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.
Saturday 26 March 2016
Mathematics - a first set of thoughts
Hilary Putnam is dead. I read through the Wikipedia page and found that he has changed his stance on the ontology of mathematics: "How real are mathematical things?"—like integral numbers, real numbers, functions, sets, deduction rules, proofs, whatever? For some reason, this makes me nervous: I thought, first of all, that this question is more or less settled—I might not know the answer, but some great philosophers—like Putnam—certainly did and do. But even if they do not, I thought that the various possible answers are so clearly described that you would not switch from one to the other. Rather, you would hold to one conviction and argue why the other ideas were wrong, or irrelevant, or not sufficient, or deficient in some other way.
And in the first place, I thought that the accepted answer is clearly that mathematical objects are in some sense "real," at least in the (or some) sense that mathematical object are "out there" even if no one is in this world or the whole universe who actually does mathematics: See e.g. Linnebo, Øystein, "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.).
But, in the second place, I know that I somehow feel confident that this is not true: That mathematics is a purely human construction, or at least a construction of minds doing mathematics. In other words, there is no mathematics without mathematicians.
This is, altogether, unsatisfactory. I am probably wrong, or vague enough so that my ideas count as wrong—but why? And what is a better answer?
I could set out to find the answer—the answers?—in the books. But I fear that I would quickly lose my own ideas, being overwhelmed by the precise arguments of the professionals. And I am not yet there. I want to know why, and how, I am wrong, not what is right.
So—I'd like to think a little bit on my own, probably with a huge bunch of implicit assumptions, at least some of which might become explicit in this process: And only then see whether this matches with someone else's ideas—and then, ideally, find refutations for these! I do not seem to be too humble, probably.
I start out with realism, about what I think Popper said: The actual material things in this world are real. Many of them (at least all the atoms) were here before the first humans and will be here after all of mankind has gone, for whatever reason. I must be careful here: That things were here does not mean that they were identifiable as things. That it is possible to classify certain of the "things in the world" as being similar, and hence be able to call them atoms, stars, trees, might already be mathematics, at its core. So I should probably rather say: The whole world, the universe was, is, and will be there independently of humans and other beings with a reflective, thinking mind. But actually, this has not solved anything: This universe changes. So even talking about a continuing existence or "behaving" of something that—by being continually there—can be seen as (or thought of?) as being something singular might be doing mathematics. So, what is this "realism" actually?
I did not think I would stumble so early. I thought that I could deal with the physical world in a paragraph or two, and then proceed to mathematics. Great. Should I now consult all the books and papers on realism to find out about how one, and we, and I should think about these "real," "observer-independent" world? Yes, I should. But I don't know where to start, and how to make sure I am not again drowned by other ideas and arguments and refutations.
Clueless.
But there is something (that's my given which I would not want to argue about). And there happens something: Some energy gets transformed—yes, we might call it a lightning stroke or a nerve impulse, but even if we would not categorize it (risk: this is already math), it was there. So I have to talk about it without categorization. I try the concept of a "description": This is some more or less precise notation that can be mapped to that world. And a description of something is "good" or "good enough" if someone else who reads the description will agree that it is a description of what it claims to describe.
Now, I have planted myself deep in a big hole full of shit: I wanted to find out what "real" means as in "something is real if it exists independently of a mindful observer"—and I end up answering this by introducing "more or less" precise "notations" that are "read" by "someone" who "agrees" to a "claim." Absolutely ridiculous, isn't it?
First, yes.
Second, I have not better idea (read the books, someone says—yes, I will, um, soon).
Third, I think I can defend at least a bit of this, at least with my target in mind, namely to show, or at least argue, that mathematics is man-made. See, descriptions, in the sense above, can be taken by suitably capable whatevers and compared to the world. And if many such whatevers do that and will e.g. converge in their predictions of what happens next, this is a good description.
Kuhn, and others, would tell me that "compare" and "converge" and "predict" are totally theory-dependent concepts. For my train of thought, however, I want to believe that they are not "totally arbitrary" in the sense that what in many discourses is "converging" will be "not converging" in ("similarly") many other discourses, but that in "most" discourses the predictions will converge. Maybe, to agree on this, it is also necessary (for the "whatevers" that do the reading and comparing and agreeing) to have the descriptions of what converging means, which again are compared and do converge, and maybe a whole hierarchy of descriptions of comparing and converging is necessary. All this would most probably be doing mathematics—but this is only an accident in my description of "realism"—it has, right now, nothing to do with my target problem of, more or less, how mathematical things come into the world. Rather, I leave it totally to the whatevers to do that as they like, and not think about what they do. But I suppose that they will agree, as we humans do over time.
