art, for example, see the writings of Kandinsky, were enormously influential, particularly in the United States via the influence of the Chicago neo-Bauhaus in the 50’s, in reformulating art education in this country. Students were made to learn highly abstract foundations before then reconstructing art from the abstract back to application.

Math curricula went through just as massive an overhaul in the postwar period as did the arts, and in a similar, modernist inspired vein. The influence of Bourbaki in France and beyond went a long way in working to standardize math education, certainly at the advanced level, as being organized around a modernist, hierarchical, foundational model. And in the US, curricula were overhauled to include bits of set theory and logic, while application and context were purged. While aspects of these have been at times brought back, math education suffers the hang-over of a modernist desire for foundations (as does art education in this country today!).

Why then do we still teach math the way we do? Or as much as we do? Whenever Republicans run for office, the only thing good they say about education is that we need higher standards (read: standardized testing) for math and science education – in the interests of national defense. But we don’t teach math to the masses to produce more rocket scientists. Rather, we do it to produce quantifiable numbers to compete with other nations, and to quantify to politicians how education is doing. Of course, most educators will tell you that standarized tests poorly teach most skills – other than math. But my question – why are we teaching so much math, and such difficult math, to those who won’t need it later in life? And when teaching it to those few who will need it later in life can be done with some rudimentary catch-up courses?

I believe there are two answers. Firstly, we teach math in secondary schools the way one would teach youngsters who want to eventually become mathematicians. Many of the skills students learn are little foretastes of topics they will only understand the need for later. For example, when I was taught matrices in high-school, the only justification for these was ‘you’ll need them in higher math’. Of course, I never took those higher math courses. And the same goes with many topics in math classes. The rationale for teaching quadratic equations is never, as math is currently taught, to solve science problems, but rather, you’ll need them for precalc! Math as a discipline believes in purity – to the extent that ‘applied mathematicians’ are looked down upon in the mathematics community. But in secondary education, math has no business being there except as applied!

Why then, in my opinion, do we really teach math the way we do? In order to make sure enough people fail high school. As someone who tutored math and science to high-school students (individual and in classes) for over a decade (it was my part-time job throughout high-school, some of college, and most of graduate school), I’ve seen this first hand. None of my students had any idea why they were learning these things, and all asked me constantly – “when will I use this later in life?!” And my only responsible answer was, “you won’t, but if you learn to play this game, it’ll get your parents off your back.” And that satisfied them. But this indicates a huge problem with the way we educate students.

And I don’t say this as someone who hates math! Rather, due to my own research over the last several years, I have needed to teach myself an enormous amount of math to proceed. In the process, I have taught myself how to read calculus, differential equations, linear algebra, vector calculus, and I’m currently working on tensors calculus! And this would’ve been more difficult if I hadn’t had high school math. But it is incidental that that math became useful later in life for me. No, I actually can enjoy aspects of math now – now that I’m not being taught to do it using tests to make sure I get the right answer (which I admittedly hated because I make silly mistakes even when I grasp the concept!). No, I really appreciate what math can do. Which gives me a certain license, I feel, to say what I’m saying here . . .

But back to the point. Math education in this country is a gate-keeper. It is an abstract game. Those you can figure out how to pass will go further in life. That’s all it is. It is a tool of power. Those who can’t figure out the game can only proceed if they can afford tutoring. And I saw over a decade of students who used tutoring, in the upper-middle class suburb where I was a tutor, use tutoring to get further in the system. And I don’t think these students really learned the ‘critical thinking and logical reasoning skills’ that pro-math educators feel they are. They learned these procedures as rote techniques. And did their best to forget them afterwards.

So, in the spirit of Jonathan Swift, I feel we’d be much more honest if we just replaced math with stupid logic puzzles. Or mazes. Or anything like that . . .

Of course, what would happen if we radically reduced math education in this country? We’d have more time to focus on skills that students really need – reading and writing. These skills open access to the entire world – and yes, even to math education! I taught myself all the math I needed from books, books written with words designed to explain math to ‘non-specialists’. Right now so many of our students graduate without the ability to read or write well. These skills are hard to teach well in any sort of systematic fashion, and are very difficult to test in a standardized manner. But if we focused more on reading and writing (and not on teaching it via literature, another modernist hold-over!), we would do students a great service

Of course, that doesn’t mean removing math, it means rather going back to what John Dewey saw as the essentials of education – problem-solving. Don’t teach anything except in problem-solving situations, in which the student’s desire to solve the problem at hand requires the learning of new knowledge. Rather than abstract or rote exercises, give increasingly difficult yet real world problems. For example, have students build a tree house properly, and learn the math needed along the way, so they see how the math actually helps. Such an approach would require melding math with other subjects, something math purists would hate to do. But its the only way to work to make math less a tool of power.

