Using modern physics, there is no way to express the concept that Ptolomy intended when he said epicycles.
As I’d mentioned elsewhere, there’s actually a pretty easy way to express that, IMO: “Ptolemy thought that planets move in epicycles, and he was wrong for the following reasons, but if we had poor instruments like he did, we might have made the same mistake”.
IF I am correct that they are caused by the socially mediated aspects of the scientific discipline and IF mathematics can avoid being socially mediated by virtue of its non-empirical nature, then I would expect that no paradigm shifts would occur.
The abovementioned non-euclidean geometry is one such shift, as far as I understand (though I’m not a mathematician). I’m not sure what the difference is between the history of this concept, and what Kuhn meant.
But there were other, more powerful paradigm shifts in math, IMO. For example, the invention of (or discovery of, depending on your philosophy) zero (or, more specifically, a positional system for representing numbers). Irrational numbers. Imaginary numbers. Infinite sets. Calculus (contrast with Zeno’s Paradox). The list goes on.
I should also point out that many, if not all, of these discoveries (or “inventions”) either arose as a solution to a scientific problem (f.ex. Calculus), or were found to have a useful scientific application after the fact (f.ex. imaginary numbers). How can this be, if mathematics is entirely “non-empirical” ?
Hmm, I’ll have to think about the derivation of zero, the irrational numbers, etc.
I should also point out that many, if not all, of these discoveries (or “inventions”) either arose as a solution to a scientific problem (f.ex. Calculus), or were found to have a useful scientific application after the fact (f.ex. imaginary numbers). How can this be, if mathematics is entirely “non-empirical”
The motivation for derivation of mathematical facts is different from the ability to derive them. I don’t why the Cartesian skeptic would want to invent calculus. I’m only saying it would be possible. It wouldn’t be possible if mathematics was not independent of empirical facts (because the Cartesian skeptic is isolated from all empirical facts except the skeptic’s own existence).
I don’t why the Cartesian skeptic would want to invent calculus. I’m only saying it would be possible.
My point is that we humans are not ideal Cartesian skeptics. We live in a universe which, at the very least, appears to be largely independent of our minds (though of course our minds are parts of it). And in this universe, a vast majority of mathematical concepts have practical applications. Some were invented with applications in mind, while others were found to have such applications after their discovery. How could this be, if math is entirely non-empirical ? That is, how do you explain the fact that math is so useful to science and engineering ?
As I’d mentioned elsewhere, there’s actually a pretty easy way to express that, IMO: “Ptolemy thought that planets move in epicycles, and he was wrong for the following reasons, but if we had poor instruments like he did, we might have made the same mistake”.
The abovementioned non-euclidean geometry is one such shift, as far as I understand (though I’m not a mathematician). I’m not sure what the difference is between the history of this concept, and what Kuhn meant.
But there were other, more powerful paradigm shifts in math, IMO. For example, the invention of (or discovery of, depending on your philosophy) zero (or, more specifically, a positional system for representing numbers). Irrational numbers. Imaginary numbers. Infinite sets. Calculus (contrast with Zeno’s Paradox). The list goes on.
I should also point out that many, if not all, of these discoveries (or “inventions”) either arose as a solution to a scientific problem (f.ex. Calculus), or were found to have a useful scientific application after the fact (f.ex. imaginary numbers). How can this be, if mathematics is entirely “non-empirical” ?
Hmm, I’ll have to think about the derivation of zero, the irrational numbers, etc.
The motivation for derivation of mathematical facts is different from the ability to derive them. I don’t why the Cartesian skeptic would want to invent calculus. I’m only saying it would be possible. It wouldn’t be possible if mathematics was not independent of empirical facts (because the Cartesian skeptic is isolated from all empirical facts except the skeptic’s own existence).
My point is that we humans are not ideal Cartesian skeptics. We live in a universe which, at the very least, appears to be largely independent of our minds (though of course our minds are parts of it). And in this universe, a vast majority of mathematical concepts have practical applications. Some were invented with applications in mind, while others were found to have such applications after their discovery. How could this be, if math is entirely non-empirical ? That is, how do you explain the fact that math is so useful to science and engineering ?