I'll say a few words on my personal journey in math.
On Tuesday, September 3, 2002, at 08:46 AM, Osher Doctorow wrote:
> One confusing point, I think, is the tendency of many mathematical
> to identify with algebra and in fact to claim that their field is a
> or outgrowth of algebra. ....
I don't place _too_ much importance on which fields is the parent and
which is the child. My personal journey in physics drifted away from
spacetime stuff almost 30 years ago, as I sought to prepare myself for
a career by moving more into solid state physics. I read the popular
accounts of guys like Kip Thorne on black holes, but thought it was far
removed from anything in my life. And aside from reading Rucker ("Mind
Tools" is one of my favorites), Gardner, Devlin, W.W. Sawyer, Halmos,
and the recent accounts of logic (Hofstadter, Smullyan), I hadn't
looked at math in many years.
This changed half a year ago when I was reading Greg Egan's "Distress,"
with the "All Topologies Model" idea, then John Baez's site, then Lee
Smolin's fabulous book "Three Roads to Quantum Gravity." And I was
starting to learn about category theory, which I had only _heard_ of
before, but had never had the slightest clue about.
As I have said here before, the stuff on category theory and
(especially) topos theory, spoke to me in a way that was compelling.
Whether it is algebra or logic or topology/geometry, it's for me the
right approach. It provides a set of tools and, more importantly, a set
of concepts for unifying a bunch of things. The unification of geometry
and logic via algebra is compelling to me.
But maybe not to others. Wei Dai asked me to recommend just one book,
so I picked Lawvere and Schanuel's "Conceptual Mathematics: A first
introduction to categories." Well, WD reported that he had not found
anything relevant to his interests, so..."different strokes for
For me, the categorification approach unifies and makes sense of causal
sets, modal logic, possible worlds semantics, probability, and the
nature of time.
And it's fun. This is by far the most important thing, that it
stimulates me to delve into so many areas.
> There are also many built-in biases in mathematical and theoretical
> and one of them in my opinion is the bias toward dissolution of
> geometry at
> the sub-Planck level. Part of this is the pre-quantum computer bias
> toward the discrete and finite or at most countably infinite and the
> vs analog computer bias (in favor of digital computers). The real
> line and
> real line segments of course are uncountably infinite and connected,
> thee would essentially be no applied mathematics or mathematical
> physics for
> example without it - and not much pure mathematics either.
I'm essentially a constructivist. I believe the abstractions about R^N
(real line, real plane, real space, etc.) are useful, and that even the
Axiom of Choice is useful, but I'm not sure any of these platonic
ideals exist outside of Reality. Brouwerian pragmatism in Cantor's
But the views are not contradictory. Topos theory gives us a set of
tools for constructing mathematical universes. Paul Taylor puts it well
in his recent book, "Practical Foundations of Mathematics," when he
says that the apparently conflicting views of Platonism vs. Formalism
can be reconciled through these modern results.
> I hope that we can resist the temptation to go into absolutes. I am
> to see that you started your reply with a tolerant and compromising
> and I will end this posting with a similar tone.
My tone may be an accident of how I was writing at that particular
time! (insert silly smiley here as you wish).
I have no reason to be partisan on these issues. I don't know what time
is, and I doubt anyone does. But I see glimpses, and new concepts,
which are giving me what I think is a more useful understanding than
any I got back when I was studying relativity intensively! (I may have
been too young, or too focussed, or too worried about a career, or just
not exposed to the wonderful mathematics I'm now aware of.)
> The discrete and the
> connected are in my opinion different theories or parts of different
> theories overall, and they are also parts of different
> interpretations. My
> view is that science progresses by tolerating different theories and
> different interpretations for competition and because many supposedly
> theories or interpretations end up much later having something useful
I agree. However, this argues for spending some more time on the
"discrete" view, as certainly the _continuous_ view of space and time
has dominated for most of the past century (time lines, time as a
river, the flow of time, Riemannian manifolds, etc.).
In particular, whether space and time "really" are discrete at the
Planck scale or "continuous all the way down" (to 10^-35 cm, 10^-50 cm,
10^-100 cm, etc.), is not what I am thinking about right now. It may
not even matter, except for the unification of QM and gravity, the main
reason for trying to resolve this issue.
Rather, I am more interested in the issue of ordered sets and lattices,
posets especially, for the study of time and causality at human scales.
(Without going into details, here, there are issues relating to the
nature of belief and trust which have to do with these causal orders
and possible worlds. I've hinted at some of these points in other posts
here...someday I'll solidify enough of the swirling ideas to summarize
I agree strongly with Lee Smolin that topos theory (and related ideas,
tools) is not just the right logic for dealing with quantum cosmology,
it is also the right logic for dealing with a huge number of other