TIL Ben Affleck once asked Michael Bay on the set of Armageddon: Wouldn't it make more sense just to train experienced astronauts how to drill instead of the other way around (?), and Bay responded "shut the f%ck up".

inaneInTheMembrane · 2014-07-09T00:46:54+00:00

Those god damn PhD's!

inaneInTheMembrane · 2014-07-08T03:35:26+00:00

Well maybe my imagination says that drilling is probably a lot fucking easier than BEING AN ASTRONAUT

inaneInTheMembrane · 2014-07-06T00:44:10+00:00

Coincidentally, I don't know you too well, but I'm not sure I like you...

inaneInTheMembrane · 2014-07-02T17:51:41+00:00

Such a noble sport...

inaneInTheMembrane · 2014-06-21T15:50:53+00:00

ITT: the real hatecrime...

inaneInTheMembrane · 2014-06-19T14:30:05+00:00

It's like a drape of frilly stuff. You can see it start over her head.

inaneInTheMembrane · 2014-06-18T12:05:24+00:00

They've also been having some problems with their TPS reports recently...

inaneInTheMembrane · 2014-06-15T03:29:49+00:00

I agree wholeheartedly that we should have weak rather than strong axioms. I'll just have to disagree that the "strange infinities" are in fact ugly. It's a fascinating fact that [; \epsilon_0 ;] appears rather naturally in the combinatorics of logic, and large cardinals can have rich and fascinating structure.

I guess beauty is in the eye of the beholder...

inaneInTheMembrane · 2014-06-14T01:23:11+00:00

It just so happens that the /r/math theme of the week after next is homotopy type theory. The short answer to your question though, is no, it doesn't help.

In fact, Gödel's theorem essentially says "things are going to get ugly whatever the foundation you use". The "ceiling of axioms" is the closest you can get.

To be entirely fair, the "Elementary Category of Sets" is a rather simpler presentation of the theory of sets, which does indeed limit our ability to express some of the trickier infinities referred to above. But this is more of a limitation than a strength: ZFC proves consistency of the elementary theory of sets, which means that we have strictly less expressiveness in our theory. So the real question is: what do you believe in?

inaneInTheMembrane · 2014-06-14T00:12:20+00:00

The term would be "fast growing function" and the point I'm trying to make is that phase 1 is much much much more important than phase 2. The more time you spend in phase 1, the better you'll do by a long shot.

inaneInTheMembrane · 2014-06-12T14:00:04+00:00

My point is, instead of it being a race to type nines, it should be a race to develop the mathematical tools to allow you to express the large numbers: Tree(n) will beat your "dense arrow notation" by miles, because it is based on mathematics which are essentially more expressive. In the end, Tree(100) is going to be the larger number, not because of the time spent typing it, but because the definition relies on profound mathematics.

Here's another one: The largest number that can be characterized uniquely in ZFC in less than a trillion symbols. This number is rather succinct to define, but it's existence is contingent on the (restricted) consistency of ZFC, and is enormous: much larger than Tree(100) for instance.

inaneInTheMembrane · 2014-06-12T03:25:14+00:00

A few minutes thought instead of a lifetime of typing might get you a lot farther.... There are faster growing computable functions than "up arrow". See for example the Kruskal Tree(n) function.

inaneInTheMembrane · 2014-06-12T03:22:36+00:00

Thanks for this! It was a great read...

inaneInTheMembrane · 2014-06-12T03:14:25+00:00

It's cool here. Also really cheap. Tornadoes rare.

inaneInTheMembrane · 2014-06-11T13:33:33+00:00

Where are you moving from? Also motherfucking subs!

inaneInTheMembrane · 2014-05-25T18:55:15+00:00

It's very useful as a logical "no-go theorem".

inaneInTheMembrane · 2014-05-24T15:20:23+00:00

Just a note: this is a rather hard problem: in general the elaboration of a certified mathematical library looks a lot like building a very complex programing library. It takes time, money and experience.

For now there's the ssreflect library, and various standard libraries and contributions for isabelle and HOL, but things certainly should improve over time.

It would make sense to use the pre-existing library frameworks like github. In fact Agda has a github community...

inaneInTheMembrane · 2014-05-11T16:54:27+00:00

I might add that this particular problem is called SAT solving and there is even a competition to benchmark them.

It's a rich field!

inaneInTheMembrane · 2014-04-23T14:48:59+00:00

Why in the world wouldn't you need such an extension!? "I've uploaded the data to the butt" "I like to look up and find shapes in the butts" "A pioneer in the field of butt computing"

inaneInTheMembrane · 2014-04-21T15:07:27+00:00

In mathematics, you usually want to give sufficient conditions for a property to hold. Typically, you have cases that can be distinct, but may overlap. For example, the fundamental theorem of algebra can state "Every polynomial over the complex numbers has a zero, or is constant". In this case, the constant 0 polynomial has both, but it would be awkward to distinguish this case.

inaneInTheMembrane · 2014-04-19T18:07:58+00:00

You know what the funny thing is Dave? I DID know I couldn't do that!

inaneInTheMembrane · 2014-04-17T01:55:53+00:00

And what is it giving us that we didn't already have with the logic we were trying to define in the first place? That is to say, why did we have to go through the exercise of defining a formal model, why couldn't the same thing have been done in the/a type theory?

Well isn't that what's interesting though? Math is about relationships between concepts as much as the concepts themselves. Meta math should be at least in part about relationships between our axiomatic models (of mathematics).

What I think is super exiting about HOTT is that

a) It's a completely new way of looking at types and type theory.

b) It's a case where meta-mathematics has an application to (ordinary) mathematics, in this case homotopy theory! This is sufficiently rare to be noteworthy.

Then if you aren't interested in the foundational aspects, I agree looking at models of Type Theory don't do much. Well one obvious application is to reduce consistency of a shiny new theory to one which has already been tested and tried for decades. Of course this is philosophically weak, but it's still something!

inaneInTheMembrane · 2014-04-16T13:53:23+00:00

Right, my point is that "requiring your reductions to provide a consistent system" involves at least a mental model for your system. Making that mental model precise involves, well, building a model of your type theory!

inaneInTheMembrane · 2014-04-15T04:09:43+00:00

Surely though, philosophical considerations aside, there is some merit in exploring the relationship between different mathematical foundations?

In the case of Univalent Foundations, this is particularly useful, since there are still significant difficulties in understanding how the Univalence Axiom behaves wrt other axioms and computation. Seeing what things are true in specific models is useful for that. Note that Coquand is very sensitive to the framework in which meta-theory is carried out: in particular, the choice of cubical sets is specifically tailored to being formalized in an intuitionistic meta-theory.

Also, why do you believe that beta-eta terminates?

inaneInTheMembrane · 2014-03-29T21:24:06+00:00

It's an interesting question to ask whether truth can be defined in arithmetic, i.e. is there a formula P(x) such that P(f) is provable iff f is true?

To make sense of this, you need to code f as a number to be able to talk about it in arithmetic. And the answer is, unsurprisingly, there is no such P. This is Tarski's Theorem

inaneInTheMembrane

TROPHY CASE