About the use of nominal definitions in physics

[cianfa72]

TL;DR Summary

About the meaning of nominal definitions employed in physics

Hi,
I was reading the interesting lecture of Feynman about Characteristics of Force -- https://www.feynmanlectures.caltech.edu/I_12.html

He basically says that nominal definitions like mathematical definitions of "abstract" objects have actually no physical meaning. For instance take the following:

Suppose we have a law which says that the conservation of momentum is valid if the sum of all the external forces is zero; then the question arises, “What does it mean, that the sum of all the external forces is zero?” A pleasant way to define that statement would be: “When the total momentum is a constant, then the sum of the external forces is zero.”There must be something wrong with that, because it is just not saying anything new.

Here he says that if we take as definition of "the sum of external forces is zero" as "When the total momentum is constant, then the sum of external forces is zero" we're actually going in circle.

The point I would try to make is that nominal definitions are indeed of the type if and only if. Thus the statement "When the total momentum is constant, then the sum of external forces is zero" should actually be of the type "the sum of external forces is zero if and only if the total momentum is constant". With this position the statement "the conservation of momentum is valid if the sum of all external forces is zero" is no longer a law (it is instead the statement defining "the sum of external forces s zero").

Did I get it correctly ? Thank you.

 

[pbuk]

Did I get it correctly ?

Not really. The point Feynman is making here is that using words to define things does not help you understand Newton's laws:

One might sit in an armchair all day long and define words at will, but to find out what happens when two balls push against each other, or when a weight is hung on a spring, is another matter altogether, because the way the bodies behave is something completely outside any choice of definitions.

Your insertion of "if and only if" doesn't change this.

 

[cianfa72]

Your insertion of "if and only if" doesn't change this.

I believe from a logical point of view if and only if is actually necessary. How else can we reasonably define a new object ?

E.g. "When the total momentum is constant, then the sum of external forces is zero" is a logical implication of type $\Rightarrow$ but from a logical point of view it might be a zero external force acting on a body/system of bodies even if the total momentum is not constant (the other way implication $\Leftarrow$).

To me the only logical reasonable way to get a nominal definition of a new object must be of the form "if and only if".

 

[PeroK]

By convention, in mathematics if and only if is not always specified in a definition.

I suspect because the equivalence in a definition is generally understood.

 

[cianfa72]

By convention, in mathematics if and only if is not always specified in a definition.

Yes, it is always implied even if not explicitly specified in the definition statement.

 

[Ibix]

The point Feynman is making here is that using words to define things does not help you understand Newton's laws:

I agree with this, @cianfa72.

If I tell you that a scribblex is a fooddely scribblex if the warghandle is zero you might (reasonably) complain that this does not help you to understand what I'm talking about. Reversing it, as Feynman does, gives you that when the warghandle is zero the scribblex is fooddely, which is no more informative. Adding "...and only if", as you do, doesn't help either.

In natural sciences there always has to come a point where it grounds in our common experience of reality. I point at something and say "this is a scribblex". Only then, once you have an example of a scribblex, will you be able to start to understand my nonsense sentence.

As you progress in your understanding of physics we no longer point at a real object, but rather we refer to abstract concepts with which you are already familiar - conservation laws and inertia in Feynman's example. But they are still grounded somewhere in your (and my and Feynman's) common knowledge that stopping something moving is hard, and it can knock you over if you aren't set.

 

[cianfa72]

If I tell you that a scribblex is a fooddely scribblex if the warghandle is zero you might (reasonably) complain that this does not help you to understand what I'm talking about. Reversing it, as Feynman does, gives you that when the warghandle is zero the scribblex is fooddely, which is no more informative. Adding "...and only if", as you do, doesn't help either.

The reverse (not the contrapositive) of your statement should actually be "the warghandle is zero if a scribblex is a fooddely scribblex" i.e. "when a scribblex is a fooddely scribblex the warghandle is zero".

Anyway I agree with you, adding "if and only if" doesn't help. However the point I was trying to make is that -- from a logical point of view -- "if and only if" is actually implied in the nominal definition statements.

 

[nasu]

Can you explain what do you mean by "nominal definition"? I did not find any specific and unique definition of this term.

