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calculus: a plea
My formal study of Mathematics, such as it was, ended at the age of thirteen. Even before then, however, it had been rather discontinuous: it was decided in the final year of primary school that Maths wasn't a priority, so for a whole year of school I took no classes in the subject. And then at thirteen, I gave up Maths along with all other Sciences.
Since then, my relationship with Maths has been distant. I hold no disdain for the discipline; it is simply that we have grown apart.
On the whole, I feel I have managed OK without Maths, with the exception of some tricky moments when it comes to calculating tips, taxes, or grades. But we get by, Maths and I, each in our own separate ways.
There are times, however, when I wish I knew more...
Reading Deleuze's Difference and Repetition, for instance, I am completely flummoxed when it comes to his long discussion of differential calculus. (It was the same when I read his book on Leibniz.) A sentence such as the following has my head spinning:
When the primitive function expresses the curve, dy/dx = -(x/y) expresses the trigonometric tangent of the angle made by the tangent of the curve and the axis of the abscissae. (172)
And there is plenty more where that comes from.
Now, I dimly remember enjoying algebra rather more than other branches of Mathematics. And I also vaguely recall that we covered differential equations. I've long forgotten how to do them, of course, but at the time I think I could. (It's not that I was so very terrible at the subject: I proudly point you to the "A" I achieved at "O"-Level.) But even then I had no idea what such equations were about: as far as I was concerned they were simply abstract exercises with letters. A sort of crossword puzzle, or Sudoku.
Hence my plea, dear reader. I have the following questions:
1. Does my ignorance of calculus matter? For my understanding of Deleuze, for my understanding of philosophy, or for anything else?
2. Is Deleuze's use of calculus a Sokalian "intellectual imposture"? Is he right about the maths? And does that matter?
3. What is calculus for or about anyways? Is there a good introduction somewhere that will explain it even to me?
Englighten me, I beg of you.
By Jon | February 16, 2006 in Doltishness, Science | Permalink
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Comments
in response to #1:
I think calculus makes clear what a "differential" means. Although Deleuze makes use of a lot of concepts, this is one of the ones which he holds on to.
its not simply understanding a curve, whatever its existence might be, but understanding what it is for a function to define a curve, and what an indefinite function is... i don't think you can get far without calculus.
As such, with respect to #3, I think any calculus textbook, insofar as they are usually formulated for college freshmen can be useful, it usually takes you from the quadratic equation to calculus proper in a chapter or two. If you want a more entertaining and perhaps more difficult route, I recommend "A Tour of the Calculus" by David Berlinsky who (strangely) provide insight into Leibniz's life... frankly I don't know what to do with that. Say what one might about Berlinsky, he has a proof of the fundamental theorem in the index, which is outstanding. additional note, there's even a greater book on set theory by Mary Tiles, "The philosophy of set theory" which provides an outline of functions and calculus and quite rightly connects them up with set theory... there you get a connection between Deleuze and Badiou (two in one). Hurrah!
Posted by: tzuchien | Feb 17, 2006 5:31:17 AM
Yeah, The Philosophy of Set Theory by Tiles is very good; it deals with exactly those mathematical concepts that you want to know about when reading Badiou - even "forcing".
Posted by: David | Feb 17, 2006 7:03:47 AM
I'd have to see the context and perhaps the French, because the passage quoted is very close to but not quite a flat, i.e., 1st semester calculus, definition of the derivative. I've been told that there are passages in Deleuze (and Deleuze and Guatarri) that are straightforward recaps of various mathematical topics that ended up mangled by translators who didn't know the math. Just what I've been told, mind you.
If you can find Michael Spivak's Calculus, (it may be out of print) I'd say go with that. Do not get his Calculus on Manifolds, which is an advanced text. Otherwise, most calculus texts are fat problem sets with poor (indeed, false) explanations. You're better off looking at textbooks on Real Analysis to get the correct proofs.
Posted by: et alia | Feb 17, 2006 8:00:18 PM
As et alia says, it's hard to tell without seeing the original passage in context, but this seems like a slightly mistranslated version of an attempt to make something simple seem complicated.
For example, "the trigonometric tangent of the angle made by the tangent of the curve and the axis of the abscissae" is a complicated way to say "the slope of the function" -- though it's unclear to me why "abscissae" should be plural.
Perhaps the argument requires reference to trigonometry and abscissae, or perhaps this wording reflects the mathematical terminology of another era or another culture, but it wouldn't be shocking to learn that it simply reflects a taste for gratuitous obscurity.
I haven't read Difference and Repetition -- what's the point of the discussion of differential calculus? Is Deleuze trying to shed light on the mathematical issues, or to use the mathematics in order to make a point about something else?
Posted by: Mark Liberman | Feb 21, 2006 7:37:46 AM
I am also ignorant of calculus. I'm pretty sure that our ignorance does not matter for understanding Deleuze's Difference & Repetition. In the chapter you quote from, chapter 4, Deleuze picks out three oddball 18th/19th century metaphysical interpreters of calculus (Maimon, Wronski, and Bordas-Demoulin, three names from "the esoteric history of differential philosophy"). They are not the best mathemeticians; they're like "wild mathematicians," as I understand ("the so-called barabaric or pre-scientific interpretations," 170). He uses their work to show that Ideas, like the symbol dx in differential calculus, are undetermined, determinable, and capable of infinite determination.
