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August 27, 2009

Feynman the Babylonian

Posted by David Corfield

When I next get around to teaching philosophy of science again I think I’ll have my students watch an episode or two from Feynman’s Messenger series of lectures. With the possibility of having studied only humanities subjects from the age of 16 it’s no surprise that there’s many a student with only the sketchiest idea of physics. Last time around when I asked the class to name any of Newton’s laws of motion, a long wait ensued before someone finally proffered an approximation to the first.

I was watching the second lecture yesterday, towards the end of which Feynman distinguishes between what he calls a ‘Babylonian’ approach and a ‘Greek’ one. His idea here is that mathematicians have a tendency to arrange their theories in the Greek style on an axiomatic basis, while this can’t work in physics, at least in a time of theoretical growth, because it is never clear which approach is basic. For example, one may view a physical theory in terms of forces or field potentials or conservation laws or paths of least action, and unpredictably any one of them might provide new insight as to how to link together physical facts.

I found myself torn between responding in two different ways:

  1. The difference is only apparent. Multiple ways of thinking about the same piece of mathematics play the same role of allowing insight into the best way to proceed there.
  2. The difference is real, but mathematics can be done in the same ‘Babylonian’ way, and this should be encouraged.

As representatives of these two responses, let’s take a look at passages by William Thurston and Pierre Cartier.

In On proof and progress in mathematics Thurston writes:

People have very different ways of understanding particular pieces of mathematics. To illustrate this, it is best to take an example that practicing mathematicians understand in multiple ways, but that we see our students struggling with. The derivative of a function fits well. The derivative can be thought of as:

  1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
  2. Symbolic: the derivative of x nx^n is nx n1n x^{n−1}, the derivative of sin(x)sin(x) is cos(x)cos(x), the derivative of fgf \circ g is fg*gf' \circ g * g', etc.
  3. Logical: f(x)=df'(x) = d if and only if for every ϵ\epsilon there is a δ\delta such that when 0<|Δx|<δ0 \lt |\Delta x| \lt \delta, |f(x+Δx)f(x)Δxd|<ϵ. \left| \frac{f(x + \Delta x) &#8722; f(x)}{\Delta x} - d \right| \lt \epsilon.
  4. Geometric: the derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent.
  5. Rate: the instantaneous speed of f(t)f(t), when tt is time.
  6. Approximation: The derivative of a function is the best linear approximation to the function near a point.
  7. Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

This is a list of different ways of thinking about or conceiving of the derivative, rather than a list of different logical definitions. Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.

I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practicing with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding.

The list continues; there is no reason for it ever to stop. A sample entry further down the list may help illustrate this. We may think we know all there is to say about a certain subject, but new insights are around the corner. Furthermore, one person’s clear mental image is another person’s intimidation:

37. The derivative of a real-valued function ff in a domain DD is the Lagrangian section of the cotangent bundle T *(D)T^{*}(D) that gives the connection form for the unique flat connection on the trivial \mathbb{R}-bundle D×D \times \mathbb{R} for which the graph of ff is parallel.

(pp. 3-4)

Now for Cartier, where in Mathemagics (A Tribute to L. Euler and R. Feynman) he writes

The implicit philosophical belief of the working mathematician is today the Hilbert-Bourbaki formalism. Ideally, one works within a closed system: the basic principles are clearly enunciated once for all, including (that is an addition of twentieth century science) the formal rules of logical reasoning clothed in mathematical form. The basic principles include precise definitions of all mathematical objects, and the coherence between the various branches of mathematical sciences is achieved through reduction to basic models in the universe of sets…

My thesis is: there is another way of doing mathematics, equally successful, and the two methods should supplement each other and not fight.