 

[cianfa72]

Maybe the better term to use for this type of definition is "mathematical definition" -- https://en.wikipedia.org/wiki/Definition

Or perhaps stipulative definition -- https://en.wikipedia.org/wiki/Stipulative_definition

 

[nasu]

I don't see the term "nominal definition" in the links. So, is there any difference between just using the term "definition" and adding "nominal" to it?

How would "stipultive definition" apply to your discussion? If it is used to define "a new or currently existing term is given a new specific meaning", what is the term and what is the new specific meaning?

 

[cianfa72]

I don't see the term "nominal definition" in the links. So, is there any difference between just using the term "definition" and adding "nominal" to it?

Adding nominal was just to "reinforce" the point that the defining statement is given in terms of other already defined terms/objects.

How would "stipulative definition" apply to your discussion? If it is used to define "a new or currently existing term is given a new specific meaning", what is the term and what is the new specific meaning?

Yes, maybe it is not the right definition's type to use.

 

[nasu]

Adding nominal was just to "reinforce" the point that the defining statement is given in terms of other already defined terms/objects.

And what will you call a definition which is not in terms of other already defined objects or terms? Do you have an example?

 

[cianfa72]

Do you have an example?

On an intuitive ground it might be based just on the "intuition" about the meaning of the term being defined; however it is not of the type we're interested in.

 

[nasu]

Then, in the end, any definition will end up with this intuitive understanding of some terms. This is one of the things Feynman is saying, I believe. You have to start with some experience about the real world, if you want to do physics. In math maybe they can use as starting point some abstract definitions that have no need for intuition. Even though, if you look at Euclid's original definitions, they don't really define these things unless you already know what they are.

 

[Ken G]

The way I would put it is, math and physics both rely on a type of surprise, but a very different type. In mathematics, our surprise is that two seemingly unrelated things end up being logically equivalent, that's a "proof." In physics, our surprise is that some dreamed up abstract notion ends up describing how things work in our experience, that's a "test." Math proves theorems, physics tests predictions. There isn't any big difference in how definitions work, but there's a big difference in how we apply them.

So you can always define forces as things that change momentum, and you can equally define momentum as the thing that forces change. What matters is, you must have one way to get forces, and another way to get momentum, based on experience in other areas, so that when you combine them in some new situation, you can succeed in predicting something you've never seen before.

 

[cianfa72]

Take $F=ma$ for example. The point Feynman made is that Newton had assumed the terms mass, acceleration and force as already defined. Regarding the force, in particular, he gave an explicit example: the law of universal gravitation.

This way $F=ma$ is actually a law between already defined terms and not just a definition of force.

 

[Ken G]

And another very important element of Newton's law is that forces are additive. So even if you define a force as something that produces an ma (and I've seen textbooks do just that), it is essential to the law that if you have more than one such thing operating at the same time, then you can just add them.

It is the same with m. I've seen the concept of inertial mass defined to be F/a, which also makes Newton's law seem like some kind of tautology, until you recognize the crucial fact that if you take one object that has one ratio F/a, and another object with a different such ratio, and attach them to each other, you get an object with a new F/a ratio that you can calculate by adding the previous two ratios.

Finally, we should mention the way "a" works as well, as that is part of the law too. Here you need a relativity to go from one frame to another, and Newton thought it was Galilean relativity (which we now know is only an approximation). Newton's law is not useful without such a principle of relativity, that tells you what to do with "a" when you have different observers watching the same motion.

All these important properties underpin the conceptual meaning of the quantities in Newton's law, and make it much more than just an equation involving three things in which any two can give you the third.

 

[cianfa72]

Newton's law seem like some kind of tautology, until you recognize the crucial fact that if you take one object that has one ratio F/a, and another object with a different such ratio, and attach them to each other, you get an object with a new F/a ratio that you can calculate by adding the previous two ratios.

Yes, and basically starting from that definition of mass as the $F/a$ ratio we discover --by means of a physical experiment-- a new physical law.

 

[gleem]

Note that in this paragraph Feynman is stressing that physical laws must be considered approximations. Feynman knows that speed cannot be infinite and that gravity is not a force. What the student is learning will eventually have to be refined.

