Fascism!

[Slothrop]

Infinite dimensions in space would not, mathematically speaking, be inconceivable. I see from a quick google that there are papers on, for example, "infinite-dimensional compact hausdorff spaces"...

Yeah, in many ways infinite dimension aren't that hard tbh. For countably infinite dimensions you can just go from thinking of coordinates (n1, n2,...,nd) to coordinates (n1,n2,...).

For uncountably infinite dimensional spaces tend to be spaces of functions. The space of continuous functions from the real numbers to the real numbers is infinite dimensional, for instance.

 

[nomadthethird]

Yeah, in many ways infinite dimension aren't that hard tbh. For countably infinite dimensions you can just go from thinking of coordinates (n1, n2,...,nd) to coordinates (n1,n2,...).

For uncountably infinite dimensional spaces tend to be spaces of functions. The space of continuous functions from the real numbers to the real numbers is infinite dimensional, for instance.

I'm not talking about dimensions in that way.

I meant: what if there was another kind of material realm that was dimensionally altogether different from ours, didn't have energy, didn't have time, or something crazy like that? I'm sure people will say "well numbers will work anywhere" but how can you be so sure that other worlds couldn't exist where they didn't?

That's like saying English is a language any human is capable of understanding, so English would still be legible/intelligible and its grammatical structure would still hold in a completely different universe where there were no atoms or molecules and instead of space and time there was something altogether different.

In a purely formal sense, yes, our formal systems would be intelligible anywhere to beings like us, but it's conceivable that there could be other kinds of Being and no beings.

Sorry not trying to sound like a total stoner but I'm thinking like those "thought experiment" philosophers were always thinking...

 

[Slothrop]

I think mathematics is a formal language that exists because we live in a world where objects exist, and not only that, where they exist in very particular way according to certain general principles.

Yeah, to some extent and on some occasions the physics pushes the maths forwards and leads to new mathematical tools being created to deal with the concepts that are arising - calculus is the classic case of this, but a lot of the quantum gravity people I've known have been working with specific sorts of manifolds and things and pushing forward the maths as well as just finding applications of the existing stuff to physics.

Another thing that's interesting is the idea of describing 'a universe' in whatever terms, rather than describing 'the universe' in terms of the laws of physics. So (for instance) the growing need to talk about theoretical computer science in 'mathematical' terms is actually leading to whole new formulations of set theory - replacing the axiom of foundation, for instance - to give a mathematical language that suits that world rather than the ZF / ZFC based language that seems to work pretty well at describing the world of physics.

To some extent, the thing that's being kept constant in all this is actually the first order predicate logic that the set theory is written in. Tbh I don't know whether you could interestingly replace that with anything else, and I think that might be the crux of the issue. My gut instinct is that there's a sort of formal description oriented Church-Turing hypothesis that says that any system with a sufficient level of power at formal description (fsvo formal description) is equivalent to any other such system. But that's a total guess.

 

[nomadthethird]

Yeah, to some extent and on some occasions the physics pushes the maths forwards and leads to new mathematical tools being created to deal with the concepts that are arising - calculus is the classic case of this, but a lot of the quantum gravity people I've known have been working with specific sorts of manifolds and things and pushing forward the maths as well as just finding applications of the existing stuff to physics.

Another thing that's interesting is the idea of describing 'a universe' in whatever terms, rather than describing 'the universe' in terms of the laws of physics. So (for instance) the growing need to talk about theoretical computer science in 'mathematical' terms is actually leading to whole new formulations of set theory - replacing the axiom of foundation, for instance - to give a mathematical language that suits that world rather than the ZF / ZFC based language that seems to work pretty well at describing the world of physics.

To some extent, the thing that's being kept constant in all this is actually the first order predicate logic that the set theory is written in. Tbh I don't know whether you could interestingly replace that with anything else, and I think that might be the crux of the issue. My gut instinct is that there's a sort of formal description oriented Church-Turing hypothesis that says that any system with a sufficient level of power at formal description (fsvo formal description) is equivalent to any other such system. But that's a total guess.

