Rota's lecture on Mathematical Snapshots
Mathematical Snapshots
Gian-Carlo Rota
This lecture is subdivided into the following sections:
1. The dark side of mathematics
2. The hidden side of mathematics
3. The bright side of mathematics
My own Walter Mitty complex has few variations. I imagine myself trying new lines of attack on problems I will never solve, or lecturing on subjects in which I am incompetent. It never occurred to me that I might one day receive the Killian Faculty Achievement Award, and thus, no drafts of this lecture were attempted in my Walter Mitty daydreaming. When David Benney phoned me last May and ordered me to fly back immediately from Strasbourg to receive the award, I was caught without a premeditated text, and with barely a year to decide what not to say in the Killian lecture.
The philosopher José Ortega y Gasset wrote that most of the tasks we undertake in the course of our lives are impossible. Nevertheless, he added, we must try to carry them out.
The Killian lecture is an example of the impossibility Ortega y Gasset probably had in mind, especially when the lecture deals with mathematics. The language and the results of mathematics seem nowadays to lie farther from the mainstream of science than they have ever been.
The only argument against this opinion would be to make the results of mathematics available in a language from which scientists will benefit. But mathematicians' inability to make themselves understood is not a recent phenomenon. It is the thoroughly documented plight of mathematics throughout history, since Pythagoras.
One reason for the mathematicians' difficulty in communicating is the mathematicians' concept of nature. The mathematicians' concept of nature is at variance with the concept of nature that is shared by other scientists.
To a scientist, nature is a primeval forest to be explored, rich in surprising and unpredictable fauna, endowed with mysterious laws that scientists bravely wrest from the jungle. Once discovered, the laws of nature are written up by scientists for the benefit of posterity, in a language that sometimes - but not always - happens to be the language of mathematics. A scientist need not be fluent in that peculiar language that is called mathematics, just as he or she need not be fluent in Urdu or Gaelic.
Mathematicians do not agree with this view.
Galileo wrote the famous sentence: "the great book of nature is written in the language of mathematics". Galileo was a great scientist, one of the greatest perhaps. But Galileo was also a practical joker. His practical jokes got him into trouble from time to time. Could it be the case that Galileo's sentence was written tongue in cheek? This insinuation will be indignantly rejected by every mathematician. To mathematicians, Galileo's sentence is sculpted in marble. Every discovery of a new scientific fact is a challenge to uncover the uderlying mathematical structure. This structure is not "abstracted" from nature, as psychologists would have us believe. It is the basic makeup of nature, it was always there, waiting to be told and staring at us all the time.
The natural laws discovered by scientists will be refined like a metal, polished like a jewel and finally stored as theorems in the archives of mathematics.
Mathematicians triumphantly point to mechanics as the example of a theory that began as an empirical science, and that eventually made its way into mathematics as a generalized geometry, geometry with time added. Mathematicians believe that every science will sooner or later meet the fate that befell mechanics.
This is the mathematicians' faith. It is also a reason why some scientists find mathematicians difficult persons to deal with.
Some students who learn higher mathematics are turned off the subject. Why? They feel that some of the mathematics they are taught belabors the obvious and pursues the preposterous.
As a matter of fact, much mathematical research done in the first half of this century was concerned with finding preposterous examples of innocent- looking definitions. Regions of the plane without an area, nowhere differentiable functions, continuous curves that fill a whole square were taken with the utmost seriousness in that bygone age.
These curios are now stored in the attic. However, the pursuit of the preposterous had a beneficial consequence. It trained mathematicians to look for unexpected instances of intuitive definitions. Some of these unexpected instances are now turning out to be downright useful.
I should like to review the discovery of such an unexpected instance. It is obtained by analyzing the everyday notion of volume, or, in abstract terms, measure. Contrary to the rules that speakers are expected to follow, I will give away the punch line. We will see that volume is characterized by four axioms, and we will find a new measure that fits these axioms, after a slight twist.
Measure is defined by two axioms. A measure v on a family of subsets, for example, subsets of ordinary space, is a real number which is assigned to subsets A, B, ... in the family, and which satisfies :
v( ) = 0, where is the empty set.
This axiom looks like a triviality, but it has unexpected payoffs.
Axiom 2.
