View Full Version : Mathematics

josef k.

14-01-2009, 04:05 PM

I recently became obsessed with mathematics. I lie awake at night, thinking of patterns. I have discovered the following things.

1) Points. Points are pure positions.

2) Lines. Lines trace relationships between points.

3) Angles. Angles describe relationships between lines.

That is as far as I've got. I am currently trying to figure out volume. I may be going insane.

vimothy

14-01-2009, 05:01 PM

Me too. Actually, I'm studying maths at the Open University. It's near the end of semester 1. Haven't got as far as volume yet.

It's not the maths that's driving me insane, though, it's the Quantitative Research Methods module I'm also taking. I lie awake at night and think about questionnaire design. How demented is that?

Mr. Tea

14-01-2009, 05:22 PM

That's where Russell and Whitehead went wrong - the fundamental basis of mathematics is not irreducible axioms but questionnaire design.

Josef, if you have three vectors A, B, C delineating a parallelopiped (http://en.wikipedia.org/wiki/Parallelopiped) (a cuboid which is allowed to be 'wonky'; the most general volumetric shape three vectors can define) then the volume is given by the magnitude of A . (B x C), where '.' is the scalar (or 'dot') product and 'x' is the vector (or 'cross') product. This seems like a reasonable place to start if you want to think about volume once you've got some idea of vectors and angles, maybe.

Edit: there's a fairly decent diagram on the Wiki page I linked to...

josef k.

14-01-2009, 05:34 PM

Hold on. I have to plot this visually. I've discovered this is the only way I can understand it.

don_quixote

15-01-2009, 07:42 AM

lines are infinitely long. wow.

don_quixote

15-01-2009, 07:43 AM

and points, fuck, there's as many points between here and the sun as there are between these two full stops ..

don_quixote

15-01-2009, 09:10 AM

massive massive book recommendation:

mathematics and the imagination by edward kasner. it's from the 30s but it's spectacular.

oooh here we go:

http://books.google.co.uk/books?hl=en&id=Ad8hAx-6m9oC&dq=mathematics+and+the+imagination&printsec=frontcover&source=web&ots=vv8LqzjrG2&sig=TW93f07UiJ3CSLmJolLDuCzxpyU&sa=X&oi=book_result&resnum=5&ct=result

since we're here... in that book it says something about terminating decimals, claiming everything we've been told in school maths is wrong - they don't terminate?

i'm sure you've heard 0.99999... = 1 right? well he claims hence 0.125 = 1.249999... so irrationals are the only numbers which never reach a recurrence.

not sure why this is lost in school though - because two results would be too confusing?

Mr. Tea

15-01-2009, 02:08 PM

i'm sure you've heard 0.99999... = 1 right? well he claims hence 0.125 = 1.249999... so irrationals are the only numbers which never reach a recurrence.

I'm not I get this: irrationals are the only non-terminating, non-repeating numbers by definition. Well actually the definiton is that they can't be written as the ratio of one finite integer to another, but it's trivial to show that this is equivalent to a non-terminating string of decimals.

I derived the fundamental theory of calculus for one of my A-level students last night, which was pretty cool. Well, for differentiation anyway, integration will have to wait til next week. :)

josef k.

15-01-2009, 02:18 PM

Curves. Curves are very interesting. Also, wiggly lines. How the fuck do you analyze a wiggly line?

IdleRich

15-01-2009, 02:29 PM

"since we're here... in that book it says something about terminating decimals, claiming everything we've been told in school maths is wrong - they don't terminate?

i'm sure you've heard 0.99999... = 1 right? well he claims hence 0.125 = 1.249999... so irrationals are the only numbers which never reach a recurrence."

I don't get what you're saying here? Recurring decimals don't terminate, they recur and any numbers which can't be written as a ratio (ie are irrational) are non-terminating and non-recurring.

vimothy

15-01-2009, 02:34 PM

How the fuck do you analyze a wiggly line?

It depends how it wiggles.

josef k.

15-01-2009, 02:43 PM

For instance, this one.

http://stereogram.wordpress.com/2009/01/15/674/

Mr. Tea

15-01-2009, 03:28 PM

For instance, this one.

http://stereogram.wordpress.com/2009/01/15/674/

Well that's just a wriggly line with no rhyme or reason - I thought you meant sine waves and stuff like that.

