Synopsis: Provides a definition of the term 'working hypothesis'; stresses that evidence is a relation between data and claims that reach beyond them; reviews three working hypothesis of celestial mechanics.
Opening line: "All right, so let me start. I use the term working hypothesis very differently from its standard use."
All right, so let me start. I use the term working hypothesis very differently from its standard use. Well, very differently, much more narrowly. My pick, what I call a working hypothesis is something that enters constitutively into evidential reasoning, such as if you don't assume it you can't run the evidential reasoning at all.
The argument simply doesn't go through. And it's a point I'll make, I'll make it for the first time tonight. It's kind of a trivial point, data is one thing, evidence is another in the way it's normally put by Chomsky and I got to the point. We had a long discussion trying to find out who made this point first, and I now, I think he does the same, footnotes the two of us in conversation.
But, data is not a relation and evidence is, evidence is a relation between data and claims that reach beyond them. So data itself, since it's not a relation as such to any claim, you know it'll straight that tonight. Something more is always needed to, now the pet phrase, turn data into evidence.
That's the phrase we can't figure out which of us use first. I used it first in print, but which I would used it first is hard to say. And, that's, to me, when you don't have a rich theory to do that job, to turn data into evidence. You need to substitute for it, and the substitute for it is usually fairly minimal working hypotheses.
That you try to protect yourself against but you start using them in just the way Ptolemy did. And there are three I put up here. The Earth is motionless, in particular, its location does not vary with respect to the stars along the Zodiac over the course of the year.
And that's almost what motion, that's almost what being at risk means here, is relative to the fixed stars. We'll worry about that more carefully later. But, that's what they were thinking, relative to the fixed stars. Second, all motion along the Zodiac is centered around the Earth. And third, all real celestial motion is compounded out of uniform or, at least, equiangular circular motions.
Now, what's interesting about those three, and the reason I pull them up and single them out, is it's the second one that's so restrictive. It's almost the theme of tonight of what happens if you relax the second one. So they're two comments. Why is it so restrictive? Answer is if all the motions are centered more or less around the Earth.
We can't do any form of triangulation to get distance. But if objects are going around the sun, and we know heliocentric longitude and geocentric longitude, we can do triangles, and you'll see that tonight. It's a dramatic change that's going to show up tonight. So, being unable to do that Ptolemy had no way whatsoever to be able to give relative distances of the planets.
And that's a major change that occurred historically with Copernicus, though Tico followed with essentially the same thing right away. So briefly, what was the reasoning that Ptolemy had to support the second? You can go read the reasoning itself, it's on supplementary material last time. But, the reason is fairly straight forward.
The sun and moon clearly go more or less in circles around the earth. Because their apparent diameters change so little during the course of a year and the sun and during the course of a month with the moon. For those who are interested this week I think is as large as the moon ever gets in full moon.
Because its very near apogy and its going to be full this week. So it's spectacular, it also makes it terrible for telescope purposes, its too bright and you can't look at the moon when its full anyway, you don't see anything, but anyway you might notice the moon and I think the number is 12% greater, In this view, than it is at other times.
Just go back. Moon and sun clearly going around the earth in circles. Stars, clearly going around the earth in circles. What's in between the stars and the moon and the sun? Planets. Why in the world would they be doing anything different, but going around the earth in circles.
But that's almost the argument that is you don't have anything like a pair of diameters to employ. You don't have any other means of doing it so you're basically just surmising. It's got to be the same as all the others. You're running it off the sun and the moon.
And, as I say, it was terribly limiting to the astronomy. I'm gonna do two more slides and then pause for questions on Ptolemy. The next slide is from a book that I intended to pass around last time. This is a remarkable book. I have put a copy on reserve in the office so people can take it put.
You now know the name Jim Evans. But he constructed this book, in which students learn. And he looked down here, he's year 1990. So that tells you he's working. He has students do current astronomy Ptolemaically, from the ground up. And they get very good and very practiced. Come on in and have a seat.
So I'll pass that book around. I've had a remarkable number of students just go out and buy that book once they look at it. But I put, that's why I now have three copies, one just to put on reserve. But it is a great great teaching book. And this is a page from it.
What I've done is simply give you the form of Ptolemy's elements that are preferable for him, they're equivalent to the ones I gave last week but they're not the same. So I gave periods. Period to complete a circuit, et cetera. That's rather useless computationally if you wanna know what's gonna happen three days from now, you wanna how far it moves in the next three days, and what you would really like therefore is the mean motion per day that then can be adjusted for any variations.
So, the actual set of parameters that are put up here are much closer to those of Ptolemy, with one notable exception. If you look at number four, the eccentricity of the deferent, denoted by E, and he says, that's the ratio of OC over R or CE over R.
Well, those aren't the same thing folks. Well those two are the same, I'm sorry, because of bisecting eccentricity. I'm misleading you. Let me state the point correctly. Ptolemy thought he what he called eccentricity was OE, it's double what we call eccentricity. We've come to talk about eccentricity actually starting it indirectly with Copernicus, as you'll see later, but primarily from looking at it as an ellipse where the eccentricity of the ellipse is the distance of either focus from the center.
Ptolemy was very much thinking, what's the distance from the observer to the equiangular point? And that's why he calls it bisection of eccentricity, because it's half the eccentricity. This way of doing it is half the eccentricity from Ptolemy's point of view. And there's a point of confusion I hope I've eliminated for the rest of the course.