If A and B are two sets, then v(A ∪ B) = v(A) +v(B) - v(A ∩ B) .
The picture shows that this axiom states that measure is additive. In particular, if sets A and B are disjoint, then v(A ∪ B) = v(A) + v(B) .
This property extends to any finite family F of sets. Let us record it:
Axiom 2'.
v( ∪{A ∈ F} A ) =
The best example of measure is the volume v(A) of a solid A in space.
Axioms 1 and 2 do not single out volume among all measures. To this end, two axioms must be added:
Axiom 3.
The volume of a set A is independent of the position of A. In other words, if a set A in three-dimensional Euclidean space can be rigidly moved onto a set B, then A and B have the same volume.
In technical language, volume is invariant under the group of Euclidean motions.
Axiom 4.
If P is a parallelotope with orthogonal sides of lengths x_{1} , x_{2} , x_{3} , then v(P) = x_{1}x_{2}x_{3}.
These four axioms, together with suitable continuity conditions, determine volume. By an approximation process such as one finds in an advanced calculus textbook, one shows that that these axioms imply that the volume of a sphere S_{r} of radius r is given by the known formula v(S_{r}) = ^{4}/_{3} π r^{3}.
A similar characterization of volume holds in n-dimensional Euclidean space for any finite dimension n. The fourth axiom is changed to
Axiom 4n.
v(P) = x_{1}x_{2} ... x_{n} whenever P is a parallelotope with orthogonal sides equal to x_{1} , x_{2} , ..., x_{n} .
The basic tools of combinatorial mathematics are the elementary symmetric functions, that is, the following polynomials in n variables:
e_{2}(x_{1} , x_{2} , ... , x_{n}) = x_{1}x_{2} + x_{1}x_{3} + ... + x_{n-1}x_{n}
...
e_{n-1}(x_{1} , x_{2} , ... , x_{n}) = x_{2}x_{3}...x_{n} + x_{1}x_{3}x_{4}...x_{n} + ... + x_{1}x_{2}...x_{n-1}
e_{n}(x_{1} , x_{2} , ... , x_{n}) = x_{1}x_{2} ... x_{n} .
e_{2}(x_{1} , x_{2} , x_{3}) = x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3} ,
e_{3}(x_{1} , x_{2} , x_{3)}= x_{1}x_{2}x_{3} .
v(P) = e_{3}(x_{1} , x_{2} , x_{3}).
a(P) = a_{2}(x_{1} , x_{2} , x_{3}) = x_{1}x_{2} + x_{1}x_{3}+ x_{2}x_{3}
Does the measure a make any sense? Of course it does. The right hand side equals, except for a factor of two, the formula for the surface area of the parallelotope P. Again, the surface areas of solids can be computed starting from axioms 1, 2, 3, and 4', by continuity considerations. The surface area of a sphere S_{r} of radius r is given - except for a factor of two - by the known formula
Emboldened by our success with two symmetric functions, we now replace axiom 4 by yet another axiom, that uses the one symmetric function that we have so far left out. We try to define a new measure that satisfies axioms 1, 2, and 3, together with
w(P) = e_{1}(x_{1}, x_{2} , x_{3}) = x_{1} + x_{2} + x_{3}
with v replaced by w. In particular, if P' is the face of the parallelotope
with sides equal to x_{1}, x_{2} , then we have
w(P') = x_{1} + x_{2}.
When a definition is proposed, the fundamental condition to be verified is its consistency. The French mathematician Henri Poincaré put it elegantly when he wrote: "in mathematics, to be is to be consistent".
Look at the two parallelotopes P_{1} and P_{2}. The first parallelotope has sides equal to x_{1,}x_{2,}x_{3}, and the second parallelotope has sides equal to y , x_{2} , x_{3} . The two parallelotopes have the common face P' . If the measure w(P) is to be consistent, then by axiom 2 we must have
w(P_{1} ∪ P_{2}) = w(P_{1}) + w(P_{2}) - w(P_{1}∩ P_{2}) = w(P_{1}) + w(P_{2}) - w(P' ) (*)
Let us check this equality.
