Alice came to a fork in the road. “Which road do I take?” she asked.
“Where do you want to go?” responded the Cheshire cat.
“I don’t know,” Alice answered.
“Then,” said the cat, “it doesn’t matter.”
— Lewis Carroll, Alice’s Adventures in Wonderland
In the later years of his life, after his journey to the interior, Basho lived in a small abandoned thatched hut near lake Biwa that he described as being “at the crossroads of unreality”*. Now, still early in our journey, we have come to our own crossroads of unreality. We are caught between dichotomies of unreal, abstract, objects. One road leads to consideration of finite collections, and properties of composition (the algebraic properties 1 through 5 from the previous entry); the other road leads to the continuum and questions of ordering and inter-relationship (properties 7 through 10 from the previous entry). The first road will lead to a new fundamental abstraction from finite collections, different from, and yet as important as, the abstraction that we call numbers; this way lies group theory and the language of symmetry that has come to underlie so much of modern mathematics and physics. The second road will lead to deep questions about the nature of reality, and, brushing past calculus along the way, lead to a new and minimalist interpretation of a continuous space through the concept of topology.
Which road do we take? As the cat said to Alice, It doesn’t matter. We are at the crossroads of unreality, and the usual rules need not apply. Which road do we take? Both.
* From the translation of Genjûan no fu by Donald Keene, in Anthology of Japanese Literature.