Mathematics Inspires Art: Aleph-1 and Goedel's Theorem

[matt grime]

http://www.guardian.co.uk/arts/news/story/0,,1699203,00.html


story about and link to someone's mathematically inspired 'art' (if printing an equation or mathematical symbol is artistic to you).

be warned the descriptions of the equations/symbols can induce speechlessness and not in a good way (example. aleph-1 is "the smallest number bigger than infinity", and a logician might go a bit potty of the consistent/complete Goedel's theorem discussion)

 

[JasonRox]

At first I thought you were talking about something one my prof creates using dynamical systems.

You can actually enjoy the work of my prof as a layman.

Having equations make you smart right?

 

[matt grime]

I thought you were one of the 'euler's equation is great' party, which is what the artist is saying.

 

[honestrosewater]

a logician might go a bit potty of the consistent/complete Goedel's theorem discussion

I wouldn't consider myself a logician, but yeah

Godel discovered a property of any logical system that truly astounded mathematicians. He began by thinking about the way rules can be used to make statements. What Godel found was that if these rules contain no contradictions then there is something strange about the statements that can be made with them: certain statements cannot be proved true using the available rules, even though they are true. Instead extra rules are needed to prove the point. So the original set of rules must be incomplete.

Godel’s theorem is that if a set of rules are consistent, they are incomplete.

isn't correct. Someone should send the artist a picture of Gödel's Completeness Theorem. Hurkyl's given several nice explanations of Gödel's First Incompleteness Theorem here if anyone's curious.

Actually, now that I think about it, I wonder what exactly makes statements like those beautiful. For example, there is something beautiful to me about

1) $$\Phi \models \phi \Leftrightarrow \Phi \vdash \phi$$

It's certainly not because I think the font is pretty (though I suppose that might have some effect). At first, I think it's a combination of the statement's meaning and the simplicity with which it is stated. But I could state the same thing even simpler by saying

$$\clubsuit$$ =_df (1).

But this doesn't make

2) $$\clubsuit$$

beautiful -- or at least not nearly as much so as (1). Stating (1) in a more complex way also eventually removes at least some of the beauty -- all of the statements preceding (1) in the chapter together basically say what (1) says. Even its closest translation into English isn't as satifying.

3) An L-formula is a logical consequence of a set of L-formulas if and only if that L-formula is deducible from that set of L-formulas.

So it seems the relationship between form and meaning that gives rise to beauty isn't so simple, even in math and logic. Anywho, I just think that's interesting; the same thing is at work in poetry (in natural languages).

 

[devious_]

Oh my... There's actually a For Sale page.

 

[Tide]

Oh my... There's actually a For Sale page.

Maybe we should start copyrighting our own creations to protect ourselves - just in case!

 

[devious_]

Maybe we should start copyrighting our own creations to protect ourselves - just in case!

I call dibs on $\int$!

 

[Cexy]

I want 0, in that case! Or perhaps 1...

I can kind of see what the author is saying - for example, I think that Stokes' Theorem stated in the language of differential forms is just fantastic:

$$\int_C d\omega = \int_{\partial C} \omega$$

but that's more because of what it says, rather than how it looks. Although I do have to admit that it looks quite pwetty :)

 

[JasonRox]

I personally think the beauty of mathematics comes from experience in mathematics. similar to an artist, an experienced artist will see the inner beauty.

Things like equations won't be considered beautiful by the general public because they just don't know what it is. Some might say it's beautiful or cool, but just like in the world of artists, their opinion is meaningless to the mathematician or artist.