there’s no way to express coherent concepts in things like Ptolomy’s epicycles or Aristole’s impetus
There are perfectly fine ways to express those things. Epicycles might even be useful in some cases, since they can be used as a simple approximation of what’s going on.
The reason people don’t use epicycles any more isn’t because they’re unthinkable, in the really strong “science is totally culture-dependent” sense. It’s because using them was dependent on whether we thought they reflected the structure of the universe, and now we don’t. Ptolemy’s claim behind using epicycles was that circles were awesome, so it was likely that the universe ran on circles. This is a fact that could be tested by looking at the complexity of describing the universe with circles vs. ellipses.
So this paradigm shift stuff doesn’t look very unique to me. It just looks like the refutation of an idea that happened to be central to using a model. Then you might say that math can have no paradigm shifts because it constructs no models of the world. But this isn’t quite true—there are models of the mathematical world that mathematicians construct that occasionally get shaken up.
My point was that trying to express epicycles in the new terminology is not possible. That is, modern physicists say, “Epicycles don’t exist.”
Obviously, it is possible to use sociological terminology to describe epicycles. You yourself said that they were useful at times. But that’s not the language of physics.
Since you mentioned it, I would endorse “Science is substantially culturally dependent”, NOT “Science is totally culturally dependent.” So culturally dependent that there is not reason to expect correspondence between any model and reality. Better science makes better predictions, but it’s not clear what a “better” model would be if there’s no correspondence with reality.
I brought all this up not to advocate for the cultural dependence of science. Rather, I think it would be surprising for a discipline independent of empirical facts to have paradigm shifts. Thus, the absence of paradigm shifts is a reason to think that mathematics is independent of empirical facts.
If you don’t think science is substantially culturally dependent, then there’s no reason my argument should persuade you that mathematics is independent of empirical facts.
And so for the curves in question, the Fourier expansion would have only a finite number of terms.
The point being that, in contrast to what was being asserted, Ptolemy’s concept is subsumed within the modern one; the modern language is more general, capable of expressing not only Ptolemy’s thoughts, but also a heck of a lot more. In effect, modern mathematical physics uses epicycles even more than Ptolemy ever dreamed.
My point was that trying to express epicycles in the new terminology is not possible.
But it is! You simply specify the position as a function of time and you’ve done it! The reason why that seems so strange isn’t because modern physics has erased our ability to add circles together, it’s because we no longer have epicycles as a fundamental object in our model of the world.
So if you want the copernican revolution to be a paradigm shift, the idea needs to be extended a bit. I think the best way is to redefine paradigm shift as a change in the language that we describe the world in. If we used to model planets in terms of epicycles, and now we model them in terms of ellipses, that’s a change of language, even though ellipses can be expressed as sums of epicycles, and vice versa.
In fact, in every case of inexpressibility that we know of, it’s been because one of the ways of thinking about the world didn’t give correct predictions. We have yet to find two ways of thinking about the world that let you get different experimental results if you plan the experiment two different ways. In these cases, the paradigm shift included the falsification of a key claim.
Rather, I think it would be surprising for a discipline independent of empirical facts to have paradigm shifts
I don’t think it’s necessarily true (for example, you can imagine an abstract game having a revolution in how people thought about what it was doing), but it seems reasonable for math, depending on how you define “math.” I think people are just giving you a hard time because you’re trying to make this general definitional argument (generally not worth the effort) on pretty shaky ground.
Thanks, that’s quite clear. Should I reference abandonment of fundamental objects as the major feature of a paradigm shift?
In fact, in every case of inexpressibility that we know of, it’s been because one of the ways of thinking about the world didn’t give correct predictions.
Yes, every successful paradigm shift. Proponents of failed paradigm shifts are usually called cranks. :)
My position is that the repeated pattern of false fundamental objects suggest that we should give up on the idea of fundamental objects, and simply try to make more accurate predictions without asserting anything else about the “accuracy” of our models.
Predictions about the world are only possible to the extent the world controls the predictions, to the extent considerations you use to come up with the predictions correspond to the state of the world. So it’s not possible to make useful predictions based on considerations that don’t correspond to reality, or conversely if you manage to make useful predictions, there must be something in your considerations that corresponds to the world. See Searching for Bayes-Structure.
Isn’t “makes accurate predictions” synonymous with “corresponds to reality in some way” ? If there was absolutely no correspondence between your model and reality, you wouldn’t be able to judge how accurate your predictions were. In order to make such a judgement, you need to compare your predictions to the actual outcome. By doing so, you are establishing a correspondence between your model and reality.
There are perfectly fine ways to express those things. Epicycles might even be useful in some cases, since they can be used as a simple approximation of what’s going on.
