Mathematics

Mr. Tea

Mr. Tea

Let's Talk About Ceps
OK, well I had a good deal of difficulty sleeping last night for some reason, and I had all these thoughts turning over in my head, some of which I might be able to remember.

Firstly, as to whether things like numbers and shapes 'exist': I said I earlier that I was leaning towards the idea that they don't really exist in any meaningful way, which presumably means they're just signs or something. But then it struck me that within the context of maths itself numbers are undoubtedly objects, in a way that they're not in tangible actuality. For instance, in the context of physical objects an empty box is practically and formally identical to box containing zero things, so here it makes sense to identify the number 0 with the concept of nothing. But in set theory, a set containing only the number 0 is not empty (symbolically, {} =/= {0}). So in this context, a set containing nothing is different from a set containing nothing! This is because although zero signifies a vanishing quantity of something, the concept of zero has an existence of its own, the same way dragons don't exist but the concept of 'dragon' - the word itself and the associations it has in people's minds - certainly does exist. Though it is of course a human construct, so if there were no humans it would not exist; by contrast the concept of 'rabbit' would also cease to exist if there were no humans around to have concepts, but of course rabbits would carry on existing regardless.

This led me to surmise that numbers exist as objects within the context of mathematics at a similar ontological level to the way words exist as objects in the context of a language. I then came to something like the following conclusions:

  • numbers, shapes, equations and so on exist as objects within the context of mathematics; it's not that they cease to exist outside this context, they just cease to be meaningful (in much the same way that if I'm speaking English to someone who doesn't understand it, it's not that my voice is inaudible to him, it's just that my words aren't recognisable as meaningful semantic objects);

  • mathematics, like a verbal language, a belief system, a set of laws or the UNIX operating system, is a human construct, essentially a set of conventions and in some sense a highly formal and symbolic language in its own right;

  • we could maybe define mathematics as "that area of human discourse where statements are true if and only if they are formally tautological".
Does this seem fair? I'd love some input here from Slothrop, Rich or anyone else who's studied mathematics at some higher level, or indeed anyone with anything to say on this topic.
 
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nomadthethird

nomadthethird

more issues than Time mag
msoes said:
it is quite well accepted that mathematics is a tautology. you create a system of rules, and then you prove that accepting these rules, another rule.
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Mr. Tea said:
OK, well I had a good deal of difficulty sleeping last night for some reason, and I had all these thoughts turning over in my head, some of which I might be able to remember.

Firstly, as to whether things like numbers and shapes 'exist': I said I earlier that I was leaning towards the idea that they don't really exist in any meaningful way, which presumably means they're just signs or something. But then it struck me that within the context of maths itself numbers are undoubtedly objects, in a way that they're not in tangible actuality. For instance, in the context of physical objects an empty box is practically and formally identical to box containing zero things, so here it makes sense to identify the number 0 with the concept of nothing. But in set theory, a set containing only the number 0 is not empty (symbolically, {} =/= {0}). So in this context, a set containing nothing is different from a set containing nothing! This is because although zero signifies a vanishing quantity of something, the concept of zero has an existence of its own, the same way dragons don't exist but the concept of 'dragon' - the word itself and the associations it has in people's minds - certainly does exist. Though it is of course a human construct, so if there were no humans it would not exist; by contrast the concept of 'rabbit' would also cease to exist if there were no humans around to have concepts, but of course rabbits would carry on existing regardless.

This led me to surmise that numbers exist as objects within the context of mathematics at a similar ontological level to the way words exist as objects in the context of a language. I then came to something like the following conclusions:

numbers, shapes, equations and so on exist as objects within the context of mathematics; it's not that they cease to exist outside this context, they just cease to be meaningful (in much the same way that if I'm speaking English to someone who doesn't understand it, it's not that my voice is inaudible to him, it's just that my words aren't recognisable as meaningful semantic objects);


mathematics, like a verbal language, a belief system, a set of laws or the UNIX operating system, is a human construct, essentially a set of conventions and in some sense a highly formal and symbolic language in its own right;


we could maybe define mathematics as "that area of human discourse where statements are true if and only if they are formally tautological".

