I decided to start my "blog" with an exposé on my thoughts about mathematics.
Math is generally a good thing. It is a pretty well thought out language that allows us to describe a lot of things that are difficult to express in pure verbal languages like English. One problem I see is that many scientists seem to think that once something is described in a mathematical form, that it is truth. There are a number of reasons why this is not the case, and I will try to outline some of them for you.
First and foremost in the refutation of math as the ultimate descriptor of reality is contained in Gödel's Incompleteness Theorem, which states that any formal descriptive system will, at some level, be self-contradictory. I will risk a statement here: "Nature cannot be self-contradictory at any level". If this statement is true, then mathematics cannot, by itself, hope to completely describe Nature.
Secondly, there are a number of concepts contained in Mathematics that simply DO NOT OCCUR in Nature, and I will try to detail some of them:
Zero. Modern thought is quite enamored with the thought of Zero. We like to think that this concept enables us to more completely understand the world around us. Mathematics is difficult to work with without this concept, but Nature has no use for it what-so-ever. Let us think about a few instances where zero is not necessary in order to describe the universe:
Vacuum. Every time we think about space, we are forced to work with the idea of a vacuum. Science has tried in vain for years to create "total" vacuums and has grudgingly accepted that they are difficult or impossible to create. I submit to you that Nature cannot acknowledge a total vacuum. Matter and energy require space to exist, we know, but Nature does not create space unless there is something to occupy it. Space in the absence of mass and/or energy does not exist.
Absolute Zero. Humans like to relate a zero temperature to a condition they can rationalize, like the freezing of water. Physicists, due to their mathematical exploration of nature, have had to define a temperature scale (degrees Kelvin) that stops at zero in order to properly define temperature based energy levels in matter. One problem, conceptually, with this is that we still tend to hold out the value of 0º Kelvin as an attainable state. This is clearly not the case. Temperature can be thought of as an expression of the amplitude of the wave property of matter and zero amplitude waves do not exist. Therefore 0º Kelvin cannot and will not ever be attained in Nature. Yet math requires the existence of this concept in order to "rationalize" the concept of temperature.
Infinity. Not only does math require infinity, but it defines many different kinds of infinity (countable and uncountable to name two). Nature has no use for infinity. Just try to define an infinite universe and you will see what I mean. It does not matter how low you set the density of an "infinite" universe, eventually you will define a volume at which your universe becomes a "black hole" which is self-contained and from which escape is impossible.
Infinitely thin boundaries. Stephen Hawking has proposed any number of wild and wonderful theories about black holes that rely on the infinitely thin boundary between our space and black hole space. Nature, however, is all about transition. Even the simple observation of the water level in a glass of water is entirely scale dependant. If one looks closely enough at the surface of the water it becomes apparent that the distinction between water and air becomes arbitrary at a certain scale. The boundary is in a constant state of flux, with some molecules of water becoming vapor and some molecules of air mixing into the water. With respect to the space/black hole boundary, if one factors in the curvature of space at the event horizon, the transition space becomes arbitrarily large and mass/energy dependant.
©2004 Robert Farries
Sept. 27, 2004