In my data mining class today we briefly talked about the challenges of visualizing data in high dimensions. The professor made a comment that we’re limited to the obvious 3 dimensions which human beings experience — but there are a few methods, like color, to expand depiction of dimensionality. This brought synthaesthesia to mind, and the possibilities of experiencing data or mathematics in very high dimensionality by not just “visualizing” data but actually fully “sensualizing” it. Our species has a propensity to rely on visual cues, likely for evolutionary reasons, but we should not be restrained to thinking only in pictures. Other modes of engagement are possible.

Sometimes when I’m very focused I get sense of something like this; that a particular concept is a slick, shiny orb with a consistency similar to water, or something dense, orange and dry. It’s been interesting to note some changes in my experience of math this term. I started out with very little experience of math at all - it was completely dead to me sensually, as I’ve always had a kind of tortured relationship with it. At the end of the semester, I start to have a sense of the size, weight and color of concepts. I wonder if there are ways to extend these experiences.

Virtual reality comes to mind as an obvious way to induce unusual experiences. For the time being, I wonder what some of these things would be like in a lucid dream. My previous experiences with mathematics in lucid dreams were limited to simple arithmetic. I have a strong conjecture that this type of “figuring” is basically impossible in these states. Mathematics engaging intuition and conceptual work rather than purely algorithmic processes could be much more open. My guess is that it may be possible, for example, to explore a quadratic equation in a dream, but not to find its roots. Maybe an experiment for the summer.

Something one of my classmates said really tickled me. Since this was the final class of the term, our professor was owning up to his biases. He tends to strongly prefer linear models, largely because of their ease of interpretation. He argues that non-linearity can be accounted for in linear models (often with some chichanery in the weights of a linear model). He also pointed out that most of the memorable equations in the history of science have been linear. One of my classmates responded that this is because humans grasp linearity easily, and are eager to apply it, and not because nature is inherently linear. Where linearity gives a tortured account, we simply add more complexity to try to minimize our error. For some reason this reminds me of a kind of aphasia or agnosia. There are boundries we traverse only haltingly, perhaps defined by our genotype. To be really wild, it makes the prospect of xenomathematics interesting — the possibility that other intelligent organisms may actually understand mathematics somewhat differently than we do. Our mutual theorems and lemma would be identical (lets hope they are), but it seems likely there would be some surprises in the different paths taken.