Paint by numbers

Written by on February 25, 2013 in Commentary with 0 Comments

I hope am I not dating myself if I ask whether you remember ‘Paint-by-Numbers’ or not? If it doesn’t ring a bell it was simply this: an illustration kit that gave you all the necessary tools to create a nice painting without any skill at all.

The idea was pure simplicity—from a pre-drawn template, all that was needed was to paint various numbered areas with a corresponding colour and voila! “A beautiful oil painting, the first time you try” , as advertised by the original Craft Master paint-kit. Amazing! real art without even a hint of genuine creativity or artistic spark whatsoever.

You have to admire the spirit of entrepreneurship. To bring the masses access to painted art through a simple idea and make a profit all the while, is pure genius. Of course Paint-by-numbers is very outdated, but on the surface it demonstrates something profound about mankind, in that we can take something so whimsical and organic like artistic ability and reduce its complexity to systematize results.

This is what Math in Science does as well. In fact much of physics is pure math these days— to the degree that if your theory cannot be expressed mathematically, it will be considered unworthy of consideration.

So for the physicist, the logic of math is an inborn attribute of nature. The challenge for science then is to tease out the logic in nature through observations, express it as an equation and hopefully what results will  lend to repeatability and ultimately future predictions.

I can’t help feeling that the universe isn’t a machine to be reverse engineered. I mean, can the analog of physical existence be purely and entirely expressed through mathematics? It would certainly seem so, especially because science has flourished over the last century under this assumption. I know it seems silly to question it, after all 1 + 1 = 2 seems to make sense – but can math be so trustworthy? What is Math exactly?

Mathematics is a language. A set of symbols that are used to describe concepts, objects and systems. Moreover, Mathematics is an abstraction of reality.

The Google dictionary defines abstraction as:

ab·strac·tion / ab’strakSHən / Noun
1. The quality of dealing with ideas rather than events.
2. Something that exists only as an idea

Take the number 1 for instance, is it a real thing? Can I go somewhere and see the number one in its natural habitat interacting with other numbers? The number 1 is an abstract idea that refers to a particular property of an object. If I have an apple and an orange on a table, you can say I have two objects and one of them is an apple. However the ‘one-ness’ of the apple says nothing at all about its entire reality; how it tastes, smells, or what colour it is. So, insofar as describing reality math can only model a limited aspect of it.

Here’s an example:

If an Apple were to fall from a tree to the ground, factoring in the speed of gravity and the height from which it fell, I imagine you would feel quite confident in predicting the time it took to hit the ground? After all its just math at that point, right? Yet, what your calculation could not possible account for are the myriad of environmental effects that are intrinsic to the experiment, i.e.; the temperature and moisture of air causes a small resistance to gravity as the apple falls, the proximity of other mass like a mountain or building will pull the Apple slightly towards it changing its direction of descent. In fact even the position of the Sun, Moon and planets in the sky will have minute effects.

If you consider the above deeply enough, it is true to say that the entire universe effects the Apple as it falls. Albeit to a ridiculous degree of subtlety but what this shows is how accurately you intend to measure anything only brings you the next decimal point, not the ultimate reality of the phenomenon. Our world is only an approximation, we are left only to decide which value is good enough for our purposes. Behold, the limitation of math and science.

These are all very practical approximations mind you, I don’t mean to diminish our sense of things, the lesson here is just don’t mistake it for absolute knowledge of the universe. You cannot take anything out of the universe in order to measure it objectively and completely, everything is contingent upon everything else. This is the interconnected oneness of the universe.

And so math is somewhat of a fair weather friend. It has no loyalty to our reality because it is not an intrinsic part of it, despite what science thinks. Math doesn’t need to be consistent with nature, it only needs to be consistent to its own logical framework and the shallow envelope of the human senses.

Here’s another example:

Suppose you want to catch a bus. Before you get there, you must get half way there. And before you get halfway, you must get a quarter of the way. Before you get quarter of the way, you must get an eighth of the way. Before you get an eighth of the way, you must get a sixteenth of the way… and so on. Do you see what’s happening here? Add up all the fractions of the distance you went, and they do not total the entire distance to the bus. So you never really reach the bus, you just get infinitely close to it.

What’s wrong with the example above is that dividing by half is perfectly acceptable in mathematics, however it doesn’t seem to fit in with our experience of reality because it seems you can do it endlessly. And this is problematic.

Infinity

Infinity is a dark truth in mathematics, as math requires definite magnitudes in order to have a foundation in logic. Infinity is not a definite or even a realistic magnitude, yet it exists within math as a natural consequence of its reasoning and operation.

Some example of infinities are:

  • The successive addition/subdivision in arithmetic – example: no matter what large number you compute, you can always add one to it. Now matter how many times you divide a number in half, you can always do it again.
  • ‘Irrational’ numbers: like the value of Pi (3.1415……) or the square root of 2 whose numbers go on endlessly after the the decimal.
  • ‘Imaginary’ numbers: like the square root of -1. No number multiplied by itself will give you a negative value, yet √-1 is a crucial  reoccurring number used in quantum calculations.

Math in History

Then was a famous Austrian logician named Kurt Godel who discovered through his Incompleteness Theorems that certain mathematical rules will provide true statements that cannot be proven.

Here’s an example to illustrate the point:

“This statement is false” – if you consider this statement as being true about its falsity, then you must conclude that it is in fact false. Yet if it is false, by consequence it nullifies its over all falsity, making it true. So which is it? Can you can see the paradox here? Each supposed condition supposes the opposite in each case. This statement is not valid and hence cannot be proved. Godel found similar anomalies in mathematical logic.

Godel’s two Incompleteness Theorems formulated in 1931, where a direct challenge to Whitehead and Russell’s famous three volume work called Principia Mathematica which attempted to show that all mathematical truths could be proven, solidifying symbolic logic as complete and incontestable. However, Godel’s theorems showed that mathematics was flawed and not consistent. This had destroyed the dream of a god-like knowledge mankind wanted over the universe.

Also, during this time, a new revolution in physics in the early part of the 20th century had very poignantly underscored our lacking grasp of the universe, something called Quantum Mechanics. It  shook the foundations of science. It had shown that the very nature of the subatomic world was illogical and uncertain, very much like our logic was. This was a point that has changed the world’s outlook on what constitutes knowledge.

–CJ–

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