This other way bears various names: symbolic method, operational calculus, operator theory… Euler was the first to use such methods in his extensive study of infinite series, convergent as well as divergent. The calculus of differences was developed by G. Boole around 1860 in a symbolic way, then Heaviside created his own symbolic calculus to deal with systems of differential equations in electric circuitry. But the modern master was R. Feynman who used his diagrams, his disentangling of operators, his path integrals… The method consists in stretching the formulas to their extreme consequences, resorting to some internal feeling of coherence and harmony. They are obvious pitfalls in such methods, and only experience can tell you that for the Dirac δ\delta-function an expression like xδ(x)x \delta(x) or δ(x)\delta'(x) is lawful, but not δ(x)/x\delta(x)/x or δ(x) 2\delta(x)^2. (p. 3)

On reflection, the Thurston response is the more relevant. Feynman’s thought about the dangers of the ‘Greek’ approach wasn’t so much about concerns of rigour excluding ways of calculating, as about obstructing ways of thinking by ordering concepts as more or less basic, so reducing heuristically valuable ways of thinking to derivatives of more basic reasoning and thereby neglecting them. But this is just what Thurston warns against within mathematics itself: “Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and explicit definitions.”

Posted at August 27, 2009 1:52 PM UTC

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Re: Feynman the Babylonian

So what does it mean if I want an idea to be understood in a variety of formal systems?

Posted by: Toby Bartels on August 27, 2009 2:20 PM | Permalink | Reply to this

Re: Feynman the Babylonian

I guess in an ideal world all forms of heuristic thinking would be presentable in at least one of your formal systems, and then you’d have the means to translate this thinking across systems. Come to think of it that’s the line being pushed for different approaches to n-categories.

I wonder if a case could be made that had category theory been better entrenched at the time that distributions, δ\delta-functions and all that would have received more quickly. Perhaps one could claim this for Finsler’s set theory.

Posted by: David Corfield on August 27, 2009 2:50 PM | Permalink | Reply to this

Re: Feynman the Babylonian

To be more specific:

I like to see precise statements as in the ‘logical’ version of the derivative, but that version is not the only possibility of that form. We have

  • f(x)=df'(x) = d if and only if for every ϵ\epsilon there is a δ\delta such that when 0<|h|<δ0 \lt |h| \lt \delta, |f(x+h)f(x)hd|<ϵ \left|\frac{f(x + h) &#8722; f(x)}{h} &#8722; d\right| \lt \epsilon (classical version, corrected form of what David cited);
  • f=gf' = g if and only if for every ϵ\epsilon there is a δ\delta such that when 0<|xy|<δ0 \lt |x - y| \lt \delta, |f(x)f(y)xyg(x)|<ϵ \left|\frac{f(x) - f(y)}{x - y} - g(x)\right| \lt \epsilon (uniform version, required in some constructive schools)
  • f=gf' = g if and only if df=gdx\mathrm{d}f = g \mathrm{d}\mathrm{x}, where x\mathrm{x} is the identity function, modulo various ways to define d\mathrm{d} (some circular, but we don't allow those);
  • f(x)=df'(x) = d if when hh is a nonzero infinitesimal, f(x+h)f(x)hd \frac{f(x + h) - f(x)}{h} - d is infinitesimal (nonstandard analysis);
  • etc.

While it would be possible to unwrap all of these further into some bare-bones language like ZFC (or ETCS, let's not blame the set theorists) and prove their equivalence there, I don't think that anyone can really argue that any of these is more basic than the others. And while there certainly are people who will argue that the full unwrapping in the language of some axiomatic set theory is more fundamental, I would dispute that; these can all be placed within an axiomatisation of analysis directly that requires no underlying set theory at all, no more than Euclidean geometry does.

By the level of rigour and logical precision, this is all mathematics in the style of Pythagoras and Hilbert, not the hand-waving of physicists. But it is Babylonian in its plurality of approaches, each of which can cast light on the others.

Posted by: Toby Bartels on August 27, 2009 2:59 PM | Permalink | Reply to this

Re: Feynman the Babylonian

My ‘logical’ definition of the derivative which you corrected reproduced a mistake from Thurston’s original paper and introduced a new one. I’ve corrected these in the post.

