Philosophy 167: Class 6 - Part 9 - A New Approach- Emphasizing Mathematical Tractability.

Smith, George E. (George Edwin), 1938-

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Synopsis: Discusses the Archimedes sparrow; spirals; and mathematical tractability.

Philosophy and science.
Galilei, Galileo, 1564-1642.
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I wanna start now with this very opening paragraph and a half on the section on naturally accelerated motion. I'm gonna read this, but I'm gonna be stopping and making comments about it, because I'm just assuming when you read it, you didn't see many things that are sitting in it.
So let's start, and first, it is appropriate to speak out and clarify the definition that best agrees with that which nature employs. Notice we're doing a definition here. Not that there is anything wrong with inventing at pleasure, some kind of motion and theorizing about it's consequent properties. In the way that some men have derived sparrow and conchoidal lines from certain motions, though nature makes no use of these.
Stop a moment. Who did that? Archimedes. The Archimedes sparrow is one of the fundamental developments in late post Euclidian geometry. Think of a ray moving at uniform speed in a circle. Think of a point starting at the bottom of that ray and moving along it at uniform speed.
What's the locus of points that object moving along the ray describes? Spiral. Different spirals by doing different ratios of speeds, okay. So he's openly referring to Archimedes by some people. I'm sure he did the conquede, too. I don't remember what a conquede is immediately, but you'll see it in Descartes, nowhere else.
Though nature makes no use of these, and by pretending these, men have laudably demonstrated their essentials from supposition. Okay, so that's perfectly okay to invent whatever math you want, whether it occurs in nature or not, because you can prove the properties. But since nature does employ a certain kind of acceleration for descending heavy bodies we decided to look into their properties.
The we here is the academician. This is all in Latin. Properties, so that we might be sure that the definition of accelerated motion, which we are about to deduce, agrees with the essence of naturally accelerated motion. And at length after continual agitation of the mind, we are confident that this has been found.
Chiefly for the very powerful reason that the essential successively demonstrated by us, correspond to and are seen to be in agreement with that which naturalia experimenta show forth to the census. I didn't translate that, because Drake translated as physical experiments, I'm not sure it's a legitimate translation, but so I left it in in the Latin.
And that doesn't mean it's not a legitimate translation, I will very often leave something in the Latin if it's near enough to the English, that it's not gonna confuse you. So notice the climb here. He's constructed, in effect, a definition from which he deduces a whole bunch of demonstrated essentials, following that definition.
And they correspond to and are seen to be in agreement with that which natural experiments show forth to the senses. Further it is as though we have been led by the hand to the investigated of naturally accelerated notion by consideration of the custom and procedure of nature herself in her other works.
In the performance of which she have habitually employs the first simplest and easiest means. And indeed, no one of judgment believes that swimming or flying can be accomplished in a simpler easier way than that which fish and birds employ by natural instinct. Okay, so two things are going on here.
Nature's simple, so this simple approach is the thing to use, develop it mathematically, compare it to nature, and see if its essentials agree. Those are two separate things. How do you balance the two? I don't have any direct answer, okay? Let me finish this. Thus when I consider that a stone falling from rest at some height successfully acquires new increments of speed, why should I not believe?
Rhetorical questions are always the clearest sign of not having a real argument, then we won't worry about that. Why should I not believe that those additions are made by the simplest and most evident rule. For if we look into this attentively, we can discover no simpler addition in increase than that which is added on always in the same way.
That is, whenever an equal times, equal additions of swiftness are added on. So equal increments of swiftness, speed, in equal times, that's the simple. Now the reason I did this whole thing is something slightly odd is gonna go on here. We're gonna develop a mathematically rigorous account of uniformly accelerated motion.
Not because we've observed retrograde loops or anything else in the sky that we have to account for, but because we're trying to get at a form of motion nature employs that gets masked by air resistance effects. And we're gonna do it in such a way that we can show that the mathematical theory agrees with experiments, whatever that means.
But also what's motivating is it's simple? Okay. And there's this play all the time through the history of mechanics again, and again, and again to such an extent that Clifford Truesdell, a name that you don't know now, but you will get to know. He's a very important figure in the historiography of mechanics, one could argue the most important figure in the 20th century in the historiography of mechanics.
He constantly insisted the way mechanics progressed is somebody sat down and did a nice, clean mathematical theory that agreed qualitatively or semi-quantitatively with experiment, and then went on from there to develop that mathematical theory and figure out how to apply it to the world most effectively. So the linear theory of elasticity is an example of this.
There are no linearly elastic bodies. You get linear elasticity by chopping off Taylor's theories after the first term, not by anything being perfectly linearly elastic. It goes place after place, that you develop this mathematical theory that seems to be motivated. The word simplicity is interesting here. It seems to be motivated by considerations of mathematical tractability, and then you look at nature to see how well it agrees.
What Truesdale actually says is mechanics is taking everyday observation, forming a mathematical theory consistent with it, then seeing how much it departs from the real world and seeing if those departures can be explained without endangering the theory. That's not what we've seen in astronomy. We saw an element of it working with circles where it's ambiguous whether they really believed in the circles or the circles were mathematically tractable.
And hereto there's a certain ambiguity about mathematical tractability versus the real world. The problem is we're not starting with any natural motions that can form the uniformly accerlated motion, because natural motions all occur in resisting media. Okay, so it's a distinctive feature. Whether this is true of Newton is a perfectly interesting question when we get there.
Whether the three laws of motion are one more example of the same thing. I have a dissertation at Stanford being written under me at the present moment on that very question. They should be construed in this same way. But I don't know that I've been persuaded by it yet.
At any rate, I just want you to see what you're gonna be reading in this 3rd day and much of the 4th day is something that's being driven in large part by mathematical tractability with a relation to the Empirical world as a secondary issue.