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Objective Beauty

BeautyI was sat contemplating yesterday. I'd been working on an article for this site and I was getting stale - trying to spot an error that would have been obvious a couple of hours before - a sure sign that it is time to stop what you are doing and do something else for a while.
For some reason my mind dove back into a philosophical question I've been pondering - is there such a thing as objective morality?
I was considering a specific point made in a debate on the issue, which asserted that there is no a-priori reason to assume an objectivity, because in similar systems of judgement there does not appear to be such a thing. The example cited was aesthetics, the assertion being that there is no absolute aesthetic - no unarguable beauty - and that it is, at least to some extent, a matter of subjective opinion. I was trying to work out if I agreed with this assertion or not. Certainly I know that physical attractiveness is at least partly subjective. Most of us have little quirks of behaviour and/or appearance that does it for us - I certainly do (and it's none of your business). But at the same time most of us can acknowledge some less subjective notion of physical attractiveness and agree to it's general validity even though it might not entirely float our boat. Most men, for example, would probably agree that David Beckham, or George Cloony are pretty damn good looking examples of our gender, even if we don't actually like much else about them. Likewise, not too many of us would class Halle Berry or Penelope Cruz as anything other than gorgeous.

Anyhoo - I pondered the matter for a while and, although I reached no conclusion, I did manage to advance my thoughts a bit closer in that direction. In the course of this, I took a mental diversion into abstract beauty, since this is surely where the most difference, the least objectivity can be found. We can all admire the Mona Lisa, even those who might, in their more pretentious moments, call it a dead example of a dying art form. Tracy Emin's "bed", however, will divide people instantly (and irrevocably in most cases).

Because I know little about art, my thoughts went to science and maths - beauty here is surely as abstract as it is possible to be. And yet most mathematicians see beauty (and ugliness) in the most abstract formulae or proof - and, more importantly, will tend to agree. I have, several times, heard a mathematician say, of something she is working on, the mathematics are just so beautiful

Most times the beauty is not something I can see - I don't have the understanding to see the relationships which must surely be at the heart of such beauty. There is one equation, however, that struck me as beautiful when I first saw it - even though I knew very little about it or what it meant. That equation is commonly known as Euler's Identity. It relates five mathematical constants in a way which is undeniably elegant (which, I propose, is itself a type of beauty). It is also mysterious - always a good thing in something aspiring to beauty. Why pi, e, i, 1 and 0 are linked is still something I have no real clue about, and why they are linked in such an elegant and concise statement as the Euler Identity is something which I am not sure anyone has a good answer for. For those unfamiliar with this identity, let me put you out of your suspense:

\[ e^{i\pi}+1=0\]


The first thing to note is the brevity. Each constant is mentioned once, and there is an economy of arithmetic. But what do the symbols mean. Let's go through them starting with the one you are almost certain to know.


\(\pi \) is a number that pops up in the most surprising places. Most people know that it defines the relationship between the diameter (d) and circumference (c) of any circle. \(c=\pi d\) . Many people will also know that pi is an irrational number. In mathematics irrational doesn't mean crazy, it means 'no ratio'.

The first great mathematicians - the Greeks, specifically Pythagoras - believed that numbers are ideal forms and the universe is a clumsy material representation of the idealised forms that exist only in the realms of number.



OK, so \(\pi\) is an irrational constant which is the relationship between circumference and diameter of any circle.   

e is another irrational number, similar to \(\pi\). e can be defined in several ways. It is the base number of natural logarithms. It is the limit of \([1+\frac{1}{n}]^n\) as \(n\rightarrow \infty\).

i is the symbol given to the imaginary number which answers the expression \(\sqrt{-1}=?\) It cannot be expressed on the normal number line. Instead we advance the usual number line, going from minus \(\infty\) to plus \(\infty\) into a second dimension, to give what we call the 'complex plane'.

complex planeHere we see an example. The real numbers are represented on the x axis and the imaginary numbers on the y axis. The rules for combining the two are logical and orderly, and by doing this to form complex numbers we open-up solutions to real-world problems that were a total surprise. In an area of interest to me - sound recording, complex numbers are used to construct complex filters to enhance and filter sound. They are also used in the basics of all digital wave processing - Fourier transformations. In other fields complex numbers have also provided astounding new tools. People working with alternating current electricity use complex numbers to model the current.


Multiplying by real numbers can be thought of as scaling. Multiplying by complex numbers can be thought of as rotating through space. This is obviously useful in any situation using 3-D or even 2-D modelling - such as CAD/CAM. You will have seen complex numbers if you have ever seen the fractal called the Mandelbrot set. So these three constants - all very useful and all which crop up in places you would not imagine, are also related, using a simple equation, to arguably the most important constants in number theory - 1 and 0. 0 is also known as the 'additive identity' which can be simply stated as n + 0 = n = 0 + n and this is true for all number sets - real, imaginary, complex, integer. You should be able to see from this that 1 is the multiplicative identity \(n1 = n = 1n\).

So what does it all mean and where does it come from? A few decades ago the best mathematicians might have answered thusly:
It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth.
Benjamin Peirce 

I must admit that I was going to leave it there, but I came across an article by a young lad called Kalid Azad on the 'better explained' website that shamed me into rethinking, His response to the quote above was:
Argh, this attitude makes my blood boil! Formulas are not magical spells to be memorized: we must, must, must find an insight.

He's right you know :-) Here is his explanation.


Deriving the identity


Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number

\(e^{ix} = cos(x) + i*sin(x)\)

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when x = π, or one half-turn (180°) around a circle: \(e^{i\pi}=cos(\pi)+i*sin(\pi)\)
Since  \(cos(\pi)=-1\)

\(sin(\pi)=0\) 

it follows that \(e^{i\pi}=-1+0i\)

which yields Euler's identity:  \[e^{i\pi}+1=0\]

 

300px ExpIPi

This is a demonstration that \(e^{i\pi}+1=0\). It uses the formula \( (1+z/N)^N \rightarrow e^z \) (as N increases). The Nth power is displayed as a repeated multiplication in the complex plane. As N increases, you can see that the final result (the last point) approaches -1, the actual value of \(e^{i\pi}\).

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