John Baez
Mar14-06, 04:00 AM
Also available at http://math.ucr.edu/home/baez/week227.html
March 12, 2006
This Week's Finds in Mathematical Physics (Week 227)
John Baez
Today I want to say a bit about physics, and then a bit about about
logic, since that's what I was studying for the last month in Marseille.
But first, the astronomy pictures of the week:
1) Endurance crater's dazzling dunes, NASA/JPL, available at
http://marsrovers.jpl.nasa.gov/gallery/press/opportunity/20040806a.html
On August 4, 2004, the Mars rover called Opportunity took these
pictures of dunes as it entered Endurance Crater. The red one is
what we'd actually see; the blue one is a false-color image designed
to bring out certain details.
Both images show show tendrils of sand less than 1 meter high stretching
from the big dunes toward the rover, and some rocks in the foreground.
The false-color image emphasizes accumulations of millimeter-sized
spheres called "blueberries" on the flat parts of the dunes. Here's
what they look like close up:
2) Mineral in Mars "berries" adds to water story, NASA/JPL, available at
http://marsrover.nasa.gov/newsroom/pressreleases/20040318a.html
Thanks to a Mossbauer spectroscope aboard the rover, which studies rocks
by firing gamma rays at them, we know these blueberries contain a lot of
hematite.
Hematite is made of ferric oxide, Fe_2O_3, otherwise known as "rust".
It's usually formed in the presence of water. For this and other
reasons, it's believed that the blueberries in Endurance Crater were
formed when Mars was wetter than today. An interesting puzzle is
whether they were formed by groundwater leaching ferric oxide from
rocks, or deposited in standing water - for example, a lake.
It's amazing how much we can learn from unmanned space probes. And
it's amazing to me how some people want to spend billions on manned
missions. To read more about what SF writers Larry Niven, Joe Haldeman,
Greg Bear and other folks including me think about the merits of manned
versus unmanned space missions, try this:
3) Meme Therapy: Life from a science fiction point of view, http://
memetherapy.blogspot.com/2006/03/brain-parade-feature-where-we-pester.html
But enough space stuff... now for some physics!
A strange thing happened around the 1980s. Before that, theorists had
been making rapid and revolutionary progress in understanding the
fundamental laws of physics for almost a century. They kept predicting
shocking new effects that were soon found in actual experiments: radio
waves, nuclear chain reactions, black holes, lasers, antimatter,
neutrinos, quarks,... up to and including the W and Z bosons. The power
of human thought never seemed greater.
Since the 1980s, most of the new discoveries in fundamental physics have
come from unexpected observations in astronomy. These observations were
mostly *not* predicted by theorists. The key examples are: dark matter,
dark energy, and neutrino oscillations. The only serious
counterexamples that come to mind are the top quark, discovered in 1995,
and Alan Guth's inflationary cosmology, dreamt up around 1979 and
*partially* confirmed by recent data - though the jury is still out.
Theorists are publishing more than ever, but most of their theories are
either not yet testable (string theory, loop quantum gravity) or seem to
have been disproved by experiment (grand unified theory predictions of
proton decay).
It's interesting to meditate on why this change has happened, what it
means, and what will happen next. I spoke about this in Marseille
shortly before coming home to Riverside. You can see my slides here:
4) John Baez, Fundamental physics: where we stand today, lecture at at
the Faculty of Sciences, Luminy, February 27th, 2006, available at
http://math.ucr.edu/home/baez/where_we_stand/
This talk was for nonphysicists, so it's not very technical. The first
part is a gentle introduction to the laws of physics as we know them.
For more info on dark matter and dark energy, try this:
5) Varun Sahri, Dark matter and dark energy, available as
astro-ph/0403324.
If you read this, you'll learn about the "cuspy core problem" - existing
cold dark matter models produce galaxies with a sharp spikes of high
density near their cores, sharper than observed. You'll learn about
"quintessence", a kind of hypothetical field that some people use to
model dark energy, thus "explaining" the accelerating expansion of the
universe. You'll learn about the "Chaplygin gas", a hypothetical
substance whose properties interpolate between those of cold dark matter
and dark energy. And, you'll learn about "phantom energy" models of
dark energy, which fit the accelerating expansion of the universe quite
nicely now but predict a "Big Rip", in which the expansion rate
eventually becomes *infinite*.
In short, you'll see how people are flailing around trying to understand
dark matter and dark energy.
For more on neutrino oscillations, try this:
6) K. M. Heeger, Evidence for neutrino mass: a decade of discovery,
available as hep-ex/01412032.
