Exploring alternate mathematical universes with hypercomputation (Wondrous Mathematics)
|Description||This talk gives a leisurely introduction to constructive mathematics, a variant of classical mathematics where we drop some of the standard axioms of ordinary reasoning. This allows us to adopt classically inconsistent "dream axioms" and explore curious alternate mathematical universes. In the talk we'll focus on a wondrous connection to models of computation, both standard ones such as ordinary programming languages and exotic models such as hypercomputation which allow for infinitely many steps in finite time and which push the laws of physics to their limits. The special properties of these alternative universes then depend on the nature of our physical reality.|
|Tags||mathematics, logic, hypercomputation|
|Language|| en - English |
en - English
|Starts at||2016/12/28 14:00|
|Ends at||2016/12/28 14:50|
The usual view is that any mathematical statement about numbers or other mathematical objects has a definite, objective meaning independent of human sentiments. More precisely, though, mathematics rests on a set of agreed-upon axioms. These are to some extent man-made and can be varied in interesting ways. The basic tenet of constructive mathematics is to drop the axiom that any statement must either be true or not true. This unlocks several new axioms which are classically plainly false but are compatible with constructive mathematics, yielding alternative mathematical universes which can even be tailored to specific applications.
It turns out that any model of computation gives rise to such an alternative universe. What do these universes look like? Which statements of classical mathematics carry over? And which new statements hold?
The answers depend on the chosen model of computation. We obtain especially interesting answers in the case that we employ models of hypercomputation, where computers can perform infinitely many calculational steps in finite time, and physical models about the real world. In the latter case, statements which are in classical mathematics simply true become non-trivial statements about the nature of the physical world.
The talk is aimed at people who enjoy mathematical thinking, but absolutely no prerequisites in formal logic are needed. You should have superficial knowledge on the Halting problem (it suffices to having read the Wikipedia entry at some point in your life). The talk reports on work by Andrej Bauer.
- Slides of the talk (will be extended with notes in the next few days)
- Unfortunately the recording failed in that there is no sound. So we can't provide you with a YouTube link. Still I'm very grateful to @timjb for his efforts!
- More on the effective topos and the smooth topos (the talk was very much inspired by and is based in parts on this beautiful paper)
- Notes on constructive mathematics and topos theory (with an application to quantum mechanics, in German)
- Slides and recording of a talk on super Turing machines: part 1, part 2
- Papers on super Turing machines (in English): expository paper, original paper
- Chaitin's uncomputable constant Ω (also check the references by Chaitin himself listed at that Wikipedia article)
- Notes on the smooth topos (for high school children, in German)
- Notes on Gödel's incompleteness theorem (for high school children, in German)
- Academic resources on topos theory: Tom Leinster's Informal introduction to topos theory, Moerdijk and Mac Lane's textbook /Sheaves in Geometry and Logic/
- Ted Chiang's short story /division by zero/
- Also check out Ted Chiang's other short stories. He's an terrific author. Story of your Life, The Merchant and the Alchemist's Gate, Exhalation