Mathematics

[josef k.]

Why must abstract objects be outside space and time?

 

[Tentative Andy]

Because they can't be identified with any spatiotemporal point, also because their necessity seems to imply that they must be transcendent. At least that's my understanding, possibly there is some way in which they don't have to be, that is sort of what I wanted to talk about.

 

[Tentative Andy]

Frege wrote about this a lot, I think (presumably in clearer fashion than me ).

 

[josef k.]

That's very interesting. What else is a abstract object?

 

[nomadthethird]

aren't there implications for mathematics from ontology?

Oh just a couple.

 

[Mr. Tea]

Why must abstract objects be outside space and time?

I like to think about this as being to do with the fact that mathematical truths - which necessarily pertain to abstract objects - are true in a way that does not depend on there being a physical universe in which these truths can be 'acted out' or 'incarnated', if you like. I can take two apples and another two apples and lo and behold, I have four apples: but this is a fact about numbers, not about apples. 2 + 2 = 4 would still be true in a universe with no apples; it's a relatively short step to accepting that it would be true if there were no universe at all. Or if there were a universe in which physical law made the existence of apples impossible even in principle (if there were no stable atoms or if all the matter were concentrated in black holes*, for example).

Facts about numbers (or sets, shapes, patterns, symmetry groups, logical propositions and so on) are fundamentally different from facts about any physical object or process, which is why I refer to them as truths rather than facts.




*which, from a thermodynamic point of view, is indescribably more likely than a universe like ours with galaxies, stars etc.

 

[nomadthethird]

I like to think about this as being to do with the fact that mathematical truths - which necessarily pertain to abstract objects - are true in a way that does not depend on there being a physical universe in which these truths can be 'acted out' or 'incarnated', if you like. I can take two apples and another two apples and lo and behold, I have four apples: but this is a fact about numbers, not about apples. 2 + 2 = 4 would still be true in a universe with no apples; it's a relatively short step to accepting that it would be true if there were no universe at all. Or if there were a universe in which physical law made the existence of apples impossible even in principle (if there were no stable atoms or if all the matter were concentrated in black holes*, for example).

Facts about numbers (or sets, shapes, patterns, symmetry groups, logical propositions and so on) are fundamentally different from facts about any physical object or process, which is why I refer to them as truths rather than facts.




*which, from a thermodynamic point of view, is indescribably more likely than a universe like ours with galaxies, stars etc.

This seems intuitively true but I don't know if I really buy it as self-evidently true.

 

[Mr. Tea]

I dunno, I've not studied ontology or metaphysics in any formal way, that's just my (intuitive, as you say) take on things from my own knowledge of maths and physics.

 

[nomadthethird]

I dunno, I've not studied ontology or metaphysics in any formal way, that's just my (intuitive, as you say) take on things from my own knowledge of maths and physics.

Well I mean I know you're probably *right* about it, I just don't want to believe that numbers and their truths are somehow a kind of universal limitation on things and thinghood in every possible scenario we can dream up. That's so boring.

 

[Mr. Tea]

Well I mean I know you're probably *right* about it, I just don't want to believe that numbers and their truths are somehow a kind of universal limitation on things and thinghood in every possible scenario we can dream up. That's so boring.

I'm not sure I quite get you. What's the alternative? Are we talking about a universe or plane of existence where the laws of mathematics and logic don't apply, where 2+2 can equal 5 or p implies not-p? Is this a worthwhile route to go down, or is it by definition pure nonsense? Hmm, that's probably not what you meant anyway...but I certainly disagree about the universality and fundamentality of mathematics being "boring".

 

[scottdisco]

FWIW Mr. Tea, i loved your post at no.66 above and you get my vote.

(my vote being meaningless as i scraped C's at GCSE Dual Award Science and Maths. but hey.)

 

[nomadthethird]

I'm not sure I quite get you. What's the alternative? Are we talking about a universe or plane of existence where the laws of mathematics and logic don't apply, where 2+2 can equal 5 or p implies not-p? Is this a worthwhile route to go down, or is it by definition pure nonsense? Hmm, that's probably not what you meant anyway...but I certainly disagree about the universality and fundamentality of mathematics being "boring".

Logic and math don't always come to the same conclusions.

I don't know if that's what I meant.

I just meant once you get past your basic "divisibility of objects in their object relationhood/relationality" then what. Sure objects are objects because they are divisible from other objects. And 'numbers' are a sort of truth about how these objects can relate to one another. But it's almost so fundmental as to seem totally given.

I just want there to be something else or in addition to numbers very badly. That ontology can grab onto I mean.

 

[Mr. Tea]

There's always God, I guess.

 

[Tentative Andy]

I like to think about this as being to do with the fact that mathematical truths - which necessarily pertain to abstract objects - are true in a way that does not depend on there being a physical universe in which these truths can be 'acted out' or 'incarnated', if you like. I can take two apples and another two apples and lo and behold, I have four apples: but this is a fact about numbers, not about apples. 2 + 2 = 4 would still be true in a universe with no apples; it's a relatively short step to accepting that it would be true if there were no universe at all. Or if there were a universe in which physical law made the existence of apples impossible even in principle (if there were no stable atoms or if all the matter were concentrated in black holes*, for example).

