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    (I just posted and cancelled a version of this which had words some in wrong order the. This version
    should contain different errors.)

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>Robert Low wrote:[/q1]
    [q2]>> In article <[email protected]> , Clark <[email protected]> wrote:[/q2]
    [q2]>> >OK. Suppose gravitational mass proportional to the square of inertial mass, so heavy objects[/q2]
    [q2]>> >fall faster. Now consider Galileo's thought experiment. Join a more massive (in either sense)[/q2]
    [q2]>> >body to a less massive body. The less massive body, falling more slowly, slows down the more[/q2]
    [q2]>> >massive one. But the joint body falls faster because it's more massive. Contradiction. So[/q2]
    [q2]>> >gravitational mass isn't proportional to the square of inertial mass.[/q2]
    [q2]>>[/q2]
    [q2]>> But this is assuming that the net effect of gravity on the compound object is the sum of its[/q2]
    [q2]>> effects on the constituent parts.[/q2]
    [q1]>[/q1]
    [q1]>Indeed. But it doesn't seem to be assuming that graviational and inertial mass are proportional. Am[/q1]
    [q1]>I missing something?[/q1]

    Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional, then
    objects of different masses will in general fall with different accelerations. The force acting on a
    body of gravitational mass M is, say, F. Then the force acting on a body of gravitational mass 2M is
    2F, but if the inertial mass of the latter is not double that of the former, then the acceleration
    will be different.

    Of course, none of this is taking place in Galilean terms: all I'm really trying to say is that
    there's a model of gravity bearing some relation to Newton's in which things need not all fall with
    the same acceleration (ignoring air resistance, of course).

    I think it was my claim that mass had to be additive that was the red herring, not the one about
    proportionality of inertial and gravitational mass.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

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

    In article <[email protected]>, Clark <[email protected]> wrote:
    [q1]>OK, I'll give you the words. It still does look as if whether Aristotelian science or Galilean[/q1]
    [q1]>science is a better description of the world is determinable a priori. Which is odd, I think. Do[/q1]
    [q1]>you disagree?[/q1]

    Partly. Firstly, I think there's more "world experience" going into the thought experiment
    than either you or Rob are quite giving credit to, so "a priori" is a little bit
    exaggerated. Only a bit, though, and I agree that Galileo's experiment is elegant and
    surprisingly powerful. Secondly, note that the Aristotelian viewpoint itself was a "thought
    experiment". Aristotle didn't *measure* the velocities of falling objects, he just relied on
    common sense. It's not *quite* so odd that a random thought is disprovable by a more
    considered thought.

    I think there is perhaps more surprise, and beauty, from the deep results that follow
    from Galileian relativity and from time/space invariance/symmetry. But that's another
    can of worms.

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

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >OK. Suppose gravitational mass proportional to the square of inertial mass, so heavy objects fall[/q2]
    [q2]> >faster. Now consider Galileo's thought experiment. Join a more massive (in either sense) body to[/q2]
    [q2]> >a less massive body. The less massive body, falling more slowly, slows down the more massive one.[/q2]
    [q2]> >But the joint body falls faster because it's more massive. Contradiction. So gravitational mass[/q2]
    [q2]> >isn't proportional to the square of inertial mass.[/q2]
    [q1]>[/q1]
    [q1]> But this is assuming that the net effect of gravity on the compound object is the sum of its[/q1]
    [q1]> effects on the constituent parts.[/q1]

    Indeed. But it doesn't seem to be assuming that graviational and inertial mass are proportional. Am
    I missing something?

    [q1]> I'm almost exactly stating that the alternative view is that gravity doesn't act that way: that[/q1]
    [q1]> the force acting on the composite body *isn't* the sum of the forces acting on the two un-joined[/q1]
    [q1]> bodies. It's the assumption that the effect of gravity on the composite object is the sum of the[/q1]
    [q1]> effects on its components that I'm getting at. It's plausible, and agrees with experience---but it[/q1]
    [q1]> isn't logically necessary.[/q1]
    [q1]>[/q1]

    Bob

    "Dr A. N. Walker" wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >I'll settle for that. Isn't that, in itself, puzzling? Given additivity of weight, it seems that[/q2]
    [q2]> >Aristotelian science is inconsistent in its treatment of gravity, and that we can know that it's[/q2]
    [q2]> >inconsistent by pure thought.[/q2]
    [q1]>[/q1]
    [q1]> Not by *pure* thought; rather, by thought tempered by experience of the real world.[/q1]
    [q1]> *Applied* thought, if you like![/q1]

    OK, I'll give you the words. It still does look as if whether Aristotelian science or Galilean
    science is a better description of the world is determinable a priori. Which is odd, I think. Do
    you disagree?

