How would you comprehend a definition in your area of expertise and a definition in an area less familiar to you? More specifically, is there a specific identifiable intellectual process specific to your individuality which is called up when needed to comprehend a definition. As a part of the process, do you use special examples and then abstract the process, or draw pictures, schematic diagrams, etc.?
Usually, the process of comprehension for new definition is similar whether the definition lies in my area of expertise or not. I try to see how the definition will exclude/include examples that I already know. For example, if the definition is about groups then I would try to see whether it clearly divides the groups that I know into those that fit and those that do not.
In the case of geometric definitions I try to imagine something a little more vague and try to give it shape according to the restrictions imposed by the definition. Schematic diagrams often help in the case when the geometric object is parametric. One can imagine a “generic” structure with some breaking of genericity at special loci.
The process is certainly easier in my area of expertise as I have more examples to draw upon. However, even in the case of definitions from other areas, I tend to continually try to mold the definition that I am reading/hearing into one that fits my area of expertise; if not in content then at least in style. This is because Algebraic Geometry (my area of expertise) has definitions molded by Category Theory whereas numerous other areas still use a Set-Theoretic approach and I prefer the CT approach.
In the course of your mathematical development, are you aware of any changes in the way you comprehend a new definition? In what ways, if any, are they different from the ones you use now?
When I became aware of the Intuitionist foundations of mathematics (as proposed by Brouwer and Heyting) I would try to exclude all definitions and results that could not be comprehended in that framework. Later I found that the Category Theory framework captured much of that part of Intutionism that applied to my area of interest (Algebraic Geometry) so I adopted it. Of late, I have also become interested in the computation and complexity issues (of Theoretical Computer Science). This makes it even more interesting to examine new and old definitions from the point of view, not just of constructibility but of computability and practical usability.
Do you have a recollection of having understood a definition or a mathematical statement in a particular way which later on resulted in a conflict? If so, how is the awareness of the conflict triggered? Does the awareness occur spontaneously or when working consciously at it?
When I first learnt algebra, I imagined all the objects constructed there as “discrete” or at best like rational numbers. This led to a conflict with the “ruler placement postulate” which, by “decimalising” the line seemed to make it discrete. More generally, I had difficulty understanding topological groups, rings and fields as these were (to my mind) “discrete”. In some way the categorical/geometric definition of the operations (as opposed to the point-wise definition) has solved this conflict for me. I similarly had a problem with the “pointwise” definition of functions which has become clearer now that I see various spaces of functions (say continuous functions) as completions of “standard” spaces such as polynomials.
There was a while when I could not convince myself of the continuity or well-defined-ness of a function except by detailed case by case analysis. Only later did I understand that the definition was not meant to be used “as such”. In fact most definitions are used to prove theorems and the theorems are used to check conformance with the definition.
Sometimes conflicts of this kind will give one “sleepless nights” without resolution. When reviewed at a later stage with a different point of view the problem will have “gone away”. At other times the counter-intuitive nature of a definition/result is resolved by calculation; one’s intuition is proved wrong in a practical way.
In your experience, if someone recasts your definition in a different way, what method(s) do you use for reconciling or understanding the new definition?
As I explained above, I had (at least) three paradigm shifts in my way of understanding definitions. When these shifts happened it was as a result of realising that this way of stating the definition was clearer and more concise. Sometimes this caused a lot of internal conflict as I had spent some time and effort internalising a different approach which would now have to be abandoned.
At other times, (as I also said elsewhere) I try to cast definitions given by others into my own way of thinking. In this I am usually reasonably successful—but this is my own opinion!
Is it possible to evolve general strategies for understanding mathematical definitions based on your research experience or teaching? To what extent are the strategies common or different across the subjects (algebra, topology, geometry, analysis,...).
Knowing enough examples always helps. Having a theoretical framework in which to fit the examples also helps. For the latter, it is necessary to understand some one area of mathematics well enough; so that is more difficult for beginners. Later, it is only in very rare cases when one has to be a total beginner and follow the definitions and results presented in a linear and dogged fashion. As soon as one has a grasp of the subject one starts building one’s own framework and fitting things into it. However, one should do enough calculations to check that this matches the author’s results and definitions.
In the algebraic approach one builds things up from elementary pieces. In the analytic approach one localises spread out objects to study them. Geometry is somewhere in between where one uses the more algebraic schema but the objects are nebulous and acquire more shape as one studies them.
It is indeed possible to evolve strategies but this has to be tailored to one’s taste. For example, I have little tolerance (or memory) for one-off results and definitions — unless I thought of them myself! Examples without a unifying theory or theory without a rich knowledge of examples don’t suit me. On the other hand famous mathematicians like Paul Erdos worked well with the former and other famous ones like Grothendieck worked well with the latter.