# Must we test whether e.g. $A=B$ and $A=C$ implies $B=C$ by experiment?

Chaper 10, conservation of momentum in "The Feynman Lectures on Physics" in the chapter entitled, the authors write that

Suppose we know from the foregoing experiment that two pieces of matter, $A$ and $B$ (of copper and aluminum), have equal masses, and we compare a third body, say a piece of gold, with the copper in the same manner as above, making sure that its mass is equal to the mass of the copper. If we now make the experiment between the aluminum and the gold, there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are. So now, by experiment, we have found a new law. A statement of this law might be: If two masses are each equal to a third mass (as determined by equal velocities in this experiment), then they are equal to each other. (This statement does not follow at all from a similar statement used as a postulate regarding mathematical quantities.)

I assume that by "a similar statement used as a postulate regarding mathematical quantities" the authors refer to the transitive axiom of algebra, that is, if $A=B$ and $A=C$, then $B=C$.

It seems to me that, using Feynman's definition of mass and the transitive axiom of algebra, one must conclude that it is the case that $B=C$ even without making an experiment. Why do Feynman et al claim that

there is nothing in logic that says these masses must be equal; however, the experiment shows that they actually are

• The transitive axiom of algebra relates symbols written down on a piece of paper. Feynman is talking about the outcomes of physical measurements. I think the broad point he is making is that one should not assume a priori that the abstract mathematics of algebra describes relationships between measurable quantities, but that it turns out experimentally that in this case (and in many others) it does. – Mark Mitchison Jul 9 '15 at 13:52
• I think you should read the whole section again. I believe Feynman is talking about you can always add more and more "laws" that work in special cases, but you have no reason to believe they will work in other cases. In a sense he is just saying be careful before making claims that something "should always happen" verse "happened in this case" – Peter Anderson Jul 9 '15 at 13:59
• what are the appropriate tags for this question? to be honest, this question is really about philosophy/epistemology of science. what knowledge can we justify etc – innisfree Jul 9 '15 at 14:02
• I'd say that assuming the masses were equal was very safe knowledge, even without the experiment. – innisfree Jul 9 '15 at 14:04
• A slightly different example, but in Aristotle's day everyone simply knew that if two unequal masses were dropped from the same height, the greater mass would fall faster than the lesser mass. There was no need to do the experiment because it was obvious. Then Galileo made the same observation as Feynman - There is nothing in logic that says the heavier mass must fall faster. So he did the experiment and found the unexpected result that they fell at the same rate. I believe that's Feynman's point - you cannot assume something just because it appears obvious - you must do the experiment. – Mark Jul 9 '15 at 17:10

Who can with certainty fathom the mind of a Feynman? :-)
However, it sounds like he is saying that there is no reason to decide with certainty that Gold, Copper and Aluminum do not have properties which affect how they behave when accelerated by a force.

An example that has some correspondence is carrying out the same experiment in air.
Mount 1 kg spheres of each metal on massless, frictionless, area less "carts" (made of unobtainium (or nonexisteum) ) such that they can be slid across a surface in a manner which is independent of the carts' properties. Now place any two spheres in contact on their carts and apply an explosion based force as before. The initial velocities will be the same as before and equal but some mysterious new force will apply that causes the spheres to act differently. Gold-Al and Gold-Cu will have different results for the non Al material. A few nano-seconds thought will allow us to see that in both cases the Al sphere behaves the same way and that there is some property of each material that affects how they behave once proceeding at a given velocity. We will have discovered air resistance and drag and the different densities will cause the different masses to have non-equal frontal areas so they will have different deceleration forces from drag.

While this is so obvious that it seems trivial, there may have been other properties of Au.Al/Cu that affected how they interacted in the initial experiment. Until we perform the experiment we cannot know what curve ball nature has or hasn't got in store.

This leads to his extremely valuable observation

   "From this example we can see how quickly we start to infer things if we are careless."


If we were unaware of air drag then expecting equal behaviour for all 3 materials after the initial explosion would leave us puzzled. That we are not puzzled in the free space case is because our expectation happened to ,match reality. But, we can never be sure that this is the case.

Real (out of this) world example:

For some while it seemed that there were 1/3 as many Neutrinos coming from the sun as calculations predicted. In due course it was posited and then experimentally "proven" that the Neutrinos 'change colour' as they amble on their way at (or just above or just below :-) ) light speed, and that only the 1/ which are the 'right' colour as they pass through our detectors get detected.

Solar neutrino problem

• Measurements of solar neutrino types were not consistent with models of the Sun's interior.

• Former Standard Model: Neutrinos should have been massless according to the then-accepted theory; this means that the type of neutrino would be fixed when it was produced. The Sun should emit only electron neutrinos as they are produced by H–H fusion. Observation

• Only one third to one half of predicted number of electron neutrinos were detected; neutrino oscillation explains the difference but requires neutrinos to have mass.

• Resolution: Neutrinos have mass and so can change type.

• Neutrino oscillation is a quantum mechanical phenomenon whereby a neutrino created with a specific lepton flavor (electron, muon or tau) can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates through space.

• First predicted by Bruno Pontecorvo in 1957, neutrino oscillation has since been observed by a multitude of experiments in several different contexts; it also turned out to be the resolution to the long-standing solar neutrino problem.)

• I don't really think this tackles the crux (as I see it) of the matter - do we have to test logical results experimentally? or is that $B=C$ safe, reliable knowledge already? – innisfree Jul 9 '15 at 14:06
• @innisfree Can you explain why you do not think it explains the issue? It seems to do to me. In maths we define certain properties and as maths is a model we= can make the model fit whatever assumptions we wish before we "turn it on". | Nature is not like that. We are unable to be certain of ANYTHING that we do not measure, and even then , we may be fooled by laws that do not apply as we expect. Au/Cu/Al might have had interactions between atoms or ...? It is only by having measured the results that we know it is not so. | Dark matter & energy are excellent candidates. Do they exist per se ... – Russell McMahon Jul 9 '15 at 14:15
• @Innisfree ... or are they attributes of mass or "gravity" (if the two can be distinguished) which we have yet to come to grips with. | Solve that and you can start on QM and gravity :-). || Rutherford said there was "Physics and stamp collecting". In reality - it's ALL stamp collecting. Physics is just descriptions of what we observe framed in maths speak. That's perfectly fine - but when something breaks (eg dark xxx) we have to re-examine some of the stamps :-). – Russell McMahon Jul 9 '15 at 14:20
• interesting ideas, not sure I agree though. surely an important aspect of science is making inferences from data, once you've collected it (i.e. it isn't all stamp/data collecting, but inferences from stamp/data collections). – innisfree Jul 9 '15 at 14:31

I think the content of his statement is that "the assignment of mass to an object as defined using the conservation of momentum experiment is a transitive property". Not all relations are transitive properties (for instance, "Wolves eat Deer, Deer eat grass, but wolves don't eat grass"). For instance, if we had defined "the mass of an object" as a weight $mg$, the transitive property might not be satisfied unless the experiment was performed at the same location on Earth.

So in my mind, the story from Feynman is a statement about how mass is well-defined using the conservation of momentum. When we say "the mass of the car is 500 kg", we hope that can mean everything that the statement "$m_{car}=500$ kg" does, and when we perform the experiment we have some evidence that it is at least transitive, as it should be.

We know how real numbers (in the mathematical sense) behave. There is no a priori reason however to assume that masses of objects behave as real numbers. The proposed experiment can be seen as a check to see if masses actually can be modeled by (a subset of the) real numbers.