    [q1]>[/q1]
    [q2]> >Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there[/q2]
    [q2]> >an impossibility/incompleteness result that stopped work on neural nets for a while until it was[/q2]
    [q2]> >worked around? Anyone know about this?[/q2]
    [q1]>[/q1]
    [q1]> Yes. Try a Google search on "Minsky perceptron" for more than you probably really want to[/q1]
    [q1]> know. Actually, AI has been bedevilled by this sort of thing. There are rival camps with[/q1]
    [q1]> strong views about how rubbishy all the other work is, which is always a bad start. Then[/q1]
    [q1]> there is lots of hype, another bad sign. Then if these things really did work [automatic[/q1]
    [q1]> translation, efficient vision, intelligent robots, heuristic reasoning, etc] they would be[/q1]
    [q1]> enormously valuable, putting further vested interests into the melting pot.[/q1]
    [q1]>[/q1]
    Thanks for that

    Bob

    Dr A. N. Walker <[email protected]> wrote:
    [q1]> Partly. Firstly, I think there's more "world experience" going into the thought experiment[/q1]
    [q1]> than either you or Rob are quite giving credit to, so "a priori" is a little bit[/q1]
    [q1]> exaggerated. Only[/q1]

    I thought I was claiming that Galileo *did* have to put a lot of real world experience into his
    gedankenexperiment! (Alas, my poor communication skills.)

    [q1]> I think there is perhaps more surprise, and beauty, from the deep results that follow from[/q1]
    [q1]> Galileian relativity and from time/space invariance/symmetry. But that's another can of[/q1]
    [q1]> worms.[/q1]

    Mmmm...Noether's theorem. Sometimes you have to eat a lot of worms to get at the fish. Or something
    like that.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> (I just posted and cancelled a version of this which had words some in wrong order the. This[/q1]
    [q1]> version should contain different errors.)[/q1]
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]

    [q2]> > Am I missing something?[/q2]
    [q1]>[/q1]
    [q1]> Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional, then[/q1]
    [q1]> objects of different masses will in general fall with different accelerations. The force acting on[/q1]
    [q1]> a body of gravitational mass M is, say, F. Then the force acting on a body of gravitational mass[/q1]
    [q1]> 2M is 2F, but if the inertial mass of the latter is not double that of the former, then the[/q1]
    [q1]> acceleration will be different.[/q1]
    [q1]>[/q1]
    [q1]> Of course, none of this is taking place in Galilean terms: all I'm really trying to say is that[/q1]
    [q1]> there's a model of gravity bearing some relation to Newton's in which things need not all fall[/q1]
    [q1]> with the same acceleration (ignoring air resistance, of course).[/q1]
    [q1]>[/q1]
    [q1]> I think it was my claim that mass had to be additive that was the red herring, not the one about[/q1]
    [q1]> proportionality of inertial and gravitational mass.[/q1]

    OK, thanks. Now I understand (I think). If we just think of *weight* (and assume it's additive),
    rather than of different characterisations of mass, have we already made the equivalence assumption?
    In a sense, I suppose we have ... but did Galileo himself make that assumption? It seems to be
    pushing it a bit to answer 'yes'.