I had a great time in Marseille. The area around there is great for
mathematicians. Algebraists can visit the beautiful nearby city of Aix
- pronounced "x". Logicians will enjoy the dry, dusty island of If -
pronounced "eef", just like a French logician would say it. And everyone
will enjoy the medieval hill town of Les Baux, which looks like
something out of Escher.
Actually the Chateau D'If, on the island of the same name, is where
Edmond Dantes was imprisoned in Alexander Dumas' novel "The Count of
Monte Cristo". It's in this formidable fortress that the wise old Abbe
Faria tells Dantes the location of the treasure that later made him
rich.
I guess everyone except me read this story as a kid - I'm just reading
it now. But how many of you remember that Faria spent his time in
prison studying the works of Aristotle? There's a great scene where
Dantes asks Faria where he learned so much about logic, and Faria
replies: "If - and only If!"
That Dumas guy sure was a joker.
Luckily I didn't need to be locked up on a deserted island to learn some
logic in Marseille. There were lots of great talks on this topic at the
conference I attended:
7) Geometry of Computation 2006 (Geocal06),
http://iml.univ-mrs.fr/geocal06/
For example, Yves Lafont gave a category-theoretic approach to Boolean
logic gates which explains their relation to Feynman diagrams:
8) Yves Lafont, Towards an algebraic theory of Boolean circuits, Journal
of Pure and Applied Algebra 184 (2003), 257-310. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
and together with Yves Guiraud, Francois Metayer and Albert Burroni, he
gave a detailed introduction to the homology of n-categories and its
application to rewrite rules. The idea is to study any sort of
algebraic gadget (like a group) by creating an n-category where the
objects are "expressions" for elements in the gadget, the morphisms are
"ways of rewriting expressions" by applying the rules at hand, the
2-morphisms are "ways of passing from one way of rewriting expressions
to another" by applying certain "meta-rules", and so on. Then one can
use ideas from algebraic topology to study this n-category and prove
stuff about the original gadget!
To understand how this actually works, its best to start with Craig
Squier's work on the word problem for monoids. I explained this pretty
carefully back in "week70" when I first heard Lafont lecture on this
topic - it made a big impression on me! You can read more here:
9) Yves Lafont and A. Proute, Church-Rosser property and homology of
monoids, Mathematical Structures in Computer Science, Cambridge U.
Press, 1991, pp. 297-326. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
10) Yves Lafont, A new finiteness condition for monoids presented by
complete rewriting systems (after Craig C. Squier), Journal of Pure and
Applied Algebra 98 (1995), 229-244. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
Then you can go on to the higher-dimensional stuff:
11) Albert Burroni, Higher dimensional word problem with application to
equational logic, Theor. Comput. Sci. 115 (1993), 43-62. Also available
at http://www.math.jussieu.fr/~burroni/
12) Yves Guiraud, The three dimensions of proofs, Annals of Pure and
Applied Logic (in press). Also available at
http://iml.univ-mrs.fr/%7Eguiraud/recherche/cos1.pdf
13) Francois Metayer, Resolutions by polygraphs, Theory and Applications
of Categories 11 (2003), 148-184. Available online at
http://www.tac.mta.ca/tac/volumes/11/7/11-07abs.html
I was also lucky to get some personal tutoring from folks including
Laurent Regnier, Peter Selinger and especially Phil Scott. Ever since
"week40", I've been trying to understand something called "linear
logic", which was invented by Jean-Yves Girard, who teaches in
Marseille. Thanks to all this tutoring, I think I finally get it!
To get a taste of what Phil Scott told me, you should read this:
14) Philip J. Scott, Some aspects of categories in computer science,
Handbook of Algebra, Vol. 2, ed. M. Hazewinkel, Elsevier, New York,
2000. Available as http://www.site.uottawa.ca/~phil/papers/
Right now, I'm only up to explaining a microscopic portion of this
stuff. But since the typical reader of This Week's Finds may know more
about physics than logic, maybe that's good. In fact, I'll use this as
an excuse to simplify everything tremendously, leaving out all sorts of
details that a real logician would want.
Logic can be divided into two parts: SYNTAX and SEMANTICS. Roughly
speaking, syntax concerns the symbols you scribble on the page, while
semantics concerns what these symbols mean.
A bit more precisely, imagine some kind of logical system where you
write down some theory - like the axioms for a group, say - and use it
to prove theorems.
In the realm of syntax, we focus on the form our theory is allowed to
have, and how we can deduce new sentences from old ones. So, one of the
key concepts is that of a PROOF. The details will vary depending on the
kind of logical system we're studying.
