Yes. You have the right meaning to (1). If the force comes from the objects, not the earth. Whether you imagine this as pushing or pulling doesn’t matter, the effect is the same. I’ve now added a parenthesis to the sentence to make that clearer.

For example, Aristotle did not say the earth projects a force that draws objects toward it. Other ancient scientists did propose that. But Aristotle argued that objects are propelling themselves toward the earth, and “accumulate steam” as it were as they push through the resisting medium in their way (thus explaining observed acceleration and terminal velocity). So in his view, there is no force coming from the earth at all. It isn’t even involved in the process.

So on Aristotle’s view, you get the cart-pulling effect, contrary to Galileo’s mistaken thought experiment: if a heavier object can push through the air faster than a lighter object and you tie them together, the lighter object will be pulled faster; the heavier object will not be slowed at all, except by the smaller amount of air resistance the smaller object adds to the combination (which may be zero if it is tied closely enough to the heavier object). Thus Galileo isn’t even describing Aristotle’s theory.

That’s his failure.

Do you get the point now?

NOTE TO READERS: This is a continuation from a more detailed discussion in Galileo’s Goofs and in comments there.

“(1) what if indeed the gravitational force was projected only by objects toward the earth and not by earth toward the objects?
(2) Then an object that weighed twice as much would generate twice the gravity.
(3) And indeed would accelerate toward the earth twice as fast.”

I think sentence (1) might mean “imagine the force of gravity pushes towards the earth (from behind)
rather than pulls toward the earth” but I don’t really know what it means.

If “twice the gravity” means “twice the gravitational force”, sentence (2) appears to be a tautology, and no premise (1) is needed to deduce it: “weighs twice as much” means “is acted on by twice the gravitational force”.

Nor do I understand how you get from twice the gravitational force to twice the acceleration. Assuming what? Assuming that Aristotle is correct (even though I don’t think Aristotle said anything about acceleration)?

I know you are not trying to say Aristotle was right after all, or Galileo’s assertion (which turns out to be supported by Newton) is wrong. I am trying to understand your arguments, and clear away what is not relevant.

I said “certain axiomatic assumptions” not merely the one you here state. You stated many more assumptions than that (reference frame logic brings a lot of baggage with it).

To get the conclusion Galileo got, you have to insert a number of assumptions about reference frames. That is what I mean. If all you did was import that single assumption you just stated, you would not get his result except the same circular way he already did: by importing assumptions about inertial mass that Aristotle rejected exactly as my article explains.

You seem here still to be confusing me as having argued against Galileo’s empirical conclusions about gravity (I did only mildly, and separately, but only to point out what you just did, e.g. he didn’t think of the center-of-mass as the target reference). The thought experiment I show is hosed and that my section is almost entirely about has nothing to do with his empirical arguments. I am ONLY saying that of the THOUGHT experiment; not any real experimental result Galileo references in the same book.

I do cut Galileo that slack. It’s what I just did in my comment above. And why I didn’t take him to task for it in my article.

You do not pay attention well.

Particularly as you again here think I argued against Galileo’s empirical arguments. My one example of a hosed thought experiment relating to gravity does not include any of his empirical arguments.

I think you need to re-read my article. And read it very carefully.

No. I’m not talking about the balls rolling down planes. Read the article you are commenting on please. I’m talking about the section where Galileo says he can, with a thought experiment alone, disprove Aristotle’s theory. And there he gives a specific argument, the one I quote, which is not about balls rolling down planes. I am not talking about anything else he argued apart from the one argument I say I am talking about. Please pay attention.

But yes, if we start with certain axiomatic assumptions about reference frames, you can get [a modified Galileo] result … But that would again be a circular argument: you are presuming your result in the constructing of your premises.

I don’t see anything in choosing the earth-body center of mass as my reference plane, that assumes bodies fall at the same rate or indeed fall at all. I don’t see anything circular in that part.

The other assumption that Galileo seems to make seems to be that there is an action and a reaction – that a fast moving body will be slowed by contact with a slow moving body and a slow moving body will be sped up (speeded up?) by a fast moving body. And this applies in general, and has nothing to do with gravity (imagine a truck colliding with a slow moving Volkswagen ahead of it on the road moving in the same direction, neither driver applies the brakes – we think we know the truck moves slower and the Volkswagen move faster after the collision).

