View Full Version : What is a Riemannian Manifold?

poetix

05-06-2009, 07:50 PM

There must be someone on here who knows, and can explain it clearly. (Assume I've already glanced at the Wikipedia page, and thought "oh, right, that looks like the sort of thing a kind and patient mathematician ought to be able to explain to me").

If there isn't, or they can't be arsed, I'm going to work it out for myself* and post the answer on this thread.

Political enlightenment and/or enhanced appreciation of Deleuze may or may not eventuate.

* I may, as the very gallant Oates said, be some time.

woops

05-06-2009, 09:03 PM

the board is a lot more active than it usually is on a friday night

Mr. Tea

05-06-2009, 10:59 PM

I'll get back to you when I've had a look over my old undergrad Maths for General Relativity notes.

This interest has been piqued by use of the term by Alain Badbwoy, I assume?

Edit: dur, Deleuze, obv.

OK, well I've had a quick look at my notes and also at Mathworld and a few Wikipedia entries, so I'm prepared to have a go.

A manifold is an N-dimensional space where every point can be locally mapped to R^N, where R means the real numbers and taking R "to the power of" N just means you're using N real numbers to describe a position - so that on an infinite, flat 2-dimensional plane, any point can be described completely with two coordinates, eg. x and y on the familiar Cartesian axes.

This flat plane is also called a 2-dimensional Euclidean space. Extend it to 3 dimensions, and it's a pretty good description of real, physical space in the absence of strong gravitational fields - i.e. the space we live in. You can add a time dimension to this 3-d Euclidean space to obtain a four-dimensional space-time, a.k.a. Minkowski space-time, which forms the basis for special relativity.

Anyway, in the middle of the 19th century Riemann, building on earlier work by Gauss, started to experiment with spaces in which the axioms of Euclidean geometry (such as "parallel lines converge at infinity" and "the shortest distance between two points is a straight line") were not assumed. This was all done in the spirit of purely mathematical enquiry, as it was not imagined at this point that there could be real physical applications of this work. So rather than just working in 'spaces', Riemann and others worked with 'manifolds', where Euclid's rules for 'flat' space don't (necessarily) hold, and the mapping of points from the manifold to R^N is perhaps non-trivial. An example that's simple to visualise is S^2, the surface of a sphere, where the x and y coordinates of 2-d Euclidean space are replaced by latitude and longitude, and the shortest line between two points is no longer straight but a geodesic, a section of a 'great circle'. Also, the surface of a sphere has intrinsic curvature, which is not shared by a flat plane. (Intrinsic curvature means that a flat sheet - a piece of paper, say - cannot be deformed onto the surface without 'crumpling': thus the cylindrical main part of a wine bottle does not have intrinsic curvature, but the 'shoulder' where this part meets the neck does have intrinsic curvature, as does a sphere.)

Now a differentiable manifold is one which is locally similar enough to Euclidean to allow calculus to be performed; in other words, it has no 'sharp' boundaries or discontinuities, ensuring that (in a given mapping to R^N) the mapping is single-valued at all points, and differentiable, i.e. continuous, or having a finite derivative with respect to some coordinate system. Another ingredient we need here is the metric, which is a function g(x1,x2) that gives the shortest distance between two points x1 and x2. Then a Riemannian manifold is a real differentiable manifold with a non-negative metric, which allows intuitive geometrical quantities like length, area, volume and angle to be meaningful.

The maths here is probably a bit mangled, because I learnt the basics of this stuff from a physicist's POV. Someone like Slothrop may be able to give a more rigorous and accurate description. Anyway, general relativity is phrased mathematically in terms of Lorentzian manifolds, which are classified as 'pseudo-Riemannian' because the metric can be negative. This may sound nonsensical, as it describes a distance, but remember we are talking here about 'distances' in space-time, not space.

Hope this has been of some help.

poetix

06-06-2009, 11:03 AM

Thanks, Tea, that is indeed helpful.

Let me go over part of what you've said so far, sticking with the example of the surface of a sphere as that's quite easy to visualise.

I have a street map of Northampton town centre, which opens out into a large rectangle. The "points" on the interior of that rectangle can be seen as an open subset of the Euclidian space R^2: if we measure in inches across the surface of the map, any point "inside" the map will be 0 < x < W inches across from the left (where W is the width of the map), and 0 < y < H inches up from the bottom (where H is the height).

If I had a very large globe map of the earth, to the same scale as my town centre map, I could find Northampton on that globe and paste my map over it so that the points on the map lined up with the points on the surface of the globe. In fact, for a rectangle so small (relative to the overall size of the globe), I probably wouldn't notice that I had to stretch or crumple the rectangle into an ever so slightly non-rectangular shape in order to make it "fit".

The surface of the globe is "locally" (e.g. within the parish bounds of Northampton) similar enough to a region of 2-dimensional Euclidian space that I can map everything within such a locality to an open set within a Euclidian space, and this mapping will be homeomorphic (in other words, if two points p and q are "near to" each other in Northampton, their mapped points p' and q' will be "near to" each other in R^2. We can give a topological definition of this "nearness" or being-in-the-neighbourhood-of in terms of membership of open sets, in which case what matters is that the mapping preserves the structure of the lattice of open sets "localising" each point).

However, "globally" the surface of the globe can't be mapped in this way - no matter how you stretch it, you can't completely completely cover the surface of a sphere with a rectangular "rubber sheet" (or not without "sewing it up" at one point. Interestingly, if you subtract just one point from the surface of a 3-d sphere - one of its poles, say - you can in fact cover it with a suitably deformed open 2-d sphere. So the "locality" for which the surface of our sphere can be mapped to R^2 is actually pretty maximally "large". If I were Badiou, I'd be making a big deal about that subtracted point right now...).

