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    Mentioning in another thread both Galileo and thought experiments reminded me of a Galileo thought
    experiment with a puzzling consequence. Puzzling to me, at least ... I wonder if anyone can explain
    away my puzzlement.

    Do heavy things fall faster than lighter things? No need to climb the Leaning Tower, or even get out
    of your chair. Just think. Consider a lighter object attached inelastically to a heavier object.
    Think of them falling together. If the heavier one falls faster than the lighter one, it will be
    slowed down by being attached to the lighter one, and so fall more slowly than if it wasn't so
    attached. But the composite object (heavier+lighter object) is heavier than the heavier object on
    its own ... so, by supposition, it will fall faster than the heavier object does alone. So,
    supposing heavier objects to fall faster than lighter ones, the composite will fall both slower and
    faster than the heavier object on its own. Contradiction. Hence, heavier objects don't fall faster
    than lighter objects.

    OK, we knew that. Why am I puzzled? Just because this seems to be a priori, whereas generally we
    don't think nowadays that you can get scientific knowledge just by sitting in your armchair and
    thinking about things.

    Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'. But,
    then, what of Galileo's thought experiment?

    Bob

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q1]
    [q1]>But, then, what of Galileo's thought experiment?[/q1]

    I think that there are a couple of deep (and 'obvious') assumptions built into this.

    1/ inertial mass is proportional to gravitational mass
    2/ mass (really both of them) is additive.

    If either of these assumptions is false then the thought experiment can fail.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Clark <[email protected]> wrote:

    [q1]>Do heavy things fall faster than lighter things? No need to climb the Leaning Tower, or even get[/q1]
    [q1]>out of your chair. Just think. Consider a lighter object attached inelastically to a heavier[/q1]
    [q1]>object. Think of them falling together. If the heavier one falls faster than the lighter one, it[/q1]
    [q1]>will be slowed down by being attached to the lighter one, and so fall more slowly than if it wasn't[/q1]
    [q1]>so attached.[/q1]

    When you attach the two objects together inelastically, you no longer have two objects. You only
    have one object - so it doesn't make sense to speak of the smaller object slowing the larger object.

    Gareth

    In article <[email protected] .uk>, [email protected] c.uk (Robert
    Low) wrote:

    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q2]
    [q2]> >But, then, what of Galileo's thought experiment?[/q2]
    [q1]>[/q1]
    [q1]> I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q1]
    [q1]>[/q1]
    [q1]> 1/ inertial mass is proportional to gravitational mass[/q1]
    [q1]> 2/ mass (really both of them) is additive.[/q1]
    [q1]>[/q1]
    [q1]> If either of these assumptions is false then the thought experiment can fail.[/q1]

    Now imagine two identical iron balls fastened to the ends of a flexible rod of negligible mass and
    air resistance and then imagine them dropped from the top of the leaning tower.

    Do this a number of times using connecting rods of varying flexibility from totally rigid to
    flexible as string.

    Does the speed of falling depend on the rigidity of the connecting rod?

    If the balls become disconnected while falling, should they slow down?

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q2]
    [q2]> >But, then, what of Galileo's thought experiment?[/q2]
    [q1]>[/q1]
    [q1]> I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q1]
    [q1]>[/q1]
    [q1]> 1/ inertial mass is proportional to gravitational mass[/q1]
    [q1]> 2/ mass (really both of them) is additive.[/q1]
    [q1]>[/q1]
    [q1]> If either of these assumptions is false then the thought experiment can fail.[/q1]

    I don't really see why if (1) has to be a hidden assuption. Probably I'm being stupid, but I wonder
    if you could explain? As for (2), yes, thanks, of course that's right. 'Acceleration proportional to
    colour' isn't contradictory, after all.

    Still, I have a faint worry ... is it the case that we can decide between Aristotelian accounts and
    Galilean accounts on a priori grounds? Seems strange, if true. (Both accounts allowing that mass is
    additive, I suppose).