The important aspect behind this notion of realism is that it is fuzzy: It is enough for me if some descriptions of some events are somewhat good. No more can be hoped for before starting with mathematics: No more is found by us and anyone else when looking into the world. And for at least thousands of years, our ideas about that reality were, for concrete aspects, completely wrong; but many were somewhat ok, and this alone did not keep us from doing math. So, it must be possible to base mathematics, at least as a human activity, on such a shaky and largely wrong view of the world. The question is whether all of mathematics must be based so. I think—or rather: feel—that yes, from these muddy waters, mathematics can be "argued up." But I have to argue this a little more.
I have to think about this.
And in the first place, I thought that the accepted answer is clearly that mathematical objects are in some sense "real," at least in the (or some) sense that mathematical object are "out there" even if no one is in this world or the whole universe who actually does mathematics: See e.g. Linnebo, Øystein, "Platonism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.).
But, in the second place, I know that I somehow feel confident that this is not true: That mathematics is a purely human construction, or at least a construction of minds doing mathematics. In other words, there is no mathematics without mathematicians.
This is, altogether, unsatisfactory. I am probably wrong, or vague enough so that my ideas count as wrong—but why? And what is a better answer?
I could set out to find the answer—the answers?—in the books. But I fear that I would quickly lose my own ideas, being overwhelmed by the precise arguments of the professionals. And I am not yet there. I want to know why, and how, I am wrong, not what is right.
So—I'd like to think a little bit on my own, probably with a huge bunch of implicit assumptions, at least some of which might become explicit in this process: And only then see whether this matches with someone else's ideas—and then, ideally, find refutations for these! I do not seem to be too humble, probably.
I start out with realism, about what I think Popper said: The actual material things in this world are real. Many of them (at least all the atoms) were here before the first humans and will be here after all of mankind has gone, for whatever reason. I must be careful here: That things were here does not mean that they were identifiable as things. That it is possible to classify certain of the "things in the world" as being similar, and hence be able to call them atoms, stars, trees, might already be mathematics, at its core. So I should probably rather say: The whole world, the universe was, is, and will be there independently of humans and other beings with a reflective, thinking mind. But actually, this has not solved anything: This universe changes. So even talking about a continuing existence or "behaving" of something that—by being continually there—can be seen as (or thought of?) as being something singular might be doing mathematics. So, what is this "realism" actually?
I did not think I would stumble so early. I thought that I could deal with the physical world in a paragraph or two, and then proceed to mathematics. Great. Should I now consult all the books and papers on realism to find out about how one, and we, and I should think about these "real," "observer-independent" world? Yes, I should. But I don't know where to start, and how to make sure I am not again drowned by other ideas and arguments and refutations.
Clueless.
But there is something (that's my given which I would not want to argue about). And there happens something: Some energy gets transformed—yes, we might call it a lightning stroke or a nerve impulse, but even if we would not categorize it (risk: this is already math), it was there. So I have to talk about it without categorization. I try the concept of a "description": This is some more or less precise notation that can be mapped to that world. And a description of something is "good" or "good enough" if someone else who reads the description will agree that it is a description of what it claims to describe.
Now, I have planted myself deep in a big hole full of shit: I wanted to find out what "real" means as in "something is real if it exists independently of a mindful observer"—and I end up answering this by introducing "more or less" precise "notations" that are "read" by "someone" who "agrees" to a "claim." Absolutely ridiculous, isn't it?
First, yes.
Second, I have not better idea (read the books, someone says—yes, I will, um, soon).
Third, I think I can defend at least a bit of this, at least with my target in mind, namely to show, or at least argue, that mathematics is man-made. See, descriptions, in the sense above, can be taken by suitably capable whatevers and compared to the world. And if many such whatevers do that and will e.g. converge in their predictions of what happens next, this is a good description.
Kuhn, and others, would tell me that "compare" and "converge" and "predict" are totally theory-dependent concepts. For my train of thought, however, I want to believe that they are not "totally arbitrary" in the sense that what in many discourses is "converging" will be "not converging" in ("similarly") many other discourses, but that in "most" discourses the predictions will converge. Maybe, to agree on this, it is also necessary (for the "whatevers" that do the reading and comparing and agreeing) to have the descriptions of what converging means, which again are compared and do converge, and maybe a whole hierarchy of descriptions of comparing and converging is necessary. All this would most probably be doing mathematics—but this is only an accident in my description of "realism"—it has, right now, nothing to do with my target problem of, more or less, how mathematical things come into the world. Rather, I leave it totally to the whatevers to do that as they like, and not think about what they do. But I suppose that they will agree, as we humans do over time.
The important aspect behind this notion of realism is that it is fuzzy: It is enough for me if some descriptions of some events are somewhat good. No more can be hoped for before starting with mathematics: No more is found by us and anyone else when looking into the world. And for at least thousands of years, our ideas about that reality were, for concrete aspects, completely wrong; but many were somewhat ok, and this alone did not keep us from doing math. So, it must be possible to base mathematics, at least as a human activity, on such a shaky and largely wrong view of the world. The question is whether all of mathematics must be based so. I think—or rather: feel—that yes, from these muddy waters, mathematics can be "argued up." But I have to argue this a little more.
I have to think about this.
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