I had an extensive discussion about the meaning of the word “space” recently. Having been (somewhat) trained in math, that biases my understanding of the word, I think. But since most of my adult life has been spent programming, I think that biases it more. The ultimate bias, however, is the medium in which our bodies sense and act. This discussion centered around “visualization” and how/why visualizations (particularly of simulations) appeal to us humans, why they help us understand the often cryptic mechanisms inside a computation. I believe they help us understand things because they appeal to the 4D spacetime in which we live. And hence, there is an intuition that’s evoked by a visualization (as well as an audialization or … a hapticalization? … how about an olfactorization? … or even an inertialization?).
Given that there are these 3 basic usage domains for the word “space” (math, compsci, and natural), when someone talks about a visualization, what do they really mean? I tend to think they mean the latter. They mean “render your abstract stuff into something in 4D so my natural senses are stimulated”. That forces me to contrast the three uses of “space” and try to establish the distinctive properties of a natural space. I won’t go through all the hemming and hawing to do that. I’ll just make my assertion. What I think makes a space natural is affine geometry, basically the preservation of parallel lines. That’s what gives us our intuition about the translation and rotation of rigid objects, distance, perspective, etc.
Of course, people are different. So, it’s reasonable to assume that some people think more in terms of graphs (networks), some in terms of sequence, order, number, etc. But I tend to think those are in the minority. I’d enjoy evidence to the contrary.
In Chapter 2 of Lee Rudolph’s Qualitative Mathematics for the Social Sciences, Rudolph asserts:
truth is what they come to believe more firmly as they function better [...] As such, truth is always conditional and subject to amendment: but it has, always and unconditionally, a net or web of meaning that anchors it pretty firmly to many places
He’s saying this in the context of 2 comments written by G.H. Hardy about the function of mathematicians, chronologically:
(1922) The function of a mathematician is [...] to observe the facts about his own intricate system of reality, [...] to record the results of his observations in a series of maps, each of which is a branch of pure mathematics.
(1940) The function of a mathematician is [...] to prove new theorems
And Rudolph makes the following statement about the first of those quotes:
Hardy wrote his Apology at the end of his mathematical career, when he was convinced, perhaps correctly, that “his creative powers as a mathematician at last, in his sixties, [had] left him.”
Rudolph’s point about the timing of the two statements is critical, the difference being before and after Hardy felt his powers had left him. But the point does not imply what Rudolph infers about mathematical truth. One of the things that all of us do, mathematicians included, is become more conservative as we age (in thought and action, not necessarily political beliefs). As whatever powers we had leave us, we are left with the fossils of the exercising of those powers. This is true of both the complete structure as well as the more sparse anchors set more firmly amongst the less firm surroundings. My guess is that those anchors seed the “crystal.” Whatever anchors we become convicted of … convinced of … while younger, tend to accumulate cruft and barnacles, leading to a stigmergic mess of arbitrarily decided and fossilized dogma … that we then carry to our graves. (Unless we have a near-death epiphany or, as more research is showing, loosen up that “crystal” in some other way.)
What Hardy successfully exhibits with his change is the path from ideological conviction to transpersonal artifact. Just like science, what matters are the artifacts we produce, the less semantically (and metaphorically) laden, the better. For math, it comes in the form of proofs. For science, it comes in the form of recipes that anyone with an equivalent sensorimotor manifold can execute. Hence, mathematical truth is not a (or many) set(s) of beliefs inside the heads of mathematicians. It is the proofs written on paper and magnetic/optical media all around us. I think we’re finally approaching a demonstration of that with the homotopy type theory (HoTT) project, whereby mathematical truth would be fully instantiated in computing machinery. Even if HoTT fails, it’ll be a huge step toward externalizing mathematics (exegesis of the esoteric … pretty much the inverse of what Rudolph concludes).