Does this sound OK so far?

Mr. Tea

06-06-2009, 01:09 PM

Yep, sounds good to me. Though I must confess your maths vocab is outpacing mine somewhat with open subsets and homeomorphisms.

Also, in your last paragraph, don't you just mean a disc, rather than a 2-sphere? The surface of the earth is itself a 2-sphere, i.e. a spherical surface with dimension 2. If you include the interior of the earth, I think in geometric terms it's then considered a 'ball'. So we can visualise a 2-sphere embedded in 3-dimensional Euclidean space - and, if we wanted to, we could conceptually wrap that sphere with a (deformed) disc, which would be a filled 1-sphere (a 1-sphere is a circle), or in other words a '1-ball'.

It's impossible to visualise a 3-sphere because, if embedded in a Euclidean space, that space would have to be 4-dimensional.

The thing about wrapping a disc around a sphere is, I think, related to the delightfully named Hairy Ball Theorem (http://mathworld.wolfram.com/HairyBallTheorem.html).

poetix

06-06-2009, 01:12 PM

Disc! That was the word I was trying to remember. Yes, a disc.

The topology stuff is me sneakily trying to shift the definition away from Deleuze's territory (differentiation, continuousness etc.) and onto Badiou's (sets, topologies, orders etc.).

nomadthethird

06-06-2009, 08:19 PM

This chapter (http://books.google.com/books?id=ooqi7o6j0-cC&pg=PA101&lpg=PA101&dq=Deleuze+riemannian+manifolds&source=bl&ots=fwjwB59eZq&sig=3JHUoXuDRUuhQBsvYyBMmQAx1UY&hl=en&ei=1ssqSpZeobG3B5D9wKQI&sa=X&oi=book_result&ct=result&resnum=1#PRA2-PA98,M1) seems good, although it's written from a critical-theoretical rather than a mathematical perspective.

This is ultimately the point of D&G's "mathematics", for what they're worth:

http://img188.imageshack.us/img188/423/bddr.jpg (http://img188.imageshack.us/my.php?image=bddr.jpg)

poetix

06-06-2009, 10:20 PM

OK, on to some more detail.

So far I've described (I think) the "atlas" of a "topological manifold". The basic idea is that for each open set U of some set of open sets covering a topological space, there's a "chart", which is a homeomorphism U -> V where V is an open subset of R^n; thus the space is "locally" Euclidean even if it's not globally Euclidean. The "charts" are like my map of Northampton, projections of regions of the globe into regions of R^2; the "atlas" is a collection of charts of regions covering the entire globe. (Obviously the globe/streetmap metaphor will only get us so far here).

Wikipedia says that the underlying topological space has to be a "second-countable Hausdorff space", but doesn't say why. For the time being I'm just going to gloss these two conditions as "non-wacky".

The other thing that a topological manifold has is "transition maps", which describe how, as we move from one open set of the manifold to another, we transition between the corresponding open subsets of R^n. Is the point about a Riemannian manifold that this transition should be "smooth" as the metric measuring the distance between points on the manifold is "smooth"? For example, moving smoothly along a geodesic between two points on the surface of a sphere, I should be able to transition smoothly between the charts of the neighbourhoods of these points, like a 2-d map scrolling from left to right on the display of a satnav...

nomadthethird

06-06-2009, 10:47 PM

"Smooth" means not "striated" as in, non-Euclidean, as in, not grid-like.

nomadthethird

06-06-2009, 11:03 PM

What's going on in parts when Deleuze uses Riemannian Manifolds (and by extension, general relativity and chaos theory), is that he is rather perversely organicizing mathematics and mathematicizing the organic by mapping terms from each over the other.

Maybe he's trying to make a Latourian hybrid out of these kinds of spaces.

(so there's smooth and striated muscle tissue, there's smooth and striated space, and they metaphorically signify the same concepts across fields for D&G without one having more authority or being more real than the other)

Agent Nucleus

07-06-2009, 12:21 AM

maybe a folded surface, or something like this?

http://www.huckmagazine.com/wordpress/wp-content/uploads/reef-6.jpg

not sure how to describe this object. Deleuze has influenced some amazing architecture (greg lynn, marcos novak). But the best theoretical interpretation of non-Euclidean space as a model for postmodernity has to be Massumi's in Parables of the Virtual (the chapter "Strange Horizon: Buildings, Biograms and the Body Topologic"). He equates manifolds/topologies/smooth space with affective/embodied space, which is why it can't be mapped along visual coordinates/grids. To me this aesthetic starts with the Merzbau (Schwitters), though, so it isn't postmodern at all. In fact you could argue that prehistoric art is topological/Riemannian since it tries to represent an interior or spiritual dimension.

http://www.digital-doa.com/jsipprell/images/news/sentosa.jpg

poetix

07-06-2009, 08:17 AM

Bear in mind that a topology on a set S can be just the pair of open sets S and {}. Topologies can be relatively "coarse" or "fine" - (S, {}) is the coarsest topology on S, P(S) (the powerset of S) is the finest.

A Euclidian space R^n is a set of points (R1, R2...Rn). The standard topology on such a space is that in which the open sets are the open balls around each point (plus their unions and finite intersections). Given a metric function g which measures the distance between two points p and q, an open ball of "radius" t around a point p is the set of all points for which g(p, q) < t. It's like a ball with its surface removed, so that it's all "inside". On the real line, R^1, the "open balls" are just the open intervals (p - t) < p < (p + t).