In the realm of semantics, we are interested in gadgets that actually
satisfy the axioms in our theory - for example, actual groups, if we're
thinking about the theory of groups. Such a gadget is called a MODEL of
the theory. Again, the details vary immensely.
In the realm of syntax, we say a list of axioms X "implies" a sentence P
if we can prove P from X using some deduction rules, and we write this
as
X |- P
In the realm of semantics, we say a list of axioms X "entails" a
sentence P if every model of X is also a model of P, and we write this
as
X |= P
Syntax and semantics are "dual" in a certain sense - a sense that can be
made very precise if one fixes a specific class of logical systems. This
duality is akin to the usual relation between vector spaces and their
duals, or more generally groups and their categories of representations.
The idea is that given a theory T you can figure out its models, which
form a category Mod(T) - and conversely, given the category of models
Mod(T), perhaps with a little extra information, you can reconstruct T.
A little extra information? Well, in some cases a model of T will be a
*set* with some extra structure - for example, if T is the theory of
groups, a model of T will be a group, which is a set equipped with some
operations. So, in these cases there's a functor
U: Mod(T) -> Set
assigning each model its underlying set. And, you can easily
reconstruct T from Mod(T) *together* with this functor.
This idea was worked out by Lawvere for a class of logical systems
called algebraic theories, which I discussed in "week200". But, the
same idea goes by the name of "Tannaka-Krein duality" in a different
context: a Hopf algebra H has a category of comodules Rep(H), which
comes equipped with a functor
U: Rep(H) -> Vect
assigning each comodule its underlying vector space. And, you can
reconstruct H from Rep(H) together with this functor. The proof is even
very similar to Lawvere's proof for algebraic theories!
I gave a bunch of talks in Marseille about algebraic theories, some
related logical systems called PROPs and PROs, and their relation to
quantum theory, especially Feynman diagrams:
14) John Baez, Universal algebra and diagrammatic reasoning, available
as http://math.ucr.edu/home/baez/universal/
I came mighty close to explaining how to compute the cohomology of an
algebraic theory... and you can read more about that here:
15) Mauka Jibladze and Teimuraz Pirashvili, Cohomology of algebraic
theories, J. Algebra 137 (1991) 253-296.
Mauka Jibladze and Teimuraz Pirashvili, Quillen cohomology and
Baues-Wirsching cohomology of algebraic theories, Max-Planck-Institut
fuer Mathematik, preprint series 86 (2005).
But alas, I didn't get around to talking about the duality between
syntax and semantics. For that Lawvere's original thesis is a good
place to go:
16) F. William Lawvere, Functorial Semantics of Algebraic Theories,
Ph.D. thesis, Columbia University, 1963. Also available at
http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html
Anyway, the stuff Phil Scott told me about was mainly over on the syntax
side. Here categories show up in another way. Oversimplifying as
usual, the idea is to create a category where an object P is a
*sentence* - or maybe a list of sentences - and a morphism
f: P -> Q
is a *proof* of Q from P - or maybe an equivalence class of proofs.
We can compose proofs in a more or less obvious way, so with any luck
this gives a category! And, different kinds of logical system give us
different kinds of categories.
Quite famously, intuitionistic logic gives cartesian closed categories.
The "multiplicative fragment" of linear logic gives *-autonomous
categories. And, full-fledged linear logic gives us certain fancier
kinds of categories. If you want to learn about these examples, read the
handbook article by Phil Scott mentioned above.
One thing that intrigues me is the equivalence relation we need to get a
category whose morphisms are equivalence classes of proofs. In
Gentzen's "natural deduction" approach to logic, there are various
deduction rules. Here's one:
P |- Q P |- Q'
------------------
P |- Q & Q'
This says that if P implies Q and it also implies Q', then it implies
Q & Q'.
Here's another:
P |- Q => R
------------
P and Q |- R
And here's another, called "modus ponens":
P |- Q Q |- R
-----------------
P |- R
There are a bunch more... and to get the game rolling we start with this:
P |- P
In this setup, a proof f: P -> Q looks vaguely like this:
f-crud
f-crud
f-crud
f-crud
f-crud
f-crud
-------------
P |- Q
The stuff I'm calling "f-crud" is a bunch of steps which use the
deduction rules to get to P |- Q.