    Bob

    In article <[email protected] .uk>, Robert Low
    <[email protected] ac.uk> wrote:
    [q1]>In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]>>Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q2]
    [q2]>>But, then, what of Galileo's thought experiment?[/q2]

    Well, it's a logical truth that gravity seems to work the way we think it does. If that
    helps! Note that G's elegant thought experiment is not *that* abstract; we could easily
    carry it out in practice, and we know [roughly] what would happen, from our everyday
    experience of how compound objects fall. It doesn't show that gravity works the way it does,
    but rather that it cannot work the way Aristotle and chums thought it should.

    AFAIK, thought experiments rather went out of fashion after Galileo, and were properly
    revived only when Einstein was thinking about relativity. You can also find a fair number in
    Eddington; and more recently they keep cropping up in that rather strange interface area
    between maths, computing, philosophy and AI.

    Galileo, BTW, in case it isn't obvious to all readers, was a genius, class 1, grade A. His
    contributions to maths, physics, astronomy and engineering are enormous. In addition, he was
    a superb writer, and by all accounts a good musician, and a class gent. His books on
    mechanics still make good reading today; you have to make *some* allowances for the state of
    the art at the time, but nevertheless his writing is full of insight and of little analogies
    and thought experiments -- and actual experiments! -- that reveal his reasoning.

    [q1]>I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q1]
    [q1]>1/ inertial mass is proportional to gravitational mass[/q1]
    [q1]>2/ mass (really both of them) is additive. If either of these assumptions is false then the thought[/q1]
    [q1]> experiment can fail.[/q1]

    Perhaps I'm missing something, but I don't understand either of these "assumptions". The
    experiment is not about inertia, gravity or mass [for which see Newton, the best part of a
    century later], but about heaviness and speed. The claim, to be refuted, was that heavy
    objects fall faster than light ones. *Now* we can use Newtonian theories to explain what
    is happening, but Galileo was concerned only with observations and common sense. Heaviness
    is an observable, like colour or circumference; its relation to mass isn't [not in G's
    day, anyway].

    He doesn't need mass to be *additive*, only for compound objects to be heavier than their
    components [so could be any monotonic law]. So a compound object "should" fall faster than
    its components, which is an affront to common sense when the components are only weakly
    connected, "QED".

    Of course, he *also* did some actual experiments, surely at Pisa, though not necessarily off
    the Leaning Tower; and you can see them being carried out at Woolsthorpe Manor [and no doubt
    elsewhere]. It's somewhat ironic that most of the experiments at Woolsthorpe are Galileian
    -- dropping balls, inclined planes, relative motion -- when it's a Newtonian museum!

    --
    Andy Walker, School of MathSci., Univ. of Nott'm, UK. [email protected]

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>Robert Low wrote:[/q1]
    [q2]>> I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q2]
    [q2]>>[/q2]
    [q2]>> 1/ inertial mass is proportional to gravitational mass[/q2]
    [q1]>I don't really see why if (1) has to be a hidden assuption. Probably I'm being stupid, but I wonder[/q1]
    [q1]>if you could explain? As for (2), yes,[/q1]

    Suppose, for example, that gravitational mass was proportional to the square of inertial mass. Then
    if you stuck two cannonballs together, you'd double the inertia but quadruple the gravitational
    force, so that heavier objects would fall faster.

    [q1]>Still, I have a faint worry ... is it the case that we can decide between Aristotelian accounts and[/q1]
    [q1]>Galilean accounts on a priori grounds?[/q1]

    I shouldn't think so. I don't see anything contradictory about a natural rest frame as opposed to a
    natural family of inertial frames, for example. (Maybe I'm missing what you'r getting at, here.)

    The problem is disentangling genuinely a priori arguments from implicit arguments about how things
    'obviously' work, I think.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Dr A. N. Walker <[email protected]> wrote:
    [q1]>Robert Low <[email protected] ac.uk> wrote:[/q1]
    [q2]>>I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q2]
    [q2]>>1/ inertial mass is proportional to gravitational mass[/q2]
    [q2]>>2/ mass (really both of them) is additive. If either of these assumptions is false then the[/q2]
    [q2]>> thought experiment can fail.[/q2]
    [q1]>[/q1]
    [q1]> Perhaps I'm missing something, but I don't understand either of these "assumptions". The[/q1]
    [q1]> experiment is not about[/q1]

    What I'm getting at is that there are logically consistent theories of gravity in which more massive
    objects fall faster than less massive ones (or slower, for that matter) and that therefore the
    'thought experiment' must be relying on some implicit assumptions about nature, not just on logic.
    (After all, once upon a time the deductions available based on the assumption that the parallel
    postulate was false were sufficiently repugnant that one might deduce that the parallel postulate
    was entailed by the others...)

