How and how much do the notations and diagrams influence our understanding of mathematical concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and Suggestions for good notation.
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Sign up to join this communityHow and how much do the notations and diagrams influence our understanding of mathematical concepts?
This question was stimulated by the MathOverflow questions Thinking and Explaining and Suggestions for good notation.
To support the last remark of Donu Arapura, the following anecdote might be helpful: The late Beno Eckmann, one of the key players of the early developments in algebraic topology in the 40ies and 50ies, was asked to explain, why the revolution in algebraic topology happened in the 50ies. You can find his answer in his "Mathematical Miniatures". In short, he explains that the idea to represent a function by an arrow, and a composition of functions by a diagram was completely unknown until the late 1940ies (!!!), when Leray introduced this notation. There seems to be no doubt that even the formulation of modern algebraic topology would have been impossible without the idea of an arrow and/or a diagram!
This example illustrates a point of view that I learnt from Dennis Sullivan. In general, when speaking of free (or projective) resolutions of $R$-modules one writes
$\cdots\to F_n\to\cdots\to F_1\to F_0\stackrel{\varepsilon}{\rightarrow} M\to 0$
where $F_n$'s are free (or projective) $R$-modules. For calculating $Ext$ or $Tor$ we use one such resolution of $M$ and then show that the resulting answer doesn't depend on the resolution chosen. The resolution above can be rewritten as a map $\varphi$ between two chain complexes of $R$-modules :
$\cdots\to F_n\to\cdots\to F_1\to F_0\to 0$
$\cdots\to 0\hspace{0.2cm} \to \hspace{0.2cm} \cdots\hspace{0.2cm}\to 0\to M\to 0$
where the only non-trivial vertical map is $\varepsilon:F_0\to M$. Notice that $\varphi$ is a quasi-isomorphism and saying that the derived functors are independent of the resolution chosen is akin to saying that any two chain complex of $R$-modules representing $M$ are quasi-isomorphic. To an algebraic topologist (and possibly for others too) this is so much more natural.
This point of view emphasizes that the non-triviality of the module $M$ gets coded into the differentials between $F_n$'s. The $F_n$'s themselves contain almost no information since the rank is the only possible invariant for $F_n$ and even that be made to change by adding a copy of $R$ to $F_{n+1}$ and $F_n$.
One example, slightly outside of mathematics, is the Feynman diagram which represent the interaction between particles in a field over a period of time. They're slightly different from space-time diagrams, but make it easier to see the contributions of particles and anti-particles to an interaction.
Another example, in chemistry and mathematics, is that a series of chemical reactions can be described by a series of equations
which does not clearly describe the "ins and outs" of this chemical cycle as well as a graph (directed graph) diagram does:
: A
: J-<-- / \ --<--B
: \/ \/
: H-->/ \---> C
: / \
: / \
: G-----------D
: / \
: F <-- -<--E
which clearly illustrates that the cyclic nature of this series of actions or the catalytic nature of some of these moities.
Sometimes, rewriting the steps of a proof simplifies understanding of it at an earlier level of our education, while it seems wholly un-necessary at later parts of our education. For example, $(p+q)^3$ is easily expanded in our heads to 1, 3, 3, 1, and $p^3, p^2q, pq^2, q^3$ and recombined into $p^3+3p^2q+3pq^2+q^3$, even though the initial and final $1$ of the binomial coefficients are effectively silent. But for beginning algebra students in high-school, putting the ones back into the expansion makes it easier to comprehend.
In the general case about encountering a new notational technique the first time may simply be about how the mind deals with novelty. In the example of Willie Wong about the Grothendieck's school's vertical arrows, does it really bring in a different mode of thinking, or does it merely indicate that the vertical arrows are viewed as being different from the horizontal arrows, and that this difference is viewed as representing a different mode of action?
In other words, unfamiliar objects or symbols are viewed as being novel, exciting certain parts of the brain more than familiar objects would. Familiar objects are already "wired up", whereas novel objects are not pre-conditioned to elicit a particular point of view, thus allowing a different approach to be considered.
For what concerns notations (and not diagrams), you can read the quote in the third page here: http://www.tug.org/TUGboat/tb24-2/tb77lawrence.pdf
It seems that one or two centuries ago, mathematicians and physicists used gothic letters in their works, and lots of students didn't manage to read them. The sad result was that they couldn't learn easily the content of the teachings, just because of the way they (I mean the teachings) looked.
I think a good answer to this comes from category theory, linear logic, diagrams and the geometry of tensor calculus (Joyal-street). We often talk about category theory through the use of diagrams which are planar graphs (objects as nodes morphisms as arrows). Written down, these diagrams can be seen as fishing nets, kind of embedded in a plane, so we don't care about how one line crosses over another. These diagrams can be rewritten in different ways that respect the topology of the network. These deformations are exactly what Joyal and Street were talking about in the geometry of tensor calculus. We know from further work that, (and excuse my poor explanation ) the geometry of tensor calculus is a model of linear logic. This would mean that the axioms of a symmetric monoidal category can support the axioms of linear logic...(please excuse the poor understanding, I have come to these thoughts with little help). The long and short is that, if we talk about category theory in terms of diagrams, then we are most likely thinking in terms of linear logic. This would be in contrast to a model of the theory of categories in Set. In that case, we have a set of objects and set of morphisms, the axioms of (some kind of ) Set theory and further the axioms of the category we want to talk about. This would be thinking in a different kind of logic.