Multiple ways of understanding

In his wonderful On Proof and Progress in Mathematics, Thurston begins his second section “How do people understand mathematics?” as follows:

This is a very hard question. Understanding is an individual and internal matter that is hard to be fully aware of, hard to understand and often hard to communicate. We can only touch on it lightly here.

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 is , the derivative of is , the derivative of is , etc.
3. Logical: if and only if for every there is a such that when

   

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 , when 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 in a domain is the Lagrangian section of the cotangent bundle that gives the connection form for the unique flat connection on the trivial -bundle for which the graph of is parallel.

These differences are not just a curiosity. Human thinking and understanding do not work on a single track, like a computer with a single central processing unit. Our brains and minds seem to be organized into a variety of separate, powerful facilities. These facilities work together loosely, “talking” to each other at high levels rather than at low levels of organization.

This has been extended on the MathOverflow question Different ways of thinking about the derivative where you can find even more ways of thinking about the derivative. (Two of the interesting pointers are to this discussion on the n-Category Café, and to the book Calculus Unlimited by Marsden and Weinstein, which does calculus using a “method of exhaustion” that does not involve limits. (Its definition of the derivative is also mentioned at the earlier link, as that notion of the derivative closest to [the idea of Eudoxus and Archimedes] 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” — this counts as another important way of thinking about the derivative.)

It has also been best extended by Terence Tao, who in an October 2009 blog post on Grothendieck’s definition of a group gave several ways of thinking about a group:

In his wonderful article “On proof and progress in mathematics“, Bill Thurston describes (among many other topics) how one’s understanding of given concept in mathematics (such as that of the derivative) can be vastly enriched by viewing it simultaneously from many subtly different perspectives; in the case of the derivative, he gives seven standard such perspectives (infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic) and then mentions a much later perspective in the sequence (as describing a flat connection for a graph).

One can of course do something similar for many other fundamental notions in mathematics. For instance, the notion of a group can be thought of in a number of (closely related) ways, such as the following:

0. Motivating examples: A group is an abstraction of the operations of addition/subtraction or multiplication/division in arithmetic or linear algebra, or of composition/inversion of transformations.
1. Universal algebraic: A group is a set with an identity element , a unary inverse operation , and a binary multiplication operation obeying the relations (or axioms) , , for all .
2. Symmetric: A group is all the ways in which one can transform a space to itself while preserving some object or structure on this space.
3. Representation theoretic: A group is identifiable with a collection of transformations on a space which is closed under composition and inverse, and contains the identity transformation.
4. Presentation theoretic: A group can be generated by a collection of generators subject to some number of relations.
5. Topological: A group is the fundamental group of a connected topological space .
6. Dynamic: A group represents the passage of time (or of some other variable(s) of motion or action) on a (reversible) dynamical system.
7. Category theoretic: A group is a category with one object, in which all morphisms have inverses.
8. Quantum: A group is the classical limit of a quantum group.

etc.
One can view a large part of group theory (and related subjects, such as representation theory) as exploring the interconnections between various of these perspectives. As one’s understanding of the subject matures, many of these formerly distinct perspectives slowly merge into a single unified perspective.

From a recent talk by Ezra Getzler, I learned a more sophisticated perspective on a group, somewhat analogous to Thurston’s example of a sophisticated perspective on a derivative (and coincidentally, flat connections play a central role in both):

37. Sheaf theoretic: A group is identifiable with a (set-valued) sheaf on the category of simplicial complexes such that the morphisms associated to collapses of -simplices are bijective for (and merely surjective for ).

The rest of the post elaborates on this understanding.

Again in a Google Buzz post on Jun 9, 2010, Tao posted the following:

Bill Thurston’s “On proof and progress in mathematics” has many nice observations about the nature and practice of modern mathematics. One of them is that for any fundamental concept in mathematics, there is usually no “best” way to define or think about that concept, but instead there is often a family of interrelated and overlapping, but distinct, perspectives on that concept, each of which conveying its own useful intuition and generalisations; often, the combination of all of these perspectives is far greater than the sum of the parts. Thurston illustrates this with the concept of differentiation, to which he lists seven basic perspectives and one more advanced perspective, and hints at dozens more.

But even the most basic of mathematical concepts admit this multiplicity of interpretation and perspective. Consider for instance the operation of addition, that takes two numbers x and y and forms their sum x+y. There are many such ways to interpret this operation:

1. (Disjoint union) x+y is the “size” of the disjoint union X u Y of an object X of size x, and an object Y of size y. (Size is, of course, another concept with many different interpretations: cardinality, volume, mass, length, measure, etc.)

2. (Concatenation) x+y is the size of the object formed by concatenating an object X of size x with an object Y of size y (or by appending Y to X).

