Topos Theory in the New Scientist

[bananas]

http://golem.ph.utexas.edu/category/2007/04/topos_theory_in_the_new_scient.html

April 14, 2007
Topos Theory in the New Scientist
Posted by John Baez

Our favorite science magazine has decided to take on Chris Isham and Andreas Döring’s work on topos theory and physics:
Robert Matthews, Impossible things for breakfast, at the Logic Café, New Scientist, April 14, 2007.

At the n-Category Café we serve only possible things for breakfast. But, many things are possible…

Over on the category theory mailing list, the renowned topos theorist Peter Johnstone writes:

Category theorists in general, and topos theorists in particular, may want to check out this week’s cover story in the New Scientist (www.newscientist.com). The author (Robert Matthews of Aston University in Birmingham) is clearly a fan of Chris Isham: it’s not clear to me whether he actually knows what a topos is, but he has committed himself to statements such as

“Topos theory could lead to a view of reality more astonishing and successful than quantum theory”

which is splashed all over page 32 of the magazine. Even if you don’t believe this (and I don’t think I do) it’s pleasant to see topos theory getting this sort of publicity.

<snip>

[On the Road]

1. I Knew Nothing about Topos Theory Before

and the Wikipedia entry, mmmm, didn't really help:

the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules. An abelian category is supposed to be closed under certain category-theoretic operations — by using this kind of definition one can focus entirely on structure, saying nothing at all about the nature of the objects involved. This type of definition traces back, in one line, to the lattice concept of the 1930s. It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work (the Tohoku paper).
http://en.wikipedia.org/wiki/Background_and_genesis_of_topos_theory

Now there is a "hand-wavy vague explanation" (their term) on this link that makes a tiny bit more sense to a non-mathematician.

Around 1963, Lawvere decided to figure out new foundations for mathematics,
based on category theory. ...Lawvere and others invented the concept of a
"topos", which is category with certain extra properties that make it a
lot like the category of sets. A topos is a category that has:

A) finite limits and colimits
B) exponentials
C) a subobject classifier

http://www.math.niu.edu/~rusin/known-math/00_incoming/topos

And then there's the treatment from philosophical history, which clarifies the context the New Scientist article:

The philosophy of physics is intimately tied into the philosophy of mathematics and logic -- especially when explaining how Quantum theory and General Relativity theory can be unified. In this case a philosophical Reductionist argument (probably rightly) claims that mathematics can be reduced to pure logic. But the question is: Which logic? Is there more than one logic?! Almost all of physics, as well as so-called Western science, is tied to Aristotle's logic. This is the Classical logic that is normally taught in universities. But, there is another kind of logic that might be more helpful. It is non-Aristotelian, invented under the name Intuitionistic logic (Intuitionistic mathematics), and has taken an evolutionary path of development in recent years. It has developed into Topos logic....So, we might call Topos a weaken kind of logic. I will explain later, that the weakness is not necessarily detrimental. For now, let us look at why we have a problem in the first place, which is: Quantum superpositions.

From what quantum theory says, before our measurement the particle is actually in more than one state at a time, e.g. two states of polarization. You can think of this as being both true and false at the same time...

How can a single particle have both true and false answers to a single mutually exclusive property? This is called a superposition -- it is a king of overlap of true and false. It is sort of in-between true and false in the middle. We will see why I call this a, middle, in a moment. How can this superposition of a particle property happen? Classical logic says the middle ground is excluded. Superposition is an excluded middle, true and false at the same time. This is not supposed to happen according to Aristotle, but it does.

And finally, there is the more layman's treatment, which is (believe it or not) on Physics web

A quantum leap for cosmology
A theory that unites quantum mechanics and general relativity claims that there was no first moment in time, but it still agrees with the predictions of classical cosmology.
One of the most challenging problems in modern physics is the application of quantum theory to the universe as a whole....

And now the payoff:

In particular, Bojowald has discovered that there is never an initial singularity (i.e. a point where the curvature of space-time becomes infinite) and therefore no first moment to time, as Alex Vilenken and others have hypothesized. Nor is there any excursion into a domain in which the universe has a boundary in "imaginary time", as hypothesized by Jim Hartle, Stephen Hawking and others. Instead the universe continues back before the moment classical cosmology predicts that it began, to a phase where it was previously expanding. This behaviour has been called a "bounce"; it suggests that the big bang arose from an event in a previous universe, either through the collapse of a black hole in that universe or from the collapse of the whole universe.

There seems to be much more, but that's my brain can fit for right now.

[meg]

2. THANKS!!

Very helpful, ribofunk!