Xamonas Chegwé wrote:Maths doesn't so much govern the universe, as explain it. The numerical relationships between objects in the 'real' world are undeniable and, in some cases, absolute.
I would here draw the distinction between "explaining" and "describing." I don't think this is a trivial distinction. It's quite at the crux of the issue, really. When we (mis)take our descriptions to be explanations, we're fooling ourselves, I think.
Maths appears in our investigations into the way things work because, at a very deep level, those things work in a very simple way that can be described mathematically.
Agreed. Described. Much as da Vinci described the model for the Mona Lisa when he painted her. No one really knows what that model looked like. Just as a photograph is a description of a subject. We have a description from a single, limited perspective and we extrapolate assumptions and approximations to fill in the gaps. That's natural, but the error creeps in when we claim certainty after having extrapolated, interpreted and approximated. That's all I'm getting at. The claim of certainty when all we really have is a coded approximation.
You claim that maths has only existed for a tiny sliver of the age of the universe. I would rephrase that. Maths has only been used for a tiny sliver of the age of the universe. Its existence is not a function of whether or not a race such as ours can solve quadratic equations. It exists as the sum of relationships between objects - Man did not invent maths, he simply discovered those relationships and invented a language with which to describe them.
Hang on. We didn't invent maths; we invented the language to describe relationships? Is not maths that very language? Is there no distinction between the system and the description of it?
Here are some examples of those relationships.
- The angles in a triangle add up to a straight line - whether you call that 180º or π radians or 96.7 g'tharphs is irrelevant - that is just the language you choose to describe it in.
- When you double the radius of a sphere, its surface area increases 4 times and its volume increases 8 times - true no matter what terminology you employ to describe the phenomena.
- The circumference of a circle is π times its diameter, ALWAYS! π is not some ethereal, ill-defined quasi-number, it is defined as exactly that ratio between the circumference and diameter of ANY circle. If we counted in multiples of π, it would be a whole number and all of our familiar integers would be never-ending, non-repeating decimals - this is the basis of the radian measure of angles. There is nothing incomplete about π and it is only irrational in purely mathematical terms - because it cannot be represented exactly as the ratio between two whole numbers.
- The action of gravity on objects can be predicted to an unbelievable degree of accuracy. Gravity is determined by 2 things only. The mass of objects and their distance from each other. It is directly proportional to the mass and inversely proportional to the square of the distance. This isn't an approximation. It is a fundamental and absolute fact of how the universe works. The "incompleteness and irrational aspects" only come in to play when we try to apply those facts to systems where there are multiple bodies of different sizes and densities all exerting gravity at the same time - systems such as... the universe or any part of it! But this doesn't mean that the underlying maths is wrong; simply that the sums are very hard!
The first example only works in Euclidean geometry, not 3-D or relativistic space-time. It is imaginary.
The second example: Where does this ideal sphere reside in the reality that we see with our own eyes or even with a telescope? It's also idealized and says more about the workings of the human brain than it does about the universe at large.
Third example: Professor Newton, let me introduce Professor Einstein.
Professor Einstein, let me introduce you to (?). The inverse square law doesn't really work much beyond the human scale. Get much smaller or bigger and the innacuracies become significant. There's nothing magical about the human scale, I'm sure you know that. Again, these formulae say more about human perception than anything absolute about the universe itself.
Applying maths to the real world may appear to be mere approximations and ideals. And so it is, for the most part. We can use the properties of spheres and gravity mentioned above to predict the behaviour of planets; but planets are never quite perfect spheres. We could compensate for that in our calculations, taking insanely accurate measurements and factoring all of it into our equations - but in practical terms, it is almost never necessary to do this in order to make predictions that are useful. We can use our knowledge of the relative masses and distances of the Earth, Moon and Sun to model the orbit of the Moon around the Earth accurate to nanometres a year! To be completely accurate, we would need to factor in the gravitational influence of the other planets, comets, asteroids and dust in our solar system, the other stars in our galaxy and every dot of dust between here and infinity - but there is no practical purpose in doing so. It is enough to appreciate the degree of approximation inherent in our model.
I wouldn't dispute any of that. I praise and admire the unprecedented accuracy and practical applicability of the mathematical code. My beef is with those who mistake the code for the reality by ignoring the approximate nature of it all and assuming that mathematics is an entity in and of itself, outside the human rational function. It's over-reaching and anthropomorphic. It's mistaking the menu for the meal. Mathematics is man-made to suit the human mind, which needs to recognize patterns. The human mind is awesomely complex and powerful, but to project aspects of the human mind on the entire universe with claims of absolute certainty is naive at best, delusional at worst. We don't know that the universe is 13.7 billion years old or that it started with a big bang or even how long a piece of string is. (cf: BBC, How Long is a Piece of String?) We have a limited amount of certainty that our mathematical code is consistent most of the time, but Godel proved that it must rest on at least one (and probably more) assumptions that can't be proven within the system, and will alway, therefore, be incomplete. An approximation. An assumption. IOW, a faith.
So, while we may never be able to model the entire universe mathematically, it is enough to know that, in theory, given infinite time and computational power, we could - because mathematical relationships underlie everything. The symbols and methods that we use to discover them are manmade - but the relationships are not.
This is an article and a declaration of faith, is it not? I'm not out to crash the whole system, mind you. I like it very much. Instead, I'm out to crash the illusion of certainty that surrounds it. We need to emphasize the tentative and approximate aspects of our claims more, so that future generations will not become so entrenched and rigid that they are incapable of revolutionary thinking. There have been more than one periods in human hisory when, for example, physics was thought to be just a matter of extending the figures out to another decimal, eh? Look how that turned out.
"A philosopher is a blind man in a dark room looking for a black cat that isn't there. A theologian is the man who finds it." ~ H. L. Mencken
"We ain't a sharp species. We kill each other over arguments about what happens when you die, then fail to see the fucking irony in that."
"It is useless for the sheep to pass resolutions in favor of vegetarianism while the wolf remains of a different opinion."
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I just watch my hands and cannot grasp that there would be no exact mathematical relationship governing the ratio of numbers of fingers to numbers of hands, completely regardless of whether anybody can formulate it or not.