Suppose we also we also have a proof
g: Q -> R
There's a way to stick f and g together to get a proof
fg: P -> R
This proof consists of setting the proofs f and g side by side and then
using modus ponents to finish the job. So, fg looks like this:
f-crud g-crud
f-crud g-crud
f-crud g-crud
f-crud g-crud
-------- --------
P |- Q Q |- R
---------------------
P |- R
Now let's see if composition is associative. Suppose we also have a
proof
h: R -> S
We can form proofs
(fg)h: P -> S
and
f(gh): P -> S
Are they equal? No! The first one looks like this:
f-crud g-crud
f-crud g-crud
f-crud g-crud h-crud
f-crud g-crud h-crud
-------- -------- h-crud
P |- Q Q |- R h-crud
--------------------- -----------
P |- R R |- S
----------------------------
P |- S
while the second one looks like this:
g-crud h-crud
g-crud h-crud
f-crud g-crud h-crud
f-crud g-crud h-crud
f-crud -------- --------
f-crud Q |- R R |- S
--------- ---------------------
P |- Q Q |- S
----------------------------
P |- S
So, they're not quite equal! This is one reason we need an
equivalence relation on proofs to get a category. Both proofs
resemble trees, but the first looks more like this:
\ / /
\/ /
\ /
|
while the second looks more like this:
\ \ /
\ \/
\ /
|
So, we need an equivalence relation that identifies these proofs
if we want composition to be associative!
This sort of idea, including this "tree" business, is very familiar from
homotopy theory, where we need a similar equivalence relation if we want
composition of paths to be associative. But in homotopy theory, people
have learned that it's often better NOT to impose an equivalence
relation on paths! Instead, it's better to form a *weak 2-category* of
paths, where there's a 2-morphism going from this sort of composite:
\ / /
\/ /
\ /
|
to this one:
\ \ /
\ \/
\ /
|
This is called the "associator". In our logic context, we can think of
the associator as a way to transform one proof into another.
The associator should satisfy an equation called the "pentagon
identity", which I explained back in "week144". However, it will only
do this if we let 2-morphisms be *equivalence classes* of proof
transformations.
So, there's a kind of infinite regress here. To deal with this, it
would be best to work with a "weak omega-category" with
sentences (or sequences of sentences) as objects,
proofs as morphisms,
proof transformations as 2-morphisms,
transformations of proof transformations as 3-morphisms,...
and so on. With this, we would never need any equivalence relations:
we keep track of all transformations explicitly. This is almost beyond
what mathematicians are capable of at present, but it's clearly a good
thing to strive toward.
So far, it seems Seely has gone the furthest in this direction.
In his thesis, way back in 1977, he studied what one might call "weak
cartesian closed 2-categories" arising from proof theory. You can read
an account of this work here:
17) R.A.G. Seely, Weak adjointness in proof theory, in Proc. Durham Conf.
on Applications of Sheaves, Springer Lecture Notes in Mathematics 753,
Springer, Berlin, 1979, pp. 697-701. Also available at
http://www.math.mcgill.ca/rags/WkAdj/adj.pdf
R.A.G. Seely, Modeling computations: a 2-categorical framework, in
Proc. Symposium on Logic in Computer Science 1987, Computer Society
of the IEEE, pp. 65-71. Also available at
http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf
Can we go all the way and cook up some sort of omega-category of proofs?
Interestingly, while the logicians at Geocal06 were talking about
n-categories and the geometry of proofs, the mathematician Vladimir
Voevodsky was giving some talks at Stanford about something that sounds
pretty similar:
18) Vladimir Voevodsky, lectures on homotopy lambda calculus,
notice at http://math.stanford.edu/distinguished_voevodsky.htm
Voevodsky has thought hard about n-categories, and he won the Fields
medal for his applications of homotopy theory to algebraic geometry.
The typed lambda calculus is another way of thinking about
intuitionistic logic - or in other words, cartesian closed categories of
proofs. The "homotopy lambda calculus" should thus be something
similar, but where we keep track of transformations between proofs,
transformations between transformations between proofs... and so on ad
infinitum.
But that's just my guess! Is this what Voevodsky is talking about??? I
haven't managed to get anyone to tell me. Maybe I'll email him and ask.
There were a lot of other cool talks at Geocal06, like Girard's talk on
applications of von Neumann algebras (especially the hyperfinite type
II_1 factor!) in logic, and Peter Selinger's talk on the category of
completely positive maps, diagrammatic methods for dealing with these
maps, and their applications to quantum logic:
19) Peter Selinger, Dagger compact closed categories and completely
positive maps, available at http://www.mscs.dal.ca/~selinger/papers.html
But, I want to finish writing this and go out and have some waffles
for my Sunday brunch. So, I'll stop here!
