a) The two really different civilizations that find each other at a distance that is conquerable: Say we found that Mars indeed has living "people" on it. When, and how, would we have found out? When Lipperhey invented the telescope, and Galileo looked at Mars? Probably it would have taken a few more years - and it is interesting whether we would have seen them first, or whether we would have shot some probe at them, unsuspecting. And how about the converse? A probe lands nicely on the Earth ...
b) Statistically, we would not meet: One of us would be dead for probably billions of years when the other one came around. What happens?--in detail, I mean. There are many SF novels where people stumble on other planets with crumbling pillars. But we will never go to other planets--we will only send probes. So what happens when that first Russian probe crashes onto Mars and, during its fall, makes a few pictures of something which looks too controlled to be "nature"? Or would that happen--would a probe that falls on Earth see anything human? Even if that happens by chance, would the second probe see houses, roads, or at least a few rectangular fields? How would those at the other end of the probe think, react, continue?
If someone knows of intelligent novels dealing with such topics, drop me a note! (if you happen to send a probe to this blog, of course--which I do not at all promote, so this might be a sort of self-referential experiment, wouldn't it?)
Monday 2 May 2016
The un/reasonable in/effectiveness of mathematics
To be read, and commented:
However, there is one thing that is really interesting: That the gravitational law is a simple square law. Why is it so that one of the easier observable patterns in this world--namely, how the moon and the planets move; it is at least so easy to observe that to our knowledge, "all people at every time and everywhere have noticed it"--can be described by mathematics which is easy to learn and seems to have been part of the earliest mathematics we know about--there are quadratic equations in very old Egyptian and Babylonian texts--, and which we can teach to all children?
There are other parts of physics that require wildly more complicated mathematics: Maxwell's laws (vector analysis), relativity (matrices), quantum physics (complex operators)--but the areas they apply to ("controlled" electricity, in contrast to things like lightning; speeds near the speed of light; effects on the atomic level) are also very hard to observe.
So, the question is: Why happen easily observable things (falling bodies, circling planets) to have easily describable formulas?
One answer: This is so by extreme cherry-picking. Almost all other easily observable things around us do not at all have simple formulas, or mathematical formulas at all that describe them. It is no wonder it took 2000 years from e.g. Aristoteles's attempts to understand the world conceptually ("scientifically") to Galileo and Kepler, who found the first tiny segments of reality that gave simple answers to the question "how?".
Still: It might have been that there is no such segment at all. Or does there have to be some "required simple and observable part" of the world?
- R. W. Hamming, The unreasonable effectiveness of mathematics
- (e.g.) D.Abbott, The Reasonable Ineffectiveness of Mathematics
However, there is one thing that is really interesting: That the gravitational law is a simple square law. Why is it so that one of the easier observable patterns in this world--namely, how the moon and the planets move; it is at least so easy to observe that to our knowledge, "all people at every time and everywhere have noticed it"--can be described by mathematics which is easy to learn and seems to have been part of the earliest mathematics we know about--there are quadratic equations in very old Egyptian and Babylonian texts--, and which we can teach to all children?
There are other parts of physics that require wildly more complicated mathematics: Maxwell's laws (vector analysis), relativity (matrices), quantum physics (complex operators)--but the areas they apply to ("controlled" electricity, in contrast to things like lightning; speeds near the speed of light; effects on the atomic level) are also very hard to observe.
So, the question is: Why happen easily observable things (falling bodies, circling planets) to have easily describable formulas?
One answer: This is so by extreme cherry-picking. Almost all other easily observable things around us do not at all have simple formulas, or mathematical formulas at all that describe them. It is no wonder it took 2000 years from e.g. Aristoteles's attempts to understand the world conceptually ("scientifically") to Galileo and Kepler, who found the first tiny segments of reality that gave simple answers to the question "how?".
Still: It might have been that there is no such segment at all. Or does there have to be some "required simple and observable part" of the world?
Mathematics - a game!
In the spirit of the title of this blog, and as I have too many things I have or want to do, I'll put down here just rough items laying out my line of thought:
- In order to show that there is no "reality" behind mathematics, i.e., that mathematical objects are not "real" in any sense that gives them existence independent of human thinkers, I argue that mathematics is practically identical to chess: A set of invented rules, whose consequences we are explore.
- This refutes the argument that solely from the way of talking that we "discover" new things in mathematics, it would follow that these things have an independent existence: We also discover new openings in chess, new moves, and new universal truths (like "two rooks and a king can win against a sole king").
- Critical thought 1: What about the concepts of "truth" and "proof," which play such an important role in mathematics? I want to argue that "truth of a statement" in math can be handled as "a position can be won" in chess: I.e., as a nontrivial mapping of a complex subset of all possible situations to a two-element set { T, F }. Of course, in chess, there is also a draw, whereas for truth, we typically assume the law of the excluded middle. However, one can either modify chess a little so that every position maps to { W, L }; or one considers non-standard logics in math.
- Critical thought 2: Math, as opposed to chess, has a usable mapping to many real-world items and problems: Rational numbers map nicely to lengths, areas and volumes exist in math as well as in real life etc. So there!--math must be something that is "much more real" than chess. I think this is not true: First, the chess rule set and its consequences is a "theoretical construct" like math, which happily maps to real world chess pieces and boards--with similar problems, e.g., that the mapping is not completely two-way (there is no mapping for pieces that straddle the boundaries of squares etc.). Still, math seems so much more "usable in unexpected ways" than the "chess model," which only maps to "real chess."
- Critical thought 3: Chess, or at least chess theory, seems to be simply a subset of mathematics. So it is not really possible to show anything interesting about mathematics by recurring to chess. Hmm. Interesting. I do not at all think that chess is a subset of mathematics; at most, it overlaps with mathematics ... but the arguing gets tricky here.
- ...etc.
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