    [q1]> He doesn't need mass to be *additive*, only for compound objects to be heavier than their[/q1]
    [q1]> components [so could be any monotonic law].[/q1]

    Hmm...yes. But the monotonic law has to be the same for both inertial and gravitational mass. So I
    guess it's really the first assumption that matters here.

    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    "Dr A. N. Walker" wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected] .uk>, Robert Low[/q1]
    [q1]> <[email protected] ac.uk> wrote:[/q1]
    [q2]> >In article <[email protected]> , Clark <[email protected]> wrote:[/q2]
    [q2]> >>Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q2]
    [q2]> >>But, then, what of Galileo's thought experiment?[/q2]
    [q1]>[/q1]
    [q1]> Well, it's a logical truth that gravity seems to work the way we think it does. If that[/q1]
    [q1]> helps! Note that G's elegant thought experiment is not *that* abstract; we could easily[/q1]
    [q1]> carry it out in practice, and we know [roughly] what would happen, from our everyday[/q1]
    [q1]> experience of how compound objects fall. It doesn't show that gravity works the way it[/q1]
    [q1]> does, but rather that it cannot work the way Aristotle and chums thought it should.[/q1]

    I'll settle for that. Isn't that, in itself, puzzling? Given additivity of weight, it seems that
    Aristotelian science is inconsistent in its treatment of gravity, and that we can know that it's
    inconsistent by pure thought. Generally we think (don't we?) that differences between scientific
    theories are at bottom empirical. Is this a genuine example of the synthetic a priori, do you think?

    [q1]>[/q1]
    [q1]> AFAIK, thought experiments rather went out of fashion after Galileo, and were properly[/q1]
    [q1]> revived only when Einstein was thinking about relativity. You can also find a fair number[/q1]
    [q1]> in Eddington; and more recently they keep cropping up in that rather strange interface[/q1]
    [q1]> area between maths, computing, philosophy and AI.[/q1]

    Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there an
    impossibility/incompleteness result that stopped work on neural nets for a while until it was worked
    around? Anyone know about this?

    Bob

    "Dr A. N. Walker" wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected] .uk>, Robert Low[/q1]
    [q1]> <[email protected] ac.uk> wrote:[/q1]
    [q2]> >In article <[email protected]> , Clark <[email protected]> wrote:[/q2]
    [q2]> >>Is it a logical truth that gravity works the way it does? I'd like to think the answer is 'no'.[/q2]
    [q2]> >>But, then, what of Galileo's thought experiment?[/q2]
    [q1]>[/q1]
    [q1]> Well, it's a logical truth that gravity seems to work the way we think it does. If that[/q1]
    [q1]> helps! Note that G's elegant thought experiment is not *that* abstract; we could easily[/q1]
    [q1]> carry it out in practice, and we know [roughly] what would happen, from our everyday[/q1]
    [q1]> experience of how compound objects fall. It doesn't show that gravity works the way it[/q1]
    [q1]> does, but rather that it cannot work the way Aristotle and chums thought it should.[/q1]

    I'll settle for that. Isn't that, in itself, puzzling? Given additivity of weight, it seems that
    Aristotelian science is inconsistent in its treatment of gravity, and that we can know that it's
    inconsistent by pure thought. Generally we think (don't we?) that differences between scientific
    theories are at bottom empirical. Is this a genuine example of the synthetic a priori, do you think?

    [q1]>[/q1]
    [q1]> AFAIK, thought experiments rather went out of fashion after Galileo, and were properly[/q1]
    [q1]> revived only when Einstein was thinking about relativity. You can also find a fair number[/q1]
    [q1]> in Eddington; and more recently they keep cropping up in that rather strange interface[/q1]
    [q1]> area between maths, computing, philosophy and AI.[/q1]

    Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there an
    impossibility/incompleteness result that stopped work on neural nets for a while until it was worked
    around? Anyone know about this?