IdleRich

07-06-2009, 01:43 PM

"Smooth" means not "striated" as in, non-Euclidean, as in, not grid-like.

I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).

Mr. Tea

07-06-2009, 01:44 PM

"Smooth" means not "striated" as in, non-Euclidean, as in, not grid-like.

No, 'smooth' in this context just means continuous and differentiable, i.e. not having any sharp discontinuities.

So a sphere is smooth, whereas the surface of a cube is manifestly not smooth as it has sharp edges and vertices. A Euclidean manifold is definitely smooth, as its derivative is identically zero at all points.

Edit:

I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).

^ Jungian synchronicity.

josef k.

07-06-2009, 02:27 PM

A Riemannian Manifold is like a box of chocolates. You never know what you're going to get.

Mr. Tea

07-06-2009, 02:40 PM

This:

http://www.huckmagazine.com/wordpress/wp-content/uploads/reef-6.jpg

looks a bit like a Calabi-Yau (http://images.google.co.uk/images?hl=en&q=calabi+yau+space&btnG=Search+Images&gbv=2) space (http://mathworld.wolfram.com/Calabi-YauSpace.html).

And this:

http://www.digital-doa.com/jsipprell/images/news/sentosa.jpg

looks like some kind of exotic tree fungus.

poetix

07-06-2009, 03:13 PM

As often turns out to be the case, it seems that you can build up to the definition of a Riemannian manifold along a sort of tree of restrictions and extensions from a more general class of objects - viz:

Set + topology -> topological space

Topological space + Hausdorff separability and countable base -> "Non-wacky" topological space

"Non-wacky" topological space + locally Euclidean (every point has a neighbourhood homeomorphic to an open set in R^n) -> Topological manifold

Topological manifold + differentiability -> Smooth manifold

Smooth manifold + metric -> Riemannian manifold

josef k.

07-06-2009, 03:48 PM

Or fold + (i) + Man.

+ (http://www.mirror.co.uk/)

nomadthethird

07-06-2009, 05:35 PM

No, 'smooth' in this context just means continuous and differentiable, i.e. not having any sharp discontinuities.

So a sphere is smooth, whereas the surface of a cube is manifestly not smooth as it has sharp edges and vertices. A Euclidean manifold is definitely smooth, as its derivative is identically zero at all points.

Edit:

^ Jungian synchronicity.

You're missing the point of Deleuze if you really think he cares about Riemannian manifolds. But point missing seems to be the strong suit of some people here.

I mean, I have no idea what smooth means in Riemannian manifolds, but I do have a decent idea what it means in D&G.

nomadthethird

07-06-2009, 05:41 PM

I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).

I dont know about the manifolds, but "smooth" is somewhat categorically used by Deleuze in opposition to "striated" and this opposition seems to loosely fit here.

"Differentiable as many times as you like" does seem consistent with "smooth" spaces a la D&G...

So you and Tea know math-- explain why RMs are relevant to general relativity?

Mr. Tea

07-06-2009, 05:44 PM

So Deleuze is basically making terms up to suit himself, borrowing words from mathematics as and when he sees fit? If he doesn't care about Riemannian manifolds, and can't be arsed to stick to the agreed definitions of mathematical terms, why doesn't he just invent a Deleuzian manifold that has whatever properties he wants to give it, without bothering to pretend to some kind of mathematical rigour?

Frankly I think it's a bit trite to tell me and Rich we've "missed the point" when we flag up an inaccuracy in some maths that turns out to be some philosopher's pseudo-maths and is merely masquerading as maths. I mean, surely you can see why this style of discourse gets up some people's noses, especially people who actually know a bit of maths?

nomadthethird

07-06-2009, 05:49 PM

So Deleuze is basically making terms up to suit himself, borrowing words from mathematics as and when he sees fit?

I really wouldn't be surprised if he were. Did he actually study mathematics anywhere? I don't think so. This is from wiki:

Similar considerations apply, in Deleuze's view, to his own uses of mathematical and scientific terms, pace critics such as Alan Sokal: "I'm not saying that Resnais and Prigogine, or Godard and Thom, are doing the same thing. I'm pointing out, rather, that there are remarkable similarities between scientific creators of functions and cinematic creators of images. And the same goes for philosophical concepts, since there are distinct concepts of these spaces."

I have more and more sympathy for Sokal and Bricmont all the time, I must say.

vimothy

07-06-2009, 05:54 PM

Doesn't everyone study maths in France?

nomadthethird

07-06-2009, 05:55 PM

Doesn't everyone study maths in France?

Everyone studies it here, too, up to calculus. But Riemannian manifolds are about err 4 or 5 years past that?

josef k.

07-06-2009, 05:57 PM

Deleuze did in fact study mathematics.

Robin McKay (who you may know as the editor of Collapse) writes:

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.

VIA (http://www.long-sunday.net/long_sunday/2006/02/calculus_a_plea.html)

So Deleuze is basically making terms up to suit himself, borrowing words from mathematics as and when he sees fit?

He must be stopped!

vimothy

07-06-2009, 06:01 PM

Yeah, of course, but it is my understanding that more people learn maths at a higher level in France, and that most French philosophers studied maths (along with other subjects) in the Grandes ecoles system. I may be misremembering this of course.

nomadthethird

07-06-2009, 06:02 PM

How can that be a "Badiousian" argument when Badiou had written basically nothing at that point, when Deleuze was writing D&R? I mean, perhaps in an anachronistic sense this is true. It's more correct, however, to say that Badiou's arguments using set theory have a Deleuzian flavor to them, given Deleuze made his 20 years earlier and they were well-known to Badiou.

