According to legend, Galileo is said to have dropped two spheres of different masses from the Leaning Tower of Pisa to demonstrate that their time of descent was the same. In fact, for Galileo, this was (at least initially) a thought experiment. The argument is simple -- imagine you have two balls of different mass. According to Aristotle (and most people's intuition), the heavier ball will fall faster. That will cause the string to pull taut as the lighter object slows the fall of the heavier object. But the system as a whole is heavier than the heavy object alone, and therefore should fall faster. It's a contradiction, and they must fall at the same speed.
Many of us repeated this experiment either by directly dropping weights or by rolling balls down ramps. If thought alone tells us the truth, why do the experiment? Because we tend to dismiss logical arguments on the basis of our preconceptions and ignore arguments and evidence that challenge them. We see things that confirm our prior beliefs and completely miss things that disconfirm them.
This is known as confirmation bias, and none of us are immune. Unchecked, this is even a aspect of a variety of neurotic or psychotic states. If one believes strongly that one is a target of other people's ill will or aggression, one is likely to be able to fit many otherwise unaccounted-for incidents into this view. This confirmation bias helps explain why people of opposing political views will look at the same events and draw very dissimilar conclusions.
What we share as a common frame of reference is what allows us to communicate with one another, and thus to form critical bonds such as families, communities, and nations. That common frame of reference is the observable physical world. So we have an almost immeasurable interest in preserving that common understanding of objects and events in the real world. Confirmation bias attacks that very fabric.
To lessen the force of cognitive bias, we can build some compensatory habits. When we see something that we automatically agree with or disagree with, we can note that some sort of confirmation is happening. We can ask "What if I thought the opposite of those ideas?" We can ask ourselves what actual evidence we have for our beliefs. When we find ourselves disagreeing, we can ask "What evidence might others have for their beliefs?"
If you disagree with the idea of wearing a mask to prevent spread of covid-19, can you open your mind enough to understand why people urge its use? The answers are readily available, and while you may not be inclined to agree, snarky posts ignoring scientific studies or the real pain of people who have lost loved ones is neither compassionate or open-minded.
At the same time, if you believe we must stay in lockdown until the risk of transmission is effectively zero, then you may be missing or refusing to acknowledge the real suffering that economic hardship causes, the many sources of secondary death, and the fragility of our food chain for many millions of people across the globe.
The truth probably includes both the idea that we need to get the economy rolling as quickly as we can, and that the fastest way to rebuild the economy includes a lot of civically-motivated people wearing masks in public. But we will take much longer to rebuild the economy, and many people will needlessly die, if we cannot move past our confirmation bias and see the reality we share.
17 comments:
I think the value in doing experiments even when we “know” what the outcome will be lies less in avoiding confirmation bias, and more in avoiding a false dichotomy. Experiments can reveal that nature has more options than we imagined. The history of science is basically a story of our learning just how limited our imaginations have been.
The Galileo thought experiment obviously doesn’t prove that light objects fall at the same rate as heavy ones (as anyone who’s dropped both a tissue and piece of wood knows). At best it shows that the rule can’t be *simply* that weight determines rate of fall. But then our everyday experience also shows us that the rule also can’t be *simply* that all objects fall at the same rate.
Aristotelian physics can be seen as a reasonable effective theory for the fluid kinematics of bodies subjected to a constant force, which is the “natural” world we find ourselves in.
The real story of the Galilean revolution is one of idealization, and figuring out how to make sense of the non-simple factors that play a role in (e.g.,) falling bodies.
Sure. Everything you say is true. I simplified a lot.
Mostly, I wanted to illustrate what confirmation bias might feel like without reaching for a political analogy.
I did have a question for you, though...
Does Galileo's thought experiment prove that inertial mass and gravitational mass must be equivalent? Or it it the Tower of Pisa experiment that makes the proof given his thought experiment. My instinct says the latter, but I can't quite seem to construct the argument that says it could be any other way. But still, I can't quite make myself believe that the violations of the equivalence principle are a logical impossibility.