Right, right. This is what I was trying to get at.

Are there any mathematicians/physicists who've tried to formalize "a universe" rather than "the universe" who spring to mind?

I really should take some math courses when I'm back in school, it's not fun to be totally clueless about these things. I think I always avoided learning about computer science/programming because that's what my father does and it was always in the background so I took it for granted. That and inorganic chemistry, which I never learned the first time because my dad did all my homework for me...

 

[poetix]

I'm not talking about dimensions in that way.

But that's pretty much the only way in which there "are" dimensions in modern physics.

I meant: what if there was another kind of material realm that was dimensionally altogether different from ours, didn't have energy, didn't have time, or something crazy like that?

It might not be able to contain beings capable of doing mathematics. But once you started filling in the details of the thought experiment, describing what properties this realm did have, my wager would be that this description would increasingly lend itself to formalisation. None of the things you've subtracted so far needs to exist in a domain for that domain to be thinkable mathematically. There's no "time" or "energy" in the domain of Euclidean solids, for example...

Since you control the description, let's suppose for a moment that you want to play a sort of mad, hysterical, radically inconsistent God. Nothing in your domain has any sort of identity: everything is already everything else, or at least indiscernible from everything else. Whenever I venture a formalisation of what you have told me so far about your universe, you get to just change the story in whatever way is necessary to break the formalisation. I can't complain that you're being contradictory, because that's just the way things are in bizarro-universe. (The only way, in fact, to formalise the description would be to develop an inerrantly predictive model of your own whims and fancies. Then it might be possible to deploy a system large enough to permit me to say to each new system-breaking move on your part, "aha! I've already thought of that...")

But even if one allows for the possibility of radically hystericised universes that say "no, that's still not it!" to every attempt at formalisation of their characteristics (and it may be that we live in just such a universe already), I don't think this really has a bearing on whether the mathematics "working" is contingent on our universe being the particular way it is; rather, it's contingent on our universe being some particular way rather than no particular way - on its being particular about the way it is. One can only give a formalised account of a stable set of particulars. Part of the appeal of something like Deleuze's conception of virtuality is that it suggests that there's always a sort of cloud of alternatives wafting around any such stable presentation. If mathematics is ontology, it is an ontology of the actual (Badiou says precisely this), and so perhaps for the Deleuzian an ontology of that which is "still not it" - "it", ultimately, being the One-All of being in its univocal becoming.

 

[poetix]

Are there any mathematicians/physicists who've tried to formalize "a universe" rather than "the universe" who spring to mind?

Er, topos theory...?

 

[Slothrop]

Right, right. This is what I was trying to get at.

Are there any mathematicians/physicists who've tried to formalize "a universe" rather than "the universe" who spring to mind?

In 'physical' terms, I'm not sure. One of the standard drives in maths is to generalize stuff, by saying 'lets take this thing that we're used to dealing with and see if there's anything that obeys most but not all of the same rules as it.' This starts with generalizing numbers and polynomials to abstract algebraic structures and generalizing intuitively 'real world' geometry to metric and topological spaces, but we're now getting to the stage where, for instance, the set theory and even the logical systems that in some sense underpin the whole lot are getting generalised on, or the notion of a bunch of structures or a spaces being abstracted to give category (nothing to do with the philosophical sense afaik) theory.

Once you've come up with some different axioms for a set theory or whatever, the obvious thing to do is to see what the sort of maths that you can do with it looks like, so I'd assume that a fair bit of that goes on. Versions that include the axiom of choice have always been known to provide 'useful' results which is why the axiom was postulated, but I think that versions that include its negation have also turned up interesting stuff. And I've been to a few category theory seminars that have involved laboriously constructing basic mathematical structures using a massively ass-backwards framework, to see if they behave normally. But I don't know if anyone's actually speculated about a universe where the laws have to be framed in these terms. The closest I can think of at the moment is people using set theories that don't use the axiom of foundation (which normally forbids sets from being members of themselves or members of members of themselves or whatever) so they can naturally model self referential data structures in theoretical computer science.