The term eccentricity is ambiguous in this course until next week, when we start with Kepler, then it becomes half of what Ptolemy called the eccentricity, okay. Otherwise, this gives you the sense of what's involved in getting the, what the elements are. What it doesn't give you and I make no effort in this course to do it, it doesn't give you any feel for what it is to learn Ptolemaic astronomy.
To learn it is to be able to compute within it. Number one. And gets sensitive enough to those computations that you can start telling how sensitive any result is to the values of the elements. Once you start computing you'll start realizing, if the eccentricity were a little different here, this would be very different.
That's what it is for people at the time to have learned Ptolemaic astronomy, particularly in the universities from 1200 to 1500, which can become reasonably adept at these calculations. Somebody like Kepler could do Ptolemaic calculations in his sleep. It's just trivial, he didn't have to pause at all.
He could just run them off. Of course what that led to was everybody who deviated from Ptolemy had learned how to do calculations there. So, when they had a new system, the new system was, in some way or another simply going to export the way in which you do calculations and Ptolemaic astronomy into a new system.
And Copernicus is very dramatic in that regard. Okay, third slide, and this is the slide that's philosophically important. There was real evidence for Ptolemaic astronomy. I'm not going to read this off to you, but I'm just going to cite the three points and make the really important ones.
But, of course, the usual way of thinking of it is the evidence came from his success in predicting salient phenomena, like stationary points, retrograde loops, maximum elongations, eclipses, et cetera. Which is what he was aiming to do, and he was, as you saw last week, spectacularly successful, spectacularly especially, versus anyone before him.
And it became the constraint for everybody who proposed any change to mathematical astronomy. They had to do at least as well as Ptolemy did with all the salient phenomenon. And it turned out that was much harder to do than you may think. In particular Copernicus spent 20 years doing it all by himself.
Then the second thing is that the theories of the seven bodies, that's a technical term. The model is generic, the theories are the model with specific values for the parameters. So the way one talks in historically in astronomy is you have a theory of the moon, a theory of Jupiter, etc.
And you can have, I'll say a little later tonight. Tico came up with two different theories of the moon. This sort of talk is standard. The theories of the seven bodies employ only five basic parameters. We have a model in common for four of the orbits but the same parameters for all of them, except the sun doesn't require all five, it only requires four.
And thereby what you're doing is reducing a very complex situation. A phrase I use up here is multiple apparent degrees of freedom down to a very few. Degrees of freedom in theories. And both of those are standard forms of evidence in philosophy in science. I'm intentionally avoiding the jargon of philosophy in science, but the first is predictive success, and the second is I guess called unification.
Is that fair, Pat? Standard unification. And they're big deals. They're very much stressed The third one is much less stressed. You can even start asking, who's been stressing it, but we'll worry about that later. Newton stresses it a good deal. The values of these parameters, this one I'm gonna read out, were determined by means of model-mediated measurements from observations.
That when repeated at different times kept yielding the same values to reasonably high precision, thereby providing evidence that the parameters are constants of nature. The key thing here isn't that they're constants. The key thing it it's well-behaved. And think of it this way. The model's saying certain parameters, should be well-behaved over time.
That's a prediction. They should be reasonably stable over time. If they're not stable, they should have a definite cyclic pattern to them that you can calculate, etc. One piece of evidence that the models are out and out wrong is when you try that and, in fact, the values of the parameters aren't stable.
So, you know, half a century later, you re-calculate all the values of the elements from observation and they're substantially different. And you start asking, what's going on here? Well, one possibility, Is the world is changing and another very serious possibility is the bloody model is no good. Okay?
So, there's no question that you make a prediction when you start doing what's now called theory mediated measurement of parameters, that those measurements should yield well behaved values and part of being well behaved values is that they're properly stable, over time, and over measurement. That is if you do multiple observations using the same inference, you should be coming back to the same thing consistently.
And the only point I'm making now, that's a prediction on any model in which you're gonna use theory mediated measurements to set as parameters. And you can, that prediction can fail. So when it doesn't fail you've got evidence some sort. Now summary comment. And I'll make a key point.
A key point. A combination of these especially the stability over time of the model mediated measurements of the parameters gave evidence that there was something fundamentally correct in Ptolemy theory. Not withstanding the distance of alternative models by virtue of Apollonius' Theorem that achieve the same as above. Now, those alternative models involve the same parameters.
They're simply reconstruing them geometrically. That's point one. More fundamental point, the third item there says they're model-mediated, but when you look at them carefully they're not necessarily presupposing every feature of the model. They're presupposing special features of the model and it's not a immediately evident what those features are.
For example, Copernicus had to discover the radius ratio, the radius of the epicycle to the deferent, lo and behold, is the radius of the orbit around the sun to the earth's orbit around the sun. Okay, but to get that, you had to see exactly what the measurement itself was presupposing, not the whole model.
So, when you look at any kind of theory-mediated measurement, you have to be very careful. At first, it looks like it's evidence for everything. But then when you look at it very carefully, especially in light of your knowledge, in this case known to every astronomer, that multiple models could fit the same thing.
As soon as you do that, then you start asking, okay, what's being presupposed as common to all those models? That's something that the stability of the measurement with the first two in place that's something that we're getting evidence there's an aspect to it that's fundamentally right, okay? And I think that's a feature that was very dominant to the success of Ptolemaic astronomy over the years.