Is Galileo's argument about falling bodies logically flawed?

Question

Galileo's famous argument against the Aristotle's theory of falling bodies goes like this (taken from THEP forum, now defunct):"Let's say heavy objects do fall faster than light ones. Then it seems the heavier weight will fall with the lighter weight acting, as it were, a bit like a parachute. In that case, the two balls will together fall more slowly than the heavy weight would on its own. On the other hand, once the two weights are tied together and held out over the parapet, they have effectively combined their weights, becoming one greater weight... they must therefore fall even faster than the heavy weight would on its own." Contradiction. Hence weight has no effect on falling rates.

Some philosophers are very fond of this argument. Gendler uses it as a prototypical example of how "reasoning about particular entities within the context of an imaginary scenario can lead to rationally justified conclusions". Snooks goes further saying "it is striking that one leaves the falling balls example with something approaching certainty for its outcome". And Brown goes all the way and claims that Aristotle's theory is "self-contradictory", and we gain a priori knowledge here. The argument does give off that flavor of "synthetic a priori" reasoning, as in geometry but without images. But is it a proof or a fallacy? Even Gendler admits that some "obvious" premises are missing, and Atkinson even calls it a "non-sequitur" for similar reasons. But Galileo's logic is not questioned it seems. Shouldn't it be?

Let's replace weight with cross-section. If objects with smaller cross section do fall faster let's strap two together (next to each other, so that the cross-sections add up) and argue as above. Ergo, cross section can not affect the falling rate either. But objects with smaller cross section do fall faster, because of air resistance, and two objects strapped together with added cross sections will fall slower than each one separately, for the same reason. Something is wrong here, but the reasoning leading to a contradiction is essentially identical to the one above.

I don't think presence/absence of air matters, not that Galileo mentions anything about vacuum. What is most suspicious about his argument is exactly that it is so general. If it works the falling rate should not depend on any (additive) characteristic of objects whatsoever, regardless of the missing extra conditions. We could give those objects electric charges and turn on electric field of our liking, and the argument still seems to go through. But leads to a wrong conclusion that the falling rate is independent of the charges.

Question: It is assumed that Galileo's argument should work with something like "free fall in a vacuum" premise properly spelled out. But I don't see where such premise could be used, or how non-vacuum would alter the conclusion, or why Aristotle's theory is self-contradictory rather than just empirically wrong. Is the problem just in unspelled premises or is the reasoning itself logically flawed? Is there a logically correct "a priori" argument?

EDIT: I believe Quentin's answer gives the right reconstruction of what the a priori aspect of Galileo's argument amounts to. I'll rephrase it somewhat. Suppose bodies move under a single "motive cause" that determines the "fastness" of motion (this formalizes fall in a vacuum). The "cause" could be Newtonian force that determines acceleration, or something else that determines velocity as Aristotle thought, etc., as long as "the stronger the cause the faster the motion" holds. Suppose further that the cause is additive, i.e. its values add up when the bodies are strapped together. Then Galileo's argument shows that the proportionality constant ("weight" in the argument) between the cause and the rate can not also be additive (a.k.a. extensive).

The weakest link is the "additivity of cause" premise. It is true of Newtonian forces, but that part is empirical, and not in an intuitive way, unlike geometry. Quentin points out that two narrowly separated bodies falling differently than two touching ones leads to a counterintuitive discontinuity. That is true, but "touching" is not the same as "strapped together". The strapping introduces rigidity that turns two bodies into a single item, it is not a priori clear that the cause should move this item just as it would two bodies that are only touching. Aristotle would probably reject this premise, especially since to him falling is a "natural" motion rather than "forced". So his theory is not exactly self-contradictory or counterintuitive, but Galileo's argument is still valid under broader assumptions than just Newtonian physics.

EDIT 2: Found this paper that analyzes Galileo's argument in detail, and reaches the same conclusion.

Answer 1

Galileo's argument shows that the magnitude which determines the speed of free fall must be intensive, not extensive.

In the case of charges, the relevant magnitude is charge per mass unit. Same in the case of cross sections: it is cross-section per mass unit. These magnitudes are intensive. When attaching two objects, you double the resistance of the air, but you also double inertial mass, so the two objects will not fall slowlier or faster than each separately (contrarily to what you are saying). Galileo's argument remains valid in this case. Idem in the case of charges attracted by a magnet: two charges won't be attracted faster when tied together, because inertial mass adds up as well as charge.

The premiss on which the argument rests is that a continuous change in the arrangement of a situation cannot produce a discontinuous change in the dynamics of these arrangement (change an arrangement a little and the dynamics should change a little). If an extensive quantity is involved in movement then the principle fails: there'd be a discontinuity between the dynamics of two spheres with an infinitely small gap and two connected spheres, which seems absurd.

In conclusion: you might quibble on the formulation, but given this premiss of continuity of change, Galileo's argument is sound.