Does this seem fair? I'd love some input here from Slothrop, Rich or anyone else who's studied mathematics at some higher level, or indeed anyone with anything to say on this topic.
Click to expand...

I'd like to nominate these two posts for some kind of prize. They are very good. I agree wholeheartedly with both of them.
 
swears

swears

preppy-kei
Mr. Tea said:
...a ghostly realm of 'pure' numbers and shapes floating around behind or above our world of physical being, mortality, imperfection and impermanence.
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This is how I thought about maths as a child, it's tempting isn't it?
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
swears said:
This is how I thought about maths as a child, it's tempting isn't it?
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I've kind of fantasised about the laws of physics and causality operating thanks to the vigil of these 'cosmic book-keepers', like little invisible elves or something, and every now and then one of them fucks up and you have something like the Marie Celeste or the Tunguska event, and they get a bollocking from head office. Maybe they're the 'machine elves' Terrence McKenna sees when he takes DMT, heh.
 
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P

poetix

we murder to dissect
If mathematics is a fiction, it's a fiction which generates impasses; the real of mathematics passes through those impasses. The human is countersigned by the inhuman.
 
Mr. Tea

Mr. Tea

Let's Talk About Ceps
poetix said:
If mathematics is a fiction, it's a fiction which generates impasses; the real of mathematics passes through those impasses. The human is countersigned by the inhuman.
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Hmm, I dunno about it being a fiction: in fiction the author usually has some choice over what they write, don't they? I mean of course any given mathematician may choose to investigate number theory or topology or whatever, but they have no choice over what conclusions to come to when they go down one particular path of investigation. Also I'm not clear what you mean by 'impass'?
 
nomadthethird

nomadthethird

more issues than Time mag
poetix said:
If mathematics is a fiction, it's a fiction which generates impasses; the real of mathematics passes through those impasses. The human is countersigned by the inhuman.
Click to expand...

I'm of two minds on this I guess. I'm interested in seeing where math can take us, but the only place I see real impasses for ontology or metaphysics when it comes to math are in its applications. Then again, were I to come across a mind-blowing aporetic set theoretical ontological endgame, I'd immediately renounce this position.

So in my mind, it's worth giving Badiou and speculative realism a shot, but so far I can't say they've delivered much of anything beyond a lot of important-sounding proclamations about the state of the game in metaphysics post-Kant.
 
P

poetix

we murder to dissect
The axiomatic starting points of set theory (for example) are products of human minds, decisions taken by people about where to begin and how to proceed. In actual fact, axiomatisation and formalisation of mathematics tends to run along slightly behind the real practice of mathematicians - it provides a kind of retroactive foundation, as well as a way of backing up when problems arise, linking those problems to some fundamental crux or crisis.

The issue I think isn't that the real world resists mathematisation - if anything, it gives in suspiciously easily - but that mathematics elaborates its own resistance (when Paul de Man said that the resistance to theory is intrinsic to theory itself, you could say he was expressing an aspiration that theory should attain to the condition of mathematics). It throws up difficulties that nobody thought of when the formal conditions under which those difficulties arise were thought of. Where do the difficulties come from? Not from the "real world", and not directly from mathematicians themselves: they "occur within" mathematics, and "occur to" mathematicians, who then have to work with and around them.
 
nomadthethird

nomadthethird

more issues than Time mag
poetix said:
The axiomatic starting points of set theory (for example) are products of human minds, decisions taken by people about where to begin and how to proceed. In actual fact, axiomatisation and formalisation of mathematics tends to run along slightly behind the real practice of mathematicians - it provides a kind of retroactive foundation, as well as a way of backing up when problems arise, linking those problems to some fundamental crux or crisis.

