Undoing to understand
Let’s begin with isomorphism. We looked at it as a means to tell if, say, two sets have the same number of elements without counting the number of elements each set has. In the process, we recognized it (isomorphism) as a generalization of inverses such as multiplying by 1/4 is the inverse of multiplying by 4, noting that doing both i.e. multiplying by 1/4 and multiplying by 4 gives back the number we began with. For example, 183 * ((1/4) * 4) = 183 or, in general a * 1/a = 1, where ‘a‘ is any number.
In other words, product of a number and its multiplicative inverse gives the multiplicative identity, which we (interpreted) generalized to mean if a process f: A -> B transforms A into B, then the inverse process is that which reverses the process by transforming B back into A. If we interpret A and B as towns, saying that a path or journey f from A to B is an isomorphism is simply saying that the journey has inverse i.e. there is a path g corresponding to the path f, which takes one back to A from B.
All of the above is not merely to highlight slightly different ways of looking at isomorphism, but to recognize the inverse or ‘undoing’ interpretation of isomorphism as the Bohr atom of understanding in the following sense, and use it as a motivation for careful study of finding ways of undoing processes more “complex” than a function
f: A -> B
from set A to set B.
We are given a beautiful sunset, understanding of which can be exhibited by showing a way to go from ‘beautiful sunset’ to ‘earth rotating around itself while revolving around the sun’, which is not to say that in the beginning there was sunset. We all agree that in beginning there was The Being which gave rise to Appearances. Of course, we have a ways to go in accounting for how we go from photons to colors; clearly, the process is taking place, understanding of which requires taking color as primitive or more generally conscious experience as primitive and trying to find that matter that gave rise to this mind.
Or simply put, given output we try to find the input that gave rise to the given output as part of understanding how inputs are transformed into outputs. In more simple terms, we are going to begin a study of ‘undoing’, beginning with isomorphisms between sets, or undoing processes in the case of sets, in the case of sets with structure, and in categories, in general. Before we leave the subject let’s recollect that we saw that some processes don’t have an inverse in the case of sets. This is true of more complex processes also, i.e. processes between objects more structured than sets, maps between structured sets, functors between categories, and so on.
However, in some cases we have the next best thing to inverse or a process as close to undoing as one can get or in technical terms: adjoint functors.
To sum up, we will start discussing isomorphisms between sets, between sets with structure, functors between categories, and finally adjoint functors.
Once we get to adjoint functors we will begin our journey of going from the general notion of adjoint functor to particular examples of adjoint functor such as limits / colimits, product / sum, etc. Here I have to admit that I am not sure if it would be advisable, pedagogically speaking, to go from abstract general to particular examples as opposed to going from examples to abstract general. The best excuse that I could come up with for going from adjoint functors to sum / product instead of from sum / product to adjoint functors: I feel that when one goes from particulars to general one’s conceptual framework is tethered to particulars and often one ends up having to ask ‘can you give me an example,’ and never really getting to experience what it is like to think like, say, Grothendieck, in thoughts unpolluted with particulars ;-) For students of conscious experience this (understanding varieties of thinking as part of the larger program of understanding conscious experience) is particularly important.
More immediately, Trygve, a regular participant in the discussions is very much interested in understanding how mathematicians think about mathematics as part of a program of engineering machines that do math like mathematicians do (I hope I am not misrepresenting Trygve’s research interests here), and the above approach might be of some interest to him among others.
I look forward to your comments and suggestions.
Happy (almost) TGIF!
P.S. I just realized: aren’t we all The Artist Formerly Known as Abstractionist–most certainly–when we speak of us?