Posted by: David Corfield on August 27, 2009 3:29 PM | Permalink | Reply to this

Re: Feynman the Babylonian

While it would be possible to unwrap all of these further into some bare-bones language like ZFC (or ETCS, let’s not blame the set theorists) and prove their equivalence there, I don’t think that anyone can really argue that any of these is more basic than the others.

Actually, I could be persuaded to make an argument not a million miles away from this claim.

There are lots and lots of bare-bones languages, and more equivalences of theories become provable when stronger axioms are available in the base. For example, locale theory and point-set topology are basically equivalent in ZFC, but cease being so in ZF – many theorems stated topologically require choice to prove, whereas their localic analogues don’t. So this constitutes a kind of argument that locale theory is more basic than point-set topology.

Even if you’ve drunk the constructive Kool-Aid, we’re not done with the argument: Sambin’s basic topology refuses impredicative definitions and the extensionality of equality at function types (both of which are okay topos-theoretically), and so must introduce distinctions that locale theory doesn’t.

I think I’d rather say that two conceptions of a theory might or might not be equivalent relative to a given choice foundations. If two intuitively-different conceptions come out the same, then it’s possible that this is a pleasing underlying symmetry, but also that the metalogic is too strong!

Posted by: Neel Krishnaswami on August 31, 2009 2:10 AM | Permalink | Reply to this

Re: Feynman the Babylonian

I have been reading this blog for a few months hoping to learn a little abstract math by osmosis. And since I understand derivatives pretty well (I know math on an engineering level, i.e. vector calculus and linear algebra), I thought I’d ask, how much of a stretch would it be for me to understand Thurston’s Derivative Interpretation #37?

I also ask because in reading the Wikipedia entry on Maxwell’s Equations recently, I ran into mention of line bundles. I tried to understand what they were but couldn’t see how they differ from vector fields. Can anyone explain?

Posted by: M.E. on August 27, 2009 2:48 PM | Permalink | Reply to this

Re: Feynman the Babylonian

M.E., there’s a really great book called “Gauge Fields, Knots, and Gravity” that (IIRC) talks a bit about bundles. Now if I could only recall who the author was…

Posted by: John Armstrong on August 27, 2009 4:19 PM | Permalink | Reply to this

Re: Feynman the Babylonian

Heh;)
by John C. Baez (Author), Javier P. Muniain (Author)
http://www.amazon.com/Gauge-Fields-Knots-Gravity-Everything/dp/9810220340

Posted by: Jonathan Vos Post on August 27, 2009 5:04 PM | Permalink | Reply to this

connections and line bundles

A connection is a way of measuring constancy of a function. (The word “parallel” that Thurston uses can be read as “constant”. The connection between the two words is that a constant function has a graph that is parallel to the x-axis.)

Here is a loose explanation. Consider the function y = f(x); we say that f is constant if y doesn’t change as x does. But suppose now that one rescales the y-variable, in a way that could depend on x. Then a function that isn’t constant before the change of variable can become constant after the change of variable; and conversely, a non-constant function may become constant after the change of variable.

Such a choice of rescaling is called a connection. And Thurston’s point 37 relates the derivative of f to the choice of connection (rescaling) needed to make f become constant.

As for a line bundle: The different between a line bundle and a vector field is the difference between the x-y plane (which is the so-called “trivial line bundle” over the x-axis) and the graph of function (which is in this language called a “section of the line bundle”).

Think of the x-y plane, where the x-axis will be the domain of our functions (or “sections”, in line bundle language) and the vertical lines (i.e. the ones parallel to the y-axis) are the ones on which we will plot the values of our functions. If you think of the x-axis as a length of wire, and the vertical lines as being strung along the wire (I imagine fence posts strung along fencing wire) you can see why we call it a line bundle: it literally looks like a bunch of lines strung along the wire.