    Bob

    "Dr A. N. Walker" wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]>, Clark <[email protected]> wrote:[/q1]
    [q2]> >OK, I'll give you the words. It still does look as if whether Aristotelian science or Galilean[/q2]
    [q2]> >science is a better description of the world is determinable a priori. Which is odd, I think. Do[/q2]
    [q2]> >you disagree?[/q2]
    [q1]>[/q1]
    [q1]> Partly. Firstly, I think there's more "world experience" going into the thought experiment[/q1]
    [q1]> than either you or Rob are quite giving credit to, so "a priori" is a little bit[/q1]
    [q1]> exaggerated. Only a bit, though, and I agree that Galileo's experiment is elegant and[/q1]
    [q1]> surprisingly powerful. Secondly, note that the Aristotelian viewpoint itself was a[/q1]
    [q1]> "thought experiment". Aristotle didn't *measure* the velocities of falling objects, he[/q1]
    [q1]> just relied on common sense. It's not *quite* so odd that a random thought is disprovable[/q1]
    [q1]> by a more considered thought.[/q1]

    That's a neat way to put it. But, even so, there still seems to be too much of 'something for
    nothing' about even the smidgen of a priori that you'll allow?
    [q1]>[/q1]
    [q1]> I think there is perhaps more surprise, and beauty, from the deep results that follow from[/q1]
    [q1]> Galileian relativity and from time/space invariance/symmetry. But that's another can of[/q1]
    [q1]> worms.[/q1]
    [q1]>[/q1]
    Care to open this can a little way? I'm interested in the notion of 'surprise', its connection with
    aesthetic appreciation, and the synthetic a priori in relation to applications of maths and logic to
    physics ...

    Bob

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q2]>> I think there is perhaps more surprise, and beauty, from the deep results that follow[/q2]
    [q2]>> from Galileian relativity and from time/space invariance/symmetry. But that's another can[/q2]
    [q2]>> of worms.[/q2]
    [q1]>Care to open this can a little way?[/q1]

    See, for example, Landau and Lifshitz, "Mechanics". [I don't know whether any more
    elementary texts take a similar fundamentalist, almost axiomatic, view of mechanics.] They
    show that in an inertial frame. a "free" particle must travel with constant velocity; that
    if the laws of motion are time- independent then a closed system conserves energy; and that
    if space is homogeneous, then a closed system conserves momentum. More generally, there is a
    "duality" between time and energy and between position and momentum that is reflected, eg,
    in Heisenberg's Uncertainty Principle. These are very general results, not consequences of
    inverse-square laws or anything specific like that. There are related consequences in the
    General Theory of Relativity.

    I don't know whether any of these results can be seen in a truly elementary way [eg that one
    could explore easily with an A-level class]. L&L start with the Principle of Least Action,
    which is scarcely elementary. But it's aesthetically pleasing that the Universe seems to be
    fundamentally lazy!

    [q1]> I'm interested in the notion of 'surprise', its connection with[/q1]
    [q1]> aesthetic appreciation, and the synthetic a priori in relation[/q1]
    [q1]> to applications of maths and logic to physics ...[/q1]

    OK. Have you read Eddington? His "Fundamental Theory" is, ermm, obscure, to say the least,
    though fun in places. But his popularisations of relativity and quantum mechanics, written
    in the '30s, are full of nice insights, inc stuff on epistemics and the like.

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

    In article <[email protected]> , Clark <[email protected]> wrote:
    [q1]>OK, thanks. Now I understand (I think). If we just think of *weight* (and assume it's additive),[/q1]
    [q1]>rather than of different characterisations of mass, have we already made the equivalence[/q1]
    [q1]>assumption? In a sense, I[/q1]

    I think that (in our terms) we have.

    [q1]>suppose we have ... but did Galileo himself make that assumption? It seems to be pushing it a bit[/q1]
    [q1]>to answer 'yes'.[/q1]

    I don't think we can fairly impute that to Galileo. It's only with the advent of Newton's theory of
    gravitation that the distinction between gravitational and inertial mass has any meaning, as far as
    I can see. Before that, the concepts didn't exist, and so couldn't really be conflated. (And as Andy
    has pointed out, the notion of 'mass' is still pretty hazy in Newton's writings. Possibly also in
    the minds of most of us, too :-))
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    Dr A. N. Walker <[email protected]> wrote:
    [q1]>More generally, there is a "duality" between time and energy and between position and momentum that[/q1]
    [q1]>is reflected, eg, in Heisenberg's Uncertainty Principle.[/q1]

    I'll go with the position/momentum thing, but the relationship between time and energy is much more
    confused (or confusing), since although there are formal similarities, time is not an observable of
    a mechanical system.