The reason people don’t use epicycles any more isn’t because they’re unthinkable, in the really strong “science is totally culture-dependent” sense. It’s because using them was dependent on whether we thought they reflected the structure of the universe, and now we don’t. Ptolemy’s claim behind using epicycles was that circles were awesome, so it was likely that the universe ran on circles. This is a fact that could be tested by looking at the complexity of describing the universe with circles vs. ellipses.
So this paradigm shift stuff doesn’t look very unique to me. It just looks like the refutation of an idea that happened to be central to using a model. Then you might say that math can have no paradigm shifts because it constructs no models of the world. But this isn’t quite true—there are models of the mathematical world that mathematicians construct that occasionally get shaken up.
My point was that trying to express epicycles in the new terminology is not possible. That is, modern physicists say, “Epicycles don’t exist.”
Obviously, it is possible to use sociological terminology to describe epicycles. You yourself said that they were useful at times. But that’s not the language of physics.
Since you mentioned it, I would endorse “Science is substantially culturally dependent”, NOT “Science is totally culturally dependent.” So culturally dependent that there is not reason to expect correspondence between any model and reality. Better science makes better predictions, but it’s not clear what a “better” model would be if there’s no correspondence with reality.
I brought all this up not to advocate for the cultural dependence of science. Rather, I think it would be surprising for a discipline independent of empirical facts to have paradigm shifts. Thus, the absence of paradigm shifts is a reason to think that mathematics is independent of empirical facts.
If you don’t think science is substantially culturally dependent, then there’s no reason my argument should persuade you that mathematics is independent of empirical facts.
This is false in an amusing way: expressing motion in terms of epicycles is mathematically equivalent to decomposing functions into Fourier series—a central concept in both physics and mathematics since the nineteenth century.
To be perfectly fair, AFAIK Ptolemy thought in terms of a finite (and small) number of epicycles, not an infinite series.
And so for the curves in question, the Fourier expansion would have only a finite number of terms.
The point being that, in contrast to what was being asserted, Ptolemy’s concept is subsumed within the modern one; the modern language is more general, capable of expressing not only Ptolemy’s thoughts, but also a heck of a lot more. In effect, modern mathematical physics uses epicycles even more than Ptolemy ever dreamed.
That’s good point; I haven’t thought about that. Go epicycles ! Epicycles to the limit !
ducks and runs away
But it is! You simply specify the position as a function of time and you’ve done it! The reason why that seems so strange isn’t because modern physics has erased our ability to add circles together, it’s because we no longer have epicycles as a fundamental object in our model of the world.
So if you want the copernican revolution to be a paradigm shift, the idea needs to be extended a bit. I think the best way is to redefine paradigm shift as a change in the language that we describe the world in. If we used to model planets in terms of epicycles, and now we model them in terms of ellipses, that’s a change of language, even though ellipses can be expressed as sums of epicycles, and vice versa.
In fact, in every case of inexpressibility that we know of, it’s been because one of the ways of thinking about the world didn’t give correct predictions. We have yet to find two ways of thinking about the world that let you get different experimental results if you plan the experiment two different ways. In these cases, the paradigm shift included the falsification of a key claim.
I don’t think it’s necessarily true (for example, you can imagine an abstract game having a revolution in how people thought about what it was doing), but it seems reasonable for math, depending on how you define “math.” I think people are just giving you a hard time because you’re trying to make this general definitional argument (generally not worth the effort) on pretty shaky ground.
Thanks, that’s quite clear. Should I reference abandonment of fundamental objects as the major feature of a paradigm shift?
Yes, every successful paradigm shift. Proponents of failed paradigm shifts are usually called cranks. :)
My position is that the repeated pattern of false fundamental objects suggest that we should give up on the idea of fundamental objects, and simply try to make more accurate predictions without asserting anything else about the “accuracy” of our models.
How can you make accurate predictions while at the same time discarding the notion of accuracy ?
I have no reason to expect that our models correspond to reality in any meaningful way, but I still think that useful predictions are possible.
Predictions about the world are only possible to the extent the world controls the predictions, to the extent considerations you use to come up with the predictions correspond to the state of the world. So it’s not possible to make useful predictions based on considerations that don’t correspond to reality, or conversely if you manage to make useful predictions, there must be something in your considerations that corresponds to the world. See Searching for Bayes-Structure.
Isn’t “makes accurate predictions” synonymous with “corresponds to reality in some way” ? If there was absolutely no correspondence between your model and reality, you wouldn’t be able to judge how accurate your predictions were. In order to make such a judgement, you need to compare your predictions to the actual outcome. By doing so, you are establishing a correspondence between your model and reality.