I don't know but I think the statement about which this thread concerns should not be analyzed. I think He had a thought about definitions which He understood but failed to convey through this statement. I say this because gravity changes momentum but no force is acting.

 

[Ken G]

I actually think Feynman, in characteristic style, has hit the nail on the head. If you say that gravity changes momentum without a force acting, I think you missed his central point: "The trick is the idealizations." We have a better theory of gravity that says gravity is not a force, and a change in (three) momentum is a change in a relationship with an observer, not a property of an object as Newton thought. Yet in schools, we still teach that gravity is a force to hundreds of times more students than we teach the better theory to, and we hardly mention relativity and the role of the observer at all. Why is this? Because the trick will always be in the idealizations, no matter what theory you decide to apply.

 

[cianfa72]

And another very important element of Newton's law is that forces are additive. So even if you define a force as something that produces an $ma$ (and I've seen textbooks do just that), it is essential to the law that if you have more than one such thing operating at the same time, then you can just add them.

ok, you mean the laws (connecting/drawing a relationship between already defined terms) into which the forces defined as the product $F:=ma$ enter, right ?

 

[f95toli]

This way $F=ma$ is actually a law between already defined terms and not just a definition of force.

No, not really,
Back before QM was discovered people still had long discussions about interpretations and definitions in physics, and F=ma is certainly not an exception, rather the opposite.
Ernst Mach's "The Science of Mechanics" goes into this in excruciating detail, without actually "solving" anything.
(for some reason I did read it for a school project, a long long time ago)

 

[cianfa72]

No, not really,
Back before QM was discovered people still had long discussions about interpretations and definitions in physics, and F=ma is certainly not an exception, rather the opposite.

Yes definitely. My point, I believe, was the one made by Feynman about the meaning of $F=ma$. For him & Newton it is actually a law connecting/drawing a relationship between already defined terms and not a just a nominal/mathematical definition of the term on the left side (the term $F$).

For instance mass $m$ is a term operationally defined and acceleration $a$ is the cinematic acceleration of a mass measured in an inertial frame.

 

[f95toli]

Yes definitely. My point, I believe, was the one made by Feynman about the meaning of F=ma. For him & Newton it is actually a law connecting/drawing a relationship between already defined terms and not a just a nominal/mathematical definition of the term on the left side (the term F).

Sure. but my point was that if you go back ~100 years or so there were plenty of people who disagreed with that viewpoint. For example, Mach was very critical of Newton when it came to the definition of mass, and proposed an alternative definition. The issue of course is that defining mass completely independently of force and acceleration is not easy. Again, there is a LOT written about this ("The science of mechanics" is not a short book)
I don't really have an opinion either way(personally I don't think it is a very interesting discussion), I was merely pointing out that the idea that there were (or are) no "philosophical issues" with classical mechanics is wrong.

 

[cianfa72]

Sure. but my point was that if you go back ~!00 years or so there were plenty of people who disagreed with that viewpoint.

You mean go back 100 years, I guess

 

[Ken G]

ok, you mean the laws (connecting/drawing a relationship between already defined terms) into which the forces defined as the product $F:=ma$ enter, right ?

Right, the law is not just an algebraic connection between three quantities, it is also a set of rules about what those quantities are and how they combine in various systems. Fortunately those rules are generally quite simple (masses add, forces add, and accelerations must be the same if the system moves as a unit).

 

[WWGD]

I believe from a logical point of view if and only if is actually necessary. How else can we reasonably define a new object ?

E.g. "When the total momentum is constant, then the sum of external forces is zero" is a logical implication of type $\Rightarrow$ but from a logical point of view it might be a zero external force acting on a body/system of bodies even if the total momentum is not constant (the other way implication $\Leftarrow$).

To me the only logical reasonable way to get a nominal definition of a new object must be of the form "if and only if".

I've been mislead into trying to prove definitions that use the " If and only If".

 

[cianfa72]

I've been mislead into trying to prove definitions that use the " If and only If".

Sorry what do you mean?

 

[WWGD]

Sorry what do you mean?

I mean, statements using an if and only if are often assumed to be theorems, results, that need to be proven. Ive mistakenly assumed , while being dostracted, at times ,that definitions using the if and only if were results requiring a proof.