I believe math education in this country is racist. Any abstract hurdle, one in which there is little real benefit afterwards, makes it more difficult for students to get that needed diploma. And those who are historically excluded from advancement in our country are from African-American and Latino backgrounds. Math is, I believe, largely the tool of this exclusion. Even those who can’t graduate from community colleges often can’t because of the continual string of remedial math courses that slow down progress to degrees in a wide variety of subject that have little to do with math. Why should a 2-year degree in English need math classes? Politics, politics, politics.

We need to radically reform the way math is taught in schools today, so that it is no longer a tool of exclusion. Or course, we need to also reform the way we teach reading and writing (and most semester I currently teach a reading/comp class along with upper-level electives). Literature is often used as another gate-keeper, but in reading/writing education. But that is another conversation for another day. For more on my thoughts on these issues, see previous entires here.

*Back to Contemporary Theory: Lacan and Badiou*

As I said earlier, I think continental philosophy needs to re-engage with math. Math is a semiotic system, and the philosophical issues involving math have largely been ignored by most ‘continental’ theorists. Why? I feel a lot of this is because of the ways in which math and logic have been adopted by the analytic tradition. I also feel that the humanities have often felt pressure to ‘scientificize’, and the humanities have survived by resisting this strongly, by arguing for the benefit of being different from science and math. But this survival strategy has become, in hindsight, a liability and limitation. And our aversion to scientific symbolization is also something that differentiates us from the analytics!

Then there is the case of Lacan, and now, Badiou. Both engage with math symbols, but in ways that, in Lacan’s case in particular, are as much mockery of mathematics as use thereof. Of course, Lacan was tutored by some of the leading mathematicians in Paris, and so, he knows quite well what he’s doing, and Lacan’s work truly does engage with math on a semiotic level. But this very real engagement with math is covered over by Lacan’s love of obscuring his teachings behind walls of linguistic play, puns, and one other tool used to keep people outside of knowledge – obscure math symbols. So Lacan limits us here.

And then there is the Sokal hoax, in which physicists Allan Sokal wrote a parody of postmodern discourse, chock with mathematical symbols and quasi-scientific language, and got it published in a major humanities journal, only to reveal it as a hoax. When I speak to scientists about philosophy, this is usually the first thing they mention, while to most philosophers, it is a cute aside at best. But it is one which has only served to increase the defensiveness of many in the humanities.

Badiou doesn’t get us much further than Lacan, however. Despite his very real mastery of set theory, topos theory, and category theory, I remain unconvinced that there is not also an aspect of mystification going on here. As someone who can read and understand the large majority of the logical symbolic writings in Badiou’s work, I understand not only what he’s saying, but why he’s saying it this way. Slogging through the entirety of Logics of Worlds, I read two very different books. The first is a brilliant work of philosophy. The second is a logical work designed to justify what is being done philosophically. But the philosophical work needs no justification from math! I remain distinctly unconvinced that he ‘proves’ anything with his math symbols. They lend consistency, but careful use of terms can provide that as well. Badiou’s symbols are a supplement to his written philosophy, but he presents his symbolic work as much, much more, verging on ‘proof,’ but certainly including axiomatic formalization. As someone who has worked hard to understand the mathematical language at work here, as well as its development within the history of math and logic, the argument hits me as largely ‘smoke and mirrors.’ While the specifics of this remain for another post (in particular, I’d like to address some of the failures of the ‘atomic logic’ sections of Logics of Worlds), I’m firmly convinced Badiou’s work would be stronger if he weakened the need to ‘demonstrate’ things in his work with math.

*Math as Semiotic*

Math is a major form of semiotic discourse, a mode of mediation in our world today. One of the few philosophers of language today working with math as semiotic is Brian Rotman, in works such as *Becoming Beside Ourselves: The Alphabet, Ghosts, and Distributed Human Beings (2008)* or *Ad Infinitum . . . The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back In (1993)*. Originally trained as a mathematician, Rotman stopped teaching university mathematics and began to publish in philosophy. He is now able to speak the multiple discourses of continental influenced ‘post-structuralist’ philosophy, and advanced philosophy of math. He’s a rare figure and his work is excellent (and he really gets networks and why they are important!.