Edit: Thanks for posting that Josef it's a good find.

Mr. Tea

07-06-2009, 06:07 PM

Which brings me to a wider question I've been meaning to explore for a while, namely: Is there a convincing argument as to why it's a good idea to model theories of social, political or psychoanalytic philosophy on mathematical theorems? Because to me it's very far from obvious that this should be the case.

nomadthethird

07-06-2009, 06:08 PM

Yeah, of course, but it is my understanding that more people learn maths at a higher level in France, and that most French philosophers studied maths (along with other subjects) in the Grandes ecoles system. I may be misremembering this of course.

Sounds right...

nomadthethird

07-06-2009, 06:12 PM

Which brings me to a wider question I've been meaning to explore for a while, namely: Is there a convincing argument as to why it's a good idea to model theories of social, political or psychoanalytic philosophy on mathematical theorems? Because to me it's very far from obvious that this should be the case.

I'm with you on this.

"But obviously it doesn't matter if a wilfully myopic pigheaded scientist doesn't get it."

How crazy of those pigheaded scientists to demand clarity and rigor of allegedly scientific work!

Sometimes I wouldn't care if "art" went up in smoke tomorrow. It seems so narcissistic and lame, and full of all of those idiotic things that come with narcissism, like nationalism, and homelandcentricity.

josef k.

07-06-2009, 06:18 PM

How can that be a "Badiousian" argument when Badiou had written basically nothing at that point, when Deleuze was writing D&R?

You're right - that was an interesting piece of rhetoric. A significant indication of the rise of Badiou, that he is being projected back in time.

How crazy of those pigheaded scientists to demand clarity and rigor of allegedly scientific work!

Granted... But nevertheless: to demand clarity and rigor from philosophical work and philosophical language seems like an imposition. None of the people Sokal and Bricmont argue against in Intellectual Impostures would have considered themselves scientists, or their work scientific.

nomadthethird

07-06-2009, 06:18 PM

So Deleuze is basically making terms up to suit himself, borrowing words from mathematics as and when he sees fit? If he doesn't care about Riemannian manifolds, and can't be arsed to stick to the agreed definitions of mathematical terms, why doesn't he just invent a Deleuzian manifold that has whatever properties he wants to give it, without bothering to pretend to some kind of mathematical rigour?

Frankly I think it's a bit trite to tell me and Rich we've "missed the point" when we flag up an inaccuracy in some maths that turns out to be some philosopher's pseudo-maths and is merely masquerading as maths. I mean, surely you can see why this style of discourse gets up some people's noses, especially people who actually know a bit of maths?

No, don't let my statements reflect on Deleuze's math, I was just trying to draw a parallel up there, between different spaces within manifolds and "smooth" versus "striated" ones in the Deleuzo-Guattarian philosophical jargonny sense of the terms. Since I know nothing about manifolds but what has been typed in this thread, and not even really that, it was just a guess. (I mean, isn't there supposed to be Euclidean and non-Euclidean space within Riemannian manifolds? If so, that's what I was talking about...smooth and striated jargon words vaguely equating to those Euclidean and non-Euclidean spaces within RM...I am probably missing something here)

What I meant to say was... I'm not sure "hybrids" of math and philosophy are entirely meaningful at this juncture.

nomadthethird

07-06-2009, 06:19 PM

Granted... But nevertheless, to demand clarity and rigor from philosophical work and philosophical language seems like an imposition.

Agreed, but aren't halfassed, sloppy "hybrids" exactly what's wrong with 'postmodern' thought? Why add to this travesty?

josef k.

07-06-2009, 06:24 PM

Which brings me to a wider question I've been meaning to explore for a while, namely: Is there a convincing argument as to why it's a good idea to model theories of social, political or psychoanalytic philosophy on mathematical theorems? Because to me it's very far from obvious that this should be the case.

I think scientific and mathematical models can be applied to social, political, psychoanalytic philosophy, as, precisely, models. In the same way that scientists invoke metaphors in order to present their own findings. To base social theories on scientific or mathematical models (held-up as the highest of all methods) seems more problematic, and has previously produced monsters.

Agreed, but aren't halfassed, sloppy "hybrids" exactly what's wrong with 'postmodern' thought? Why add to this travesty?

I support Hybrid Rights. I think the point for Serres is that rigor (non-sloppiness) comes with the subordination to a discipline, which is what he wants to avoid... he is the philosopher of the crossroads, which is to say, the half-ass.

Mr. Tea

07-06-2009, 06:36 PM

He must be stopped!

Well personally I couldn't care less if some philosophy professor gets a hard-on for writing about Riemannian manifolds, but it would seem to call into question his credibility if he does so in a sloppy, capricious way.

Obviously this is something Sokal can't understand. But obviously it doesn't matter if a wilfully myopic pigheaded scientist doesn't get it.

Zing!

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).

Oh, the horror. Could it be that maths teachers are using maths lessons primarily to teach maths - or, at the very least, the ability to pass a maths exam - rather than philosophy?

I'm sure there are plenty of things that are not "satisfying to a philosopher" but which 99.99% of people don't lose sleep over, if they're even aware of them.

vimothy

07-06-2009, 06:43 PM

Oh, the horror. Could it be that maths teachers are using maths lessons primarily to teach maths - or, at the very least, the ability to pass a maths exam - rather than philosophy?

This is almost certainly true, and probably a bad thing.

nomadthethird

07-06-2009, 06:43 PM

Yeah, the constant jockeying for the Importance of Philozzawwzophy is getting tired.

nomadthethird

07-06-2009, 06:50 PM

I'm sure there are plenty of things that are not "satisfying to a philosopher" but which 99.99% of people don't lose sleep over, if they're even aware of them.