    Bob

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >Robert Low wrote:[/q2]
    [q2]> >> I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q2]
    [q2]> >>[/q2]
    [q2]> >> 1/ inertial mass is proportional to gravitational mass[/q2]
    [q2]> >I don't really see why if (1) has to be a hidden assuption. Probably I'm being stupid, but I[/q2]
    [q2]> >wonder if you could explain? As for (2), yes,[/q2]
    [q1]>[/q1]
    [q1]> Suppose, for example, that gravitational mass was proportional to the square of inertial mass.[/q1]
    [q1]> Then if you stuck two cannonballs together, you'd double the inertia but quadruple the[/q1]
    [q1]> gravitational force, so that heavier objects would fall faster.[/q1]

    OK. Suppose gravitational mass proportional to the square of inertial mass, so heavy objects fall
    faster. Now consider Galileo's thought experiment. Join a more massive (in either sense) body to a
    less massive body. The less massive body, falling more slowly, slows down the more massive one. But
    the joint body falls faster because it's more massive. Contradiction. So gravitational mass isn't
    proportional to the square of inertial mass.

    Where's the mistake in that? Sure enough you can escape the contradiction if the gravitational mass
    of the joint body isn't necessarily greater than that of the more massive single body ... that is,
    if weight isn't additive in some way or other (needn't be monotonic). But I still don't see how
    negating the eqwuivalence of the two kinds of mass gets us off the hook.

    [q1]>[/q1]
    [q2]> >Still, I have a faint worry ... is it the case that we can decide between Aristotelian accounts[/q2]
    [q2]> >and Galilean accounts on a priori grounds?[/q2]
    [q1]>[/q1]
    [q1]> I shouldn't think so. I don't see anything contradictory about a natural rest frame as opposed to[/q1]
    [q1]> a natural family of inertial frames, for example. (Maybe I'm missing what you'r getting at, here.)[/q1]
    [q1]>[/q1]
    [q1]> The problem is disentangling genuinely a priori arguments from implicit arguments about how things[/q1]
    [q1]> 'obviously' work, I think.[/q1]

    I agree. I'm trying to see where I'm going wrong. The point about Aristotle vs Galileo: Both allow
    additivity of weight, if you like (monotone or not, let's say, just two bodies together weigh more
    than each one separately). Aristotle says that heavy bodies fall faster. Galileo says not. And it
    seems (still) that we can see that Galileo is right a priori. Which is puzzling. I'm still loking
    for the implicit assumption that requires empirical testing, in other words.

    Bob

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>OK. Suppose gravitational mass proportional to the square of inertial mass, so heavy objects fall[/q1]
    [q1]>faster. Now consider Galileo's thought experiment. Join a more massive (in either sense) body to a[/q1]
    [q1]>less massive body. The less massive body, falling more slowly, slows down the more massive one. But[/q1]
    [q1]>the joint body falls faster because it's more massive. Contradiction. So gravitational mass isn't[/q1]
    [q1]>proportional to the square of inertial mass.[/q1]

    But this is assuming that the net effect of gravity on the compound object is the sum of its effects
    on the constituent parts. I'm almost exactly stating that the alternative view is that gravity
    doesn't act that way: that the force acting on the composite body *isn't* the sum of the forces
    acting on the two un-joined bodies. It's the assumption that the effect of gravity on the composite
    object is the sum of the effects on its components that I'm getting at. It's plausible, and agrees
    with experience---but it isn't logically necessary.

    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >Robert Low wrote:[/q2]
    [q2]> >> I think that there are a couple of deep (and 'obvious') assumptions built into this.[/q2]
    [q2]> >>[/q2]
    [q2]> >> 1/ inertial mass is proportional to gravitational mass[/q2]
    [q2]> >I don't really see why if (1) has to be a hidden assuption. Probably I'm being stupid, but I[/q2]
    [q2]> >wonder if you could explain? As for (2), yes,[/q2]
    [q1]>[/q1]
    [q1]> Suppose, for example, that gravitational mass was proportional to the square of inertial mass.[/q1]
    [q1]> Then if you stuck two cannonballs together, you'd double the inertia but quadruple the[/q1]
    [q1]> gravitational force, so that heavier objects would fall faster.[/q1]

    OK. Suppose gravitational mass proportional to the square of inertial mass, so heavy objects fall
    faster. Now consider Galileo's thought experiment. Join a more massive (in either sense) body to a
    less massive body. The less massive body, falling more slowly, slows down the more massive one. But
    the joint body falls faster because it's more massive. Contradiction. So gravitational mass isn't
    proportional to the square of inertial mass.

