That is, the concept cannot be coherently expressed using the new terminology. For example, there’s no way to express coherent concepts in things like Ptolomy’s epicycles or Aristole’s impetus.
I’m not seeing how the second sentence is an example of the criterion in your first sentence.
Using modern physics, there is no way to express the concept that Ptolomy intended when he said epicycles. More casually, modern physicists would say “Epicycles don’t exist” But contrast, the concept of set is still used in Cantor’s sense, even though his formulation contained a paradox. So I think the move from geocentric theory to heliocentric theory is a paradigm shift, but adjusting the definition of set is not.
I’m also not seeing what the attributes “empirical” and “non-social” have to do (causally) with the ability to form coherent concepts.
I’m using the word science as synonymous with “empirical studies” (as opposed to making stuff up without looking). That’s not intended to be controversial in this community. What is controversial is the assertion that studying the history of science shows examples of paradigm shifts.
One possible explanation of this phenomena is that science is socially mediated (i.e. affected by social factors when the effect is not justified by empirical facts).
I’m asserting that mathematics is not based on empirical facts. Therefore, one would expect that it could avoid being socially mediated by avoiding interacting with reality (that is, I think a sufficiently intelligent Cartesian skeptic could generate all of mathematics). 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.
This whole reference to paradigm shifts is an attempt to show a justification for my belief that mathematics is non-empirical, contrary to the original quote. If you don’t believe in paradigm shifts (as Kuhn meant them, not as used by management gurus), then this is not a particularly persuasive argument.
WP lists “non-euclidean geometry” as a paradigm shift, BTW.
If Wikipedia says that, I don’t think it is using the word the way Kuhn did.
WP lists “non-euclidean geometry” as a paradigm shift, BTW.
If Wikipedia says that, I don’t think it is using the word the way Kuhn did.
For Kuhn, the word was, if anything, a sociological term—not something referring to the structure of reality itself. (Kuhn was not himself a postmodernist; he still believed in physical reality, as distinct from human constructs.) So it seems to me that it would be entirely consistent with his usage to talk about paradigm shifts in mathematics, since the same kind of sociological phenomena occur in the latter discipline (even if you believe that the nature of mathematical reality itself is different from that of physical reality).
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 ?
My position, FWIW, is that all of science is socially mediated (as a consequence of being a human activity), mathematics no less than any other science. Whether a mathematical proposition will be assessed as true by mathematicians is a property ultimately based on physics—currently the physics of our brains.
Using modern physics, there is no way to express the concept that Ptolomy intended when he said epicycles. More casually, modern physicists would say “Epicycles don’t exist” But contrast, the concept of set is still used in Cantor’s sense, even though his formulation contained a paradox. So I think the move from geocentric theory to heliocentric theory is a paradigm shift, but adjusting the definition of set is not.
I’m using the word science as synonymous with “empirical studies” (as opposed to making stuff up without looking). That’s not intended to be controversial in this community. What is controversial is the assertion that studying the history of science shows examples of paradigm shifts.
One possible explanation of this phenomena is that science is socially mediated (i.e. affected by social factors when the effect is not justified by empirical facts).
I’m asserting that mathematics is not based on empirical facts. Therefore, one would expect that it could avoid being socially mediated by avoiding interacting with reality (that is, I think a sufficiently intelligent Cartesian skeptic could generate all of mathematics). 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.
This whole reference to paradigm shifts is an attempt to show a justification for my belief that mathematics is non-empirical, contrary to the original quote. If you don’t believe in paradigm shifts (as Kuhn meant them, not as used by management gurus), then this is not a particularly persuasive argument.
If Wikipedia says that, I don’t think it is using the word the way Kuhn did.
For Kuhn, the word was, if anything, a sociological term—not something referring to the structure of reality itself. (Kuhn was not himself a postmodernist; he still believed in physical reality, as distinct from human constructs.) So it seems to me that it would be entirely consistent with his usage to talk about paradigm shifts in mathematics, since the same kind of sociological phenomena occur in the latter discipline (even if you believe that the nature of mathematical reality itself is different from that of physical reality).
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 ?
Hmm, “justified” generally has a social component, so I doubt that this definition is useful.
So this WP page doesn’t exist? ;)
My position, FWIW, is that all of science is socially mediated (as a consequence of being a human activity), mathematics no less than any other science. Whether a mathematical proposition will be assessed as true by mathematicians is a property ultimately based on physics—currently the physics of our brains.