(part 1)
***Does Galileo's thought experiment prove that inertial mass and gravitational mass must be equivalent?***
Well, some easy points before tackling your more specific (headache-inducing) question:
There’s no logical requirement for there to a property of mass at all. In fact, it’s probably safe to say that no one before Newton had our current concept of mass.
It’s presumably logically possible that dirt would fall faster than water, and that air and fire would go up and not down.
Similarly, Galileo supposes that the lighter (slower) object would have to slow down the heavier (faster) object if they were tied together. But surely that is not a logical truth. We can easily conceive of both objects falling faster once they are joined by a cord. We just find the situation unintuitive. Unlikely to be true.
Or, an Aristotelian might reply to Galileo by saying that the important question for the rate of fall is the substantial form of the object in question. It may be that simply tying two objects together isn’t enough to produce some third new object (in which case the lighter object acts as a parachute), but if you did form a new object by imposing a new substantial form, then the heaviness of that new object would determine its rate of fall.
Galileo assumes that “heaviness” is simply additive as we take mass to be. But that too isn’t a logical truth. Perhaps mixing together the lightness of air and fire with the heaviness of water and earth is much more complicated process. Who’s to say how heavy the new object would turn out to be?
And we know this is possible, because in a way, it’s actual. Measure the heaviness of a helium tank and a large empty balloon. Put them on a scale. Now, inflate the balloon and tie it to the tank. How heavy is the new combination? Put them on a scale (if they haven’t floated away already).
So Galileo is moving towards a notion of mass that is conserved and is simply additive, which, of course, turned out to be an extremely useful notion. (Though, it’s worth noting, there’s no such thing in the actual world. In relativity theory mass isn’t simply additive; for example, binding energy adds to the mass of an object, and mass can be created and destroyed.)
(part 2)
But to move to your more difficult question: Suppose we do have this property of mass that is simply additive under rigid connections, so that any time we join two bodies, we can consider the union of the two to be a third body with a mass that’s the sum of its parts (or, I suppose, any well-defined mass greater than either of the original bodies’ masses will do for this thought experiment).
And we’ll also suppose that we have the Newtonian notions of force and inertial mass (which, of course, Galileo did not).
As a preliminary, I think it’s obvious that in a Newtonian context there’s no reason that active gravitational mass (the mass that produces the gravitational field) needs to be equal to passive gravitational mass (the mass that tells us how much force gravity produces on an object). This actually leads me to wonder whether relativity or quantum considerations force this connection. I’m not sure.
But your question is whether we can conclude that passive gravitational mass has to be equivalent to inertial mass given Galileo’s thought experiment.
We’re assuming that both are well-behaved (conserved and additive) in the same way, presumably; or else the thought experiment really doesn’t get off the ground. And we’re assuming that there’s a single gravitational force that all passive masses feel the same way. And we’re assuming that there are no other forces at work
In that case, I think it does follow that inertial mass is equivalent to (or at least proportional to) passive gravitational mass. Think about it this way: suppose inertial mass were half of gravitational mass. Then things would fall more quickly. But that would be the same as the gravitational field being stronger. We would just fold it into the Newtonian gravitational constant and say that the two masses are equal.
So it seems to me that all the real work is being done by our assumptions about the principles of composition (inertial mass and gravitational mass get put together the same way) and assumptions about there only being a single force at work – and there being no other properties that are relevant.
I should probably mention that there’s been a substantial discussion of thought experiments (which my better half has contributed to) in the philosophical literature, and some of those discussants would claim that there are important things I’m leaving out. But I either don’t know what those things are, or I apparently don’t think they’re worth mentioning.
(end)
Hmmm...
Thanks for all that. I'll sleep much easier tonight. Assuming I don't float away. ;-)
Thinking about it a bit more, I’m not sure the Galileo thought experiment shows the equivalence of inertial mass and gravitational mass even given the above stipulations.
Suppose gravitational mass were like electric charge. More charge means more force, more acceleration. But more inertia means less acceleration.
Two bodies, equal mass, one with more charge than the other.
The more strongly charged one accelerates more (“falls faster”). Tying them together will slow the acceleration, because now there’s more mass to accelerate. The speed of the joined body (at any time) will be in between those the two would have had if they had been separate.
