Imagine that over the last 11 thousand years (that is, the period of stable climate following upon the last ice age which allowed the human civilisation to develop)the atmospheric conditions on Earth were different: the skies were always covered, even in the absence of clouds, by a very light haze, not preventing the developmentof agriculture, but obscuring the stars and turning the sun and the moon into amorphous light spots. Would mathematics have had a chance to develop beyond basic arithmetic and geometry sufficient for measuring fields and keeping records of harvest? I doubt that. Civilisations which developed serious mathematics also had serious astronomy (it was an equivalent of our theoretical physics). But I claim even more: the movement of stars in the sky was the paradigm of precision andreproducibility, the two characteristic features of mathematics. Where else could humans learn the concept of absolute precision?I agree with that. Without astronomy, we would not have much civilization today.
Monday, March 29, 2021
Civilization was inspired by Astronomy
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"Where else could humans learn the concept of absolute precision?"ReplyDelete
Where else indeed.
No, no, and no. Today's mathematicians love to pat themselves on the back for things they didn't do, claim knowledge of what they don't know, and have the gall to think themselves so informed without actually taking the time to be informed. This is hubris, not understanding, and can be cured by putting down their abaci and picking up a god damn book or two about history.
First, lets be clear on the actual terminology, Astronomy had nothing to do with most of what is being retroactively laid claim to, Astrology did. That's right, watching the stars was the desperate business of quacks (or astrologers) who were frantically trying to predict disaster ('bad star') and human behavior with astrological charts, largely to court favor with royalty and the wealthy. Much like today's scientists, astrologers realized they could acquire fame, wealth and political influence far beyond their low born status if they predicted dire events and catered to the fears and conceits of the high born, rich and powerful.
Calendar development was far more important and vital to the advancement of human civilization through agriculture, as farmers needed to know how to plan, they needed to know when to plant and harvest, when the Nile would flood, the rainy season begin, etc, and this was determined by days and cycles of the moon, not stars or mysterious 'planeta'. There were many places that had frequently cloudy skies that still needed calendars, and could count the days perfectly well without the help of court stargazers. Further precision of time, or hours, became essential beyond dawn, noon, and sunset as efficient planning of human work schedules became more necessary and complex, leading to the development of clocks of ever increasing precision. I would posit that star gazing via the Sextant had far more impact on navigation at sea (allowing for greater trade, travel, and the exchange of ideas) than the development of 'absolute precision' drivel. There were also lots of opportunities for mathematics to have far more practical use of precision outside of astrology, largely in the pursuit of building and making things.
Just because you can analyze something that happened in the past mathematically, does not mean it was developed by mathematicians.
"I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail."
I appreciated your comment. Very, very sensible.
Thanks for pointing out the paper. ... I don't know why, but whenever mathematicians write such papers (I mean the papers which are written informally and a bit expansively, may be with "carelessness"), my first reaction while going through the text invariably is a bit of a frown. Which, soon enough, turns into a smile. And that, in turn, is because it's always fun to see how mathematicians, when left all to themselves (i.e. without physicists / engineers guiding their thoughts) typically fail to get on the top of the actual problem. The purer the mathematician, the worse he typically is on this count. They don't even get a good handle on the problem let alone generate a good line of attack for it. But they still often are intelligent, and always are sincere, and sometimes also are engaging. So, the "fun" part.
In particular, coming to this paper. Let me ask you this simple question: Did you manage to remember the title of this paper by the time you finished all the diversions and sub-threads contained in the text? And then, think: what would be the best way to attack the problem in the title?
As to me, I would have anyway tried to tackle the problem by starting with some simplest examples. (I didn't know it, but looks like I was always following Gelfand's advice!) The upside with such an approach (examples) is: You can get to some definite thing with which you can work. The downside is: You can't be sure how general your ideas / solutions are; you've to work on that part separately.
With that caveat, how best to tackle the title problem?
IMO, we have to try and take some contrasting but typical examples. What kind of problems are solved in biology, as in contrast to in physics? How to go about doing that? Start with school time...
In school, while in physics we are being taught Pascal's law of pressure, geometric optics, heat etc., and then, Newton's laws of motion, what topics appear in biology? Answer: Characteristic of life the phenomenon; various forms of living organisms, their morphological / structural features; distinguishing details of their living processes / functioning; taxonomy.
Maths works in the first, but not in the second. Why? What makes for the real distinction?
[Contd in the next comment...]
[Contd from the previous comment...]ReplyDelete
Physics studies only such problems where the sizes of objects (or sizes of their properties, attributes, or actions) are of paramount importance in capturing the operative causal relationships. It is for this reason that in physics, to arrive at a statement of a natural law is to make a quantitative statement.
In contrast, biology studies those problems in which phenomenological isolation of properties / attributes / characteristics, and organization of objects / processes into classes and long-range relationships are of paramount importance.
For instance, the sheer quantity of food that a class of organisms eats doesn't help us characterize it so well (some animals hibernate), but the fact that classes of living beings can be organized as naturally forming a food chain, is important.
In biology, qualitative statements predominate. Which means, differences aren't crucial unless they lead to categorical (class-wise) differences. (Indeed, the very existence of categories is important; the precise locus of the boundary lines between them isn't! So, the issue of the "distance" from the "center" of a class/regime to the boundary simply does not arise!)
Now are there any *mathematical* methods which can be used in capturing qualitative / categorical differences?
Answer: Yes. Nonlinear dynamics, in particular, catastrophe theory.
Indeed, these two, taken together, allow us to span the quantitative (i.e. size-wise measures) with the qualitative.
OK, so that was what *I* thought of, after some 10--15 minutes worth of thought. (And I admit, I make for a bad mathematician.)
Now, check out what our pure mathematician proposes (despite having touched on V.I. Arnold's works in the same paper): (1) stochasticity, (2) multi-dimensional combinatorics.
Of course, I could easily be wrong. What I thought about (above) may not be the most productive line of attack to deal with the title question. But I'm damn sure it's much, much better than either stochasticity, or multi-dimensional combinatorics, or both! Why do I say that? Because, even physicists (and not just mathematicians) have tried using those lines of attack already, but with little success to show for their effort. That's why!
...See why I find it fun to read such papers by mathematicians? the purer the better?