Pornography, calculus, cinema, metaphor

[Monte Davis]

> Pynchonwiki says that delta-t is 'An increment of time 
> represented 
> spacially, as on a graph'. Wikipedia says that delta-t is 'the time 
> difference obtained by subtracting Universal Time from 
> Terrestrial Time.' 
> Could you please elaborate a bit on the notion of delta-t and the way 
> Pynchon uses it? 

Delta is just the difference between two quantities, or the change in value
of a quantity. It can be represented spatially, but doesn't have to be: I've
heard geeky business and technology types say" What's the delta over the
last two quarters?" when they meant "How much did sales [or profits, or
production quantity, or whatever] change in thoise six months?" So the
Wikipedia citation is just one instance familiar to astronomers, maybe a
"definition" for them but not more broadly.

Pynchon's uses in GR (working from memory here) all circle around calculus,
the tool --set of tools, really -- that Newton and Leibniz developed to
analyze changing motion. Imagine a graph of time as the horizontal axis,
distance traveled as the vertical axis. Motion at a constant speed would be
a straight line with a constant slope -- say, 20 miles (vertical units) per
hour (horizontal unit). You get the same slope whether you measure over a
short or long time interval (delta-t) if you're given only the time since
starting, simple multiplication gives you the distance traveled. Likewise,
given a constant speed, it's easy to figure out time of travel from the
distance covered.

If the speed is constantly changing -- say, an object falling in gravity --
then the line on the graph is a curve with constantly changing slope. Since
we derived the idea of slope from "how many units of change on this axis per
unit on that axis,"you could even ask, is the idea of slope *at a point*, of
speed *at an instant of time* (rather than over an interval) even
meaningful? (Zeno's paradoxes and all that.) Even if you know exactly how
the speed is changing -- e.g., gravity at sea level adds about 9.8 meters
per second to the speed of fall every second -- how do you do the "how far
in delta-t" or "how long for delta-d" calculations above?

Calculus was the answer. Leibniz and Newton found that for many such curves
-- equivalent to many of the mechanical-force situations here on earth,
gravitaitonal-force situations in the heavens, and (as it would later turn
out) electromagnetic and nuclear forces too -- there *is* a way to derive
the slope (speed) over any arbitrarily small delta-t, *and* a way to
calculate the total distance traveled over any total time. Doesn't have to
be just motion; the same math worked for all sorts of smoothly varying
quantities. Hey presto -- modern physical science took off.

After a lot of nationalistic feuding between Newton fans and Leibniz fans,
the latter's notation became standard. In that notation, delta-v divided by
delta-t (change in velocity per change in time) is traditionally used to
represent accelerated motion. Then the teacher sneaks up on the calculus by
conceptually "shrinking" both to "dv/dt" -- the expression for
*instantaneous* acceleration. It's the "closer and closer to the truth"
answer you get as delta-t gets smaller without limit  -- but never zero,
nonono Tyrone never zero, because dividing by zero is formally undefined,
meaningless, bad shit, gives "infinity" as the answer no matter what
quantity is being divided.

OK, this is all very sloppy and non-rigorous (which is OK, it took nearly
200 years to make calculus rigorous enough for most mathematicians, and some
think the job isn't finished -- not that the scientists cared, because it
*worked*). What Pynchon <ahem> zeroes in on, what provides such fertile
ground for metaphor in GR, is the weirdness as delta-t (a measurable time
interval) becomes dt (an "infinitesimal" one). Every new student of calculus
feels it, every teacher of calculus should. There's an inescapable  sense of
getting away with something, almost of *cheating* Time and Nature and maybe
God. You approach very very near to doing something meaningless/forbidden.
and instead of coming away with near-nonsense, as you might expect, you come
away with (a) the right answer, and (b) an incredibly powerful and general
tool for analyzing and predicting the physical world.

That is Real Working Magic, and it's woven right into the methodological
heart of all the scary mechanistic inhuman technological blah blah blah that
the Romantics and their heirs try to tell me is so bleakly un-magical.