Rotman argues that much of contemporary math can be thought of as attempts to deal with questions of time (counting, algebra) and space (geometry). We have developed a wide symbology to deal with abstract ways to manipulate symbolic analogues of time and space, but ultimately, that’s what we’re doing. When you solve an algebraic equation, you use an ‘x’ to signify a quantity you ‘don’t yet know’, just as in geometry, you deal clearly with issues of space. Number and figure are the grounding instances of the complex semiotics of mathematical entities.

For those who have followed the history of abstract math, these issues come into plain relief. It is in fact during the ‘modernist’ period of mathematics, the period of Hilbert, Noether, Klein, etc., that we see the desire to make math a unified system. Many approaches were taken, but it is in fact the approach made by Emmy Noether – one of the first female mathematicians to be allowed to make a name for herself – that perhaps interests us most here. Noether’s field was abstract algebra, often called ‘group theory,’ a field which looks for symmetries, or invariants, amongst series of ‘objects’ – anything from physical objects to mathematical equations. Noether systematized the approach that was to have the greatest influence on Bourbaki (and ultimately, category theory), namely, the idea that ‘mathematical objects’, regardless of content, can be seen as a series of morphisms. Math is about objects under transformation.

Objects under transformation. It is hard to not hear the echo of Levi-Strauss, and in fact, structuralism in French theory was widely influenced by what is often called ‘structuralist’ mathematics as propounded by Bourbaki (the French collective of mathematicians widely credited with producing structuralist math in France in the postwar period). And yet, with the critique of structuralism put forth by post-structuralist theory, the integration of math and semiotics in contemporary theory has unfortunately waned dramatically since the days of Lacan.

And Lacan is precisely the point here, despite his obfuscations. For in fact, he treated math the way abstract algebraists (a topic with little in common with ordinary algebra) see it – as series of rule governed combinatories between objects. The difference between much of what algebraists do and what Lacan did, however, is that algebraists seek mathematical objects to be highly precise and well defined. Lacan’s objects were intentionally polyform – signifiers. His mathemes are in some sense a rigorous joke at the expense of the very mathematicians he was pilfering. But it is also shrewd semiotic/philosophical commentary. Math need not only be made of precisely defined symbols – and often symbols work fine, in math and elsewhere, when not ‘rigorously’ defined (calculus worked just fine before it was axiomatized, for example!). Lacan’s mathemes produced a combinatory that any abstract algebraist would recognize. But he did it for a field incredibly different, namely, psychoanalysis. Continental philosophy has yet to catch up to Lacan in understanding the potential ramifications of his gestures. Which is not to say we need proliferate mathemes and simply imitate Lacan. But we need to understand why and how Lacan did what he did.

I feel we need to return to math at precisely this semiotic level. The level of Lacan and his mathemes, the level of the creation of signs. But in order to do this, we need a way to demystify math. Most of us trained in philosophy see math as a field which requires years of study to gain any proficiency with. But as my recent research has shown me, math is little else than a foreign language. It requires time and effort to learn to read, but no more than reading French or any other foreign language. But now that I can read things like, say, differential equations and increasingly, tensor calculus, I find a few things strike me.

Firstly, there are so few decent texts to explain these things. And there is a great need, if we are to bring math, semiotics, and contemporary theory into discourse, for clear texts to explain math to non-specialists, particularly those with philosophical/semiotic interests. Secondly, now that I can read many of these texts, my response is often – well, what’s the big deal? Once the coating in math symbols is made transparent, often the philosophical issues are fascinating, but hardly difficult. The math is often more than anything an armor, and if we are to be psychoanalytic, a defense-mechanism.

Or a power play. We need to demystify math. And we do that by teaching it as a foreign language. I firmly believe that our Phd programs need to start allowing that option as a viable one alongside non-English ‘natural’ languages. I’m tempted to write a book called ‘Math for Philosophers’, but I’ve already got enough things on my plate. But I do plan to use a lot of math in my next book, and one of the side goals of this book will be to make this math understandable.