3. (Iteration) x+y is formed from x by incrementing it y times.

4. (Superposition) x+y is the “strength” of the superposition of a force (or field, intensity, etc.) of strength x with a force of strength y.

5. (Translation action) x+y is the translation of x by y.

5a. (Translation representation) x+y is the amount of translation or displacement incurred by composing a translation by x with a translation by y.

6. (Algebraic) + is a binary operation on numbers that give it the structure of an additive group (or monoid), with 0 being the additive identity and 1 being the generator of the natural numbers or integers.

7. (Logical) +, when combined with the other basic arithmetic operations, are a family of structures on numbers that obey a set of axioms such as the Peano axioms.

8. (Algorithmic) x+y is the output of the long addition algorithm that takes x and y as input.

9. etc.

These perspectives are all closely related to each other; this is why we are willing to give them all the common name of “addition”, and the common symbol of “+”. Nevertheless there are some slight differences between each perspective. For instance, addition of cardinals is based on perspective 1, while addition of ordinals is based on perspective 2. This distinction becomes apparent once one considers infinite cardinals or ordinals: for instance, in cardinal arithmetic, aleph_0 = 1+ aleph_0 = aleph_0 + 1 = aleph_0 + aleph_0, whereas in ordinal arithmetic, omega = 1+omega < omega+1 < omega + omega.

Transitioning from one perspective to another is often a necessary first conceptual step when the time comes to generalise the concept. As a child, addition of natural numbers is usually taught initially by using perspective 1 or 3, but to generalise to addition of integers, one must first switch to a perspective such as 4, 5, or 5a; similar conceptual shifts are needed when one then turns to addition of rationals, real numbers, complex numbers, residue classes, functions, matrices, elements of abstract additive groups, nonstandard number systems, etc. Eventually, one internalises all of the perspectives (and their inter-relationships) simultaneously, and then becomes comfortable with the addition concept in a very broad set of contexts; but it can be more of a struggle to do so when one has grasped only a subset of the possible ways of thinking about addition.

In many situations, the various perspectives of a concept are either completely equivalent to each other, or close enough to equivalent that one can safely “abuse notation” by identifying them together. But occasionally, one of the equivalences breaks down, and then it becomes useful to maintain a careful distinction between two perspectives that are almost, but not quite, compatible. Consider for instance the following ways of interpreting the operation of exponentiation x^y of two numbers x, y:

1. (Combinatorial) x^y is the number of ways to make y independent choices, each of which chooses from x alternatives.

2. (Set theoretic) x^y is the size of the space of functions from a set Y of size y to a set X of size x.

3. (Geometric) x^y is the volume (or measure) of a y-dimensional cube (or hypercube) whose sidelength is x.

4. (Iteration) x^y is the operation of starting at 1 and multiplying by x y times.

5. (Homomorphism) y → x^y is the continuous homomorphism from the domain of y (with the additive group structure) to the range of x^y (with the multiplicative structure) that maps 1 to x.

6. (Algebraic) ^ is the operation that obeys the laws of exponentiation in algebra.

7. (Log-exponential) x^y is exp( y log x ). (This raises the question of how to interpret exp and log, and again there are multiple perspectives for each…)

8. (Complex-analytic) Complex exponentiation is the analytic continuation of real exponentiation.

9. (Computational) x^y is whatever my calculator or computer outputs when it is asked to evaluate x^y.

10. etc.

Again, these interpretations are usually compatible with each other, but there are some key exceptions. For instance, the quantity 0^0 would be equal to zero [ed: I think this should be one —S] using some of these interpretations, but would be undefined in others. The quantity 4^{1/2} would be equal to 2 in some interpretations, be undefined in others, and be equal to the multivalued expression +-2 (or to depend on a choice of branch) in yet further interpretations. And quantities such as i^i are sufficiently problematic that it is usually best to try to avoid exponentiation of one arbitrary complex number by another arbitrary complex number unless one knows exactly what one is doing. In such situations, it is best not to think about a single, one-size-fits-all notion of a concept such as exponentiation, but instead be aware of the context one is in (e.g. is one raising a complex number to an integer power? A positive real to a complex power? A complex number to a fractional power? etc.) and to know which interpretations are most natural for that context, as this will help protect against making errors when manipulating expressions involving exponentiation.

It is also quite instructive to build one’s own list of interpretations for various basic concepts, analogously to those above (or Thurston’s example). Some good examples of concepts to try this on include “multiplication”, “integration”, “function”, “measure”, “solution”, “space”, “size”, “distance”, “curvature”, “number”, “convergence”, “probability” or “smoothness”. See also my blog post below in which the concept of a “group” is considered.

I plan to collect more such “different ways of thinking about the same (mathematical) thing” in this post, as I encounter them.

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