Though you could still differentiate that curve, or integrate it, or express it as a Fourier series, or draw tangents to it at selected points...

Edit: actually you couldn't, as it goes back on itself at a couple of points. You could still parameterise it as a trajectory or something, I dunno.

IdleRich

15-01-2009, 03:40 PM

"It depends how it wiggles."

And what you mean by analyze. I suspect that you mean find the function that generates it - which in the case of the line that you've just linked to would be, as Mr Tea says, rather hard and possibly pointless because any function that gave that line would be completely ad hoc. I guess you could find something that approximates it though if that's what you wanted.

josef k.

15-01-2009, 03:44 PM

I have put it in a cage... I am now examining it.

PS - By analyze, I just mean figure out its relationship to other stuff. As in, wobbly lines exist. How do people understand them mathematically? What mathematical operations are performed on them in order to try and understand them? I don't really care about this particular line. But please don't tell it I said that.

Mr. Tea

15-01-2009, 03:51 PM

I have put it in a cage... I am now examining it.

PS - By analyze, I just mean figure out its relationship to other stuff. As in, wobbly lines exist. How do people understand them mathematically? I don't really care about this particular line, although please don't tell it I said that.

Hm, I can see what you mean, but in a way it's a bit like a kid drawing a totally surreal fantasy animal and the asking an adult what real animals it's related to, if you get me. There's no reason why it should have a relationship to other stuff. In the same way that hfyeuieoffjsgrepl could be considered a 'word', since I can write it using the same letters I use spell real words that have a meaning - but that certainly doesn't endow it with a meaning of its own.

josef k.

15-01-2009, 03:56 PM

Interesting...

So, is it that, wobbly lines do not exist? Or that, when they do, it is never arbitrary, as this one is.

Take the path of a kite through a sky on a rainy day. The path traced by the kite. How do people mathematically analyze this?

josef k.

15-01-2009, 04:12 PM

ps - in the case of the fantasy animal, experts conclude: it is three-headed, lamalian, behorned and tigroid, scale-necked and beastish, amphibious, zebrine, duck-billed and bewinged.

Mr. Tea

15-01-2009, 04:12 PM

Interesting...

So, is it that, wobbly lines do not exist? Or that, when they do, it is never arbitrary, as this one is.

Take the path of a kite through a sky on a rainy day. The path traced by the kite. How do people mathematically analyze this?

Of course they exist! In the same way the nonsense string of letters typed exists. It's a question of whether it means anything deeper.

To what extent is anything 'arbitrary'? The path of a kite is determined by gusts of wind and tugs on the string, just as the shape of that line was determined by the movement of the muscles of your hand, in turn produced by electrical signals in your brain and spinal column. These things can all be analysed according to well understood physical laws, which of course have a rigorous mathematical basis. That doesn't mean your kite is going to trace a perfect ellipse in the sky or that you're going to draw a nice neat sine wave; the processes are extremely complex (stochastic, loosely 'chaotic') and cannot, for practical purposes, be predicted - in other words it's non-deterministic.

Mr. Tea

15-01-2009, 04:13 PM

ps - in the case of the fantasy animal, experts conclude: it is three-headed, lamalian, behorned and tigroid, scale-necked and beastish, amphibious, zebrine, duck-billed and bewinged.

Is it partly rugose and partly squamous?

vimothy

15-01-2009, 04:22 PM

There are lots of wiggly lines in the social sciences.

josef k.

15-01-2009, 04:26 PM

Yes, absolutely. It is:

Monstrine & lamalian,

Leggaled and armine. Angeloid,

Green-billed. Cashinal, pervoid.

Scale-necked, amphextorous

Olfractic, dreamal & ioned.

**

"It's a question of whether it means anything deeper."

A wobbly line has a relationship to other mathematical properties - area, volume, angle, point, curve. Basically, I'm interested in the relationships between mathematical properties, in figuring out how they fit together.

IdleRich

15-01-2009, 04:32 PM

"By analyze, I just mean figure out its relationship to other stuff. As in, wobbly lines exist. How do people understand them mathematically? What mathematical operations are performed on them in order to try and understand them?"

