“Analyze This”, is a very funny 1999 sendup of the Mafia and psychiatry with Robert DeNiro and Billy Crystal. For some reason the diagram on p. 7 of Barrett O’Neill’s book “Elementary Differential Geometry” revised 2nd edition 2006 made me think of it.
O’Neill’s book was highly recommended by the wonderful “Visual Differential Geometry and Forms” by Tristan Needham — as “the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excpted !”
So O’Neill starts by defining a point as an ordered triple of real numbers. Then he defines R^3 as a set of such points along with the ability to add them and multiply them by another real number.
O’Neill then defines tangent vector (written v_p) as two points (p and v) in R^3 where p is the point of application (aka the tail of the tangent vector) and v as its vector part (the tip of the tangent vector).
All terribly abstract but at least clear and unambiguous until he says — “We shall always picture v_p as the arrow from point p t0 the point p + v”.
The picture is a huge leap and impossible to axiomatize (e.g. “Axiomatize This”). Actually the (mental) picture came first and gave rise to all these definitions and axioms.
The picture is figure 1.1 on p. 7 — it’s a stick figure of a box shaped like an orange crate sitting in a drawing of R^3 with 3 orthogonal axes (none of which is or can be axiomatized). p sits at one vertex of the box, and p + v at another. An arrow is drawn from p to p + v (with a barb at p + v) which is then labeled v_p. Notice also, that point v appears nowhere in the diagram.
What the definitions and axioms are trying to capture is our intuition of what a (tangent) vector really is.
So on p. 7 what are we actually doing? We’re looking at a plane in visual R^3 with a bunch of ‘straight’ lines on it. Photons from that plane go to our (nearly) spherical eye which clearly is no longer a plane. My late good friend Peter Dodwell, psychology professor at Queen’s University in Ontario, told me that the retinal image actually preserves angles of the image (e.g. it’s conformal). 1,000,000 nerve fibers from each eye go back to our brain (don’t try to axiomatize them). The information each fiber carries is far more processed than that of a single pixel (retinal photoreceptor) but that’s another story, and perhaps one that could be axiomatized with a lot of work.
100 years ago Wilder Penfield noted that blood flowing through a part of the brain which was active looked red rather than blue (because it contained more oxygen). That’s the way the brain appears to work. Any part of the brain doing something gets more blood flow than it needs, so it can’t possibly suck out all the oxygen the blood carries. Decades of work and zillions researchers have studied the mechanisms by which this happens. We know a lot more, but still not enough.
Today we don’t have to open the skull as Penfield did, but just do a special type of Magnetic Resonance Imaging (MRI) called functional MRI (fMRI) to watch changes in vessel oxygenation (or lack of it) as conscious people perform various tasks.
When we look at that simple stick figure on p. 7, roughly half of our brain lights up on fMRI, to give us the perception that that stick figure really is something in 3 dimensional space (even though it isn’t). Axiomatizing that would require us to know what consciousness is (which we don’t) and trace it down to the activity of billions of neurons and trillions of synapses between them.
So what O’Neill is trying to do, is tie down the magnificent Gulliver which is our perception of space with Lilliputian strands of logic.
You’ve got to admire mathematicians for trying.
Chemiotics II 