The issue I think isn't that the real world resists mathematisation - if anything, it gives in suspiciously easily - but that mathematics elaborates its own resistance (when Paul de Man said that the resistance to theory is intrinsic to theory itself, you could say he was expressing an aspiration that theory should attain to the condition of mathematics). It throws up difficulties that nobody thought of when the formal conditions under which those difficulties arise were thought of. Where do the difficulties come from? Not from the "real world", and not directly from mathematicians themselves: they "occur within" mathematics, and "occur to" mathematicians, who then have to work with and around them.
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Right. But if they only occur within mathematics, do they matter to anybody? Do they really make a difference? I know this is the standard argument that people make about why philosophy in general isn't relevant to the "real world", so it's tiresome and stupid... But do we agree that there's something dodgy about the idea that the more formal you get, the more 'absolute' the truths?

In some ways the speculative realist arguments remind me of the race discussion we had here months ago. Correlationism boils down to the "if a tree falls in the forest, and no one is there to hear it, it doesn't make a sound" cliche. But that tree *does* make a sound, because when it falls, it causes molecules to move in the form of sound waves, whether anyone is there to hear or not.

Race is not "real" in that it doesn't exist outside of a "correlation", it is not really "there"-- correlationists are the ones who love to declare things like race entirely "real" just because they are intersubjectively "true." It's funny because I do wonder whether speculative realists would actually want to empty out the ontological realm politically in the way the seem to propose.
 
P

poetix

we murder to dissect
nomadthesecond said:
Right. But if they only occur within mathematics, do they matter to anybody? Do they really make a difference?
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They are differences. But, for example, they only make a difference to philosophy if philosophy places itself under condition of some mathematical procedure, as Badiou's does. I think it's possible to argue that other philosophers employ semi-formal conceptual arrangements that are mathematically tractable, and that some of the problems that appear within those conceptual arrangements can be restated as mathematical problems (and then subject to the kind of retroactive formalisation I mentioned, which may or may not clarify the situation). I don't think a thoroughgoing subordination of philosophy to mathematics is necessary, or even possible.

I know this is the standard argument that people make about why philosophy in general isn't relevant to the "real world", so it's tiresome and stupid... But do we agree that there's something dodgy about the idea that the more formal you get, the more 'absolute' the truths?
Click to expand...

Well, formalisation is a kind of absolutisation. It's not the universal key to all mythology, but it is a procedure that enables you to get a certain sort of traction over certain kinds of problems.

In some ways the speculative realist arguments remind me of the race discussion we had here months ago. Correlationism boils down to the "if a tree falls in the forest, and no one is there to hear it, it doesn't make a sound" cliche.
Click to expand...

Yes and no - that cliche is really about primary versus secondary qualities (the idea being that moving molecules are physical realities, while "sound" is something experienced by an auditor). Correlationism effectively insists that absent some subject that organises (transcendentally or otherwise) a field of experience containing entities such as tree, forest, moving molecules etc., none of these entities really exists, since their mode of existence is intrinsically and irrevocably relational. Experience is experience <em>of</em> something (correlationism giveth), but it is only ever <em>experience</em> of something (and correlationism taketh away).

(Mathematics, rather usefully from the point of view of someone trying to frame a counter-argument to this, is not the experience of anything at all).

Race is not "real" in that it doesn't exist outside of a "correlation", it is not really "there"-- correlationists are the ones who love to declare things like race entirely "real" just because they are intersubjectively "true." It's funny because I do wonder whether speculative realists would actually want to empty out the ontological realm politically in the way the seem to propose.
Click to expand...

Since I've been reading Graham Harman's very splendid book on Latour recently, I feel emboldened to suggest that race has powerful allies in reality, and that anti-racism has to separate race from those allies (thereby diminishing the intensity with which it appears in the world) and not merely dismiss it as an irritatingly persistent illusion. Somewhere in Junky, Burroughs describes a confederate as looking like a ghost that has materialised in someone's old overcoat. Race has a dense cloud of confederates, among them cops, prisons, newspapers and superannuated white geneticists - it certainly "makes a difference", to which anti-racists must seek to make a difference in their turn.
 
