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Galileo's famous argument against the Aristotle's theory of falling bodies goes like this. "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.

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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.

  • Your argument assumes that air resistance is a function of mass only and shape doesn't affect it. Or you are assuming that the heavy and light balls are exactly the same shape and that drag coefficients don't exist. Galileo's original experiment was with a large (heavy) ball and a small (light) ball. His thought experiment breaks down because the mass is linearly proportional to the volume, but the volume is not linearly proportional to the crossectional area. Also, tying them together will change the drag coefficient because the air path is changed (no air can flow past the joint). – ProfessorFluffy Dec 2 '14 at 15:46
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    He mostly got lucky that the balls were relatively similar and the drop was relatively short so the difference was too small to notice. If we did the same thing today with a pebble size ball and a car sized ball and dropped them from an airplane, there would be a clear difference - and Galileo would be very confused. – ProfessorFluffy Dec 2 '14 at 15:54
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    I am not assuming this at all, I observe that air resistance is a force, and the resultant acceleration is a=F/m (not F alone), that's the point. That makes acceleration proportional to an intensive magnitude (per mass unit) not extensive like mass. Galileo's argument is perfectly correct: acceleration cannot be proportional to extensive quantities, in pain of contradiction. – Quentin Ruyant Dec 2 '14 at 16:00
  • Your point about dragging forces is, basically, irrelevant. Simply add "assuming dragging forces are negligeable" for the sake of the argument, and you can show that if speed or acceleration are made proportional to extensive quantities, two readings of the same experiment (one where a composite system is considered, one where its parts are considered) lead to different results which is not acceptable. – Quentin Ruyant Dec 2 '14 at 16:14
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    Just did some additional research outside of the OP statement, and it looks like Galileo meant for this example to be in a vacuum. I had assumed the context was natural air (because it was unstated in the OP). So, with air there are problems, but neglecting air his argument is sound. – ProfessorFluffy Dec 2 '14 at 18:07
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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.

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    I also think that the flaw is in arguing "if one ball is slower than the other then it drags on it" in the first part of the argument, the second part leads to the correct conclusion (under Aristotle's theory). But I can't quite put my finger on what the logical issue is exactly. – Conifold Dec 1 '14 at 20:48
  • Are you looking for a formal fallacy? There may be one here, but the more obvious one seems to be the False Premise informal fallacy. Falling balls do not, in reality, act like a parachute in the way Galileo is claiming, so his conclusions are invalid, informally. – ProfessorFluffy Dec 1 '14 at 21:16
  • I see, but what is wrong with the parachute reasoning exactly since it seems intuitively convincing to many people? – Conifold Dec 1 '14 at 23:48
  • I assume that at this point we have left the logical fallacy argument and you are now asking about physics/math. In a vacuum, this whole conversation doesn't make sense, but Aristotle probably was drawing from observations like steel balls falling faster than feathers or paper in regular air. The reason a ball hits first is because it has a higher terminal velocity than the feather. Terminal velocity exists because the earth pulls you down but air exerts a force pushing you up (drag). Drag increases as a function of geometry - more aerodynamic shapes have higher terminal velocities. – ProfessorFluffy Dec 2 '14 at 0:36
  • Mass also contributes to terminal velocity. Higher mass pushes harder on the air and allows a faster terminal velocity. Terminal velocity is the point where gravity and air resistance are equal. Ok, so a parachute slows things down by decreasing the terminal velocity by using a huge shape to created air resistance. This air resistance is a negative force acting against gravity. That is why things get slower – because of the difference in the shape, not the weight. – ProfessorFluffy Dec 2 '14 at 0:37
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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.

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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.

  • Yes, but Newton's logical gaps can be filled in, and were by Lagrange and Hamilton. I don't think the same can be done with this particular argument of Galileo's, it is fallacious even "in spirit" as it were. – Conifold Dec 2 '14 at 2:31
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    Physics is not logically flawed. This is logic what has gaps. I.e. logic itself, especially Aristotle's, is incomplete and insufficient to reason in physics domain. – Dims Dec 2 '14 at 16:06
  • @conifold:can you explain the 'fallaciousness' in spirit here? I'd say that this is essentially an argument 'in spirit' in terms of a limiting argument (take the spheres to a small limit), and also from symmetry (the two balls being identical as are the circumstances) - these are two very important principles in physics (and mathematics); that is in 'spirit'. – Mozibur Ullah Dec 6 '14 at 10:19
  • @Dim: yes, but did Newton 'know' that? After all it was thought after the development of mathematical logic by Boole that mathematics could be put on a logical foundation - and we know how that worked out ie Gödel. To put the calculus on a rigorous foundation took three centuries - ie Robinsons non-standard calculi and Brouwers/Kocks synthetic geometry – Mozibur Ullah Dec 6 '14 at 10:22
  • @Dim: logic that has gaps in it, in my book, is 'logically' flawed; in the precise sense that not every step can be justified; I agree though that physics 'logic' is different to the precise sense of logic established by Aristotle where every step needs justification, or contemporary logic. – Mozibur Ullah Dec 6 '14 at 10:30

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