As far as I'm concerned, the most interesting philosophy is already going on in cosmology. Not that I can understand it, because I can't. But it's there.

I'm a neo-Aristotelian.

Mr. Tea

07-06-2009, 06:59 PM

This is almost certainly true, and probably a bad thing.

You mean the bit about passing exams? Well yes, it's kind of missing the point rather. But this has been discussed at length in other threads.

Obviously I'm not against teachers/lecturers encouraging their students to think about what things mean, as opposed to merely how things are. That said, an exhaustive analysis of Newtonian vs. Leibnitzian conceptions of infinitesimal quantities is probably not a prerequisite for a class of 16-year-olds to get to grips with the basics of calculus.

josef k.

07-06-2009, 07:00 PM

Oh, the horror. Could it be that maths teachers are using maths lessons primarily to teach maths - or, at the very least, the ability to pass a maths exam - rather than philosophy?

I don't think Deleuze (and/or Serres) is demanding the destruction of maths and gulagization of maths teachers... I think he (they) is trying to find a way of thinking about maths philosophically... exploring its connections to other disciplines... Which will naturally involve stepping-outside of mathematical norms and deforming mathematical concepts a little... he (and G.) says in "What in Philosophy" that philosophy is about inventing concepts, science is about inventing functions, and art is about inventing something or another.

I'm sure there are plenty of things that are not "satisfying to a philosopher" but which 99.99% of people don't lose sleep over, if they're even aware of them.

Perhaps nothing* is satisfying to a "true" philosopher... philosophy being, after all, a really stupid thing to want to do.

*EDIT: Beyond normal run of the mill things like dressing up like a women, and plotting the terms of Aroundball...

*EDIT2: And stapling pieces of paper together...

nomadthethird

07-06-2009, 07:07 PM

philosophy being, after all, a really stupid thing to want to do.

Or maybe it's a really decadent thing to want to do. When the philosophers start getting massive recognition and obsequious praise isn't that usually when it's widely acknowledged that the empire is doomed?

Sort of how it is decadent that I can spend all day reading or listening to whatever I can steal online while watching The Anatomy of a Pandemic and eating candy.

vimothy

07-06-2009, 07:09 PM

Obviously I'm not against teachers/lecturers encouraging their students to think about what things mean, as opposed to merely how things are. That said, an exhaustive analysis of Newtonian vs. Leibnitzian conceptions of infinitesimal quantities is probably not a prerequisite for a class of 16-year-olds to get to grips with the basics of calculus.

But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely procedural. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this.

Mr. Tea

07-06-2009, 07:12 PM

But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely instrumental. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this.

Fair enough. Do you think more emphasis on the history and philosophy of the subject would facilitate students' conceptual understanding of it? It certainly seems plausible, I guess.

poetix

07-06-2009, 07:14 PM

Well personally I couldn't care less if some philosophy professor gets a hard-on for writing about Riemannian manifolds, but it would seem to call into question his credibility if he does so in a sloppy, capricious way.

But it's liberated to be sloppy and capricious. And authoritarian to be otherwise. The great totalitarian monsters of the 20th century were all noted for their keen, analytic intelligence and ruthless dedication to mastering complex technical domains! As they marshalled their theorems, so did they command men! Vast, methodically drilled armies of indoctrinated rationalists, sweeping across Europe, eliminating all traces of empathy, creativity and fun...

vimothy

07-06-2009, 07:17 PM

Fair enough. Do you think more emphasis on the history and philosophy of the subject would facilitate students' conceptual understanding of it? It certainly seems plausible, I guess.

Not necessarily. I'm thinking in much less grand terms -- though contextualisation would surely help students appreciate the value of the subject -- more time understanding stuff (making mistakes, understanding mistakes), less time racing through procedures.

nomadthethird

07-06-2009, 07:18 PM

Nah, totalitarians spent very little time in armchairs...

josef k.

07-06-2009, 07:18 PM

Or maybe it's a really decadent thing to want to do. When the philosophers start getting massive recognition and obsequious praise isn't that usually when it's widely acknowledged that the empire is doomed?

Sort of how it is decadent that I can spend all day reading or listening to whatever I can steal online while watching The Anatomy of a Pandemic and eating candy.

Praising famous men is clearly a pretty dangerous idea, and a perpetual danger in all fields of achievement, often exacerbated by certain rhetorical fly-trap moves which people leave in their text (perhaps unconsciously) for precisely this reason... then again, every generation throws a hero up the pop charts, and society is perpetually manufacturing idols. The image of the philosopher as icon is really just a surrogate image of Athene in this sense... then again, there is also the way in which the culture industry encourages this attitude, through outlandish, authority-condensing blurbs and other forms of marketese.

nomadthethird

07-06-2009, 07:19 PM

Not necessarily. I'm thinking in much less grand terms -- though contextualisation would surely help students appreciate the value of the subject -- more time understanding stuff (making mistakes, understanding mistakes), less time racing through procedures (our data are full of teachers saying things like "you don't need to understand this, let's move on").

Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.

josef k.

07-06-2009, 07:20 PM

But it's liberated to be sloppy and capricious. And authoritarian to be otherwise. The great totalitarian monsters of the 20th century were all noted for their keen, analytic intelligence and ruthless dedication to mastering complex technical domains! As they marshalled their theorems, so did they command men! Vast, methodically drilled armies of indoctrinated rationalists, sweeping across Europe, eliminating all traces of empathy, creativity and fun...