Oh hang on, and I think a few people have started using "quantum logic", where the AND operator is replaced by AND THEN in the hopes of making the time dependancy of quantum into a feature of the mathematical language. But I'm not sure whether they're getting anywhere.

I really should take some math courses when I'm back in school, it's not fun to be totally clueless about these things.

Maybe go for set theory and mathematical logic and the foundations of computer science (Turing machines, the lambda calculus, that sort of stuff) over straight up maths? Maths is great, but it does take quite a commitment of time and effort before you get to anything really interesting, and the philosophical side you only really pick up from talking to other mathematicians. Maybe my perspective on what's 'really interesting' is skewed by being at the back end of eight years of study, though.

FWIW and IMO this is a really good introduction to mathematical logic. It was originally written as a series of maths lectures aimed at philosophy students, so it's in the mathematical style but doesn't assume that you're already familiar with that way of working. In any case, I used it when I was doing undergrad set theory and it was handy, and has some really good stuff about incompleteness, the failure of set theoretic reductionism, models and so on.

Sorry, that was an epic pile of maths geekery. And I might be offline for a few days so I probbaly won't be able to get back to people immediately...

 

[nomadthethird]

(The only way, in fact, to formalise the description would be to develop an inerrantly predictive model of your own whims and fancies. Then it might be possible to deploy a system large enough to permit me to say to each new system-breaking move on your part, "aha! I've already thought of that...")

LOL. How much time do you have?

But even if one allows for the possibility of radically hystericised universes that say "no, that's still not it!" to every attempt at formalisation of their characteristics (and it may be that we live in just such a universe already), I don't think this really has a bearing on whether the mathematics "working" is contingent on our universe being the particular way it is; rather, it's contingent on our universe being some particular way rather than no particular way - on its being particular about the way it is. One can only give a formalised account of a stable set of particulars. Part of the appeal of something like Deleuze's conception of virtuality is that it suggests that there's always a sort of cloud of alternatives wafting around any such stable presentation. If mathematics is ontology, it is an ontology of the actual (Badiou says precisely this), and so perhaps for the Deleuzian an ontology of that which is "still not it" - "it", ultimately, being the One-All of being in its univocal becoming.

So math in one form or another is always (or at least, most probably) going to be the most radically abstract formalization possible-- while also the most actual, which is to say the least virtual-- in any universe where there are beings/objects becoming, even a Deleuzean one that vaguely gestures toward alternatives that can't yet be formalized.

But yeah I would say math doesn't even come close to formalizing the One at the Spinozan core of Deleuzian ontology.

I suppose it's not inconsistent with Deleuzian ideals to think of "multiplicities" as the single most significant feature of a world/universe, but for Deleuze this notion is in stark contrast to "One-Multiple" organization of dualism/Cartesian geometry, as he makes abundantly clear.

 

[nomadthethird]

Sorry, that was an epic pile of maths geekery. And I might be offline for a few days so I probbaly won't be able to get back to people immediately...

A welcome one.

 

[nomadthethird]

The closest I can think of at the moment is people using set theories that don't use the axiom of foundation (which normally forbids sets from being members of themselves or members of members of themselves or whatever) so they can naturally model self referential data structures in theoretical computer science...

Sounds interesting. At least more promising than this:

Oh hang on, and I think a few people have started using "quantum logic", where the AND operator is replaced by AND THEN in the hopes of making the time dependancy of quantum into a feature of the mathematical language. But I'm not sure whether they're getting anywhere.

...which sounds like a minor semantic tweak rather than a solid innovation. (to the admittedly clueless third party)

 

[nomadthethird]

Er, topos theory...?

I meant in the hard sciences but this works. In fact, I already ordered it once from Amazon used books and it was never delivered. Or maybe it was lost in the between-homes ether.

(Those Amazon "related" fora suggested to Badiou readers under the sales pitch look like hours of fun...)