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Mr. Tea

Mr. Tea

Let's Talk About Ceps
poetix said:
The issue I think isn't that the real world resists mathematisation - if anything, it gives in suspiciously easily
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Ha, I like this line - isn't there some quote from a famous scientist (Einstein?) to the effect that "the most incredible thing about the universe is that we can understand any of it at all"? If there isn't, there should be...

poetix said:
- but that mathematics elaborates its own resistance (when Paul de Man said that the resistance to theory is intrinsic to theory itself, you could say he was expressing an aspiration that theory should attain to the condition of mathematics). It throws up difficulties that nobody thought of when the formal conditions under which those difficulties arise were thought of. Where do the difficulties come from? Not from the "real world", and not directly from mathematicians themselves: they "occur within" mathematics, and "occur to" mathematicians, who then have to work with and around them.
Click to expand...

Do you mean things like Goedel's incompleteness theorem? That seems to be the most obvious example of this kind of "resistance", though there may be others.
 
nomadthethird

nomadthethird

more issues than Time mag
poetix said:
Well, formalisation is a kind of absolutisation. It's not the universal key to all mythology, but it is a procedure that enables you to get a certain sort of traction over certain kinds of problems.
Click to expand...

This is a good way of putting it.

poetix said:
Yes and no - that cliche is really about primary versus secondary properties (the idea being that moving molecules are physical realities, while "sound" is something experienced by an auditor). Correlationism effectively insists that absent some subject that organises (transcendentally or otherwise) a field of experience containing entities such as tree, forest, moving molecules etc., none of these entities really exists, since their mode of existence is intrinsically and irrevocably relational. Experience is experience <em>of</em> something (correlationism giveth), but it is only ever <em>experience</em> of something (and correlationism taketh away).

(Mathematics, rather usefully from the point of view of someone trying to frame a counter-argument to this, is not the experience of anything at all).
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I don't know if I buy the difference between "primary" and "secondary" properties, though. M. has quite a bit more convincing to do before I'm going to decide that sound is different from sound waves (moving molecules).** These are the sorts of distinctions that, again, matter more in theory than in practice, they sound great as a metaphysical vocabulary but if you tried to make a scientific distinction between them you'd be hard pressed.

poetix said:
Since I've been reading Graham Harman's very splendid book on Latour recently, I feel emboldened to suggest that race has powerful allies in reality, and that anti-racism has to separate race from those allies (thereby diminishing the intensity with which it appears in the world) and not merely dismiss it as an irritatingly persistent illusion. Somewhere in Junky, Burroughs describes a confederate as looking like a ghost that has materialised in someone's old overcoat. Race has a dense cloud of confederates, among them cops, prisons, newspapers and superannuated white geneticists - it certainly "makes a difference", to which anti-racists must seek to make a difference in their turn.
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I agree completely. It's never an either/or sort of proposition, Ontology or ontologies, but I do feel a tinge of guilt spending so much energy worried about certain things when certain other things are so much more of a problem for other people. Re Prince of Networks, I emailed GH and asked for this (thnx btw, I wouldn't have known about it if your blog hadn't tipped off) and it's excellent and certainly deserves it's own thread. Latour is a new obsession for me, and luckily, unlike some of the soc of science guys, his work is actually translated into English.

**Of course, I realize that M thinks he's a realist, and therefore that he doesn't fall into the same intellectual traps as the correlationists, but he does want to "rehabilitate" primary and secondary metaphysical properties in AF.
 
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nomadthethird

nomadthethird

more issues than Time mag
Another thing I disagree completely with Meillassoux (and maybe even Harman/Latour) about? Heidegger being a correlationist. It's especially funny the quotes that are used to make this argument.

But that's another thread.
 
vimothy

vimothy

yurp
"It is a profound puzzle that on the one hand mathematical truths seem to have a compelling inevitability, but on the other hand the source of their "truthfulness" remains elusive"
 
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