Fractal Geometry and the Foundations of Maths

[cback]

Hey Everyone,

I just wanted to ask for a bit of help on this research assignment I have to do. I have to show how Fractal Geometry contributes to the theory that Mathematics was invented. I have been looking into fractal dimensions and the fact that the dimensions we have labelled (1,2 and 3) don't actually exist within nature, thus in that way we are inventing maths (shown by the fact that we were wrong) to try and explain nature.

Anyway, I am a little stuck on this project and any help that you could provide would be very helpful!

Thanks in advance.

 

[Evo]

This isn't Philosophy. Please pay attention to where you post.

 

[chiro]

This isn't Philosophy. Please pay attention to where you post.

Fractal analysis is a real area of mathematics with serious research and results. It is not just philosophy and his question is very specific.

 

[Pythagorean]

Fractal analysis is a real area of mathematics with serious research and results. It is not just philosophy and his question is very specific.

Evo was referring to the post itself when she said "this isn't philosophy"

@ Evo, The question of whether mathematics is invented/discovered is philosophical. However, the poster is really asking the mathematical question about how to go about implying an answer to the philosophical question, so I think when it comes down to it, the mathematical question is the one he wants answered.

 

[chiro]

Hey Everyone,

I just wanted to ask for a bit of help on this research assignment I have to do. I have to show how Fractal Geometry contributes to the theory that Mathematics was invented. I have been looking into fractal dimensions and the fact that the dimensions we have labelled (1,2 and 3) don't actually exist within nature, thus in that way we are inventing maths (shown by the fact that we were wrong) to try and explain nature.

Anyway, I am a little stuck on this project and any help that you could provide would be very helpful!

Thanks in advance.

Hey cback and welcome to the forums.

Theres a few sides to fractal mathematics. The one that most people are aware of, even in the public amongst non-mathematicians concerns that usually of geometric "pictures" like the Koch-snowflake or things like the Mandelbrot set, or Julia set.

However there is a solid mathematical foundation behind fractals and one part of this deals with the quantification of dimension in a fractal representation. The notion of fractional dimensions in systems was made clear and studied within these kinds of systems.

The basic idea is that to describe a system that has say a dimension of 1.6, you don't need need one or two independent variables but somewhere "in-between". If you think of a line then this is one dimensional. You can treat it like a piece of string: you can bend the string and move it anyway you want, but the string is still one-dimensional. For two-dimensions, this corresponds to a piece of paper. Again you can take the paper and bend it and translate it but it is still two-dimensional.

Also fractal like geometry is found quite a lot in nature and this has been picked up by quite a number of people, so the idea that it doesn't exist is a little bit of a misnomer.

If you want to look into fractional dimensions associated with fractals I suggest you look at:

http://en.wikipedia.org/wiki/Hausdorff_dimension

This is very detailed and mathematical discussion, but the images and summary should give you a bit of an idea. Also:

http://en.wikipedia.org/wiki/Fractal_dimension

Also with dimensions, you have to realize that for most purposes, a dimension refers to the number of variables that you need to specify to describe some object. As with what was said above, a line is 1-dimensional while a surface (piece of paper) is two-dimensional. You could also have something like a filled in circle in two dimensions.

 

[Curious3141]

Fractal analysis is a real area of mathematics with serious research and results. It is not just philosophy and his question is very specific.

I dunno. Some parts of the OP like:

...contributes to the theory that Mathematics was invented.

and

the fact that the dimensions we have labelled (1,2 and 3) don't actually exist within nature,

and

we are inventing maths (shown by the fact that we were wrong)

just set my Crank-senses tingling. Evo's skeptical reticence is probably well-justified, IMHO.

 

[chiro]

To be fair to the OP, mathematics is a work in progress and new additions to it are being created all the time so the idea of new mathematics being invented is not as far from the truth as it can be made out to be.

Hausdorff did actually put fractional dimension on a formal basis, just like Euler started off ideas related to topology when he considered the Konigsberg (with an umlaut o) bridge problem.

I don't think the OP personally has a lot of experience with mathematics and they may be in high school which might demonstrate a lack of understanding of terminology and conciseness of expression, but his post had specifics along with misunderstandings that could be easily corrected.

 

[Curious3141]

To be fair to the OP, mathematics is a work in progress and new additions to it are being created all the time so the idea of new mathematics being invented is not as far from the truth as it can be made out to be.

Hausdorff did actually put fractional dimension on a formal basis, just like Euler started off ideas related to topology when he considered the Konigsberg (with an umlaut o) bridge problem.

I don't think the OP personally has a lot of experience with mathematics and they may be in high school which might demonstrate a lack of understanding of terminology and conciseness of expression, but his post had specifics along with misunderstandings that could be easily corrected.

Fair enough. Benefit of the doubt and all that. At any rate, you've already given the OP some good introduction and references to read up on, so the ball is in the OP's court to clarify his exact meaning and requirements should he require further help.

 

[cback]

Thank you very much, especially Chiro, for your help :) I am sorry about the whole wrong forum issue, I just thought that since it dealt with the question of whether Mathematics was invented (or discovered), that it ought to be placed in the Philosophy section.

Also, I am in high school as you assumed, so sorry about not using correct terminology in my question.

Nevertheless, I read all the links Chiro, and I think I have a decent understanding on Fractal dimensions/geometry now. I still fail to see how it contributes to the belief that Mathematics was invented.

If you could assist me a little more on that I would be really grateful.

 

[Pythagorean]

Nevertheless, I read all the links Chiro, and I think I have a decent understanding on Fractal dimensions/geometry now. I still fail to see how it contributes to the belief that Mathematics was invented.

I don't think you really learned enough about it if you think that half a day is a decent understanding. People spend a long time developing their understanding by actually working with fractal systems: doing the mathematics and playing with values to see how they affect the system and playing with more thane one such system.

You may have a very basic conceptual understanding now, but it's bound to be superficial until you've spent some time working at it and slept on it.

 

[apeiron]

Nevertheless, I read all the links Chiro, and I think I have a decent understanding on Fractal dimensions/geometry now. I still fail to see how it contributes to the belief that Mathematics was invented.

It sounds like this could be referring to the fact that computers were needed to really appreciate fractal geometry. Simpler fractals like the Koch snowflake were known about already of course, but it was Benoit Mandelbrot's access to IBM mainframes that got the field going.

So as objects they had to be constructed, they could not simply be imagined from looking at the algorithms that generated them.

The general issue of computer-assisted proofs is quite controversial and may be relevant to your question - http://en.wikipedia.org/wiki/Computer-assisted_proof

 

[HallsofIvy]

But Gaston Julia defined and drew Julia sets (to which the Mandelbrot set is a kind of "index") in the early twentieth century- long before computers were invented.

 

[apeiron]

But Gaston Julia defined and drew Julia sets (to which the Mandelbrot set is a kind of "index") in the early twentieth century- long before computers were invented.

As I said, fractal objects like Koch snowflakes had already been constructed. And geographers drawing maps of British coastline had noted the regularity of the irregularity centuries earlier.

But the usual telling of the story is that the rich structure of Julia sets, for instance, was only glimpsed before the advent of computers. The philosophical debate can then commence whether this is just an irrelevant difference between hand calculation and computer assisted visualisation, or whether in fact the construction of the objects is somehow necessary/fundamental to the maths.

The OP was unclear about the actual thesis he is meant to argue here. But this may be the controversy he is tapping into. Certainly, the dimensionality thing does not seem the issue.