What is more common in nature, regularities or irregularities?
Let's take symmetry of functions as an example of regularity, others may be their periodicity, etc. If one analyzes it, one can conclude that even functions, which are by definition those for which f(-x)=f(x), examples are polynomials consisting of even powers of x, and odd functions, for which f(-x)=-f(x), examples of these are polynomials consisting of odd powers of x, are actually exceptions, rather than a rule, ie that functions are generally speaking asymmetric objects with respect to the x=0 axis, that do not necessarily have anything to do with those that are symmetric. However, the fact is quite the opposite, every asymmetric function can be represented as a sum of an even and odd part, like this:
f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(even) + f(odd)
So, for even functions, odd part equals to zero, and vice versa. That may be surprising, that such a simple logic shows the truth that may seem counterintuitive. Interesting is however, that symmetry in a micro world, for example in the world of elementary particles, is exact, while in a macro world, for example in biology, is only approximate. Why is it so?
f(x) = [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2 = f(even) + f(odd)
So, for even functions, odd part equals to zero, and vice versa. That may be surprising, that such a simple logic shows the truth that may seem counterintuitive. Interesting is however, that symmetry in a micro world, for example in the world of elementary particles, is exact, while in a macro world, for example in biology, is only approximate. Why is it so?
Comments (5)
I think the fundamental reason is that some properties of elementary particles (described by quantum mechanics) are intrinsically "discrete", whereas properties of objects in the macro world are continuous.
This is due to the fact that the only observable states of wave functions are the "eigenfunctions" of some symmetry operators. In other words we can only see states that have some exact geometric symmetries.
So, in a sense, the shapes of micro world are made of a limited number "perfect" mathematically defined forms.
It is not that symmetry in the world of elementary particles is exact, but rather that physicists have attempted to model the world of elementary particles in terms of exact symmetry. They have got useful results, but there are many things that don't fit, and it is not clear at all that this world could be perfectly modeled in terms of exact symmetry. In fact, because particle physicists have forced exact symmetry in their models they have to invoke a fundamental phenomenon of "symmetry breaking" for the model to be accurate at all.
I would say regularity is in the eye of the beholder. You could look at a glass of water and see it as a regular shape. Then you might look much closer with a microscope and see tiny impurities in a jittery, irregular motion. Then you might model water and the impurities as being made of extremely tiny particles interacting with one another in a regular way, and you could explain the irregular motion of a tiny impurity as being the result of all the regular interactions between it and the particles that make up water (which are not at rest but move in various directions with various velocities).
We can see regularities in irregularities, and irregularities in regularities.
Correct ! And an interesting speculation in that respect is about the concept of 'orde-disorder' involved in the second law of thermodynamics. In my opinion, that issue supports the view that 'time' is a psychological construct bound up with the human cognitive urge to 'predict and control'.