This post is a digression from the topic of the previous parts (1 2 3 4). We’ll move away from discussing how high a tennis ball should bounce when dropped on top a basketball, and into some metadiscussion of the arguments made in the first four parts. It’s a long post as well, but it’ll be good for you, because half the words are Galileo’s, not mine, and he’s a dude worth reading.
Last time, I cited Galileo as our source for understanding uniformly accelerated motion – the motion of a ball dropped or thrown in the air.
Before introducing his idea of what uniformly accelerated motion is, Galileo gives us an extended prelude. It’s long, but I think it’s worth seeing all at once, rather than piece-by-piece.
For anyone may invent an arbitrary type of motion and discuss its properties; thus, for instance, some have imagined helices and conchoids as described by certain motions which are not met with in nature, and have very commendably established the properties which these curves possess in virtue of their definitions; but we have decided to consider the phenomena of bodies falling with an acceleration such as actually occurs in nature and to make this definition of accelerated motion exhibit the essential features of observed accelerated motions. And this, at last, after repeated efforts we trust we have succeeded in doing. In this belief we are confirmed mainly by the consideration that experimental results are seen to agree with and exactly correspond with those properties which have been, one after another, demonstrated by us. Finally, in the investigation of naturally accelerated motion we were led, by hand as it were, in following the habit and custom of nature herself, in all her various other processes, to employ only those means which are most common, simple and easy.
For I think no one believes that swimming or flying can be accomplished in a manner simpler or easier than that instinctively employed by fishes and birds.
When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody?
Galileo is mixing two approaches, and they appear to be intrinsically intertwined in his mind. The first is the ultra-skeptical pure empiricism viewpoint. This line of thought says that the only way to know about a thing is to confirm it by experiment. All scientific theories are to be tested against nature. If the theory and experiment agree, we fail to reject the theory. If the theory and experiment disagree, we reject the theory. Many modern scientists cite this as the true scientific viewpoint. (Note that from this point of view, you never confirm a scientific theory. Many scientists will agree with this – you never prove anything to be true in science. Also, I have called this viewpoint “empiricism”, a term which is sometimes used slightly differently in epistemology, where it refers to the belief that knowledge comes from sensory experience in general, rather than scientific experimentation in particular. Nonetheless, the cores of scientific and epistemological empiricism are similar.)
But, along with his statement that his knowledge of falling bodies comes from experiment, Galileo also has curious references to simplicity, in particular some out-of-place stuff about swimming fish and flying birds. This, to me, is the germ of a new idea – an idea that what we learn about nature ought to make sense to us on a deep level, once we’ve learned it. Greek philosophers (so I hear, not having read them) believed the Universe ought to make sense, and that they could therefore understand it with a priori reasoning. This is not quite what Galileo seems to believe. He holds himself responsible to experiment, unlike Aristotle, but I think that if experiment gave strange or unusual results that Galileo couldn’t understand, he’d be extremely dissatisfied. He feels a deep need to take the mathematical results, back them up with data, but then do even more. He needs them to make sense.
Two New Sciences is written as a dialogue (or, there being three interlocutors, a trialogue?), with Sagredo and Simplicio, two men who haven’t learned the new sciences, questioning Salviati, who has learned them and is explaining them to his friends. Galileo uses this device to explore intuition. He has Sagredo and Simplicio raise all manner of interesting objections to Salviati’s ideas, just so Salviati can find interesting answers to allay their unease. (This format is out of style in modern physics text, with rare exceptions like Spacetime Physics, a book I enjoy much more today than I did when first learning special relativity from it six years ago.)
For example, Sagredo thinks there is a problem with saying that a body dropped from rest has a speed proportional to the time fallen. He objects,
…we must infer that, as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.
He thinks there is a disconnect between the math and experiment, because the math says that when you drop something, it has almost no speed after falling a short distance, but Sagredo thinks that when you drop a heavy thing it starts falling quickly immediately. Maybe you don’t have this difficulty of intuition, but if you do, Salviati replies by appealing to an experiment.