What about Galileo’s argument that the joined body should go faster because now there’s more charge (weight)?
Well, it neglects the fact that the speed doesn’t depend only on charge (gravitational mass), but also on inertial mass (which we’re now supposing can be different).
So the thought experiment only works against a very simplistic notion of how speed is related to “heaviness” and how weights combine.
Or, at least, that’s what I’m thinking now.
So the logical argument is a reaction to the Aristotelian assertion that lighter objects fall slower, so I'll frame the argument that way. But I think the sign does not really matter.
The thought experiment is to take 2 weights with minimal air resistance and drop them. I think we can assume, though it was not well formulated, a society that knew about sails and had walked into a stiff sea breeze, knew the wind resistance was a thing and was distinct from what they were trying to measure, even if they could mot conceive of a perfect vacuum. Or at least that we do.
In the experiment you tie those two masses together. Assume the larger mass falls faster. The string pulls taught and now the smaller mass slows the larger mass, and the both move more slowly. But no -- at the point where the rope becomes taut the become one mass falling and larger masses move faster. So the joined mass must fall faster because it is still more massive.
That is the logical impossibility. You know all that, I'm really just defining terms. The point is that it is logical -- unifying the mass cannot make it fall both faster and slower.
According to http://csep10.phys.utk.edu/ojta_samples/course1/synthesis/galileo/inertia_tl.html, Perhaps Galileo's greatest contribution to physics was his formulation of the concept of inertia. So it seems like he knew what he was proposing -- that a mass is accelerated by gravity in exact proportion to how it is retarded by inertial mass (in our time, we say the constant of proportionality is G). In other words, it is the equivalence principle. And, even though it was motivated by his experiments with inclined planes, once you have the two concepts of inertia and acceleration due to gravity, the equivalence principle seems to be a strictly logical outcome.
Your counter is not clear to me -- you ask what if "gravitational mass were like electric charge..." I think you are asking "what if we could add gravitational mass to an object without changing its inertial mass" If so, then that is logically precluded by the thought experiment. It fails in the same way. Take two identical objects and sprinkle magic gravity dust on one so it fall faster. You end up with the same logical paradox I think.
Galileo's argument does not neglect the fact that acceleration depends on both gravitational mass and inertial mass - it proves that it it must (i.e., the equivalence principle).
I think it would even holds if acceleration due to gravity did not vary with the square of mass -- let's say it went with mass cubed. If there is a such a thing as acceleration due gravity, there has to be some resistance or things would go to infinite speed instantly under infinitesimal influences. If there is a resistance, that is inertia. Unless inertial mass is proportional to gravitational mass, you have Galileo's paradox.
I think it is robust even if weights do not combine additively, as long as they combine monotonically. It does fail if there is some magical agent that can determine if the weights are unified by a tensile connection versus a rigid connection. But absent that particular form of pixie dust, it seems to be remarkably powerful logical construction.
Maybe it's just me, but there seems to be a little foreshadowing of "spooky action at a distance" in the idea that the weights could somehow know if they were tethered by a rigid or tensile connection.
(part 1)
***unifying the mass cannot make it fall both faster and slower.***
Very true. However, the arguments for the claims (1) “the smaller mass slows the larger mass,” and (2) “at the point where the rope becomes taut they become one mass falling,” both depend on premises that an Aristotelian needn’t accept.
If a taut rope between falling objects isn’t enough to make them a single object that is heavier than the individual ones, then there’s no contradiction. Alternatively, if objects falling slowly never impede the motion of heavier falling objects (and, to the contrary, can add to that falling motion), then there’s no contradiction.
The question of whether Galileo had the concept of inertia will depend on what we mean by inertia. (As Kuhn pointed out, it’s generally a mistake to trust science textbooks to be accurate when it comes to history; their goal is get you to latch onto our current concepts, which often were not exactly what the “great men of science” had in mind.)
Galileo did not have the notion of a gravitational force. That only came with Newton. Galileo was still thinking in terms of falling being a “natural motion” towards the Earth. Galileo’s example of inertial motion was a ball rolling on a table that wraps all the way around the Earth. It would just keep going and going.