Well, like Mr Tea said you could do stuff like measure its curvature at a given point or work out its length or whatever and that would be an analysis of a kind I think. You could then compare its curvature to another line or whatever if you're interested in relationships to other things. I don't think that you could come up with a single neat function to describe it but that doesn't mean that it doesn't exist, why would it?

josef k.

15-01-2009, 04:46 PM

Because maths knows only functions?

poetix

15-01-2009, 05:03 PM

A wiggly line can be described by a continuous function between the set of real numbers occupying the interval 0 <= x <= 1 and a set of points in a Euclidian space. The function takes 0 to the start of the line, 1 to the end of it, and continuously maps all of the points in the interval to points along the wiggly path.

Some lines wiggle more than others. A Koch curve, for instance, has wiggles on its wiggles, and so on ad infinitum. But that makes it more, rather than less, "mathematical".

Mr. Tea

15-01-2009, 05:10 PM

Because maths knows only functions?

Maths knows all sorts of things: functions, functionals, operators, tensors, maps, sets, groups, rings, symmetries, homotopies...it's kind of difficult to know where to start. In part because a lot of the stuff that forms the real conceptual basis of maths is a lot more difficult to get to grips with than the stuff you do at school, much of which has a more fundamental origin that you don't learn about unless you study it to degree level.

IdleRich

15-01-2009, 05:15 PM

"Because maths knows only functions?"

I don't think that's true. It's certainly not true that it only knows "single neat functions" which is what I said.

"A wobbly line has a relationship to other mathematical properties - area, volume, angle, point, curve. Basically, I'm interested in the relationships between mathematical properties, in figuring out how they fit together."

Well, the line in question obviously has no volume or area in itself, you could have a shape of which that line made up one edge and you could ask the same kinds of questions about the shapes that resulted. Again they might not be simply describable (nb you can fairly easily work out the volume of most shapes by seeing how much water they displace but I guess that's not the kind of answer you mean).

The line has different curvature at different points and its tangents would be at various angles to each other, again it would be hard to sum them up simply.

I suppose the point is that it is hard to analyse something except in the specific if it is not generated by some kind of rule. It's kind of like in the Borges story where the guy (Funes) develops a new number system that doesn't repeat at all and cannot grasp why it is useless to everyone - except him who has a perfect memory and does not need to have a generating rule to help him comprehend the bigger picture.

"A wiggly line can be described by a continuous function between the set of real numbers occupying the interval 0 <= x <= 1 and a set of points in a Euclidian space. The function takes 0 to the start of the line, 1 to the end of it, and continuously maps all of the points in the interval to points along the wiggly path."

Yes of course but the point is that it's hard to say anything more about it than that. That's exactly what I meant by ad hoc.

"Some lines wiggle more than others. A Koch curve, for instance, has wiggles on its wiggles, and so on ad infinitum. But that makes it more, rather than less, "mathematical"."

I don't think anyone is saying that that other line is unmathematical. You can have far more complex lines that are nowhere continuous or nowhere differentiable or whatever but I think that if you have a rule for generating it there is some sense in which you have an understanding of it that you don't have with the random wiggle (of course it's perfectly possible that any given apparently random wiggle may be generated by a generalisable rule but I'm not aware of any systematic method to discover what that rule is).

poetix

15-01-2009, 05:24 PM

"Ad hoc" here seems to mean something a bit like Kolmogorov randomness (http://en.wikipedia.org/wiki/Kolmogorov_complexity#Kolmogorov_randomness).

poetix

15-01-2009, 05:27 PM

I would imagine that most wiggly lines I could draw by hand on the back of an envelope would turn out to be quite compressible - it would be quite a challenge to produce one that couldn't be broken down into a handful of frequencies.

don_quixote

15-01-2009, 05:36 PM

I don't get what you're saying here? Recurring decimals don't terminate, they recur and any numbers which can't be written as a ratio (ie are irrational) are non-terminating and non-recurring.

i'm saying there are no terminating numbers

Mr. Tea

15-01-2009, 05:40 PM

i'm saying there are no terminating numbers

What about...say...1?

poetix

15-01-2009, 05:50 PM

0.999999...

poetix

15-01-2009, 05:52 PM

But that's just another decimal representation of 1.

0.1 in base 3 is 0.33333... in base 10.

0.5 in base 10 is 0.11111... in base 3.

