Questions about Feynman's contrasting Definitions and Laws in physics

[mark2142]

https://www.feynmanlectures.caltech.edu/I_10.html
(From the paragraphs after equation 10.5)
'It is not just a definition to say the masses are equal when the velocities are equal, because to say the masses are equal is to imply the mathematical laws of equality, which in turn makes a prediction about an experiment.'

I am trying to understand about laws which Feynman is trying to explain.
1. What are mere definitions he is mentioning and what are laws?
2. Can anybody give me more examples like this where its not just definition but law where through mathematical laws there are predictions about an experiment which would produce new laws?

 

[mark2142]

He is just saying as I understand is that if say we define something like if the velocities are equal after explosion then the masses are equal too. Now if we experiment with A and B and then with A and C and establish that they are equal masses then by mathematical law of equality it is suggesting to check with an experiment that whether B=C. And it is. So there is a new law which says if two masses are equal to a common mass then they are equal to each other. And this new law is obtained by our definition. So that is not just a definition any more but a physics law.

 

[Hall]

Okay, do you remember this statement:
“Things which are equal to same things are equal to one another”.
?

 

[mark2142]

Okay, do you remember this statement:
“Things which are equal to same things are equal to one another”.
?

Hi, Yes I remember.

 

[Hall]

Hi, Yes I remember.

Do you also remember that it was one of Euclid's axioms? It was not a proven result, right?

 

[mark2142]

Do you also remember that it was one of Euclid's axioms? It was not a proven result, right?

Yeah! Axiom 1. I don't know if its proven or not but its a common sense.

 

[malawi_glenn]

I don't know if its proven or not

Axioms are never proven, they are taken to be true apriori

 

[Hall]

Yeah! Axiom 1. I don't know if its proven or not but its a common sense.

No, it’s not common sense. If I and (had I had a sister) my sister are equal to my parents (we both would have been their children, so, equal love) would that have made me equal to my sister? (Though President Lincoln used it very exquisitely)

$A=B; A=C \implies B=C$ applies only to mathematical objects. Euclid said it to assume in order to begin mathematics.

What Feynman was saying, assume that masses are equal only when the explosion (the explosion of which he was talking about) throws them with equal velocities. Now, Feynman uses this axiom (or observed law) to prove that axiom of Euclid’s, commonly called law of transitivity. Here is what Feynman is doing:
Suppose object A and B are thrown with same velocities in that explosion, so they have same mass by definition. Now, we take another object C whose mass is equal to B, and we also know $\text{mass}(A)= \text{mass}(B)$. Put C and A into that explosion, it is found that they fly away with same velocity, hence establishing $\text{mass}(A) = \text{mass}(C)$. So, this proves the law of transitivity experimentally:
$\text{mass}(A)= \text{mass}(B);~ \text{mass}(B) = \text{mass}(C)$ $\implies \text{mass}(A)= \text{mass}(C)$

You must notice that we assumed a definition for proving that axiom, that is, “masses are equal when they fly off with same velocities”.

 

[mark2142]

Though President Lincoln used it very exquisitely)

President Lincoln did not really understood it. So he gave a wrong example.

Feynman uses this axiom (or observed law) to prove that axiom of Euclid’s, commonly called law of transitivity

Sorry but I don't agree. Feynman is saying look its not just a definition(masses are equal when velocities are) but a law because mathematics suggests us an experiment and when done it produces a new law(A=B=C).

 

[PeroK]

President Lincoln did not really understood it. So he gave a wrong example.

Sorry but I don't agree. Feynman is saying look its not just a definition(masses are equal when velocities are) but a law because mathematics suggests us an experiment and when done it produces a new law(A=B=C).

The mathematical property in question is called being transitive. It is a required property for an equivalence relation. Not all relations are equivalence relations.

Whether $A = B$ and $A = C$ implies $B = C$ depends in how $=$ is defined.

 

[jbriggs444]

No, it’s not common sense. If I and (had I had a sister) my sister are equal to my parents (we both would have been their children, so, equal love) would that have made me equal to my sister?

Equality can be seen as a particularly strong form of equivalence. Two objects are equivalent if they have the same attribute. (e.g. their remainder when divided by five). Two objects are "equal" if all of their attributes are identical.

Having one attribute in common (same amount of love from a particular parent pair) would amount to equivalence, not equality.

 

[mark2142]

I don't want to go into mathematical definitions. I want to know what is a definition and what is a law that Feynman is talking about. Does law means that it happens in reality which describes our world whereas definition is just...( I don't know)?

 

[jbriggs444]

I don't want to go into mathematical definitions. I want to know what is a definition and what is a law that Feynman is talking about. Does law means that it happens in reality which describes our world whereas definition is just...( I don't know)?

Definitions are just words. If they do not improve your understanding and you do not need to communicate with anyone using the defined words, you are free to ignore them.

 

[sophiecentaur]

President Lincoln did not really understood it. So he gave a wrong example.

Sorry but I don't agree. Feynman is saying look its not just a definition(masses are equal when velocities are) but a law because mathematics suggests us an experiment and when done it produces a new law(A=B=C).

President Lincoln was just another danged politician. He he said what he said in order to achieve an effect. We are getting a lot of that in the UK at the moment and one should never use what a politician says to support your arguments. Same goes for philosophers and salesmen.

 

[mark2142]

Definitions are just words. If they do not improve your understanding and you do not need to communicate with anyone using the defined words, you are free to ignore them.

But definitions are attached to this world that is around us. A definition is not a fiction, like a law. One defines something that is already present in this world like a machine or book or Tv or ball or iron or pillow. Can you give me an example please?

 

[jbriggs444]

But definitions are attached to this world that is around us. A definition is not a fiction, like a law. One defines something that is already present in this world like a machine or book or Tv or ball or iron or pillow. Can you give me an example please?

A "square" is a rectangle where all four sides are equal.

That says nothing whatsoever about anything. It is purely words. It does not enrich geometry in any way. It merely abbreviates it. It allows us to say "square" instead of "rectangle with four equal sides".

We can proceed to a more physically grounded example. Newton's second law...

"The rate of change of momentum is proportional to the force applied" Or, in mathematical notation with a well chosen set of units, and an assumption of mass conservation: $F=ma$.

If we do not have a good definition of "force" already in hand, this can be seen as a definition of force. It says nothing about the world around us. It just says that if we can make quantitative measurements of mass and of acceleration then we can compute their product, arrive at a numeric result and call that result "force".

[This is the modern interpretation of the second law]

On the other hand, if we do have a notion of "force" already in mind and a way to measure it then this same statement can be seen as a law of motion rather than as a definition.

[This is the interpretation of the second law that I felt most comfortable with as a first year physics student]