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
March 12, 2006
This Week's Finds in Mathematical Physics (Week 227)
John Baez
Today I want to say a bit about physics, and then a bit about about
logic, since that's what I was studying for the last month in Marseille.
But first, the astronomy pictures of the week:
1) Endurance crater's dazzling dunes, NASA/JPL, available at
http://marsrovers.jpl.nasa.gov/gallery/press/opportunity/20040806a.html
On August 4, 2004, the Mars rover called Opportunity took these
pictures of dunes as it entered Endurance Crater. The red one is
what we'd actually see; the blue one is a false-color image designed
to bring out certain details.
Both images show show tendrils of sand less than 1 meter high stretching
from the big dunes toward the rover, and some rocks in the foreground.
The false-color image emphasizes accumulations of millimeter-sized
spheres called "blueberries" on the flat parts of the dunes. Here's
what they look like close up:
2) Mineral in Mars "berries" adds to water story, NASA/JPL, available at
http://marsrover.nasa.gov/newsroom/pressreleases/20040318a.html
Thanks to a Mossbauer spectroscope aboard the rover, which studies rocks
by firing gamma rays at them, we know these blueberries contain a lot of
hematite.
Hematite is made of ferric oxide, Fe_2O_3, otherwise known as "rust".
It's usually formed in the presence of water. For this and other
reasons, it's believed that the blueberries in Endurance Crater were
formed when Mars was wetter than today. An interesting puzzle is
whether they were formed by groundwater leaching ferric oxide from
rocks, or deposited in standing water - for example, a lake.
It's amazing how much we can learn from unmanned space probes. And
it's amazing to me how some people want to spend billions on manned
missions. To read more about what SF writers Larry Niven, Joe Haldeman,
Greg Bear and other folks including me think about the merits of manned
versus unmanned space missions, try this:
3) Meme Therapy: Life from a science fiction point of view, http://
memetherapy.blogspot.com/2006/03/brain-parade-feature-where-we-pester.html
But enough space stuff... now for some physics!
A strange thing happened around the 1980s. Before that, theorists had
been making rapid and revolutionary progress in understanding the
fundamental laws of physics for almost a century. They kept predicting
shocking new effects that were soon found in actual experiments: radio
waves, nuclear chain reactions, black holes, lasers, antimatter,
neutrinos, quarks,... up to and including the W and Z bosons. The power
of human thought never seemed greater.
Since the 1980s, most of the new discoveries in fundamental physics have
come from unexpected observations in astronomy. These observations were
mostly *not* predicted by theorists. The key examples are: dark matter,
dark energy, and neutrino oscillations. The only serious
counterexamples that come to mind are the top quark, discovered in 1995,
and Alan Guth's inflationary cosmology, dreamt up around 1979 and
*partially* confirmed by recent data - though the jury is still out.
Theorists are publishing more than ever, but most of their theories are
either not yet testable (string theory, loop quantum gravity) or seem to
have been disproved by experiment (grand unified theory predictions of
proton decay).
It's interesting to meditate on why this change has happened, what it
means, and what will happen next. I spoke about this in Marseille
shortly before coming home to Riverside. You can see my slides here:
4) John Baez, Fundamental physics: where we stand today, lecture at at
the Faculty of Sciences, Luminy, February 27th, 2006, available at
http://math.ucr.edu/home/baez/where_we_stand/
This talk was for nonphysicists, so it's not very technical. The first
part is a gentle introduction to the laws of physics as we know them.
For more info on dark matter and dark energy, try this:
5) Varun Sahri, Dark matter and dark energy, available as
astro-ph/0403324.
If you read this, you'll learn about the "cuspy core problem" - existing
cold dark matter models produce galaxies with a sharp spikes of high
density near their cores, sharper than observed. You'll learn about
"quintessence", a kind of hypothetical field that some people use to
model dark energy, thus "explaining" the accelerating expansion of the
universe. You'll learn about the "Chaplygin gas", a hypothetical
substance whose properties interpolate between those of cold dark matter
and dark energy. And, you'll learn about "phantom energy" models of
dark energy, which fit the accelerating expansion of the universe quite
nicely now but predict a "Big Rip", in which the expansion rate
eventually becomes *infinite*.
In short, you'll see how people are flailing around trying to understand
dark matter and dark energy.
For more on neutrino oscillations, try this:
6) K. M. Heeger, Evidence for neutrino mass: a decade of discovery,
available as hep-ex/01412032.