You are an idiot.

poetix

07-06-2009, 07:21 PM

The problem in my experience with maths teaching (and curricula) in the UK is not so much the lack of "philosophical" context as the neglect of "motivation" - it's seldom apparent what the problem, or class of problems, is that a particular technique is supposed to give you some purchase on. It's just: "These are matrices. Learn them". I still don't know what matrices are for, outside of the rather limited context of composing transformations to apply to vector graphics in silly computer games.

vimothy

07-06-2009, 07:25 PM

Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.

The time constraints are a function of a different phenomenon -- the "audit culture" of testing and measurement -- they are not naturally occurring. If we want to change the way teaching and learning occurs, one of the things we need to think about is our system of examinations.

josef k.

07-06-2009, 07:26 PM

Yeah, people always say this stuff, but in the end time constraints seem to be a huge enemy of this kind of conceptually comprehensive meta-education that everyone seems to agree would be best.

The outstanding question is: what is education (broadly defined) for? It seems clear that educational institutions are devoted at least in part to the furtherance of the interests of that institution, and its supporting networks of institutions (e.g. university socializes people into the middle class, public schools-Oxbridge (or the Ivy League in the US) provide recruitment facilities for the elites... from a different angle, teachers and special interest groups (faith schools) courted as voters in various ways that do not necessarily coincide with the interests of their students, society-at-large, or humanity, whatever that is...

vimothy

07-06-2009, 07:27 PM

It's just: "These are matrices. Learn them". I still don't know what matrices are for, outside of the rather limited context of composing transformations to apply to vector graphics in silly computer games.

Exactly -- connections are not made, and instead teaching is the transmission of procedures.

nomadthethird

07-06-2009, 07:27 PM

I think in some ways, you have to learn (lower level) math by trial and error and then it will slowly become conceptually clear to you. It's hard to work backwards from the concepts down to the rules, isn't it? I remember thinking wtf is trig supposed to apply to, and I had no idea why the rules worked, but they did. And the more I worked them the more trig made sense. Then when I took calculus I understood why they made us take geometry and trig.

Tentative Andy

07-06-2009, 07:27 PM

Vimothy was right - past a certain point, all threads do converge.

vimothy

07-06-2009, 07:30 PM

The outstanding question is: what is education (broadly defined) for?

That question even breaks down into two questions, because there is another question hidden inside it, viz. what should education be for? Or (I prefer), what does education do? And what should education do?

poetix

07-06-2009, 07:32 PM

Vimothy was right - past a certain point, all threads do converge.

Threads are locally Euclidean neighbourhoods on the surface of a larger, hideously involuted structure.

josef k.

07-06-2009, 07:32 PM

Exactly -- connections are not made, and instead teaching is the transmission of procedures.

And routines... Which is why the practice of philosophy as a "nomad science" of making unusual and unanticipated connections is valuable... and why philosophy is required to break or mutate disciplinary models (transgressing police boundaries) in order to do this... and why any attempt by a philosopher is to set-itself as a uber-policeman equipped with a masterful method and meta-language is threatening.

vimothy

07-06-2009, 07:33 PM

Vimothy was right.

Of course!

josef k.

07-06-2009, 07:34 PM

That question even breaks down into two questions, because there is another question hidden inside it, viz. what should education be for? Or (I prefer), what does education do? And what should education do?

There are at least two different sides to education... one being the disciplinary machining of bodies into particular roles and identities, the other being a kind of savage education which transcends roles and identities...

nomadthethird

07-06-2009, 07:34 PM

Vimothy was right - past a certain point, all threads do converge.

There's only one thread.

It's a function of the fact that there are only a dozen regulars on this board, who, like old marrieds, can read one anothers' minds and therefore know that each thread will basically write itself, with everyone in character and the requisite amount of bickering.

It's only the new people who infuse /d/ with fresh blood. Right now you're still new, which is why I read everything you write and enjoy your fresh-faced reactions that to the rest of us are like unique snowflakes falling onto dirt.

Mr. Tea

07-06-2009, 07:35 PM

Vimothy was right - past a certain point, all threads do converge.

http://mathworld.wolfram.com/StrangeAttractor.html

josef k.

07-06-2009, 07:35 PM

Vimothy was right - past a certain point, all threads do converge.

The point of convergence is always quite random and unpredictable... a related set of feeder threads tend to converge into a kind of torrent...

nomadthethird

07-06-2009, 07:36 PM

The point of convergence is always quite random and unpredictable... a related set of feeder threads tend to converge into a kind of torrent...

Oh please this shit writes itself!

josef k.

07-06-2009, 07:36 PM

like unique snowflakes falling onto dirt.

or pieces of my lungs slowly falling to my feet, like snow.

nomadthethird

07-06-2009, 07:40 PM

or pieces of my lungs slowly falling to my feet, like snow.

Bladerunner quotations or variations on them always work rather well.

josef k.

07-06-2009, 07:42 PM

Oh please this shit writes itself!

See also: http://www.youtube.com/watch?v=HrfCixsd2N8

nomadthethird

07-06-2009, 07:42 PM

And routines... Which is why the practice of philosophy as a "nomad science" of making unusual and unanticipated connections is valuable... and why philosophy is required to break or mutate disciplinary models (transgressing police boundaries) in order to do this... and why any attempt by a philosopher is to set-itself as a uber-policeman equipped with a masterful method and meta-language is threatening.

If you'll note, "nomad" is also a pallindrome of "monad"...

nomadthethird

07-06-2009, 07:45 PM

The time constraints are a function of a different phenomenon -- the "audit culture" of testing and measurement -- they are not naturally occurring. If we want to change the way teaching and learning occurs, one of the things we need to think about is our system of examinations.