    Where's the mistake in that? Sure enough you can escape the contradiction if the gravitational mass
    of the joint body isn't necessarily greater than that of the more massive single body ... that is,
    if weight isn't additive in some way or other. But I still don't see how negating the equivalence of
    the two kinds of mass gets us off the hook.

    [q1]>[/q1]
    [q2]> >Still, I have a faint worry ... is it the case that we can decide between Aristotelian accounts[/q2]
    [q2]> >and Galilean accounts on a priori grounds?[/q2]
    [q1]>[/q1]
    [q1]> I shouldn't think so. I don't see anything contradictory about a natural rest frame as opposed to[/q1]
    [q1]> a natural family of inertial frames, for example. (Maybe I'm missing what you'r getting at, here.)[/q1]
    [q1]>[/q1]
    [q1]> The problem is disentangling genuinely a priori arguments from implicit arguments about how things[/q1]
    [q1]> 'obviously' work, I think.[/q1]

    I agree. I'm trying to see where I'm going wrong. The point about Aristotle vs Galileo: Both allow
    additivity of weight, if you like (let's say just that two bodies together weigh more than each one
    separately). Aristotle says that heavy bodies fall faster. Galileo says not. And it seems (still)
    that we can see that Galileo is right a priori. Which is puzzling. I'm still loking for the implicit
    assumption that requires empirical testing to decide between the two theories, in other words.

    Bob

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there an[/q1]
    [q1]>impossibility/incompleteness result that stopped work on neural nets for a while until it was[/q1]
    [q1]>worked around? Anyone know about this?[/q1]

    Only to confirm that I have a similarly vague memory: I think it was in the context of something
    called a 'perceptron', but don't recall any more than that either.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    In article <[email protected] c.uk>, Robert Low
    <[email protected] ac.uk> wrote:
    [q1]>What I'm getting at is that there are logically consistent theories of gravity in which more[/q1]
    [q1]>massive objects fall faster than less massive ones[/q1]

    Wait! What do you mean [in terms that Galileo knew!] by "more massive"? The observable
    property is that some objects are heavier than others [ie, tend to fall with more power, ie
    cause a balance to tilt when compared with lighter objects]. There is a possibly-separate
    observable of inertia [how hard it is to push something], but this is irrelevant to
    Galileo's experiment.

    [q1]> (or slower, for that matter) and that therefore the 'thought[/q1]
    [q1]> experiment' must be relying on some implicit assumptions about[/q1]
    [q1]> nature, not just on logic.[/q1]

    I think we're getting too hung up on the "thought" of the experiment. We can't do TE's "in a
    vacuum"; we can follow G's argument, and Einstein's, only because we have seen how the world
    works and can imagine how the experiment would turn out if we really did do it.

    So G is not relying on implicit assumptions, he is merely suggesting to us that if we
    imagine a certain situation, then we will see why Aristotle is wrong.

    For something more concrete, imagine the experiment carried out with electric charge. So
    yes, we could imagine a planet where gravity was negligible, but we were all charged and so
    apples fell [up or down!] to the charged ground. In this case, electric mass [charge] is
    quite divorced from inertia, so some apples will indeed fall faster than others. The
    experiment still works enough to show that an apple-pair falls with an intermediate
    velocity, not with an additive velocity. So, tendency to fall is not monotonic with charge
    [or with mass], and electric-Aristotle is still wrong.

    The assumptions that Galileo has to make are not to do with inertial mass, etc., but are
    rather that connecting two masses together does not of itself make them fall faster, that if
    a slow object and a fast one are connected inextensibly then the fast one is slowed down and
    the slow one speeded up, and, yes, the "additive" one that compound objects are heavier than
    their components. None of these are logical necessities, which is why this is not *just*
    armchair science, but requires contact with the real world.