EDIT: perhaps the argument is even more general and does not depend on continuous change. Consider a composite system S1+S2. The extensive magnitudes of S1 and S2 will add up while the speeds of S1 and S2 won't (you'll take average speed for the composite system). Insofar as nature's behaviour does not depend on how you name and group things, extensive magnitudes cannot be directly relevant for movement.

Answer 2

Galileo's logic is correct, but an important part of his reasoning is not so explicit, that it need to be.

The main statement he bases on is that weight is additive. He uses this statement when regards compound body, made from two glued bodies, as the same as two bodies, connected by wire.

What we see as body's weight is it's attraction force towards Earth, which is guided by Newtonian law of gravitation. Newtonian law states, that the force of attraction is proportional to the body's mass F = K * m, where F is an attraction force (weight), m is a body's mass and K is some constant.

This proportionality is the physical cause of Galileo's law. The heavier bodies are attracted stronger, but they are more inert by the same factors (this is from another Newtonian laws -- three famous laws).

So.

From the physics point of view, it is quite possible, that gravitation law was different, than Newtonian. I.e. the force of attraction to Earth could not be proportional to body's mass, but be guided by other rules.

For example, it could be guided so as Aristotle though, causing heavier bodies fall faster. In this case the picture would be exactly as Galileo described it: lighter body, tied to heavier one, would act like a parachute.

But these other rules would automatically cause the violation of weight additivity. We couldn't not use weighters to know the objects composition, for example, the amount of gold inside coins. Entire civilization would be different.

Galileo knew weight is additive and he based his brilliant qualitative reasoning basing on this fact. Any nowadays qualitative proof would be more complex.

Answer 3

Take the example of momentum and apply Galileo's logic. Momentum equals mass x velocity. If you double the mass, you double the momentum. A ball with twice the mass will have twice the momentum. If you take two balls, each having the same velocity, and tie them together, the new ball has more momentum than the heavier ball alone. The smaller ball does not act as a "parachute" to momentum.

Now, if you take the same example and say that the velocities are exactly opposite (equal magnitude but pointed in the opposite directions), then the resulting momentum would be in the direction of the heavier ball, but it would be less than the heavier ball alone due to the "parachute" effect of the smaller ball.

In the case of dropping the ball, Galileo is being contradictory in his decription of the smaller ball. One way is to say "this ball is fast and that ball is faster" and the other way is to say "this ball is fast and that ball acts against motion creating a negative force". The former adds the speed of the ball, the latter subtracts it.

Conceptually, he is equating "less" with "negative". This would be a fallacy of equivocation.

*Edit The above is only true when considering a real-life example with air resistance and other modern common sense knowledge. If you neglect air resistance (as Galileo did for in this example), then Galileo's statement is sound. Aristotle's view was so primative that it's hard to think about it in terms of his restricted knowledge. If a mass has a mystical set speed, then it makes sense that it would slow down a larger mass, but then they are the same mass and should be going faster, so the conclusion Galileo make is that this can't be true - which is correctly reasoned.

Answer 4

Most of physics is logically flawed - the example I usually go for is that of the calculus which Newton used for various arguments in his Physics; it was Bishop Berkeley that pointed out the logical in the arguments; not really to kick physics down but to point out there are various kinds of 'reasoning' - which ought to be obvious when reflected on - the literary imagination is different from the logical; and the theological is different from the physical. I'm speculating here; as I do not know the history very well, but I suspect he was defending traditional Christian epistemology from various attacks which originated in a purely materialist doctrine.

Trying to put physics on a logically coherent foundation is a large question and a large project - there are various axiomatisations of Newtonian physics, and quantum field theory - its an ongoing project.

One of the biggest 'logical' holes in Physics is the use of Occams Razors; for example there is no logic that can prove there are only four forces. Perhaps, at the incredibly high energy levels when the universe is compressed to the size of a golf-ball a new powerful field comes into existence - who knows; possibly and most probably we can never know.

Answer 5

It is true that that the gravitational charge could be different from the inertial mass, and there is no logical contradiction in that theory. It would pretty much be identical to the Coulomb force theory. In such theories, different objects would fall at different rates, but the rates would be unrelated to their inertial mass.

The statement that more massive objects fall faster is clearly stronger than that, because it makes the gravitational accelerations dependent on the inertial mass using an increasing function a(m).

Suppose you join two objects of different inertial masses together using a glue-force. The system must move with a common acceleration. When the system is in free fall, the accelerations due to gravity of the two components are different (by our assumption). If there was no glue force, then the distance between the components would increase.

But the glue force is an attractive force between the components which opposes this, by adding an acceleration to the heavier component in the opposite direction of its gravitational acceleration. Hence the common acceleration of the system is lower than that of the heavier component. Hence we reach the contradiction because the system has a higher inertial mass than the components, and yet falls slower.

So, while there are no contradictions with making the gravitation charge independent of the inertial mass, there is a contradiction with the idea of making the gravitational acceleration dependent on the inertial mass using an increasing function a(m).

In a theory where we make the gravitational charge independent from the inertial mass, the gravitational acceleration would be directly proportion to the charge and inversely proportional to the mass.