You can have some fun with this here (http://www.easysurf.cc/cnver17.htm).

Mr. Tea

15-01-2009, 05:53 PM

0.999999...

Which can be written as 1, right? '1' is the most intuitive and 'ordinary' way to express the concept of unity - to write it as 0.9999999... is to deliberately express it as a non-terminating number, which is unnatural and (for most purposes) pointless. So if an argument can be made that 1 can be artificially and unnaturally written as 0.9999....., I'd say a more powerful counterargument is to say that 0.99999.... can be naturally and easily as 1.

poetix

15-01-2009, 06:02 PM

Every number has a non-terminating representation, in a given base. Some numbers also have terminating representations in that base, while some do not. I'm not sure I understand what the fuss is about.

IdleRich

15-01-2009, 06:10 PM

"Ad hoc" here seems to mean something a bit like Kolmogorov randomness.

A bit like I think.

"I would imagine that most wiggly lines I could draw by hand on the back of an envelope would turn out to be quite compressible - it would be quite a challenge to produce one that couldn't be broken down into a handful of frequencies."

I'm not sure. I don't think that "most" could be broken down into a way that would count as an analysis of the kind that Josef K was requesting.

"I'm saying there are no terminating numbers"

Do you mean that terminating numbers can be written in a form in which they don't terminate? That might be true but I don't see how that means that there are no terminating numbers.

Mr. Tea

15-01-2009, 06:20 PM

Every number has a non-terminating representation, in a given base. Some numbers also have terminating representations in that base, while some do not. I'm not sure I understand what the fuss is about.

Yeah, OK I accept that - there's nothing inherently special about 0.5 ( = 1/2) that's different from 0.3333.... ( = 1/3). But of course irrationals are irrational in any base.

IdleRich

15-01-2009, 06:28 PM

"But of course irrationals are irrational in any base."

Not if you use the number in question as the base presumably.

Mr. Tea

15-01-2009, 06:46 PM

Not if you use the number in question as the base presumably.

Well I was thinking that, but I can't really see why you'd do that. But maybe someone has all the same...I guess most people can't see why anyone would do maths in the first place.

IdleRich

15-01-2009, 06:51 PM

"Well I was thinking that, but I can't really see why you'd do that."

I think that the only reason you would do it (off the top of my head) is if you wanted to make, say, pi rational but you would be sacrificing an awful lot in terms of simplicity for most things. I don't see why it shouldn't be possible though.

don_quixote

15-01-2009, 08:06 PM

woaaaah using pi as a base? i'm not entirely sure you can do that. at least im really struggling to visualise it. how would you count in it for a start?

don_quixote

15-01-2009, 08:10 PM

hold on, thinking about place value in an irrational base; well each place in a number in base pi for example would have infinite number of possible digits; right?

IdleRich

16-01-2009, 10:04 AM

"woaaaah using pi as a base? i'm not entirely sure you can do that. at least im really struggling to visualise it. how would you count in it for a start?"

Well, I guess you would do it in the same way as any other base. When you're talking about base 10 you have a number of digits, the final one represents the the number of zeroth powers of 10 - which is 1 as anything to the power zero is one. The column next to that represents first powers of 10 ie tens, then you have ten to the power 2 ie hundreds and so on so any integer in base ten can be written:

a x (10 to the power 0) + b x (10 to the power one) + c x (10 to the power 2) + d x (10 to the power 3) + .... and so on. Supposing that we are talking about a number between a thousand and ten thousand ie one where all of the letters after d in the sequence I have just described are 0 then the number would be written dcba in base 10.

(if it wasn't an integer then you would need minus powers as well ie 10 to the minus one is one divided by ten ie 0.1 and so on)

Similarly in base pi you would have a column for 1s, a column for pi's, a column for pi squareds and so on. Suppose you wanted to write, say, the number which is 21 in base 10 in base pi then you would need to work out how many for each column. pi cubed is going to be more than 21 so you don't need any in the "thousands column", pi squared is between nine and ten so you are going to need two in the hundreds column (because 21 is between 2 and 3 times pisquared), 21 minus 2(pi squared) is less than pi so you are going to have zero in the tens column, 21 minus 2(pi squared) is between 1 and 2 so you are going to have a 1 in the units column. So the number is going to be 201 point something. You continue this process, moving into negative powers, to work out all the digits although obviously it will be non-recurring and non-terminating because 21 is irrational in base pi.