We need to bring math down to size. It is a wonderful tool, but so often one used to maintain the status quo of power systems. It is time to democratize math, at many levels of scale, from continental philosophy to secondary education. For math, no matter how complex, is ultimately about manipulating symbols by series of rules. And once you get those rules, you actually begin to see just how physical math is. All these symbols are designed to deal with real situations in space and time. Euler, Lagrange, Laplace – all the pioneers of math solved problems at the cusp of math and physics. And while of course discoveries are made in pure math, very often these are found to have highly practical applications later on. Math is a highly physical enterprise (which is why Rotman subtitled Ad Infinitum ‘Taking God Out of Mathematics and Putting the Body Back In).

It is time to retake math for the theorists and philosophers. Because if there is one thing continental thought does well, it is debunk ideologies. And most philosophy of math today, Rotman’s continental approaches excluded, is a form of ‘platonism’ (their word), ultimately, a form of modernism. Post-structuralism and post-modernism, whatever we take these words to mean, have yet to hit philosophy of math (save the case of Rotman!). Lacan’s forays, as brilliant as they are, need to be dealt with on math’s term’s as much as that of literature and philosophy.

It is time to retake math for the philosophers, and begin to debunk the illusions. Both to annoy the mathematicians, and to start the process of the democratization of math. It is also time for this democratization to happen on other fronts as well. And this is why I strongly feel massive educational reform is needed.

It is time we retake math. After all, its just another semiotic, another form of mediation, just objects under transformation . . .

*UPDATE: RESOURCES FOR SELF-STUDY OF ADVANCED MATH, ADVANCED PHYSICS:*

So, a reader responded to me by email after this post, asking exactly which books I’d used to get my math skills up on my own to be able to ‘read’ as a foreign language. Of course, this project is really ongoing, since I’m not a trained mathematician, but it is possible! But in a post which argues the need to democratize access to math, and recommending taking on math by philosophers, it only makes sense to include the list below. Here’s what I’d recommend for the intrepid types, but this is, however, only what worked for me:

1. Without doubt, the best book by far to teach yourself advanced math is The Princeton Companion to Mathematics, by Gowers, et.al. Yes, its $71 to buy new, but its 1000 pgs, hardcover, and one of the most incredible compendiums of knowledge I’ve seen yet – and its CLEAR. It assumes nothing more than high school math, maybe some rudimentary calculus, and you can do the rest from there.

2. Two great general, if slightly older texts are Morris Kline’s Mathematics for the Nonmathematician, and Richard Courant’s What is Mathematics?. While older, these books are still some of the best out there, and have the benefit of being really cheap.

3. Also in the ‘cheap and wonderful’, but this time not old, is the work of Ian Stewart. Anything he does is amazing. Concepts of Modern Mathematics can’t be beat for hitting lots of bases in everyday language. Its particularly strong on things like topology and sets. I also really like his Does God Play Dice?: The New Mathematics of Chaos, an introduction to a variety of math topics via complexity studies, but its touches really on a ton of stuff. His From Here to Infinity is also quite good, though some of his books start repeating topics after a while. Most of his books are also pretty cheap.

4. For most historical introductions to these topics, the best book by far is Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire. A page turner, it takes you from basic high school algebra to the seemingly unrelated developments in ‘modern’ algebra, stuff like Galois and Group theory. Great book. For a really good history of math’s search for certainty, check out Morris Kline’sMathematics: The Loss of Certainty. Its particularly good with history of the development of calculus. For a similar book which emphasizes more the development of non-Euclidean geometries, check out Jeremy Gray’s mammoth mathematical history Plato’s Ghost: The Modernist Transformation of Mathematics.A book that does a good job of describing the history of Bourbak and structuralism in mathematics, and which explains some of the math along the way, is Maurice Marshaal’s Bourbaki: A Secret Society of Mathematicians.

5. If you want to teach yourself the relativity theory and quantum mechanics in a way that doesn’t exclude the math, what worked for me is to start with popular science stuff, make sure I knew the big picture, then went to the popular science stuff with some math, then slowly upped the ante. The best non-mathematical book on relativity is Lewis Carroll Epstein’s Relativity Visualized (very visual!), while one of the best books on quantum physics and relativity is About Time: Einstein’s Unfinished Revolution (very clear!). Having read a LOT of books on these topics, I’d definitely go to these first, everything else will be easy with that foundation.