I had a great time in Marseille. The area around there is great for
mathematicians. Algebraists can visit the beautiful nearby city of Aix
- pronounced "x". Logicians will enjoy the dry, dusty island of If -
pronounced "eef", just like a French logician would say it. And everyone
will enjoy the medieval hill town of Les Baux, which looks like
something out of Escher.
Actually the Chateau D'If, on the island of the same name, is where
Edmond Dantes was imprisoned in Alexander Dumas' novel "The Count of
Monte Cristo". It's in this formidable fortress that the wise old Abbe
Faria tells Dantes the location of the treasure that later made him
rich.
I guess everyone except me read this story as a kid - I'm just reading
it now. But how many of you remember that Faria spent his time in
prison studying the works of Aristotle? There's a great scene where
Dantes asks Faria where he learned so much about logic, and Faria
replies: "If - and only If!"
That Dumas guy sure was a joker.
Luckily I didn't need to be locked up on a deserted island to learn some
logic in Marseille. There were lots of great talks on this topic at the
conference I attended:
7) Geometry of Computation 2006 (Geocal06),
http://iml.univ-mrs.fr/geocal06/
For example, Yves Lafont gave a category-theoretic approach to Boolean
logic gates which explains their relation to Feynman diagrams:
8) Yves Lafont, Towards an algebraic theory of Boolean circuits, Journal
of Pure and Applied Algebra 184 (2003), 257-310. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
and together with Yves Guiraud, Francois Metayer and Albert Burroni, he
gave a detailed introduction to the homology of n-categories and its
application to rewrite rules. The idea is to study any sort of
algebraic gadget (like a group) by creating an n-category where the
objects are "expressions" for elements in the gadget, the morphisms are
"ways of rewriting expressions" by applying the rules at hand, the
2-morphisms are "ways of passing from one way of rewriting expressions
to another" by applying certain "meta-rules", and so on. Then one can
use ideas from algebraic topology to study this n-category and prove
stuff about the original gadget!
To understand how this actually works, its best to start with Craig
Squier's work on the word problem for monoids. I explained this pretty
carefully back in "week70" when I first heard Lafont lecture on this
topic - it made a big impression on me! You can read more here:
9) Yves Lafont and A. Proute, Church-Rosser property and homology of
monoids, Mathematical Structures in Computer Science, Cambridge U.
Press, 1991, pp. 297-326. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
10) Yves Lafont, A new finiteness condition for monoids presented by
complete rewriting systems (after Craig C. Squier), Journal of Pure and
Applied Algebra 98 (1995), 229-244. Also available at
http://iml.univ-mrs.fr/~lafont/publications.html
Then you can go on to the higher-dimensional stuff:
11) Albert Burroni, Higher dimensional word problem with application to
equational logic, Theor. Comput. Sci. 115 (1993), 43-62. Also available
at http://www.math.jussieu.fr/~burroni/
12) Yves Guiraud, The three dimensions of proofs, Annals of Pure and
Applied Logic (in press). Also available at
http://iml.univ-mrs.fr/%7Eguiraud/recherche/cos1.pdf
13) Francois Metayer, Resolutions by polygraphs, Theory and Applications
of Categories 11 (2003), 148-184. Available online at
http://www.tac.mta.ca/tac/volumes/11/7/11-07abs.html
I was also lucky to get some personal tutoring from folks including
Laurent Regnier, Peter Selinger and especially Phil Scott. Ever since
"week40", I've been trying to understand something called "linear
logic", which was invented by Jean-Yves Girard, who teaches in
Marseille. Thanks to all this tutoring, I think I finally get it!
To get a taste of what Phil Scott told me, you should read this:
14) Philip J. Scott, Some aspects of categories in computer science,
Handbook of Algebra, Vol. 2, ed. M. Hazewinkel, Elsevier, New York,
2000. Available as http://www.site.uottawa.ca/~phil/papers/
Right now, I'm only up to explaining a microscopic portion of this
stuff. But since the typical reader of This Week's Finds may know more
about physics than logic, maybe that's good. In fact, I'll use this as
an excuse to simplify everything tremendously, leaving out all sorts of
details that a real logician would want.
Logic can be divided into two parts: SYNTAX and SEMANTICS. Roughly
speaking, syntax concerns the symbols you scribble on the page, while
semantics concerns what these symbols mean.
A bit more precisely, imagine some kind of logical system where you
write down some theory - like the axioms for a group, say - and use it
to prove theorems.
In the realm of syntax, we focus on the form our theory is allowed to
have, and how we can deduce new sentences from old ones. So, one of the
key concepts is that of a PROOF. The details will vary depending on the
kind of logical system we're studying.