Leaving aside for the moment, or forever, what that means (I can sort of understand it in relation to Kant, but I can't explain it), let's just say he uses calculus to demonstrate the problematic character of ideas: "In short, dx is the Idea, the Platonic, Leibnizian, or Kantian Idea, the 'problem' and its being" (171).
On p 180 and 181, Deleuze says that calculus isn't even the basis of math (set theory is). And he has already said that within calculus, these three weirdos aren't the most scientific interpreters of calculus. But calculus "has a wider universal sense," it "designates the composite whole that includes Problems or dialectical Ideas."
Sorry to have gone on so long & so repetitively. Calculus is illustrative for Deleuze, in a way that's different from what set theory is for Badiou. There, ignorance of math does matter.
Posted by: Diana | Feb 23, 2006 11:52:27 AM
You've got a (somewhat humanist?) fan, Jon (that's a link to a review of Einstein's Heroes).
Posted by: BTS | Feb 24, 2006 11:10:46 AM
From the probably unique position of (a)someone who's (supposed to be) writing a PhD on Deleuze and Mathematics and (b)someone who spent much of their life ignorant of what calculus was and why it might be important:
>1. Does my ignorance of calculus matter?
>For my understanding of Deleuze, for my
>understanding of philosophy, or for
>anything else?
Important for Deleuze: Yes, because from D&R onwards he bases his logic of difference and problems on concepts drawn directly from modern mathematical analysis (whose foundation is calculus). Without an understanding of analysis I think one can only achieve a vague discursive idea of Deleuze's argument (like most secondary texts). Implicitly Deleuze's -somewhat Badiousian- argument is that philosophical thought needs to take account of the conceptual manouevres contained in mathematical analysis, relating to universality, generality, singularity, if it is to escape various classical images - just as modern analysis finally escaped the long reign of aristotelianism, so must philosophy. The new mode of thought Deleuze is proposing bases itself on the historical thought-events of modern analysis. Briefly, to appreciate the mathematical side deepens your historical and philosophical understanding of what Deleuze is doing.
For an understanding of philosophy: Yes, because the more you look into this the more you will see that every major philosopher has something to say about calculus (even Engels!) - Why? (and this is the answer to the final question): because it is the single most important enabling conceptual mechanism for modern science. Put simply, it provides a mathematical handle on the large majority of physical phenomena which are not rectilinear. It constitutes a singular meeting-point of the problems of physics, mathematics and metaphysics (hence the interest in the 'esoteric history' - Whereas for Badiou mathematics eventually sloughs off all philosophical problematics clinging to it, Deleuze is interested in recovering these metaphysical problems and the relation of mathematics to philosophy and physics that they suggest.)
>2. Is Deleuze's use of calculus a Sokalian
>"intellectual imposture"? Is he right about
>the maths? And does that matter?
No, it's not simply a rhetorical misuse. But it is difficult to read and interpret. It's not an 'example' or an 'illustration' or a 'metaphor' either: deleuze's discussion is about the relationship between history, mathematics, physics and philosophy, from a (french philosophy of science) point of view which attempts to fathom the articulation of different orders (historical order, logical order, epistemological order. Obviously this is something Sokal can't understand. But obviously it doesn't matter if a wilfully myopic pigheaded scientist doesn't get it.
>3. What is calculus for or about anyways?
>Is there a good introduction somewhere that
>will explain it even to me?
It's used to give a formal treatment of problems which involve relationships between continuously varying quantities, ie. almost any physical, economical, astronomical, etc. problem you care to mention. (see Kline book, below, for simplest possible examples)
The problem I've found, as other comments suggest, is that pedagogical texts, even high school ones, never give any conceptual explanation that would be satisfying to a philosopher (which is rather shocking given the historical importance of calculus).
I would recommend firstly the calculus chapter in Kline's 'mathematics in western culture', which is the simplest account I have read of why calculus matters.
Secondly Boyer's 'The Concepts of the Calculus' is a fascinating book on the history (Deleuze read it).
More technically, Bruce Exner's book 'Inside Calculus' is the only academic maths book I have read which seems thoughtful on a conceptual level, and which gives discursive expositions before introducing massive equations; it describes very clearly the modern (epsilon-delta) form of calculus.
Posted by: robin | Jul 24, 2006 6:12:28 AM
Just a quick one - a good nontechnical introduction to the main ideas of Calculus is a book titled "Calculus for Cats", although I forget the author's names. It is written by an English professor and a mathematician.
Posted by: Phillip Grebe | Dec 22, 2006 11:41:44 PM
Phillip-- Delighted to see that you mentioned my book, Calculus for Cats. It is written by Kenn Amdahl (me) and Jim Loats, Ph.D. If the original poster remains unsatisfied with his relationship with the subject, I would heartily concur that this book might help a bit. I also have a thorny relationship with all things numerical, and contributed to the book to help others who share my unease. Either way, thanks for mentioning the book. Kenn Amdahl
Posted by: Kenn Amdahl | Jun 12, 2007 6:37:43 PM
The best book I've found that describes what various branches of mathematics are about, without getting bogged down in technical detail, and without merely skimming over the surface in search of pretty pictures, is Mathematician's Delight by W. W. Sawyer. I have it in a Pelican edition, and it seems to be intended for people who have a dread for mathematics but might otherwise be good at it, or, maybe, teachers who have to teach such people.
Posted by: Hendrik Boom | Jan 30, 2009 2:27:49 PM
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