ℝolliℵg M∀th Thr∑a∂

I mean the point is that by choosing a well-ordering you can give the rational numbers a notion of "adjacent to," but this notion has nothing to do with and need not be compatible with (indeed, CANNOT be compatible with) the natural notion of "between", and it's this lexical slippage that creates the apparent inconsistency.

By "compatible with" I might mean something like "if A is adjacent to B and B is adjacent to C and C is not A, then B is between A and C" which seems natural given the English words, but, y'know, slippage.

(more seriously: so this is where constructive vs. nonconstructive proofs come into play, i think? if you give me two concrete reals that are not equal, i have a procedure that constructs a new real that is between them. but if you give me a single concrete real, i only have a non-constructive proof that there is a number that is the "next higher" and not a construction of it. since i can't get my hands on that "next" number, then i can't feed it into my first procedure and construct something in between.

i only know that it would be a contradiction for this next number not to exist, but to actually get my hands on it is hopeless.)

― big WHOIS aka the nameserver (s.clover), Sunday, January 3, 2016 3:57 PM (5 hours ago) Bookmark Flag Post Permalink

...but it's also a contradiction for it to exist

ie, for any next number candidate you can say, give me the one epsilon closer

I mean the point is that by choosing a well-ordering you can give the rational numbers a notion of "adjacent to," but this notion has nothing to do with and need not be compatible with (indeed, CANNOT be compatible with) the natural notion of "between", and it's this lexical slippage that creates the apparent inconsistency.

― Guayaquil (eephus!), Sunday, January 3, 2016 7:09 PM (2 hours ago) Bookmark Flag Post Permalink

ok yeah this is otm

a non-ilx internet friend just posted this quote by terry tao to on twitter

There is a tradeoff between +∞ and negative numbers.

If one wants to keep many useful laws of algebra then one can use infinity, xor negative numbers, but it is difficult to have both at the same time.

Once one adopts the convention +∞ · 0 = 0 · +∞ = 0, then multiplication becomes upward continuous (i.e.: when both multiplicands increase, the product is what you would expect) but not downward continuous—so 1÷n → 0 works but 1÷n · +∞ ↛ 0 · +∞ fails.

This asymmetry ultimately forces us to define integration from below rather than from above, which leads to still other asymmetries, and finally to two versions of measure and integration theory.

Terence Tao, Intro to Measure Theory

ie, for any next number candidate you can say, give me the one epsilon closer

― flopson, Sunday, January 3, 2016 9:52 PM Bookmark Flag Post Permalink

right, but since the proof it exists is nonconstructive, you can't pick any particular candidate and actually execute that construction. that's why there's not a contradiction.

or i think i'm tangling myself here. that's why explicit infinitesimals are not a contradiction in a system like synthetic differential geometry without excluded middle. with regards to standard analysis, ignore all this :-)

hey so, math thread, as a programmer/computer science type and not a mathematician at all I stopped fucking with continuous domains after taking multivariable calculus in high school, so the diffeq and analysis and algebra sequence is pretty much unknown to me at any level of sophistication. Is there a good book/resource/PDF/set of lecture notes out there where I can learn, like, some "greatest hits" of analysis without tons of additional prerequisites? Like idk precisely what I'm asking for, I just have this sense that there's some Fun Facts About the Reals that I could get a sense of with appropriate scaffolding but without having to like consume three semesters' worth of course material.

i only took up to analysis 4 but you could always download a pdf of Rudin and read the definitions and theorems skipping the proofs

does anyone know if there's an R equivalent to STATA's .do files? i'm switching over to R from STATA cuz they don't own a license to the latter at my job and i feel like i've exhausted what i can get out of vba/excel, and i really liked those .do files when i was in school

If you are using R and you are only sort of a programmer, using R Studio will probably help you out a lot. It looks like .do files are just scripts, so yes, there is an equivalent, just save R commands to a text file and then run Rscript on it or load it into your interactive session with source().

sweet, thx silby

yep, also ask for a raise, you're a programmer now

Lol at the Rudin recommendation upthread.

i wanna make a pdf of that, like a Jefferson bible of Real Analysis

hey so, math thread, as a programmer/computer science type and not a mathematician at all I stopped fucking with continuous domains after taking multivariable calculus in high school, so the diffeq and analysis and algebra sequence is pretty much unknown to me at any level of sophistication. Is there a good book/resource/PDF/set of lecture notes out there where I can learn, like, some "greatest hits" of analysis without tons of additional prerequisites? Like idk precisely what I'm asking for, I just have this sense that there's some Fun Facts About the Reals that I could get a sense of with appropriate scaffolding but without having to like consume three semesters' worth of course material.