I hope that makes sense. I've written it quite quickly as I'm busy at the moment. I hope there are no errors. When I get a chance I'll write it out more generally.

don_quixote

16-01-2009, 10:05 PM

no i understood that, i'm just not sure about recurring sequences in an irrational base. obviously numbers formed from the pure base would not recur, but any other number, including most rationals would. and how would (the equivalent of) decimal places etc. even work? i'm seriously struggling with the concept in the context that we're talking about

don_quixote

16-01-2009, 10:08 PM

oh hold on, yeah, i get it!!

wow, that's a really powerful example of why there are far more irrational numbers than rational - every number that isn't related to pi creates a brand new irrational number; do you see?

don_quixote

16-01-2009, 10:12 PM

let me explain that better.

ok every rational number in that base is irrational, right? now think of those numbers as written in a decimal base - this is a one to one relationship, so every rational number has an irrational partner by writing it in base pi.

further, every irrational number which cannot be written as a sum of products of pi is going to be irrational in base pi - so there's a huge amount of these, including those we've already written.

hell, they're probably trancendental too.

Slothrop

17-01-2009, 12:03 AM

I think that the only reason you would do it (off the top of my head) is if you wanted to make, say, pi rational but you would be sacrificing an awful lot in terms of simplicity for most things. I don't see why it shouldn't be possible though.

The most obvious thing that you're sacrificing is the fact that sums and products of 'rational numbers' are 'rational numbers'. Which means that whether or not things are rational kind of stops being of so much interest... as far as I can see the nice thing about rational numbers is less the 'terminating or recurring decimal' part than the 'quotients of two integers' thing.

IdleRich

17-01-2009, 11:14 AM

"no i understood that, i'm just not sure about recurring sequences in an irrational base. obviously numbers formed from the pure base would not recur, but any other number, including most rationals would. and how would (the equivalent of) decimal places etc. even work? i'm seriously struggling with the concept in the context that we're talking about"

Are you saying that numbers that are rational in base ten would have a recurring decimal expansion in base pi? If so you're wrong, they (except for 1,2 and 3) would be irrational in base pi and therefore would have a non-terminating and non-recurring expansion.

Decimal places would work in exactly the same way as in base ten, the first digit after the point would represent the number of (1/pi)s and the next digit would represent the number of (1/(pi squareds)) and so on.

"ok every rational number in that base is irrational, right? now think of those numbers as written in a decimal base - this is a one to one relationship, so every rational number has an irrational partner by writing it in base pi.

further, every irrational number which cannot be written as a sum of products of pi is going to be irrational in base pi - so there's a huge amount of these, including those we've already written"

This does not in itself constitute a proof that there are "more" irrational numbers than rational ones (though there are in the sense that you mean) - you have to demonstrate that there can't be a one-to-one mapping, it's not enough to suggest one mapping that exhausts the rational numbers without exhausting the irrationals.

poetix

17-01-2009, 12:18 PM

Yes, the point is that the rationals are countably infinite - there are in fact "just as many" rationals as there are natural numbers (because a one-to-one mapping can be established between the rationals and the natural numbers), whereas the irrationals are uncountably infinite.

I'll tell you what does my head in a bit, and that's the fact that ZFC implies that there's a well-ordering of the reals...

Mr. Tea

28-01-2009, 04:47 PM

How's the programme going, Josef? Have you solved Goldbach's conjecture (http://en.wikipedia.org/wiki/Goldbach%27s_conjecture) yet?

josef k.

28-01-2009, 04:51 PM

No, but I did write an epic poem about my adventures.

It runs:

O child of my scribble!

Eccentric line that wiggles

What art thou to the other numbers?

Wherefore doth thou logic slumber?

**

That's it, so far.

Mr. Tea

28-01-2009, 04:54 PM

Like it, like it.

A graduate student at Trinity

Computed the square of infinity;

But writing the digits

Gave him the fidgets,

So he dropped Maths and took up Divinity.

Edit: shouldn't 'thou' be 'thine'?

josef k.

28-01-2009, 05:08 PM

Yes, I am convinced that the problem of the wiggly line harbors hidden profundity. Consider:

1) It cannot be reduced to another, more fundamental function.