The left side is computed by observing that the parallelotope P_{1}∩ P_{2} has sides equal to x_{1}, x_{2} and x_{3} + y. Therefore , Axiom 4" tells us that
w(P_{2}) = x_{1} + x_{2} + y w(P') = x_{1} + x_{2} ,
Again, continuity considerations enable us to compute the measure w(A) when A is any reasonable solid in ordinary space. The limiting process required to carry out such computation is only a little more complex than the limiting processes we teach in 18.02.
What is the meaning of the new measure w ?
The definition of w(P) for a parallelotope P has a geometric interpretation. When multiplied by 4, it equals the perimeter of the parallelotope P, that is, the sum of the lengths of all the edges of the parallelotope P.
But, one may object, w(P) makes sense for a parallelotope P, because a parallelotope has a well defined perimeter. What about w(A) when A is a solid that does not have a well defined perimeter, a sphere for example? Defining the perimeter of a sphere seems to fly in the face of common sense.
Einstein wrote : "Common sense is the residue of the prejudices that were instilled into us before the age of seventeen". Since the new measure w is well defined, common sense will have to adjust to reality.
The measure w is called the mean width, a misnomer that has been kept for historical reasons. The formula for the mean width of a sphere of radius r is
a(T) = ^{1}/_{2} x^{2}√3 ,
w(T) = ^{1}/_{π} 3α x,
where cos α = -^{1}/_{3}.
In n dimensions, each of the n elementary symmetric functions leads similarly to a generalization of volume. We thereby obtain n invariant measures in n-dimensional space. These measures are called the intrinsic volumes.
The intrinsic volumes are independent of each other, except for certain inequalities they satisfy which remain to be discovered. We know little about the intrinsic volumes , because they have not been around for long.
I know of no person who has an intuitive feeling for the mean width, similar to the intuitive feeling we have for volume and area. The closest we can come to date to an intuitive interpretation of the mean width is the following probabilistic interpretation in the very special case of convex sets.
Take two convex sets A and B in three dimensional Euclidean space, and suppose that A is contained in B. Again, let us begin by belaboring the obvious. Suppose that we take a point at random belonging to the larger set B. What is the probability that the point shall belong to the smaller set A? The answer is obvious: such a probability equals the volume of A divided by the volume of B.
Now let us take a leap of reason. Instead of choosing a point at random, let us choose a straight line at random in space. Assuming that such a straight line meets the larger set B, what is the probability that such a straight line will also meet the smaller set A?
The answer is satisfying. Such a probability equals the surface area of the set A, divided by the surface area of the set B.
You can tell what is coming next. We take a random plane in space. Assuming that the plane meets the larger set B, what is the probability that it will also meet the smaller set A? Such a probability equals the mean width of A, divided by the mean width of B.
It is likely that when scientists become aware of the existence of the mean width, they will find interpretations and applications of this measure.
We can visualize this fact by another picture.
But wait a minute: are we telling the truth? If the two arrows are what they are supposed to be, then we must specify which of the two arrows goes underneath the other. After we decide which arrow goes underneath, then we see that the iteration of a transposition is no longer the identity permutation. One of the strings keeps winding around the other as we iterate.
We are not entitled to asssume that the iteration of a transposition gives the identity permutation. This is the unwarranted assumption that we made for a hundred years. What happens when we take the bold step of dropping this assumption? Why, permutations are replaced by new objects , called braids, as in the figure.
Products of braids are taken in the same way as products of permutations, by placing the diagram of a braid underneath the other. The inverse of a braid is obtained by flipping the diagram of the braid, and the product of a braid with its inverse is the identity braid.
The theory of symmetry is now being revamped after the advent of braids. This is the cutting edge of mathematics. New theories are sprouting up: quantum groups, the Yang-Baxter equations, monoidal categories, and what not. The end is nowhere in sight. It is not known at present what will replace the old symmetry classes. In technical terms, no one has yet determined the irreducible representations of the braid group, and even the concepts of group and representation may get overhauled.
It is likely that these new theories will have a domino effect on our picture of the physical world. Other ingrained prejudices about space will be dealt a fatal blow. A new world is in the making.
When we meet again in the new world ten years from now, we will marvel at how we could have ever entertained such prejudices, while the truth was always there, waiting to be told and staring at us all the time.
Thank you.
JOC/EFR March 2006
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