In celebration of the decision of CalTech to put The Feynman Lectures On Physics online, I thought I'd reproduce something he wrote in his wonderful little lecture QED: The Strange Theory of Light and Matter. The full lecture can be found here. As far as I know, this is the only lecture on quantum field theory aimed at a popular audience. I first read this book years ago, and I often find myself using Feynman's visualization tricks often. For instance, Feynman's visualization of complex numbers as being little clocks on points helped me make sense of complex analysis. I've found that even the obvious fact that \( e^z \) can pass from positive to negative without hitting zero can be a stumbling block for people raised on real numbers

Naturally, this lecture series starts out with a sort of mission statement and declaration of purpose. Reviewers prefer to skip these, instead reviewing the book/lecture/etc that they want instead of what they are really getting. And I'm sure that some will be disappointed that this book doesn't go into detail about, for instance, the mathematics of path integrals. But this throat clearing is an important part of engaging with an audience, even an audience that wants to see you. In so doing, Feynman actually produces an interesting thesis on the philosophy of science:

"The Maya Indians were interested in the rising and setting of Venus as a morning 'star' and as an evening 'star' - they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their 'nominal years' of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya invented a system of bars and dots to represent numbers (including zero), and had rules by which to calculate and predict not only the risings and settings of Venus, but other celestial phenomena, such as lunar eclipses.

In those days, only a few Maya priests could do such elaborate calculations. Now, suppose we were to ask one of them how to do just one step in the process of predicting when Venus will rise as a morning star - subtracting two numbers. And let's assume that, unlike today, we had not gone to school and did not know how to subtract. How would the priest explain to us what subtraction is?

He could either teach us the numbers represented by the bars and dots and the rules for 'subtracting' them, or he could tell us what he is really doing: 'Suppose we want to subtract 236 from 584. First, count out 584 beans and put them in a pot. Then take out 236 beans and put them on one side. Finally, count the beans left in the pot. That number is the result of subtracting 236 from 584.'

You might say, 'My Quetzalcoatl! What

*tedium*- counting beans, putting them in, taking them out - what a job!'

To which the priest would reply, 'That's why we have the rules for the bars and dots. The rules are tricky, but a much more efficient way of getting the answer than by counting beans. The important thing is, it makes no difference as far as the answer is concerned: we can predict the appearance of Venus by counting beans (which is slow, but easy to understand) or by using tricky rules (which is much faster, but you must spend years in school to learn them).'"

I think this is a wonderful metaphor for mathematical modeling, and it is a very good approach to teaching students. In my teaching, I very frequently encounter students completely uninterested in my subject - it would be more correct to say that I occasionally find students interested in math. What I do to engage the students is to encourage them to think that what I am teaching is not abstracta to be vomited onto a test, but tools they can use as scientists and engineers. I notice that I get much more engagement when I do this (it seems to also get higher grades, but I haven't done a regression or anything...).

Incidentally, Feynman earlier admits that his history of QED is a Whig history, and mentions a couple of other issues in then contemporary philosophy of science. Feyerabend's claim that he was philosophically ignorant was always complete horseshit, based on Feyerabend's unwillingness to confront new scientific and philosophical difficulties. For instance, Feynman's anti-foundationalism shows up in a later part of this section - to him, it's modeling all the way down. I don't know whether this is trivially true or exaggerated. Just wanted to beat that dead horse a little more.

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