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

    [q2]> > Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional,[/q2]
    [q2]> > then objects of different masses will in general fall with different accelerations. The force[/q2]
    [q2]> > acting on a body of gravitational mass M is, say, F. Then the force acting on a body of[/q2]
    [q2]> > gravitational mass 2M is 2F, but if the inertial mass of the latter is not double that of the[/q2]
    [q2]> > former, then the acceleration will be different.[/q2]

    Don't you have it backwards: The different masses falling with different accelerations comes first,
    then hypotheses as to how and why follow from the observed behavior.

    Oh, you are talking about 'G' - universal gravitational constant and (g) - I'm going to bed, but,
    will read on in future. Too long out of education.

    Rich

    "Virgil" <[email protected]> wrote in message news:[email protected]...
    [q1]>[/q1]
    [q3]> > > Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional,[/q3]
    [q3]> > > then objects of different masses will in general fall with different accelerations. The force[/q3]
    [q3]> > > acting on a body of gravitational mass M is, say, F. Then the force acting on a body of[/q3]
    [q3]> > > gravitational mass 2M is 2F, but if the inertial mass of the latter is not double that of the[/q3]
    [q3]> > > former, then the acceleration will be different.[/q3]
    [q1]>[/q1]
    [q1]> Don't you have it backwards: The different masses falling with different accelerations comes[/q1]
    [q1]> first, then hypotheses as to how and why follow from the observed behavior.[/q1]

    Virgil <[email protected]> wrote:
    [q2]>> > Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional,[/q2]
    [q2]>> > then objects of different masses will in general fall with different accelerations. The force[/q2]
    [q2]>> > acting on a body of gravitational mass M is, say, F. Then the force acting on a body of[/q2]
    [q2]>> > gravitational mass 2M is 2F, but if the inertial mass of the latter is not double that of the[/q2]
    [q2]>> > former, then the acceleration will be different.[/q2]
    [q1]>Don't you have it backwards: The different masses falling with different accelerations comes first,[/q1]
    [q1]>then hypotheses as to how and why follow from the observed behavior.[/q1]

    That would be a different discussion, along the lines of 'what is a good theory to explain these
    observations?'.

    This discussion is 'is Galileo's thought experiment to argue that all objects fall at the same speed
    really a priori, or is he using real world knowledge implicitly?'

    Exhibition of a logically consistent (if disgusting) theory of gravitation in which all objects
    to not fall at the same speed is evidence that Galileo was using real world experience, not just
    pure logic.
    --
    Rob. http://www.mis.coventry.ac.uk/~mtx014/

    I'm thick, but, what about the fact the gravity is a constant ? What about a feather super-glued to
    a cannon ball ? Or, should I take pi tv2 into consideration ?

    Thicky, Rich

    "Virgil" <[email protected]> wrote in message news:[email protected]...
    [q1]>[/q1]
    [q3]> > > Maybe I am...but as far as I can see, if inertial and gravitational mass aren't proportional,[/q3]
    [q3]> > > then objects of different masses will in general fall with different accelerations. The force[/q3]
    [q3]> > > acting on a body of gravitational mass M is, say, F. Then the force acting on a body of[/q3]
    [q3]> > > gravitational mass 2M is 2F, but if the inertial mass of the latter is not double that of the[/q3]
    [q3]> > > former, then the acceleration will be different.[/q3]
    [q1]>[/q1]
    [q1]> Don't you have it backwards: The different masses falling with different accelerations comes[/q1]
    [q1]> first, then hypotheses as to how and why follow from the observed behavior.[/q1]

    Thom Baguley <[email protected]> wrote:
    [q1]>Isn't it a mistake to have two (implicit) contradictory definitions of mass?[/q1]

    It's a mistake in the sense that it disagrees with experience, not in the sense that it's logically
    inconsistent. (The original question was whether Galileo's argument was strictly a priori, or
    whether it requires 'real world' information.)