Now imagine that the x-axis (our wire strand) formed a loop, instead of just an infinitely long straight strand. (So we will be considering functions whose domain is a circle, rather than the real line.) The vertical line are still staying as vertical lines. This is the trivial line bundle over circle.

Now choose a point on the circle; call it the origin. As you move around the circle away from the origin, start tilting the vertical lines, more and more as you move around, so that when you are opposite the origin they are horizontal, and so that when you come around a full 360 degrees they have tilted all the way over so that they become vertical again (but what was the “top” of the line is now the “bottom”). You have now made a non-trivial line bundle over the circle. It looks like a bundle of fence posts strung around a circular piece of wire that are falling over more and more as they go around the circle, so much that they have fallen all the way upside down when you get back to the origin.

The picture also looks just like a Mobius strip, which is in fact what it is.

If you plot a graph on this non-trivial line bundle, you are giving an example of what is called a section of it. One thing you can try to prove: whereas it is easy to make a no-where zero section of the trivial bundle (just choose a non-zero constant function), you can’t find a non-zero section of the non-trivial bundle: any graph you try to draw will be forced to cross the x-axis (i.e. the circle around which all the lines are strung).

You can make more complicated bundles by working with more complicated, higher dimensional spaces in place of the x-axis, and bundling together higher dimensional vector spaces (say planes, 3-spaces, etc.) instead of just lines. Vector fields are examples of sections of such more general “vector bundles”. (Usually, “vector field” refers to a section of the so-called tangent bundle of the base manifold — “the base manifold” being whatever more complicated space you put in place of the x-axis.)

Posted by: Matthew Emerton on August 29, 2009 6:01 PM | Permalink | Reply to this

Re: Feynman the Babylonian

It remains to parse “Lagrangian section of the cotangent bundle” part of Thurston’s definition. After mucking around with tautological 1-form α\alpha and the resulting symplectic form dαd\alpha, one discovers that “Lagrangian section” means closed 1-form. For the real line, any closed 1-form is exact, so “Lagrangian section” becomes dgd g for some function gg. The second part of the definition guarantees that g=f+cg =f +c, where cc is a constant.

Posted by: Eugene Lerman on August 31, 2009 3:41 AM | Permalink | Reply to this

Re: Feynman the Babylonian

I’ll just (as “that’s interesting”) that in your two viewpoints you talk about doing mathematics whilst most of the subsequent discussion centres upon understanding of a given preexisting piece of mathematics. (For instance, when you’re concentrating on what student’s think you naturally want them to understand “the established ideas” rather than necessarily attack the issue another way (which might benefit from different “intuitions” and/or axiomatizations): if you’re teaching a class on analytical integration strategies you’re unlikely to be impressed if someone develops the rigorous theory of how to numerically compute integrals.) In this situation of “learning an established area”, it’s likely that you’re going to get a different view on things than if you look at how people (mathematicians, physicists) behave when they’re working on some issue where the theory isn’t known (or at least, established as “the” way to do things).

What would be interesting data would be to know about mathematical areas where the theory didn’t really develop until “effective” axioms were created. (I guess communication theory might count as such an area, although it was more defining/naming the key concepts rather than axioms. But I’m not an expert.)

Posted by: bane on August 27, 2009 6:18 PM | Permalink | Reply to this

Re: Feynman the Babylonian

1. Infinitesimal: the ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function.
—————————————–

The calculation of Pi has been a preoccupation of mathematicians since the days of the elder gods. My favorite conceptual way is the Archimedes’ method of averaging the circumscribing and inscribing perimeters of a circle, r=1, and then averaging them. For huge n of the polygons, the 3 lines tend to merge into one line, no longer distinguishable by the eye. Since in this case the perimeter of the circle is equal to the area of the circle, the approximation of the area of the circle grows toward 99.999…% of the true platonic value of Pi. The remaining area of the circle continues to inversely shrink/vanish as the value n, of the number of sides of the polygons, tends towards infinity. This can also be done be constructing triangles within subtended arcs.

http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html

“The truly unique aspect of Archimedes’ procedure is that he has eliminated the geometry and reduced it to a completely arithmetical procedure, something that probably would have horrified Plato but was actually common practice in Eastern cultures, particularly among the Chinese scholars.”