    --
    Andy Walker, School of MathSci., Univ. of Nott'm, UK. [email protected]

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>I'll settle for that. Isn't that, in itself, puzzling? Given additivity of weight, it seems that[/q1]
    [q1]>Aristotelian science is inconsistent in its treatment of gravity, and that we can know that it's[/q1]
    [q1]>inconsistent by pure thought.[/q1]

    Not by *pure* thought; rather, by thought tempered by experience of the real world.
    *Applied* thought, if you like!

    [q1]>Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there an[/q1]
    [q1]>impossibility/incompleteness result that stopped work on neural nets for a while until it was[/q1]
    [q1]>worked around? Anyone know about this?[/q1]

    Yes. Try a Google search on "Minsky perceptron" for more than you probably really want to
    know. Actually, AI has been bedevilled by this sort of thing. There are rival camps with
    strong views about how rubbishy all the other work is, which is always a bad start. Then
    there is lots of hype, another bad sign. Then if these things really did work [automatic
    translation, efficient vision, intelligent robots, heuristic reasoning, etc] they would be
    enormously valuable, putting further vested interests into the melting pot.

    --
    Andy Walker, School of MathSci., Univ. of Nott'm, UK. [email protected]

    Dr A. N. Walker <[email protected]> wrote:
    [q1]>Robert Low <[email protected] ac.uk> wrote:[/q1]
    [q2]>>What I'm getting at is that there are logically consistent theories of gravity in which more[/q2]
    [q2]>>massive objects fall faster than less massive ones[/q2]
    [q1]>[/q1]
    [q1]> Wait! What do you mean [in terms that Galileo knew!] by "more massive"? The observable[/q1]
    [q1]> property is that some[/q1]

    I'm not sure I can carry out the discussion at all purely in terms Galileo knew! The potential
    distinction between gravitational and inertial mass is only really possible post-Newton. I'm only
    addressing the question of whether his deduction was entirely a priori, or whether it uses real
    world knowledge. I *think* we agree that the latter is the case.

    [q1]>than their components. None of these are logical necessities, which is why this is not *just*[/q1]
    [q1]>armchair science, but requires contact with the real world.[/q1]

    Agreed entirely: armchair science can't answer the question; it requires contact with the
    real world.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Clark <[email protected]> wrote in uk.education.maths:
    [q1]>Do heavy things fall faster than lighter things? No need to climb the Leaning Tower, or even get[/q1]
    [q1]>out of your chair. Just think. Consider a lighter object attached inelastically to a heavier[/q1]
    [q1]>object. Think of them falling together. If the heavier one falls faster than the lighter one, it[/q1]
    [q1]>will be slowed down by being attached to the lighter one, and so fall more slowly than if it wasn't[/q1]
    [q1]>so attached.[/q1]

    Err -- if they're attached, then they're one object, which is heavier than either of the originals.

    --
    Stan Brown, Oak Road Systems, Cortland County, New York, USA http://oakroadsystems.com/ "My theory
    was a perfectly good one. The facts were misleading." -- /The Lady Vanishes/ (1938)

    In article <[email protected] c.uk>, Robert Low
    <[email protected] ac.uk> wrote:
    [q2]>> Wait! What do you mean [in terms that Galileo knew!] by "more massive"? The observable[/q2]
    [q2]>> property is that some[/q2]
    [q1]>I'm not sure I can carry out the discussion at all purely in terms Galileo knew![/q1]

    OK, well, I didn't *quite* mean that [my Latin is v rusty, and my Italian is confined to
    musical terms]!

    [q1]> The potential distinction between gravitational and inertial mass is only[/q1]
    [q1]> really possible post-Newton.[/q1]

    Did the concept of "mass" exist pre-Newton? I think not, not in any meaningful sense; and
    even after Newton it was a pretty silly concept:

    Principia, Definition 1: [Motte's translation] The quantity of matter is the measure of the same,
    arising from its density and bulk conjointly.

    THUS AIR of a double density, in a double space, is quadruple in quantity; [...]. It is this
    quantity that I mean hereafter everywhere under the name of body or mass.