But notice that this motion is circular, not rectilinear. Again, not until Newton do we get the idea that things travel in a straight line, and that gravity is a force that pulls them away from that natural straight motion.
Since Galileo did not have the Newtonian concept of force, and didn’t know F=ma, I’d say his notion of “inertial mass” was fuzzy at best.
It’s been a long time since I’ve looked at Galileo, and I’ve never been a scholar of his work, but I believe that he was mostly thinking of the constant acceleration of falling bodies being a form of natural motion. I don’t think he had the idea that heavy bodies had a large inertial mass that counterbalanced the increased gravitational force they felt due to their gravitational mass.
Galileo’s picture was that things could be supported, in which case they would keep rolling around the Earth unless disturbed, or they could be allowed to fall, in which case they would accelerate downward. And, of course, you could combine these two natural motions as you do with pendulums and inclined planes.
So (again, with the caveat that while I know the history better than most physics teachers, I don’t know it better than most historians of science), I don’t think he had enough of an understanding of inertia to be able to consider the equivalence principle.
(part 2)
But, leaving history and moving to the force of the thought experiment itself: I took it that the question we were asking was whether it was a logical possibility that the gravitational mass and inertial mass of a body might differ.
Of course, if one were to stipulate, “There’s only one property – mass – that both determines how a body accelerates under a force and determines how much gravitational force that body feels,” then we’ve stipulated that the equivalence holds.
So presumably we’re supposed to consider a case where the gravitational mass of a body can be different than its inertial mass.
So lets say we have two bodies both with a gravitational mass of 10g. Put them on a balance beam and it’s level. But, we suppose that they have a different inertial mass. We subject them both to the same force, and one accelerates at twice the rate as the other.
Now, is there a contradiction lurking here?
I gather that you’re saying that the Galileo thought experiment reveals a contradiction: separated, one would fall more slowly (and so would retard the other’s motion if they were tied together), but if we consider the joined pair as a single body then that larger body would have more gravitational mass and would have to fall faster.
But once we move beyond the simplistic notions“heavier/lighter” and “faster/slower,” and pay more attention to acceleration and inertia, I think we can see that there’s need be no contradiction.
Let’s take our two bodies with identical gravitational masses and different inertial masses. The one with less inertial mass accelerates faster when dropped, because it’s feeling the same force, and F=ma.
What happens when we tie them together? Well, the slower one (with more inertial mass) will feel a force from the rope that will accelerate it even more than if it were falling freely. Likewise, the faster one (with less inertial mass) will feel a force from the rope that decelerates it. So the two will accelerate together downward at an intermediate rate.
What about if we consider them as a single body? Well, now we add up their gravitational mass to see how much force is acting on the body. It will be double the force each individual body felt. Then we ask about the rate of acceleration of that body. Now we add up their inertial masses, and it will be less than the gravitational mass. (let’s say it’s 15 inertial grams, because the second body has only 5 inertial grams of inertial mass, even though it has 10 grav grams of gravitational mass).
When we calculate the acceleration using F=ma, we find that acceleration of the larger composite body is between that of the faster body falling freely and that of the slower body falling freely. And it’s exactly what we got when we calculated the acceleration of the two bodies exerting a force on each other through the rope.
So, no contradiction.
(part 3)
And I think this is fairly obvious when you think about the fact that mass is a charge for the gravitational force rather like electric charge is for the electromagnetic force.
So the situation that we’re imagining with gravitational mass being different than inertial mass is rather like what happens in the actual world when objects with different magnetic charges are both attracted in the same magnetic field.
Do away with gravity (by moving to space, or putting things on an air table) and take two magnets of equal strength. Now double the mass of one of the magnets by putting it on a larger block of wood. Turn on an electromagnet at the end of the table to attract both objects.
The lighter one will accelerate faster, because both feel the same force, but the one has less inertial mass.
Now, what happens when we tie them together? Exactly as in our gravitational case, they will accelerate at an intermediate rate. And we can analyze this either in terms of the forces in the string, or by treating them as a single body with a total magnetic charge and inertial mass given as a simple sum of the component charges and masses. (Assuming we’ve set things up so the precise locations of the charges don’t matter.)
