How to explain this kind of reasoning?

Question

First, sorry for the vague title, but I couldn't think of a descriptive title for this question.

I was teaching some physics to my nephew and eventually we got to the formula $$distance = speed \times time$$ I explained it to him and he agreed that the formula "makes sense" intuitively.

Then he tried to solve the following exercise: If we walk a distance of 6km in 2hours what is our speed?

He couldn't solve it. So I explained to him that he could use the formula above to deduce that $$speed = \frac{distance}{time}$$

But then he said: That makes no sense! You told me that $distance = speed \times time$, but you gave me no formula for the speed.

Well, that got me thinking. First I couldn't understand what was his doubt but then I finally got it. Here is what he meant:

Imagine you have a particle moving with constant speed on a line. If we watch it for 10 seconds we know that the particle has traveled an actual distance with certain fixed speed. We know that $d = s \times t$, where $d=$distance, $s=$speed, $t=$time. That way we can know the distance the particle has traveled in 10 seconds. Suppose that this formula is all we know from physics. I know that the number $s$ equals $s = \frac{d}{t}$, but how do I know that this $s$ is the ACTUAL distance traveled by the particle? In other words, we know that mathematically $d = s \times t$ is equivalent to $s = \frac{d}{t}$, but how do we know that this last formula "is valid in physics"?

Maybe this looks like a physics question but it's not. I understand that his doubt is an example of the following question:

Imagine we have a problem we want to study (i.e, particle moving on a line) and we interpret it mathematically. How do I know that the mathematical conclusions I get about my interpretation of the problem actually reflects some property of my original problem? For example, imagine I'm trying to solve a problem about real numbers $a,b$ such that $a^2+b^2=1$, so I can interpret the problem about the pair $(a,b)$ lying on the circle of radius $1$, from that fact I can use properties of the circle to learn something about my original problem, but how do I know that properties of the circle reflect properties of my original problem?

I've been doing this kind of reasoning my whole life, so I couldn't understand why someone would not understand it, I'm just used to it. How can I explain this in a clear manner? You may answer this question using another example of such reasoning, not necessarily this physics problem.

Thank you.

Answer 1

This is an interesting philosphical question:

Imagine we have a problem we want to study (i.e, particle moving on a line) and we interpret it mathematically. How do I know that the mathematical conclusions I get about my interpretation of the problem actually reflects some property of my original problem?

This is classical applied mathematics or (in a way) theoretical physics. You build a mathematical model of a physical system with properties that are physically (i.e. experimentally) correct, make mathematical deductions, and hope that the mathematical conclusions match the real world so can be checked by experiment. If your model and your mathematics are good enough you've done useful work. If not, it's back to the drawing board - improve the model or do better mathematics.

But I agree with some of the other commenters that in your particular problem the issue isn't the suitability of the model. I think it's in the definition of speed. Essentially, what you measure is distance and time. Then you use mathematics (i.e. division) to define the (average) speed. That means the statement $$ s = \frac{d}{t} $$ is just a definition telling you how to use the word "speed". It's not a physical law, nor is the consequence that $$ d = s \times t . $$ That's just a tautology.

Of course when the speed is not constant (that is, when measured distance isn't proportional to measured time) you have to invent calculus.

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An anecdote:

Years ago in the first lecture of a course on the mathematics of quantum mechanics Professor George Mackey began by drawing two big circles on the board, with arrows back and forth between them, all the while saying (but not writing) that one of the circles represented the real world and the other a mathematical abstraction. The arrow going one way was model construction, the other arrow model predictions.

Later in the lecture he returned to the metaphor, turned to point to the picture on the board and wondered (with no irony) "now which circle is the real world?"

Answer 2

His problem is probably not as philosophical as you make it out to be. Try giving him an example: suppose a car travels 110 miles in two hours at constant speed. How do you find the speed?

Answer 3

but how do we know that this last formula "is valid in physics"?

Because mathematics is the language that we use to express models of physical (or even social) situations. The consistent mathematical rules that have been developed, originally to model "real-world" situations, have shown to give results that are in accord with the behaviour of the real world (within acceptable tolerance). So our confidence has grown that the rules for manipulating mathematical expressions will always produce results consistent with the real world. However, mathematics has grown beyond the real-world and as long as it is internally consistent a mathematical system is deemed to be correct (e.g. non-commutative algebras) even if the maths has seemingly no application to the "real world". Sometimes the "abstract" mathematical tools later come to have real world application such as group theory applied to particle physics and many others.

Update I am reading this accessible essay by the mathematician Michael Atiyah on the relationship between mathematics and the physical world.