You say the experiment appears to show that immediately after a heavy body starts from rest it acquires a very considerable speed: and I say that the same experiment makes clear the fact that the initial motions of a falling body, no matter how heavy, are very slow and gentle. Place a heavy body upon a yielding material, and leave it there without any pressure except that owing to its own weight; it is clear that if one lifts this body a cubit or two and allows it to fall upon the same material, it will, with this impulse, exert a new and greater pressure than that caused by its mere weight; and this effect is brought about by the [weight of the] falling body together with the velocity acquired during the fall, an effect which will be greater and greater according to the height of the fall, that is according as the velocity of the falling body becomes greater. From the quality and intensity of the blow we are thus enabled to accurately estimate the speed of a falling body. But tell me, gentlemen, is it not true that if a block be allowed to fall upon a stake from a height of four cubits and drives it into the earth, say, four finger-breadths, that coming from a height of two cubits it will drive the stake a much less distance, and from the height of one cubit a still less distance; and finally if the block be lifted only one finger-breadth how much more will it accomplish than if merely laid on top of the stake without percussion? Certainly very little. If it be lifted only the thickness of a leaf, the effect will be altogether imperceptible. And since the effect of the blow depends upon the velocity of this striking body, can any one doubt the motion is very slow and the speed more than small whenever the effect [of the blow] is imperceptible? See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary. (brackets added by translator)
I get the feeling, while reading this passage, that Galileo cites this experiment simply because it gives him pleasure to do so. But in this case, even the experiment is not enough for him. He continues
But without depending upon the above experiment, which is doubtless very conclusive, it seems to me that it ought not to be difficult to establish such a fact by reasoning alone. Imagine a heavy stone held in the air at rest; the support is removed and the stone set free; then since it is heavier than the air it begins to fall, and not with uniform motion but slowly at the beginning and with a continuously accelerated motion. Now since velocity can be increased and diminished without limit, what reason is there to believe that such a moving body starting with infinite slowness, that is, from rest, immediately acquires a speed of ten degrees rather than one of four, or of two, or of one, or of a half, or of a hundredth; or, indeed, of any of the infinite number of small values [of speed]?
Here we see the second approach to nature. The idea that, once we’ve formulated a theory and tested it, we’re still not done. We need to reason about it, too. We need to go back, take the solution, and make it ours. We need to convince our grandmothers, who don’t know math, that this is the way it ought to be. And both these processes are intertwined. You can use the idea that nature ought to be simple to figure out what the laws are, but if you do, you’re still subject to testing them by experiment. Conversely, you can use experiment to figure out the laws, but if you do, you’re still subject to figuring out why things came out that way.
Galileo is the earliest source I’ve seen with this new, sophisticated attitude. Naturalists wanted to observe, discover, and document what happened around us. Philosophers wanted to talk about it in the abstract and explain its deeper logic. But Galileo wanted to do both. And it’s only when you do both that you’ve accomplished the real goal – understanding.
I’m not saying this attitude sprung up in Galileo’s work with no precedent, but I do think it’s clearly evident here, and since Two New Sciences is a landmark work in terms of the physical ideas it presents, it’s important to examine in terms of the philosophical ones is presents, too.
This Galilean principle still guides us today. Science isn’t about testing hypotheses and controlling experiments and statistical significance. Science is about figuring things out. The methods of modern science evolved over time as the problems scientists dealt with demanded them. (A great deal of statistics was invented specifically to study genetic inheritance, for example). Galileo didn’t have our textbook scientific method, but ultimately he didn’t need it to make great progress.
Today we need things like careful laboratory conditions and error propagation formulas to keep us from screwing up when things get tricky and hard to interpret. But the core of my world outlook, which I am not afraid to claim is also the core of the scientific one, is that you are just trying to figure things out, subject to checking what really happens, and then, once you do that, trying to understand.
Next time, I’ll take a look at one of Galileo’s arguments that didn’t work. That’s the other thing about science that I like. Nobody’s perfect, and you’re expected to screw up at least once in a while.