    OK, so mass is densityxvolume. And what is density, if not mass/volume? It's a meaningless
    definition. Of course, you can make progress ... You can compare densities of solids and fluids by
    seeing if the solid floats. And if we assume that mass is an intrinsic property of *a* body [as
    opposed to mass being body, as in the definition], then we can use known compressions to measure it.
    But Newton continues in a rather interesting way:

    And the same is known by the weight of each body, for it is proportional to the weight, as I
    have found by experiments on pendulums, very accurately made, which shall be shown
    hereafter.

    No comment! [But for the fact that he uses a conjunction to start a sentence with.]

    But my point was that I don't think "mass" was an interesting property to Galileo. Weight,
    yes; inertia, at least possibly; momentum, at least possibly. That's enough to start him
    thinking his experiment, and he could certainly have done experiments using inclined planes
    and collisions to determine the relationship between weight and inertia, impulse and
    momentum, and so on. Mass only becomes interesting when the relation between force and
    momentum becomes clearer.

    [q1]> I'm only addressing the question of whether his deduction was entirely a priori, or[/q1]
    [q1]> whether it uses real world knowledge. I *think* we agree that the latter is the case.[/q1]

    Yup.

    Following on a little from this discussion, I hate the way in which maths and physics are so
    often taught as givens -- "Here are N's Laws, now find out what happens in this situation."
    When I first read "Principia" and "Dialogue", it was a complete revelation. One comes to
    understand the circumstances in which these laws were discovered, the writing is brilliant,
    and the whole topic comes alive. I currently teach a module [see
    "http://www.maths.nott.ac.uk/personal/anw/G1ASOC"] a large part of which is this stuff, and
    I think the students find it a revelation too. Simply seeing a time-line from Galileo to
    Newton to Euler to Lagrange to Gauss, etc., gives them a new understanding of maths that
    they never [seem to] get at school or in the usual mechanics/analysis/algebra lectures.

    --
    Andy Walker, School of MathSci., Univ. of Nott'm, UK. [email protected]

    Dr A. N. Walker <[email protected]> wrote:
    [q1]>Robert Low <[email protected] ac.uk> wrote:[/q1]
    [q3]>>> Wait! What do you mean [in terms that Galileo knew!] by "more massive"? The observable[/q3]
    [q3]>>> property is that some[/q3]
    [q2]>>I'm not sure I can carry out the discussion at all purely in terms Galileo knew![/q2]
    [q1]>[/q1]
    [q1]> OK, well, I didn't *quite* mean that [my Latin is v rusty, and my Italian is confined to[/q1]
    [q1]> musical terms]![/q1]

    Heh. I didn't really mean his language, but his world view. I know I don't understand that well
    enough to argue within
    it.

    [q1]> Did the concept of "mass" exist pre-Newton? I think not, not in any meaningful sense; and[/q1]
    [q1]> even after Newton it was a pretty silly concept:[/q1]

    Yep: sure takes some sorting out.

    [q1]>OK, so mass is densityxvolume. And what is density, if not mass/volume?[/q1]

    Same problem with mass as force/acceleration, and force as mass times acceleration (even avoiding my
    pet peeve of rate of change of momentum).

    Circularity abounds.

    [q1]> But my point was that I don't think "mass" was an interesting property to Galileo. Weight,[/q1]
    [q1]> yes; inertia, at least possibly; momentum, at least possibly. That's[/q1]

    Agreed.

    [q1]> Following on a little from this discussion, I hate the way in which maths and physics are so[/q1]
    [q1]> often taught as givens -- "Here are N's Laws, now find out what happens in this situation."[/q1]

    Being able to treat mechanics as a deductive system appeals, I think, to the pure mathematician that
    lurks at the heart of some physicists. And of course, one can give competence at 'solving mechanics
    problems' that way.

    But trying to get across the idea of how to get at the concept of mass and force without circularity
    is relatively hard, even if it is at least as interesting.

    [q1]>comes alive. I currently teach a module [see[/q1]
    [q1]>"http://www.maths.nott.ac.uk/personal/anw/G1ASOC"] a large[/q1]

    Looks like fun stuff. It's almost paradoxical that one gets to spend more time pondering the
    important questions with non-specialists, since the specialists have to spend so much effort in
    becoming proficient at 'doing' it.

    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/
 
 
 
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