In the realm of semantics, we are interested in gadgets that actually
satisfy the axioms in our theory - for example, actual groups, if we're
thinking about the theory of groups. Such a gadget is called a MODEL of
the theory. Again, the details vary immensely.
In the realm of syntax, we say a list of axioms X "implies" a sentence P
if we can prove P from X using some deduction rules, and we write this
as
X |- P
In the realm of semantics, we say a list of axioms X "entails" a
sentence P if every model of X is also a model of P, and we write this
as
X |= P
Syntax and semantics are "dual" in a certain sense - a sense that can be
made very precise if one fixes a specific class of logical systems. This
duality is akin to the usual relation between vector spaces and their
duals, or more generally groups and their categories of representations.
The idea is that given a theory T you can figure out its models, which
form a category Mod(T) - and conversely, given the category of models
Mod(T), perhaps with a little extra information, you can reconstruct T.
A little extra information? Well, in some cases a model of T will be a
*set* with some extra structure - for example, if T is the theory of
groups, a model of T will be a group, which is a set equipped with some
operations. So, in these cases there's a functor
U: Mod(T) -> Set
assigning each model its underlying set. And, you can easily
reconstruct T from Mod(T) *together* with this functor.
This idea was worked out by Lawvere for a class of logical systems
called algebraic theories, which I discussed in "week200". But, the
same idea goes by the name of "Tannaka-Krein duality" in a different
context: a Hopf algebra H has a category of comodules Rep(H), which
comes equipped with a functor
U: Rep(H) -> Vect
assigning each comodule its underlying vector space. And, you can
reconstruct H from Rep(H) together with this functor. The proof is even
very similar to Lawvere's proof for algebraic theories!
I gave a bunch of talks in Marseille about algebraic theories, some
related logical systems called PROPs and PROs, and their relation to
quantum theory, especially Feynman diagrams:
14) John Baez, Universal algebra and diagrammatic reasoning, available
as http://math.ucr.edu/home/baez/universal/
I came mighty close to explaining how to compute the cohomology of an
algebraic theory... and you can read more about that here:
15) Mauka Jibladze and Teimuraz Pirashvili, Cohomology of algebraic
theories, J. Algebra 137 (1991) 253-296.
Mauka Jibladze and Teimuraz Pirashvili, Quillen cohomology and
Baues-Wirsching cohomology of algebraic theories, Max-Planck-Institut
fuer Mathematik, preprint series 86 (2005).
But alas, I didn't get around to talking about the duality between
syntax and semantics. For that Lawvere's original thesis is a good
place to go:
16) F. William Lawvere, Functorial Semantics of Algebraic Theories,
Ph.D. thesis, Columbia University, 1963. Also available at
http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html
Anyway, the stuff Phil Scott told me about was mainly over on the syntax
side. Here categories show up in another way. Oversimplifying as
usual, the idea is to create a category where an object P is a
*sentence* - or maybe a list of sentences - and a morphism
f: P -> Q
is a *proof* of Q from P - or maybe an equivalence class of proofs.
We can compose proofs in a more or less obvious way, so with any luck
this gives a category! And, different kinds of logical system give us
different kinds of categories.
Quite famously, intuitionistic logic gives cartesian closed categories.
The "multiplicative fragment" of linear logic gives *-autonomous
categories. And, full-fledged linear logic gives us certain fancier
kinds of categories. If you want to learn about these examples, read the
handbook article by Phil Scott mentioned above.
One thing that intrigues me is the equivalence relation we need to get a
category whose morphisms are equivalence classes of proofs. In
Gentzen's "natural deduction" approach to logic, there are various
deduction rules. Here's one:
P |- Q P |- Q'
------------------
P |- Q & Q'
This says that if P implies Q and it also implies Q', then it implies
Q & Q'.
Here's another:
P |- Q => R
------------
P and Q |- R
And here's another, called "modus ponens":
P |- Q Q |- R
-----------------
P |- R
There are a bunch more... and to get the game rolling we start with this:
P |- P
In this setup, a proof f: P -> Q looks vaguely like this:
f-crud
f-crud
f-crud
f-crud
f-crud
f-crud
-------------
P |- Q
The stuff I'm calling "f-crud" is a bunch of steps which use the
deduction rules to get to P |- Q.