SH: In Turing’s early paper, “On Computable Numbers” Pi is featured as infinite number which is nonetheless computable (one finite digit at a time). So that in the life of this universe a PC can compute a finite number of digits for the value of Pi; but this value is dwarfed by the potential of the ideal TM, which can compute finitely unbounded values for the value of Pi which would encompass the calculations of myriad physical lifetimes of universes, since it has no resource limitations, neither time nor storage.

Posted by: Stephen Harris on August 27, 2009 8:26 PM | Permalink | Reply to this

Re: Feynman the Babylonian

I like Cartier’s quote, but Gian-Carlo Rota has also written some very interesting papers on this subject, some of which I quoted here. I think all of the quotes are relevant to this discussion, so I’ll only single out one:

The axiomatic method of mathematics is one of the great achievements of our culture. However, it is only a method. Whereas the facts of mathematics, once discovered, will never change, the method by which these facts are verified has changed many times in the past, and it would be foolhardy to expect that it will not change again at some future date.

Posted by: Qiaochu Yuan on August 28, 2009 1:39 AM | Permalink | Reply to this

Re: Feynman the Babylonian

To be honest I find neither ‘Babylonian’ nor ‘Greek’ approaches particularly appealing. They seem much too extreme to me. I would like to think that I practice neither one, but something else. I hesitate to say that it is somewhere in between, since there is a certain 1-dimensionality to such as statement. I don’t entirely agree with Cartier’s quote either: it’s not one approach or the other, it’s something else that working mathematicians actually practice in the privacy of their heads. How they write things up is another story.

Posted by: Eugene Lerman on August 28, 2009 2:56 AM | Permalink | Reply to this

Re: Feynman the Babylonian

This aspect of Feynman’s second lecture was also discussed a little at Terry Tao’s blog.

I agree with Terry’s comment in his post that the impression Feynman obtained about pure mathematics (some of which at least seems to have been formed while he was a graduate student) differs from how modern mathematics is performed. It’s useful to remember that the first half (roughly speaking) of the last century was a time in which a lot of topological ideas (among others) were being consolidated axiomatically. One had the axiomatic foundations of (what is now called) point-set topology, followed by the algebraic foundations of (what is now called) algebraic topology (e.g. in the book of Eilenberg and Steenrod, one of the goals of which, as they explained in their introduction, was to replace a mixture of algebraic and geometric arguments of Babylonian nature — to use Feynman’s terminology — by arguments founded on the notion of exact sequence and a collection of algebraic axioms). My guess is that it is this particular mathematical environment that informed Feynman’s view of mathematics. (In “Surely your joking” Feynman writes about the Banach–Tarski paradox, and uses it to contrast mathematical idealism with the concreteness of reality; he also says that he learnt topology as far as homotopy groups.)

I think that Feynman correctly sensed a certain dogmatic tendency in mathematics. One sees it in the passage from pre-Eilenberg–Steenrod to Eilenberg–Steenrod algebraic topology. It also occured in number theory, with the efforts of Chevalley, Artin, and Tate to remove all vestiges of analytic argument from the proofs of class field theory. It stems from the desire to have well-founded theories. (And here I don’t mean foundations in the narrow sense of set theory, but in the broader sense of being deduced from an internally coherent, aesthetically pleasing set of axioms.) I think that it is actually an important engine for driving mathematical progress; it is obviously strongly coupled to mathematicians’ aesthetic sense.

On the other hand, too much dogma can be stifling, and my feeling is that contemporary mathematics too some extent is exhibiting a reaction against the axiomatizing dogma of the previous century. As one illustration, methods of symplectic geometry and complex geometry, which lie outside the purely algebraic language of schemes that forms the most comprehensive foundation of algebraic geometry, have played an important role in recent progress in that subject. There are many others. (One that might be particularly relevant to this post is the resurgence of interactions between mathematics and physics.)