I was just trying on a Mr. Tea character for a second.

Mr. Tea

07-06-2009, 07:46 PM

Threads are locally Euclidean neighbourhoods on the surface of a larger, hideously involuted structure.

"Not in the spaces we know, but between them..."

A differentiable manifold is a topological manifold equipped with an atlas whose transition maps are all differentiable. More generally a Ck-manifold is a topological manifold with an atlas whose transition maps are all k-times continuously differentiable.

A smooth manifold or C∞-manifold is a differentiable manifold for which all the transitions maps are smooth. That is derivatives of all orders exist; so it is a Ck-manifold for all k.

An analytic manifold, or Cω-manifold is a smooth manifold with the additional condition that each transition map is analytic: the Taylor expansion is absolutely convergent on some open ball.

'C' quite clearly standing for 'C*****u' here. By cosmic coincidence, my friend I'd lent my copy of Cyclonopedia gave it back to me today...

Tentative Andy

07-06-2009, 07:47 PM

It's only the new people who infuse /d/ with fresh blood. Right now you're still new, which is why I read everything you write and enjoy your fresh-faced reactions that to the rest of us are like unique snowflakes falling onto dirt.

Hehehe. It's not often that I get to be a fresh-faced innocent in relation to anything these days. Seriously, I'm enjoying being a noob here again, in relation to both the music and the weightier things that are discussed (and the terms of how they are discussed, etc.).

josef k.

07-06-2009, 07:48 PM

I was just trying on a Mr. Tea character for a second.

Mr. Tea is a dangerous prophet who might unite with other mystagogues, get power, and overturn the order of modernity.

nomadthethird

07-06-2009, 07:48 PM

Nobody read Cyclonopedia from start to finish. Let's be honest.

nomadthethird

07-06-2009, 07:49 PM

Hehehe. It's not often that I get to be a fresh-faced innocent in relation to anything these days. Seriously, I'm enjoying being a noob here again, in relation to both the music and the weightier things that are discussed (and the terms of how they are discussed, etc.).

But seriously, you're the best one we've had in years.

Tentative Andy

07-06-2009, 07:50 PM

Awww shucks, thanks.

josef k.

07-06-2009, 07:51 PM

The point of convergence is always quite random and unpredictable... a related set of feeder threads tend to converge into a kind of torrent...

This man is an idiot.... no, but on the other hand: it is interesting to think about the architecture of thread conversion I think. Perhaps it is what Badiou calls an "event."

nomadthethird

07-06-2009, 07:54 PM

Almost certainly not but I'll give you $100 if you don't type the word Badiou for the rest of the week.

josef k.

07-06-2009, 07:59 PM

What about "Boudu"?

Mr. Tea

07-06-2009, 08:00 PM

Nobody read Cyclonopedia from start to finish. Let's be honest.

*demurely raises hand*

poetix

07-06-2009, 08:01 PM

I read all of it, but backwards.

nomadthethird

07-06-2009, 08:01 PM

What about "Boudu"?

Nope that and "rhetoric" are out.

I'll give up "D&G" and "science".

nomadthethird

07-06-2009, 08:03 PM

*demurely raises hand*

How many months did this take?/How many hits of acid?

No, I don't believe either of you.

josef k.

07-06-2009, 08:04 PM

I'll swap "rhetoric" for "science"

nomadthethird

07-06-2009, 08:07 PM

I'll swap "rhetoric" for "science"

No swaps, no fingers crossed, no take-backs.

Mr. Tea

07-06-2009, 08:11 PM

How many months did this take?/How many hits of acid?

No, I don't believe either of you.

Couple of weeks.

No acid, but I did read a bit of it while otherwise intoxicated - dunno if it make any more sense, but I don't think it made any less.

josef k.

07-06-2009, 08:15 PM

No swaps, no fingers crossed, no take-backs.

You drive a hard bargain, white man.

nomadthethird

07-06-2009, 08:16 PM

Couple of weeks.

No acid, but I did read a bit of it while otherwise intoxicated - dunno if it make any more sense, but I don't think it made any less.

I was gonna say, don't waste good acid!

I just thought within the first 30 pages-- ok I get it. Maybe I just have ADD, or did at the time.

Mr. Tea

07-06-2009, 08:56 PM

I suppose Advanced Dungeons & Dragons tournaments do eat into one's time somewhat.

vimothy

07-06-2009, 09:14 PM

eat into one's time

Mr. Tea

07-06-2009, 09:25 PM

"Strictly to be opened in an inertial reference frame ONLY!"

http://img142.imageshack.us/img142/4415/timeh.jpg

IdleRich

07-06-2009, 10:20 PM

"But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely procedural. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this."

But didn't the operations come first historically? The (rigorous) philosophy of it came later when people realised that it was a handy tool and the explanations as they stood weren't adequate. So in a sense people are learning it in an order that is somewhat justfiable.

I read Cyclonopedia to the end - although I didn't really enjoy it so I'm not sure why really.

vimothy

07-06-2009, 11:33 PM

I'm not arguing for Philosophy, just conceptual understanding. It's a question of connecting one procedure to the others, so that you can understand it as a concept -- a mathematical, not philosophical concept -- as well as a procedure. But there are time constraints imposed because everyone needs to know enough procedures to pass the exam.

IdleRich

08-06-2009, 12:06 AM

"I'm not arguing for Philosophy, just conceptual understanding. It's a question of connecting one procedure to the others, so that you can understand it as a concept -- a mathematical, not philosophical concept -- as well as a procedure. But there are time constraints imposed because everyone needs to know enough procedures to pass the exam."