Suppose we also we also have a proof
g: Q -> R
There's a way to stick f and g together to get a proof
fg: P -> R
This proof consists of setting the proofs f and g side by side and then
using modus ponents to finish the job. So, fg looks like this:
f-crud g-crud
f-crud g-crud
f-crud g-crud
f-crud g-crud
-------- --------
P |- Q Q |- R
---------------------
P |- R
Now let's see if composition is associative. Suppose we also have a
proof
h: R -> S
We can form proofs
(fg)h: P -> S
and
f(gh): P -> S
Are they equal? No! The first one looks like this:
f-crud g-crud
f-crud g-crud
f-crud g-crud h-crud
f-crud g-crud h-crud
-------- -------- h-crud
P |- Q Q |- R h-crud
--------------------- -----------
P |- R R |- S
----------------------------
P |- S
while the second one looks like this:
g-crud h-crud
g-crud h-crud
f-crud g-crud h-crud
f-crud g-crud h-crud
f-crud -------- --------
f-crud Q |- R R |- S
--------- ---------------------
P |- Q Q |- S
----------------------------
P |- S
So, they're not quite equal! This is one reason we need an
equivalence relation on proofs to get a category. Both proofs
resemble trees, but the first looks more like this:
\ / /
\/ /
\ /
|
while the second looks more like this:
\ \ /
\ \/
\ /
|
So, we need an equivalence relation that identifies these proofs
if we want composition to be associative!
This sort of idea, including this "tree" business, is very familiar from
homotopy theory, where we need a similar equivalence relation if we want
composition of paths to be associative. But in homotopy theory, people
have learned that it's often better NOT to impose an equivalence
relation on paths! Instead, it's better to form a *weak 2-category* of
paths, where there's a 2-morphism going from this sort of composite:
\ / /
\/ /
\ /
|
to this one:
\ \ /
\ \/
\ /
|
This is called the "associator". In our logic context, we can think of
the associator as a way to transform one proof into another.
The associator should satisfy an equation called the "pentagon
identity", which I explained back in "week144". However, it will only
do this if we let 2-morphisms be *equivalence classes* of proof
transformations.
So, there's a kind of infinite regress here. To deal with this, it
would be best to work with a "weak omega-category" with
sentences (or sequences of sentences) as objects,
proofs as morphisms,
proof transformations as 2-morphisms,
transformations of proof transformations as 3-morphisms,...
and so on. With this, we would never need any equivalence relations:
we keep track of all transformations explicitly. This is almost beyond
what mathematicians are capable of at present, but it's clearly a good
thing to strive toward.
So far, it seems Seely has gone the furthest in this direction.
In his thesis, way back in 1977, he studied what one might call "weak
cartesian closed 2-categories" arising from proof theory. You can read
an account of this work here:
17) R.A.G. Seely, Weak adjointness in proof theory, in Proc. Durham Conf.
on Applications of Sheaves, Springer Lecture Notes in Mathematics 753,
Springer, Berlin, 1979, pp. 697-701. Also available at
http://www.math.mcgill.ca/rags/WkAdj/adj.pdf
R.A.G. Seely, Modeling computations: a 2-categorical framework, in
Proc. Symposium on Logic in Computer Science 1987, Computer Society
of the IEEE, pp. 65-71. Also available at
http://www.math.mcgill.ca/rags/WkAdj/LICS.pdf
Can we go all the way and cook up some sort of omega-category of proofs?
Interestingly, while the logicians at Geocal06 were talking about
n-categories and the geometry of proofs, the mathematician Vladimir
Voevodsky was giving some talks at Stanford about something that sounds
pretty similar:
18) Vladimir Voevodsky, lectures on homotopy lambda calculus,
notice at http://math.stanford.edu/distinguished_voevodsky.htm
Voevodsky has thought hard about n-categories, and he won the Fields
medal for his applications of homotopy theory to algebraic geometry.
The typed lambda calculus is another way of thinking about
intuitionistic logic - or in other words, cartesian closed categories of
proofs. The "homotopy lambda calculus" should thus be something
similar, but where we keep track of transformations between proofs,
transformations between transformations between proofs... and so on ad
infinitum.
But that's just my guess! Is this what Voevodsky is talking about??? I
haven't managed to get anyone to tell me. Maybe I'll email him and ask.
There were a lot of other cool talks at Geocal06, like Girard's talk on
applications of von Neumann algebras (especially the hyperfinite type
II_1 factor!) in logic, and Peter Selinger's talk on the category of
completely positive maps, diagrammatic methods for dealing with these
maps, and their applications to quantum logic:
19) Peter Selinger, Dagger compact closed categories and completely
positive maps, available at http://www.mscs.dal.ca/~selinger/papers.html
But, I want to finish writing this and go out and have some waffles
for my Sunday brunch. So, I'll stop here!
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