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i think this website's days as a bulletin board are numbered. lets slip this into google docs

Yo, 1968 is on the line from San Francisco, it wants its collaborative terminal back.

(I'm on terminal 13 in Menlo Park.)

Much less laggy, bloated Javascript than ****ty google docs

It wasn't clear before why there are aleph_n "transcendentals", referring to my claim earlier about the weirdness of transcendentals. So I defined general notions of both "irrational numbers" and "polynomials" to show that the equivalent of "transcendental" amounts to an aleph_n sized object.

I don't know enough about computability to know if I can extend to this to computable numbers, but it seems the patterns are aligning well enough that my conjecture is "yes".



So this has a very interesting implication, one that I find very, very exciting. If this is the case, then you can extend R to contain roots of power series. If I'm not mistaken, my proof implies you can extend R by an aleph_1 amount of elements to give every power series a root.

All from basically pure set theory and group theory. The fuuuck man.

Does anybody really understand "pure set theory"?

As long as you understand the difference between a small category and Set, sure

I'm curious to learn more about the motivation for your work on algebra. It seems to me that you are interested in computability (hence the appearance of stuff like infinite cardinals and recursive functions), which I understand as being connected to the study of groups in a very general setting. But on the other hand, I see the topic of free groups as being linked closed to geometric group theory (have you browsed this book?), which in my mind is somewhat linked to algebraic topology. But to me, geometric group theory is very well motivated, and has connections to dynamics and some very beautiful relationships between grammars / computation / dynamics, symmetry / algebra, and geometry.

Maybe the initial craziness and unwieldy nature of set theory was all along the result of the fact that when you try to study the infinite using formal language, you really are studying some kind of dynamics. And if you really want to understand dynamics, you should be studying geometry, which is what geometric group theory gets you. Or something like that?

Which would be funny, because from what I've heard, geometric group theory gets a lot of its power by applying the tools of analysis to problems in groups. So whereas in the beginning, people thought that by formulating analysis itself in the language of sets, they might better understand the "continuum" (whatever that means), it really turned out that analysis would be more useful as a tool for understanding the kind of crazy dynamics that such formal language opened up in the first place (now understood as a part of geometry and dynamics, through algebra).

I was thinking about the essence of abstract algebra, which I view as the study of finitely many binary operations. Specifically I mean, in algebra, there is no concept of a "limit", we can't take the limit of b^n as n->∞.

I started thinking then about how that works with infinite products. For instance, any finite product of Z_2 has a generating set given by:

(1,0,...,0)
(0,1,...,0)
...
(0,0,...,1)

However, this fails in the countable case: the element (1,1,1,1,...) can't be given as a sum of the elements:

(1,0,0,0,...)
(0,1,0,0,...)
(0,0,1,0,...)

Which means, the generating set must be much larger. It was a question of mine exactly how many elements must be in the generating set. I realized what is generated by the "naive" generating sets I've listed is just the direct sum. So I thought, maybe you could "upgrade" the relationship of direct sum to direct product in higher cardinalities. And sure enough, I was able to prove it works out. As a result, it seems many of the "how many" results about the types of numbers can be reinterpreted through this framework for higher cardinals as well.

It was all just a result of thinking about the essential nature of group theory.



I'm interested in computability in the sense that it gives us some teeth to begin speaking about certain transcendentals, like pi and e, maybe not so much in the theory itself.



I have not seen that book, but I have it saved and will look through it. In fact, my department has people in geometric group theory, I should probably just discuss it with them.

I know not even square 1 about dynamics, so I don't have a comment about any relation there.

If geometric group theory takes algebra and adds limits in an abstract way, then I'd have to recant my statement above as naive. That could very well be, and I should probably find that information out.

Everything I said is incredibly vague and based on things I heard from other people at university. :downs:

I'm glad you're taking a more precise approach to this stuff than me.

You see weird stuff come up in it. It's a very interesting subject. You wouldn't expect formal grammars to have to do with dynamics, but it's there.

http://www.scholarpedia.org/article/Symbolic_dynamics

Algebra and computability connected through automata theory. And my understanding is that algebra in general has to do with symmetry in dynamics, but it also describes formal operations and rules of some kind.