 

[Mr. Tea]

Hmmm. But I think math is really about a very specific universe in which things are divisible, they exist, they exist in time and space in just such a way...it's difficult to conceive of yet in specific terms but generally conceivable that there could be other universes, worlds, whatever.

I'd have to disagree more or less completely with this...the unique thing about maths is that is applies even in the absence of any universe at all. There's no possible universe in which 2+2 isn't going to make 4. Furthermore, one can (and mathematicians certainly do) investigate all kinds of universes that are radically different from ours - a good example of this is working out what the laws of physics and cosmology would look like in universes with fewer or more than three space dimensions, or with dimensions that are non-trivially connected or have other weird geometric and topological properties. There are even serious mathematicians who've investigated multiple time dimensions.

 

[poetix]

That 2+2=4 is a fact about 2, +, = and 4, rather than a fact about the universe.

 

[Mr. Tea]

That 2+2=4 is a fact about 2, = and 4, rather than a fact about the universe.

Sure, that's kind of what I was getting at: mathematics conceptually precedes the material universe because it doesn't require the existence of one in order to make demonstrably true statements. In fact I would go so far as to say that it's in mathematics that one can derived or prove truths, whereas any true statement that's contingent on the material universe is a fact. So: that Mount Everest is the highest point on earth and that electrons have a negative charge are facts, but that the interior angles of a triangle in Euclidean space sum to 180 degrees is a truth.

 

[nomadthethird]

ugh

There are possibly universes where there are no objects or at very least where objects are not experienced the same way we experience them here.

Like I said before, you can contend that all formal systems hold in any possible universe, but it's possible that they might not be entirely relevant to the way things are in other kinds of universes.

 

[nomadthethird]

So: that Mount Everest is the highest point on earth and that electrons have a negative charge are facts, but that the interior angles of a triangle in Euclidean space sum to 180 degrees is a truth.

Doesn't what's "true" mathematically get a little less clear/straightforward when it's applied to physics?

 

[nomadthethird]

That 2+2=4 is a fact about 2, +, = and 4, rather than a fact about the universe.

Was thinking about this and about the comment you made about computers not being farther up or down the 'Great Chain of Being' than humans...

Is there a sense in which McLuhan's "extension of man" idea can be thought without recourse to any qualitative differentiation between beings, but in a sort of chronological way that takes into account the fact that without human beings of a certain sort who evolved in a certain way there'd quite possibly never have been computers? It does seem significant that math and computers appeared quite a while after beings with consciousness like ours did.

 

[nomadthethird]

Check out the last couple of paragraphs on this page and the next few pages for AB's take on Deleuze and formalization w/r/t Being:

http://books.google.com/books?hl=en...=X&oi=book_result&resnum=3&ct=result#PPT41,M1

 

[poetix]

Was thinking about this and about the comment you made about computers not being farther up or down the 'Great Chain of Being' than humans...

Is there a sense in which McLuhan's "extension of man" idea can be thought without recourse to any qualitative differentiation between beings, but in a sort of chronological way that takes into account the fact that without human beings of a certain sort who evolved in a certain way there'd quite possibly never have been computers? It does seem significant that math and computers appeared quite a while after beings with consciousness like ours did.

I love Ray Brassier's line about how there's no obvious way in which the theoretical conception we currently have of, say, quantum physics is an extension of our ability to use hammers. There might still be some very subtle, tortuously complex way in which the one was a development of the other, though. Perhaps not so much an extension as an extrusion, a metastasis. Historically speaking, we got here (quantum physics) from there (tool use). There is a chronology involved, for sure: mathematical concepts don't just fall from the sky.

On the other hand, shit just happens: just because the shit that happened later happened after the shit that happened earlier, doesn't mean that the later shit is in any sense logically or metaphysically dependent on the earlier shit, inherits its fundamental characteristics and limitations or is otherwise linked to it in any way except through belonging to the same overall history - and that history might be one of corruption and deviation rather than linear filiation, such that its later moments preserve nothing of the image or essence of the earlier ones.

