A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
So it’s abstracting, but then it’s not about the real world anymore.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.
What do cardinals correspond to?
I suppose it’s about the language, not about the model. Still.
Screw it. I’ll just go do a PhD thesis on how abstraction works, and then anyone who wants to actually understand can read that.
A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
Ok, so you would say inaccessible cardinals, versions of the continuum hypothesis etc. are “about the real world.”
At this point, I am bowing out of this conversation as we aren’t using words in the same way.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I’m actually only using it for the former.
You may be thinking of Scott Aaronson’s review of “A new kind of science”.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
(Of course there are also cardinals and cardinals.)
What was the point of writing that? Do you think I was talking about “3”?
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.