I know you've said stuff about the word problem and its relation to combinatorial group theory (precursor to geometric group theory, involving Cayley's work, and early algebraic topology, which seems to be involved with a bunch of stuff involving computability and free groups.

Check out Teichmuller theory. Related to geometric group theory and geometric topology (many geometric topologists moved to geometric group theory after Perlman's proof). Super advanced though.

http://matrixeditions.com/TeichmullerVol1.html

If anybody is being vague here it is me. I doubt you have to change anything.

I'm pretty sure the numbers I've stumbled upon are actually the hyperreals, where each Laurent series is representable by a hyperreal, in fact they're constructed in many cases as series of real numbers, so there's a strong case here.

I've also made assumptions about the continuum hypothesis, if anyone cares.

You should read Conway's ONAG!

Reid you wanted to know why programmers hate math

http://abstrusegoose.com/206

OK last one

http://abstrusegoose.com/432

Its too hard

One thing about those comics is that you need to read the tooltips.

Not sure if that one was serious though.

http://ieeexplore.ieee.org/document/1056648/
http://www.ai.soc.i.kyoto-u.ac.jp/~xin/surveyMAPF.pdf
https://en.wikipedia.org/wiki/Linear_complementarity_problem -> https://en.wikipedia.org/wiki/Quadratic_programming#Complexity

"The ray-tracing problem is a decision problem: given an optical system (namely, a finite set of reflective or refractive objects), a light ray's initial position and direction, and some fixed point p, does the ray eventually reach p? ... Ray tracing in three-dimensional optical systems consisting of a finite set of reflective and partially reflective surfaces represented by a system of rational linear inequalities is PSPACE-hard."
Computability and Complexity of Ray Tracing (J.H. Reif et al.)

and for AI:
https://arxiv.org/abs/1203.1895
by, among others, all-around super fascinating dude Erik Demaine.

just off the top of my head.

I have nothing polite to say about that kind of computer science undergraduate.

I see that now. It is trying to be XKCD, so I should have thought to read the tooltip. I don't read XKCD though so I can probably excuse myself for not doing so. Moral of the story: Abstruse Goose is trying to be XKCD.

People who don't attempt to learn any of the complicated but powerful areas of their field are going to be relegated to what is, to put it bluntly, a lower tier of artisan. As the comic points out, the best video games are produced by people who paid attention in lecture.

Abstruse Goose is pretty good. I haven't read it since CM convinced me to switch from bookmarks to an RSS reader though. lol. technology.

Edit: Abstruse Goose is a lot more academic than xkcd. The former is written by a practicing physicist. The latter is written by someone with a physics undergrad.

Heh. That is so true it hurts.

I assume that, in this metaphor, the frame represents not caring about overfitting.

I would ask: "why didn't anybody warn me how difficult grad school is?"

But they did. Like everybody warned me.

Would it help if I warned you again

Sure, go ahead.

Grad school is really hard, man.

Thanks. There is actually a sort of comfort in knowing that this is supposed to be as hard as it is.

As cheesy as it is, I'm also reminded of the JFK speech where he said we choose to go to the moon because it's hard.

JFK was a psycho. The moon program announcement was a calculated distraction from the Bay of Pigs fiasco. Don't be like JFK. Don't do things because they're hard or because they're distracting, do things because they're worth doing.

Yeeeh I just get that one line stuck in my head. He was a good speechgiver. Definiely not a role model person though.

Or at least do things because they're there.

As a kid we had the Grolier 1995 Multimedia Encyclopedia. When you opened the program, it played a cool little sound collage of famous things from history, including the "ask not..." bit from JFK.

A couple years later in middle school, we had to give a speech presentation on somebody we admired. I remembered how noble sounding that JFK line was, so I picked him.

Anyway, I won the class contest because my teacher was a big fan of JFK. However, I lost the next round to a girl who picked Jesus as the person she admired, because I actually had no reason at all for admiring JFK other than his speech sounding admirable, and when I told the judges this they weren't impressed.