I think there's something hopelessly wrong about the notion that all of mathematics is an extension of some kind of book-keeping ability, that it all starts with counting apples or comparing dick size or whatever and works itself up from there. From really very early on in the history of mathematics, the bodily and haptic context starts falling away: problems appear (impasses of formalisation) that are not in any recognisable sense the problems of bodies.

The problem Pythagorus had with irrational numbers would be one example: you take a right-angled triangle, with two sides of unit length, and it turns out that the length of the hypotenuse is the square root of 2, a number that can't be found amongst either the whole numbers or the "rational" fractions. Nothing in the human-animal experience of being (in) a body really correlates to a number like that: an apple, yes; half an apple, yes; three quarters of an apple, OK; a quantity of apple that can't ever be arrived at by cutting an apple into equal-sized pieces and taking a whole number of the pieces - um...and it only gets worse from there.

Mathematics is where something like a reality principle (the symbolic real, the impasse of symbolic formalisation) shows up right in the middle of the collection of simple cognitive tricks that a bunch of monkeys have developed to facilitate the reliable sizing up of medium-sized dry goods, and systematically perverts their thinking from the inside out. Like I said, not extension so much as extrusion, metastasis, perversion or corruption. The butchering open of thought.

 

[nomadthethird]

I love Ray Brassier's line about how there's no obvious way in which the theoretical conception we currently have of, say, quantum physics is an extension of our ability to use hammers. There might still be some very subtle, tortuously complex way in which the one was a development of the other, though. Perhaps not so much an extension as an extrusion, a metastasis. Historically speaking, we got here (quantum physics) from there (tool use). There is a chronology involved, for sure: mathematical concepts don't just fall from the sky.

On the other hand, shit just happens: just because the shit that happened later happened after the shit that happened earlier, doesn't mean that the later shit is in any sense logically or metaphysically dependent on the earlier shit, inherits its fundamental characteristics and limitations or is otherwise linked to it in any way except through belonging to the same overall history - and that history might be one of corruption and deviation rather than linear filiation, such that its later moments preserve nothing of the image or essence of the earlier ones.

I think there's something hopelessly wrong about the notion that all of mathematics is an extension of some kind of book-keeping ability, that it all starts with counting apples or comparing dick size or whatever and works itself up from there. From really very early on in the history of mathematics, the bodily and haptic context starts falling away: problems appear (impasses of formalisation) that are not in any recognisable sense the problems of bodies.

The problem Pythagorus had with irrational numbers would be one example: you take a right-angled triangle, with two sides of unit length, and it turns out that the length of the hypotenuse is the square root of 2, a number that can't be found amongst either the whole numbers or the "rational" fractions. Nothing in the human-animal experience of being (in) a body really correlates to a number like that: an apple, yes; half an apple, yes; three quarters of an apple, OK; a quantity of apple that can't ever be arrived at by cutting an apple into equal-sized pieces and taking a whole number of the pieces - um...and it only gets worse from there.

Mathematics is where something like a reality principle (the symbolic real, the impasse of symbolic formalisation) shows up right in the middle of the collection of simple cognitive tricks that a bunch of monkeys have developed to facilitate the reliable sizing up of medium-sized dry goods, and systematically perverts their thinking from the inside out. Like I said, not extension so much as extrusion, metastasis, perversion or corruption. The butchering open of thought.

This is very elegantly put, one of the better posts I've read on here.

Mathematics is where something like a reality principle (the symbolic real, the impasse of symbolic formalisation) shows up right in the middle of the collection of simple cognitive tricks that a bunch of monkeys have developed to facilitate the reliable sizing up of medium-sized dry goods, and systematically perverts their thinking from the inside out. Like I said, not extension so much as extrusion, metastasis, perversion or corruption.

Now this I can get behind.

Thinking about math as another tool in techne's arsenal, but an especially disruptive one, brings us full-circle back to Heidegger, in whose margins D&G and AB are scribbling (I've always figured).