Mathematics

josef k.

Dangerous Mystagogue
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

IdleRich
"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?
 

poetix

we murder to dissect
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

Shub-Niggurath, Please
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

IdleRich
"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

we murder to dissect
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.
 

poetix

we murder to dissect
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.
 

Mr. Tea

Shub-Niggurath, Please
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

we murder to dissect
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

IdleRich
"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

Shub-Niggurath, Please
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.
 

Mr. Tea

Shub-Niggurath, Please
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

IdleRich
"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.
 
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