6. If you want to ease into the math in these subjects, I’d start with an understanding of Maxwell’s equations, and the best intro to these (and vectors) is the incredibly clear (no math beyond HS needed!) A Student’s Guide to Maxwell’s Equations, by Daniel Fleisch. And for a book that can get you from HS through the most difficult of Einstein’s tensors (really!), the only book you’ll need is The Mathematics of Relativity for the Rest of Us, by Louis Jagerman. I’ve been stumped by tensors for ages, but this book manages to really, really explain it! Its a really impressive achievement. Of course, if you can get through that, the best place to go is the mammoth tome The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose. I don’t know anyone who’s read the whole thing, and I’m not sure its possible, and not just cause its over 1000pgs long. Its got everything in it, and its designed, at least in theory, to be understandable by anyone with HS math and basic calculus. There are, however, some points in here that will be impossible to get through without other resources, particularly when he gets to tensors. But used in conjunction with these other texts, its an amazing resource. There are some other books to supplement these as you go, ones that make particular bits of the puzzle crystal clear: Deep Down Things: The Breathtaking Beauty of Particle Physics, by Bruce Shumm (particularly good with multiple dimensions and Lie algebras), Force of Symmetry by Vincent Icke (particuarly good with symmetry and Feynman diagrams), and Timeless Reality: Symmetry, Simplicity, and Multiplcity by Victor Stenger (general all-around good, integrates many of these issues with philosophical issues). Roland Omnes’ Quantum Philosophy is probably the best for linking these issues with philosophy.

7. For math and philosophy, I really like Rueben Hersch’s What is Mathematics, Really?, but really the master is Brian Rotman (texts mentioned earlier in the post). I also like James Robert Brown’s book, and of course Wittgenstein’s lectures on philosophy and math are also great.

This is one of the most cogent calls for education reform I’ve ever read, and I wholeheartedly agree.

I’m a student and teacher interested in political theory, now applying for graduate school. As a result I’m preparing for the maze -err, I mean test – that is required to get me into all of the programs in the U.S. in which I’m interested (interestingly, the Canadian school in which I’m interested does not).

Reviewing basic algebra and geometry for the GRE has actually been a lot of fun, excepting the fear that mistakes on the test will lead to very real losses of educational opportunity. That said, I’m one of the relatively privileged few who managed to have a great teacher in public school, who lifted me over the hurdle of mathematics, into the realm where it begins to make intuitive sense. I also suffered through tutoring lessons in calculus, (doing well enough on the AP exam to skip Calc. 1 in college, only to nearly fail Calc. 2 in college). As much as I love them, I have never once used differential equations in my every day life.

I can see from experience how mathematics systematically excludes those who can’t afford such early and continuous investments in mathematic learning, not because minorities are any worse at the subject (which should be so obvious as not to need stating), but because mathematical abstraction is difficult for nearly everyone, and the educational system (as part of a broader regressive social, economic, and political system) concentrates rewards/support rather than distributing them.

Additionally, your point about mathematics as a language is well made. For the past three years I have been lucky enough to study Chinese in China, and I can see how access to this (or any language) opens up domains of knowledge that are simply inaccessible to others, except through scarce translations.

I would like to offer the analogy that the books you mentioned are a kind of translations from an abstract mathematic discourse into more familiar terms (or keys/ciphers to a symbolic code). Education, at base, is transforming such knowledge into experience and vice-versa, which involved translating/transforming knowledge between domains and scales. Unfortunately, altering the educational system involves a political struggle about the nature of those domains, and control over the transformative power of knowledge.

Readers interested in what you wrote may also be interested in writers like George Lakoff (Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being), Douglas Hofstadter (GEB; I Am a Strange Loop), or Michel Foucault’s (The Order of Things), just to name a few.

Avi said this on September 20, 2011 at 9:39 pm |

Incidentally, neither Lakoff nor Hofstadter seem to fully understand or elaborate the social (and therefore political) consequences of their theories. While Lakoff goes further in various works exploring the construction of political concepts and values through embodied processes, (and I think you can read between the lines), neither go far enough to explain how the physical world and embodied symbolic processes mutually (re)arrange, (re)produce, and (re)enforce each other (dialectically). What’s more, they don’t seem to see (or articulate) how individuals and groups of people (lets call them “mediators”) wield power by creating and controlling systems at the interface of the symbolic and material world (i.e. by creating artistic, literary, and mathematical canons; standardized tests; etc.). That’s why I recommended reading more explicitly political and sociological works such as Foucault and Bourdieu. (BTW, I don’t mean to create my own canon, here. These are just two authors who most clearly conveyed the above points to me).

(Cheers, and thanks for such an educational post!)

Avi said this on September 21, 2011 at 4:43 pm |