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

    Thom Baguley <[email protected]> wrote:
    [q1]>Having two contradictory definitions of the same concept must be logical inconsistent (which is not[/q1]
    [q1]>to say the argument is). The[/q1]

    That's not the same as having two different notions of mass (in particular, gravitational 'charge'
    and inertial mass) which need not, in principle, have anything to do with each other. It was a
    fundamental problem for a long time to explain the observed strict proportionality between
    gravitational and inertial mass.

    [q1]>argument breaks down unless the "size" of the the joined and unjoined objects is conceptually the[/q1]
    [q1]>same thing.[/q1]

    No. And, changing domains of discourse, it's not even true in modern physics that the mass of a
    composite object is the sum of the masses of the component parts, if they interact strongly. The
    mass of a deuteron is not the sum of the masses of a proton and a neutron. It is logically possible
    that the inertial masses of the two could add while the gravitational masses have a small binding
    energy deficit (or vice versa). Observationally, this does not happen. But it is not inconsistent
    that it should.

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

    Clark 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.[/q1]
    [q1]> But the joint body falls faster because it's more massive. Contradiction. So gravitational mass[/q1]
    [q1]> isn't proportional to the square of inertial mass.[/q1]
    [q1]>[/q1]
    [q1]> Where's the mistake in that? Sure enough you can escape the contradiction if the gravitational[/q1]
    [q1]> mass of the joint body isn't[/q1]

    Isn't it a mistake to have two (implicit) contradictory definitions of mass? You can only define a
    small mass object and large mass object (in this examples) as such if the mass is an essential
    property of the individual object (that doesn't change when you change the objects, by joining,
    dividing etc.). Mass is better considered a continuous quantity applied to collections of matter.
    Joing two objects creates a new collection that has a new mass?

    Thom

    This may not be what the original poster meant.

    Minky and Papert proved that the perceptron (a simple precursor to neural networks with no hidden
    layers) couldn't learn very simple things such as XOR (exclusive or). This killed neural network
    research almost stone dead until the 80s wwhen McClelland & Rumelhart (and others) started producing
    hidden layer networks.

    Thom

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> In article <[email protected]> , Clark <[email protected]> wrote:[/q1]
    [q2]> >Do I recall (very vague, this) that a priori reasoning has had some bad effects here? Was there[/q2]
    [q2]> >an impossibility/incompleteness result that stopped work on neural nets for a while until it was[/q2]
    [q2]> >worked around? Anyone know about this?[/q2]
    [q1]>[/q1]
    [q1]> Only to confirm that I have a similarly vague memory: I think it was in the context of something[/q1]
    [q1]> called a 'perceptron', but don't recall any more than that either.[/q1]
    [q1]> --[/q1]
    [q1]> Rob. http://www.mis.coventry.ac.uk/~mtx014/[/q1]

    Robert Low wrote:
    [q1]>[/q1]
    [q1]> Thom Baguley <[email protected]> wrote:[/q1]
    [q2]> >Isn't it a mistake to have two (implicit) contradictory definitions of mass?[/q2]
    [q1]>[/q1]
    [q1]> It's a mistake in the sense that it disagrees with experience, not in the sense that it's[/q1]
    [q1]> logically inconsistent. (The original question was whether Galileo's argument was strictly a[/q1]
    [q1]> priori, or whether it requires 'real world' information.)[/q1]

    Having two contradictory definitions of the same concept must be logical inconsistent (which is not
    to say the argument is). The argument breaks down unless the "size" of the the joined and unjoined
    objects is conceptually the same thing.

    Thom

    Thom Baguley <[email protected]> wrote:
    [q1]>glue or string used). I meant something slightly different though. The original verbal description[/q1]
    [q1]>says "small" and "large" object. The appeal to their different speeds of falling when joined[/q1]
    [q1]>involves an essentialist notion of "size" as opposed to some function of the original "sizes". They[/q1]
    [q1]>can't both be true unless two different meaning of "size" are used (in which case the argument is[/q1]
    [q1]>underdetermined).[/q1]

    Ah, I see what you're getting at. I think that I (and probably the others) had simply mentally
    translated 'small' and 'large' to 'light' and 'heavy' (in the context of cannonballs, or whatever)
    without even realising it. It really was the mass/weight that we were talking about, rather than the
    size as such: honest.

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