2) It bears a precise mathematical relationship to the universe - but this is strictly impossible to calculate, even roughly, without performing a series of ultimately pointless and futile operations.

3) Those operations may themselves be related to each other, and an entire complex geometrical edifice built on top of the wiggly line - without the empirical essence of the line itself ever being known.

BareBones

28-01-2009, 06:04 PM

this thread is hugely interesting but i don't really understand any of it, and it's scaring me off that astrophysics course i was considering applying for. God, i wish i understood this stuff more.

josef k.

28-01-2009, 06:10 PM

You are not alone. I don't even understand my own contributions to it.

Tentative Andy

28-01-2009, 06:18 PM

Is this an appropriate place to ponder what the ontological implications of mathematics are? - esp, does the existence of mathematical objects contradict materialism?

I'm only capable of discussing this in a rambling, confused way, but that doesn't usually stop me.....

josef k.

28-01-2009, 06:27 PM

proceed!

Tentative Andy

28-01-2009, 06:30 PM

Well, as I understand it, if you think of sets and numbers as objects, rather than formal constructions, then they have to be abstract objects, which by definition exist outside of space and time. Is this not a part of what is at stake in the dispute between nominalism and mathematical realism?

As you can see I ain't no philosopher. :confused:

PeteUM

28-01-2009, 06:35 PM

Ditto with rambling and confused but aren't there implications for mathematics from ontology? I speak as someone who suffered "extra maths", i.e quite thick and with a bone to pick.

Tentative Andy

28-01-2009, 06:38 PM

I had to take maths in a seperate year from all my other Highers to make sure I could pass it. Good times, good times.

josef k.

28-01-2009, 06:39 PM

Why must abstract objects be outside space and time?

Tentative Andy

28-01-2009, 06:42 PM

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

28-01-2009, 06:44 PM

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

josef k.

28-01-2009, 07:15 PM

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

nomadthethird

28-01-2009, 07:54 PM

aren't there implications for mathematics from ontology?

Oh just a couple.

Mr. Tea

29-01-2009, 06:48 PM

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

29-01-2009, 11:10 PM

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

29-01-2009, 11:45 PM

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

29-01-2009, 11:56 PM

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

30-01-2009, 12:34 AM

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

30-01-2009, 12:39 AM

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

30-01-2009, 01:02 AM

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

30-01-2009, 01:12 AM

There's always God, I guess. ;)

Tentative Andy

30-01-2009, 01:17 AM

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

30-01-2009, 01:28 AM

There's always God, I guess. ;)

Dammit.

Mr. Tea

30-01-2009, 01:32 AM

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

30-01-2009, 01:34 AM

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

30-01-2009, 01:41 AM

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

30-01-2009, 01:52 AM

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

30-01-2009, 05:15 AM

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.

Mr. Tea

01-02-2009, 02:07 PM

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.

nomadthethird

01-02-2009, 08:40 PM

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.

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.

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

01-02-2009, 10:31 PM

...a ghostly realm of 'pure' numbers and shapes floating around behind or above our world of physical being, mortality, imperfection and impermanence.

This is how I thought about maths as a child, it's tempting isn't it?

Mr. Tea

02-02-2009, 12:17 AM

This is how I thought about maths as a child, it's tempting isn't it?

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 (http://en.wikipedia.org/wiki/Tunguska_event), and they get a bollocking from head office. Maybe they're the 'machine elves' Terrence McKenna sees when he takes DMT, heh.

poetix

02-02-2009, 07:39 AM

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

02-02-2009, 12:11 PM

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.

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

02-02-2009, 08:41 PM

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.

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.

poetix

02-02-2009, 09:13 PM

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

02-02-2009, 09:32 PM

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.

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.

poetix

02-02-2009, 10:05 PM

Right. But if they only occur within mathematics, do they matter to anybody? Do they really make a difference?

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?

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.

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.

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.

Mr. Tea

02-02-2009, 10:24 PM

The issue I think isn't that the real world resists mathematisation - if anything, it gives in suspiciously easily

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...

- 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.

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

02-02-2009, 10:26 PM

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.

This is a good way of putting it.

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).

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.

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.

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.

nomadthethird

02-02-2009, 11:07 PM

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.