## Wednesday, May 30, 2007

### Nothing is True, Everything is Permitted

Permit me to explain a bit of the motto of this blog. Occult enthusiasts will immediately recognize it as the words of Aleister Crowley, who himself took them from Nietzsche's Thus Sprach Zarathustra, who in turn took it from the 11th-century Ismaili Hassan-i Sabah. But what does it mean?

It's simplistic to say that what Sabah meant was that there was no such thing as right or wrong. Not so. What he, Nietzsche and Crowley meant was that absolute, capital-T Truth exists only within our perceptions, and when removed from said perceptions it falls completely apart. If that's the case, then why do people still insist on there being absolute truth?

Because that's what gets you what you want. In our society, assertion is necessary, or you go unfulfilled. I'm sure there's some interesting evolutionary biology in there somewhere.

What applications does this have in the real world? Welcome to the wonderful world of topos theory.

A traditional axiomatic foundation of mathematics is set theory, in which all mathematical objects are ultimately represented by sets (even functions which map between sets.) More recent work in category theory allows this foundation to be generalized using topoi; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternate topoi. A standard formulation of the axiom of choice makes sense in any topos, and there are topoi in which it is invalid. Constructivists will be interested to work in a topos without the law of excluded middle. If symmetry under a particular group G is of importance, one can use the topos consisting of all G-sets.

Unfortunately, if you don't know some category theory, the above definition will be mysterious and will require a further sequence of definitions to bring it back to the basic concepts of category theory - object, morphism, composition, identity. Instead of doing all that, let me say a bit about what these items A)-C) amount to in the category of sets:
A) says that there are:

an initial object (an object like the empty set)
a terminal object (an an object like a set with one element)
binary coproducts (something like the disjoint union of two sets)
binary products (something like the Cartesian product of two sets)
equalizers (something like the subset of X consisting of all elements x such that f(x) = g(x), where f,g: X → Y)
coequalizers (something like the quotient set of X where two elements f(y) and g(y) are identified, where f,g: Y → X)
In fact A) is equivalent to all this stuff. However, I should emphasize that A) says all this in an elegant unified way; it's a theorem that this elegant way is the same as all the crud I just listed.

B) says that for any objects x and y, there is an object yx, called an "exponential", which acts like "the set of functions from x to y".

C) says that there is an object called the "subobject classifier" Ω, which acts like {0,1}, in that functions from any set x into {0,1} are secretly the same as subsets of x. You can think of Ω as the replacement for the usual boolean "truth values" that we work with when doing logic in the category of sets.

Learning more about all these concepts is probably the best use of your time if you wants to learn a little bit of topos theory. Even if you can't remember what a topos is, these concepts can help you become a stronger mathematician or mathematical physicist!

This one is from the New Scientist (sorry, to read the rest you need a subscription):

CHRIS ISHAM has a problem with truth. And he suspects his fellow physicists do too. It is not their honesty he doubts, but their approach to understanding the nature of the universe, the laws that govern it and reality itself. Together with a small band of allies, Isham is wrestling with questions that lie at the very core of physics. Indeed they run even deeper, to such basic concepts as logic, existence and truth. What do they mean? Are they immutable? What lies beyond them?

After years of effort, Isham and his colleagues at Imperial College London and elsewhere believe they can glimpse the answers to these profound questions. They didn't set out to rethink such weighty issues. When they started nearly a decade ago, the researchers hoped to arrive at a quantum theory of the universe, an ambitious enough task in itself. Yet in the process they might have bagged something bigger.

For if their results stand up, Isham and his colleagues appear to have found a new way of making sense of reality using concepts even more fundamental than mathematics and logic. Not only could their insights be good news for quantum theory, they could lead to a whole new way of constructing theories of reality.

Since its emergence around a century ago, quantum theory has become one of the cornerstones of modern science. It underpins everything from the behaviour of quarks and semiconductors to the power of medical scanners. And it has passed virtually every test thrown at it, its predictions agreeing with experiment to many decimal places.

With a track record like that, quantum theory might seem ideal for casting light on the ultimate questions about the universe, such as why it exists at all. Not so. In fact, it runs into very big trouble very quickly, because quantum theory has a problem with truth.

With hindsight, perhaps this shouldn't be so surprising. Right from the start, quantum theory has had a reputation for giving odd answers to even seemingly simple questions.

It goes on to describe some very interesting things. Since I'm something of a simpleton when it comes to math, this is what I understand from it:

You can build different mathematical systems from different points of view. Imagine what math is like as Schroedinger's Cat or on the inside of a black hole, and you can build an entire new system from scratch. This means that there may be different parallel mathematical realities occupying the same space as this one, we just can't see them from the Euclidean POV.

Think of the questions this raises. Is there life in these parallel math realities? Can we communicate with them? Do we already participate with them at the unconscious level? What does this mean for our current definitions of "possible" and "impossible?"

Topos theory gives us but a glimpse of the reality we've been missing out on. So, take a moment to remind yourself that this universe you're in may not be the only one, and maybe decide to start or end your day today by reciting the motto above.

Nothing is true, everything is permitted.