Posted by: Matthew Emerton on August 28, 2009 4:16 AM | Permalink | Reply to this

Re: Feynman the Babylonian

“ If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories.…… The mathematician will have also to take into account not only of those theories coming near reality, but also , as in geometry, of all logically possible theories. He must be always alert to obtain a complete survey off all conclusions derivable from the system of axioms assumed. “ David Hilbert

Posted by: Serifo on August 28, 2009 6:34 PM | Permalink | Reply to this

Re: Feynman the Babylonian

Actually Cartier has a new short text on categories vs sets, where he sets out the project of formalizing the proofs of category theory. What do you make of it?

Posted by: newanon on August 29, 2009 10:12 AM | Permalink | Reply to this

Re: Feynman the Babylonian

To Thurston’s list of different ways of thinking about the derivative should be added the interesting “limit-free” definition of Marsden and Weinstein from their undergraduate textbook Calculus Unlimited.

It is that notion of the derivative closest to Euclid and Archimeedes idea of “the tangent line touches the curve, and in the space between the line and the curve, no other straight line can be interposed”, or “the line which touches the curve only once”.

Definition. Let ff and gg be real-valued functions with domains contained in \mathbb{R}, and x 0x_0 a real number. We say that ff overtakes gg at x 0x_0 if there is an open interval II containing x 0x_0 such that:

  • xIx \in I and xx 0x \neq x_0 implies xx is in the domain of ff and gg.
  • xIx \in I and xx less than x 0x_0 implies f(x)f(x) less than g(x)g(x).
  • xIx \in I and xx greater than x 0x_0 implies f(g)f(g) greater than g(x)g(x).

Given a function ff and a number x 0x_0 in its domain, we may compare ff with the linear functions l m(x)=f(x 0)+m(xx 0)l_m(x) = f(x_0) + m(x-x_0).

Definition of the derivative. (look mom, no limits!) Let ff be a function defined in an open interval containing x 0x_0. Suppose that there is a number m 0m_0 such that:

  • mm less than m 0m_0 implies ff overtakes l ml_m at x 0x_0.
  • mm greater than m 0m_0 implies l ml_m overtakes ff at x 0x_0.

Then we say that ff is differentiable at x 0x_0, and that m 0m_0 is the derivative of ff at x 0x_0.

Posted by: Bruce Bartlett on August 29, 2009 3:34 PM | Permalink | Reply to this

Students Difficulties in Calculus; Re: Feynman the Babylonian

Students’ Difficulties in Calculus
Plenary presentation in Working Group 3,
ICME, Québec, August 1992
David Tall
Mathematics Education Research Centre
University of Warwick
COVENTRY CV4 7AL

1. The Calculus

It should be emphasised that the Calculus means a variety of different things in different countries in a spectrum from:

1. informal calculus – informal ideas of rate of change and the rules of differentiation with integration as the inverse process, with calculating areas, volumes etc. as applications of integration
to

2. formal analysis – formal ideas of completeness, e–d definitions of limits, continuity, differentiation, Riemann integration, and formal deductions of theorems such as mean-value theorem, the fundamental theorem of calculus etc.,

with a variety of more recent approaches including
3. infinitesimal ideas based on non-standard analysis,

4. computer approaches using one or more of the graphical, numerical, symbolic manipulation facilities with, or without, programming.

In some countries the first of these is taught in secondary school and the second to mathematics majors in college. In others a subject somewhere along the spectrum between the two is taught as the first major college mathematics course. In a few countries (e.g. Greece), the formal ideas are taught from the beginning in secondary school.

The details of these approaches, the level of rigour, the representations (geometric, numeric, symbolic, using functions or independent and dependent variables), the individual topics covered, vary greatly from course to course….

Posted by: Jonathan Vos Post on September 3, 2009 7:12 PM | Permalink | Reply to this

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