I think Galileo says if you take this empirical “fact” and apply it to falling bodies you get a result about how gravity must work. I am very suspicious of thought experiments without empirical input but I am unsure whether Galileo did that here. It seems Galileo took something that was accepted by his contemporaries and drew from it an unexpected conclusion about gravity.

Galileo should more accurately have written, “If we were to follow Aristotle’s model, then” Galileo’s statement about the iron balls would be true. – RC

So I think we could cut Galileo some slack here, and assume that what he meant by “Aristotle says” was “if we were to follow Aristotle’s model” as you say. The passage comes from a fictitious conversation, and it might be the sort of thing Galileo’s contemporaries would say. I suppose we would have to check the original Latin or Italian (beyond my pay grade), and the punctuation used, to see if it was intended as a literal quote from Aristotle.

Galileo describes experiments he seems to claim to have done, about balls rolling down an inclined plane – I don’t know if people doubt he did this inclined plane experiment. But the fact that Galileo describes that experiment in detail suggests to me that he was not trying to assert the superiority of philosophical deduction over experimental work. Even if he “literally boasted” he didn’t need experiment, he appears to know he needs experimental confirmation.

There are thousands of corrections one could make it one had better instruments that Galileo had (for example air friction, rolling friction, and if one performed the experiment at different locations on the earth, the different local acceleration due to gravity and different centrifugal acceleration of the spinning earth).

I have often wondered if it was a correct argument that all bodies necessarily accelerate the same, at a given point in a given gravitational field (I first came across it in a book by Ernest Mach, a book which is said to have influenced Einstein’s work on gravity). I remain unsure about it, since your discussion of various reasons why the time of falling of different bodies might differ minutely seems irrelevant, or a little angels-on-the-head-of-a-pin-ish. But this probably means that I don’t understand philosophical arguments.

Note Aristotle never said that.

Galileo is putting words in his mouth. That quotation is a fabrication of Galileo’s.

Galileo should more accurately have written, “If we were to follow Aristotle’s model, then” Galileo’s statement about the iron balls would be true. And that’s closer to true, but still a bit off, because Aristotle said this of the acceleration and that it would reach a terminal velocity (and thus the acceleration is not constant), so Galileo’s statement is strictly speaking false—Aristotle would have disagreed with its specifics even conceptually—but the gist remains correct, as in, it seems Aristotle would have imagined some overall faster time of fall in such a situation. But he never describes such a situation (Aristotle never says anything about dropping iron balls, much less of such disparate masses—and he would likely have insisted the greater air resistance would slow the heavier ball being the larger, though Galileo does anticipate that objection in the text later on).

Note Aristotle’s empirical error appears to have already been noticed within two generations of Aristotle (see my references to Strato and Hipparchus). But what was done with that discovery (and what replaced his model) is unknown. But also note I am not objecting to Galileo’s empirical arguments; but to his attempt to prove an empirical fact from a logical argument (his “thought experiment,” not any actual experiment). That is what he hosed. And yes, he did claim he could prove his conclusion by the thought experiment alone (literally boasting he didn’t need the empirical observation).

Though also note, it is doubted Galileo ever performed the experiment he describes. He appears to be paraphrasing John Philopon, a medieval Christian commentator on Aristotle (Galileo draws on Philopon a lot). It is not clear Philopon ever conducted the experiment either. Philopon’s version appears to just be a casual everyday observation, that has been “telephone gamed” into Galileo’s more specific version. This also illustrates Galileo’s tendency toward overconfident substitution of traditions he prefers for actual testing them with experiments. This time he just by accident was (more or less) right. He often was not.

Neither Galileo nor anyone he thought he was criticizing had conceived of “an inertial reference frame.” So that can’t be interpretively relevant. (The classical relativity Galileo did describe was already well known by the Roman era—Galen discusses it—and didn’t affect the question of whether objects can toward the earth at different rates or not.)

But yes, if we start with certain axiomatic assumptions about reference frames, you can get Galileo’s result (just as Galileo could with his axiomatic assumptions). But that would again be a circular argument: you are presuming your result in the constructing of your premises. The question “are your premises true” remains unanswered, and thus so does the question whether your conclusion is true.

Hence the problem is fundamentally conceptual, before we even get to asking what actually is the case—which must be empirical. It cannot be answered with thought experiments. Reductio can eliminate some options a priori (by demonstrating their logical inconsistency); but Aristotle’s wasn’t one of them.