Facts about numbers (or sets, shapes, patterns, symmetry groups, logical propositions and so on) are fundamentally different from facts about any physical object or process, which is why I refer to them as truths rather than facts.




*which, from a thermodynamic point of view, is indescribably more likely than a universe like ours with galaxies, stars etc.

First of all - sorry to abandon thread, computer has been playing up like a bastard all day.
I thought this was very well put. I certainly agree with it, and it's something I've heard before from people that are serious about maths, albeit not always so clearly and confidently. However, again I would ask - what does this imply? Does it mean that, in addition to the physical realm we inhabit, there is also a non-physical realm of numbers? What would such a thing be like? Or am I thinking about this in the wrong way?

 

[nomadthethird]

There's always God, I guess.

Dammit.

 

[Mr. Tea]

It is tempting to postulate something like that, isn't it? Plato's realm of 'forms', in other words. As far as I remember Josef K is very hostile to this and related 'transcendental' formulations of being. In a way though I think we're imagining there's a problem there when there isn't, or it's just our way of thinking about things (a hangover from religion, superstition, belief in an 'otherworld', 'afterlife' or 'Kingdom of God', perhaps?) that makes it sound as if this implies a ghostly realm of 'pure' numbers and shapes floating around behind or above our world of physical being, mortality, imperfection and impermanence. To me this seems (neo-)Platonist, Christian-Gnostic, mystical, alchemical. Now that I get down to it, I'm not sure I think things like numbers really exist at all. Perhaps it merely seems that truths like 2+2=4 exist 'outside' or 'before' material reality because it's impossible to postulate a universe, no matter how weird-n-wacky, in which adding two things to two other things could fail to make four things.

By contrast, it's perfectly possible to imagine that there could be (in some obscure corner of the multiverse) a universe with 107 spatial dimensions, or where the speed of light depends on the day of the week, or in which gravity is universally repulsive. Because all these things could, in principle, be described by consistent laws of physics, albeit very different ones from those we know, which in turn 'work' because they are formulated in the language of mathematics, which doesn't know how NOT to work. I'm reminded of a conversation years ago with a mate of mine who did pure maths at Cambridge (i.e. not merely pure, but actually a smokable, freebase form of maths - 'crystal math', if you will) where we arrived at the humorous conclusion that any question in a maths exam that says "Show such-and-such" could be answered with "Well it's obvious, isn't it?". Perhaps maths is inherently tautological? Could we define mathematics as "that which is necessarily and self-evidently true"? It's certainly the only discipline in which you can really prove results, I think, as opposed to merely finding empirical evidence to support them, or providing a convincing plausibility argument that someone could then counter with an equally convincing counter-argument, and people taking sides depending on their particular agenda or intellectual allegiance.

One last thing: the idea of a universe where laws of logic and arithmetic don't apply reminds me of

Most men have one testicle smaller than the other; however, each of Chuck Norris's testicles is bigger than the other.

 

[nomadthethird]

It is tempting to postulate something like that, isn't it? Plato's realm of 'forms', in other words. As far as I remember Josef K is very hostile to this and related 'transcendental' formulations of being. In a way though I think we're imagining there's a problem there when there isn't, or it's just our way of thinking about things (a hangover from religion, superstition, belief in an 'otherworld', 'afterlife' or 'Kingdom of God', perhaps?) that makes it sound as if this implies a ghostly realm of 'pure' numbers and shapes floating around behind or above our world of physical being, mortality, imperfection and impermanence. To me this seems (neo-)Platonist, Christian-Gnostic, mystical, alchemical. Now I get down to it, I'm not sure I think things like numbers really exist at all. Perhaps it merely seems that truths like 2+2=4 exist 'outside' or 'before' material reality because it's impossible to postulate a universe, no matter how weird-n-wacky, in which adding two things to two other things could fail to make four things.

By contrast, it's perfectly possible to imagine that there could be (in some obscure corner of the multiverse a universe with 107 spatial dimensions, or where the speed of light depends on the day of the week, or in which gravity is universally repulsive. Because all these things could, in principle, be described by consistent laws of physics, albeit very different from those we know, which in turn 'work' because they are formulated in the language of mathematics, which doesn't know how NOT to work.

Bingo.

 

[Tentative Andy]

Mr Tea for prez?
In seriousness though, that seems very close to what I want to believe - accepting the necessary truth of mathematics but not making this a reason to accept Platonism of a mystical kind. I feel I just need to flesh this view out a bit to myself though, at the moment it seems a bit like a hunch. Take your point also that we may be imagining there to be more of a problem than there really is, because of hang-ups from religion and so forth.

 

[Mr. Tea]

Bingo.

Yay, I win the Nomad Prize. *

Take your point also that we may be imagining there to be more of a problem than there really is, because of hang-ups from religion and so forth.

"Like riding an ox, in search of the ox..."

Right, I really am going to bed now. Sleep well, chums.



*absolutely the last time I use that obnoxious little icon in this thread, scout's honour.

 

[msoes]

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.