[jarvist]

What do you not like about Feynman's "little arrows" / rotating clock hands in the QED book? I can't think of a more simple metaphor for the exponential of a complex phase, exp(i omega t). I suppose you could try and do it with more commonplace trigonometric functions, but then you lose the simple vector interpretation of adding the contributions. Or are you arguing that you should always try and teach complex numbers and the Euler identity to avoid strained analogies?

[hifrote]

> What do you not like about Feynman's "little arrows" / rotating clock hands in the QED book?

It’s difficult to articulate, but two aspects are:

The amount of times I have only confused people more by trying to explain even modular arithmetic by calling on the clock analogy.

And the fact that the little “clock hands” are a complete abstraction from both the physics being described and the mathematical models that describe that physics. ~“Quantum physics is just about adding clocks?”

> I can't think of a more simple metaphor for the exponential of a complex phase, exp(i omega t).

As I noted in the gp I think code implementations or numerical methods should be the goal.

The solution to the confusion about referencing clocks when talking about modular arithmetic was just to write down a complete numerical example, ie all natural numbers mod 6 up to 10, and use that as the abstraction for further discussion: negatives, reals, periodicity, infinities, applications, et al.

[klabb3]

> As I noted in the gp I think code implementations or numerical methods should be the goal.

I’m 100% with Feynman on this one. I loved the book because of the intuition it gave me about quantum physics. He even has this amazing analogy for how to teach arithmetic without numbers. Now, you could absolutely claim that he fails in his analogies (I’m not among the .1% of people if not less who can debate that), but I can still say claim confidently that math is not the goal. Abstraction is not intuition.

[zarzavat]

In mathematics, geometric and algebraic explanations are complementary.

If you plot a function, you can observe many properties easily, for instance where does it cross the axes? Is it symmetrical? How quickly does it grow?

However there are also many properties that are easier to observe algebraically. For example if you plot x^n you can see if n is odd or even, but you can’t see what value n has because x^10 looks very much like x^12. But if you have the algebraic representation you can read it off.

The issue with Feynman’s clocks is that he only provides the geometric explanation (what physicists would call “intuition”), and not the algebraic explanation.

This only helps two kinds of people: 1) people not capable of understanding the algebra, 2) people who already know the algebra and want to develop intuition.

For the third group of 3) people are capable of understanding the algebra but haven’t learned it yet, only talking about clocks is a bit dizzying.

[lupire]

I strongly disagree. The geometric explanation lets you understand the main concept without the hassle of algebra. The algebra isn't needed for these fundamental topics.

There's no requirement to do opaque algebra before approaching intuition.

Feynman invented Feynman Diagrams, which are a major contribution to Physics because they avoid algebra, and physicists are certainly capable of algebra.

[marcosdumay]

> people are capable of understanding the algebra but haven’t learned it yet

Those people are perfectly capable of taking any other textbook. What value is there in another book that explains the subject in the same exact way?

Besides, almost all of those people will have a much easier time picking the other textbook if they read the intuitive explanation first.

[red75prime]

And after you'll learn negatives, reals, periodicity, etc., you'll find that a rotating clock hand is a completely fine analogy. So, maybe it's not that bad to have this analogy from the beginning to not lose the forest behind the trees.

[Tao3300]

> The amount of times I have only confused people more by trying to explain...

That's not Feynman's fault!

[hifrote]

> Or are you arguing that you should always try and teach complex numbers and the Euler identity to avoid strained analogies?

I think it’s okay to be explain complex numbers. I think it’s just best to additionally explain why. That is, show why (real, imaginary) is a better numerical system than the more broadly taught (x,y) of the 2 dimensional space being explored.

As for the Euler identity I suppose you could include that when explaining why we use the exp() function, which is because it plays nicer with integration and derivation than other numerical representations.

I want the analogies to be representative of the work rather than my own mental model of it.