In this set up, the magnets work as the “magic gravity dust.” We can add or subtract magnets as we please, and we can increase or decrease the inertial mass with the size of our wood blocks.
This makes me think there can’t be a contradiction lurking someplace subtle in the case where gravitational masses don’t match inertial masses. But perhaps I’m missing something.
***there seems to be a little foreshadowing of "spooky action at a distance" in the idea that the weights could somehow know if they were tethered***
There is indeed. In fact, Galileo was a big proponent of the "mechanical philosophy" that hoped to do away with the sort of "occult forces" that the science of his day accepted.
It's interesting to see how the dreams of the atomists and mechanical philosophers were both gloriously vindicated by modern physics, and at the same time completely shattered by the very same advances.
There's an unfinished book chapter that discusses this that I could send you, if you wanted to waste even more time on the topic.
Well, I was actually planning on wasting more time with Nozick this afternoon. But there's always tomorrow...what could be more fun than an unfinished book chapter? That would be wonderful!
I should be clear, I'm relatively less interested in what Galileo thought...I totally agree that concepts we take as certain were fuzzier for him. I have not been especially clear about that, but I'm more interested in what a modern read of the thought experiment reveals.
Your math for a composite body is rather elementary. But I think you are letting it cloud your vision. Again, I'm going to ignore sign and assume the Aristotelian position. If you make the mass larger by adding to it, it must fall faster. But if you make the mass larger by attaching a smaller mass to it the smaller mass must retard the first mass and that same mass falls more slowly. Both cannot be true. But surely "making the mass larger" and "making the mass larger by attaching a smaller mass to it" are not physically distinct ideas.
Your air table and magnet approach is more compelling to me though. I think that has pushed me over the edge to accepting that you need the physical experiment. But since I don't have any magic gravity dust, I'm at a loss for how you would even do the experiment.
I don't read Italian, so I must rely on second hand accounts of Galileo's experiments. I read that he rolled a ball down an inclined plane and then back up another and determined the final height was the same as the original height. Inertia is one way to describe that result. Conservation of energy is another. So roughly speaking, he defined matter as those things that obey conservation of energy (yes, ignoring matter-energy transformations such as in quantum mechanics).
So, 1) if we found something that did not have that property, would we even call it matter? If not, is equivalence of gravitational mass and inertial mass just a tautology?
And 2) fast forward a few years and observe that light (energy) is affected by gravity. It never occurred to me that this is a matter-energy duality that is seen independent of quantum mechanics wave-particle duality.
OK. Clearly the heat is getting to me.
*** the smaller mass must retard the first mass ***
Why would an Aristotelian (with no notion of inertia) accept this claim? Why wouldn't they just assume that the first mass would continue in its natural motion downward, and the lighter body would then have a forced downward motion in addition to its natural downward motion, and so fall faster?
(Just so we're not crossing wires, let me mention that I'm becoming convinced that the Aristotelian would likely deny the first horn of the dilemma and say that lighter falling bodies never slow down heavier falling bodies. However, when we're talking about the possibility of inertial mass being unequal to gravitational mass we're rejecting Aristotelian physics and instead considering modified Newtonian physics. In that case, it seems that the anti-equivalence-principle advocate will reject the second horn of the dilemma and say that lighter bodies always slow heavy bodies, even when they join to become an even heavier body.)
It was obvious to Galileo and many others that probably *something* was conserved in the motions of pendulums, colliding bodies, and so on. However, they were struggling to figure out what that something was. Descartes thought it was speed. Later there was a big debate over the question of whether the conserved quantity was vis viva (living force, kinetic energy) or vis mortu (dead force, momentum).
I would have to look to see how Galileo thought of it, but he wasn't very close to our notion of energy conservation.
Pretty much everyone at this time (and I think Galileo too) was pretty comfortable with thinking that there might be very different kinds of matter that would naturally behave in very different ways. Obviously the celestial motions were quite different than anything we see down here at the bottom of the universe.
In the 19th century the energeticists claimed that energy is the fundamental stuff of the universe that everything is made up of. So at that point I think there probably were people pushing for the position you suggest. But probably not in the 17th century, I suspect.
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