Well, when we started doing differentiation we did derive it in a way that made sense. It's just that at university we did it in a different way as we had a more subtle understanding of various types of limits and things. But my understanding is that that derivation came later so I don't have a problem with seeing them that way round. What I'm saying is that I certainly don't feel as though the methodology of calculus was just plucked out of the blue when I was doing my a-levels - although we had a very good teacher, maybe that wasn't the usual experience.

I didn't think that you were arguing for a philosophical description as such but rather something that went further than "this seems to work and although we're not totally sure why and the derivation is a bit dodgy who cares?" - and I agree with you, but I felt that I did get the first steps of that.

vimothy

08-06-2009, 12:20 AM

That's cool. Obviously, it varies. I guess at A level it's the prior knowledge of the class versus the constraint of the exam. A really good, homogeneous class can probably do a lot.

IdleRich

08-06-2009, 12:38 AM

"That's cool. Obviously, it varies. I guess at A level it's the prior knowledge of the class versus the constraint of the exam. A really good, homogeneous class can probably do a lot."

Well, it's interesting, I did maths and further maths at a-level and in the further maths group there were only about six or eight of us who were all quite bright (if I can say that) and everyone was obviously interested enough to do maths twice; on the other hand the normal maths group was very mixed ability and about thirty people and it was in this that we first studied differentation obviously. I think the justification was most likely presented along the lines of "you may not stricly need to know this for the exam but here it is". It was certainly a far cry from "You don't need to know this so shut the fuck up".

In fact, I never really found that was the caes studying maths. In a desperate attempt to find some focus to my "career" I started studying CIMA (accountancy) and found this attitude rife - I gave up very quickly 'cause that made it so fucking boring.

Mr. Tea

08-06-2009, 01:27 AM

I'm not arguing for Philosophy, just conceptual understanding. It's a question of connecting one procedure to the others, so that you can understand it as a concept -- a mathematical, not philosophical concept -- as well as a procedure. But there are time constraints imposed because everyone needs to know enough procedures to pass the exam.

Yes, I've seen this quite often with my students - I'll say something like "Oh come on, I know you can do this, it's just a fraction!" and they respond "Oh, I didn't know this was a fractions question...". As if maths (or any subject) is naturally divided up into nice discrete, non-overlapping topics. Which is, probably unavoidably, how it's taught of course.

I'm also hearing you on procedure vs. understanding. They'll get stumped on some really trivial problem because it's not presented to them in exactly the format they're used to, and they don't know what to do - then when I give them a nudge in the right direction they can (usually) solve it using their familiar techniques.

I think I have once had a student say those terrible words "But I don't have to understand it though, do I?". :mad: Poor girl, I mean it's just a symptom of her teachers' (and the entire system's) approach. Though she was an unusually bad case - rich parents, ergo possibly a lot of pressure to get those precious grades at that expensive school.* Edit: though I should add this cropped up more often in her physics work than in the maths.

*not to imply that my less well-off students' parents didn't want their kids to do well too, of course - they presumably wouldn't have hired me in that case.

Mr. Tea

08-06-2009, 01:43 AM

If you'll note, "nomad" is also a pallindrome of "monad"...

...rhymes with "gonad"...

nomadthethird

08-06-2009, 03:18 AM

I suppose Advanced Dungeons & Dragons tournaments do eat into one's time somewhat.

But you don't understand, this is the only place I can spazz out and be a dork. My friends have all read theory and stuff but they don't really care about it--beyond a few extremely cracked out/impressionistic remarks here and there-- because they are artists, which means they mostly care about a) being hot/looking good b) fucking other scenesters and c) maintaining a base-level of abject poverty that doesn't quite match their earning potential.

These days I prefer to be alone where I can wallow in depression without having to bother with full make-up or cover charges. I'd rather dole out exp points here than live like that anyway. :D

pajbre

08-06-2009, 05:03 AM

erm, i read all of cyclonopedia (during the day, mostly, at night the girlfriend would look at some of the illustrations and say 'don't go to the darkside!')... one thing that helped was realizing that if one emailed reza with questions, he would respond with almost ridiculously generous explanations/provocations.

vimothy

08-06-2009, 03:26 PM

Mr Tea: that's pretty much exactly what I mean.

nomadthethird

08-06-2009, 08:02 PM

erm, i read all of cyclonopedia (during the day, mostly, at night the girlfriend would look at some of the illustrations and say 'don't go to the darkside!')... one thing that helped was realizing that if one emailed reza with questions, he would respond with almost ridiculously generous explanations/provocations.

I was just reading through again and it's sort of like Deleuzian madlibs.

nomadthethird

12-06-2009, 03:49 AM

from Theoretical Writings:

http://img19.imageshack.us/img19/134/theoreticalwritings.jpg (http://img19.imageshack.us/i/theoreticalwritings.jpg/)

poetix

12-06-2009, 08:54 PM

It occurs to me that Lyotard's "great ephemeral skin" is a similar figure; which is unlikely to be a coincidence.

josef k.

12-06-2009, 08:58 PM

You should read that book again Dominic.

nomadthethird

12-06-2009, 09:31 PM

Well, both are 'schizoid' reinscriptions of the phallic order through deterritorialization and then newly mapping (zillions of) 'locales' over one another, so I think yeah, there's a parallel to be drawn and a clear and direct influence of Lyotard on Deleuze...

josef k.

12-06-2009, 10:27 PM

What she said.

nomadthethird

13-06-2009, 06:46 PM

What she said.

That wasn't grammatical at all was it. Well, Artaud probably comes before Lyotard.

Edit: looks like LE was published after MP. So I'm not sure which way the influence was flowing.