Michael Spivak’s Calculus and Needham’s Visual Complex Analysis

I like “Classic” Rudin, but “Baby” Rudin is a fantastic way to discourage students from studying the subject.

Yes, exactly.

i said to just read the theorems! and i was talking about classic Rudin. presumably silby already knows most of the results in baby Rudin from calculus.

does spivak prove anything past multivariable calculus? that's kind of the thing with analysis, just dainty ways of proving the stuff you took for granted when you took calculus. i think measure theory and functional analysis was the first time i felt like i was actually learning something new

still never took complex though

still never took complex though

Complex Analysis is different in the United States. It’s commonly taken after an Introduction to Analysis course (i.e. “Baby” Rudin), but before a Real Analysis course (i.e. “Classic” Rudin and Royden’s Real Analysis). It’s weird.

(i.e. I suspect you wouldn’t learn too much!)

isn't Complex Analysis where you learn what holomorphic means? i don't know what it means

i took complex analysis in college because i was interested in spectroscopy, it turned out to not really be an analysis class, we just went over taylor and maclaurin series and fourier transforms in depth

nice riddle and solution

http://fivethirtyeight.com/features/should-you-pay-250-to-play-this-casino-game/

Suppose a casino invents a new game that you must pay $250 to play. The game works like this: The casino draws random numbers between 0 and 1, from a uniform distribution. It adds them together until their sum is greater than 1, at which time it stops drawing new numbers. You get a payout of $100 each time a new number is drawn.

For example, suppose the casino draws 0.4 and then 0.7. Since the sum is greater than 1, it will stop after these two draws, and you receive $200. If instead it draws 0.2, 0.3, 0.3, and then 0.6, it will stop after the fourth draw and you will receive $400. Given the $250 entrance fee, should you play the game?

Specifically, what is the expected value of your winnings?

slick solution:

https://twitter.com/octonion/status/715995679060000768

i'd say yes.

1M simulations gets me $21,829,600 of winnings

isn't Complex Analysis where you learn what holomorphic means? i don't know what it means

LOL (Yes!)

R studio RULES

Yeah, it’s dope. But it’s just a gateway drug to Jupyter. :D

It's not! I really like Jupyter notebook, but I wish there were a development environment as good as R Studio for Python! In fact, I just wish R studio would include full python support (technically you can run python from inside it, but not very well). I was so spoiled coming from R, I assumed there was something as good as R Studio for every language.

There's some exciting R/Python crossover going on. Hadley and Wes McKinney just created Feather together: http://blog.rstudio.org/2016/03/29/feather/

I've tried Rodeo, but it's just a pale facsimile of R Studio.

It's not! I really like Jupyter notebook, but I wish there were a development environment as good as R Studio for Python! In fact, I just wish R studio would include full python support (technically you can run python from inside it, but not very well). I was so spoiled coming from R, I assumed there was something as good as R Studio for every language.

PyCharm (maybe)?

(I like RStudio too! I just find it difficult to organize projects.)

There's some exciting R/Python crossover going on. Hadley and Wes McKinney just created Feather together: http://blog.rstudio.org/2016/03/29/feather/

They re-implemented HDF? I guess? Nonetheless, I like Hadley and Wes. :D

Oh, Spyder is worth a look too. https://pythonhosted.org/spyder/

fuck i'm getting crucified on StackOverflow for my noob R questions lol

i'm finding weening myself off of for-loops is pretty difficult... all my intuitions about how to approach a problem kind of go out the window. but i'm slowly getting the hang of the 'just put it in a list and lapply' philosophy. someone on SO directed me to this which i read last night stoned in my room at like 2 am and it blew my mind. my scripts are gorgeous now, even though it takes me twice as long to write them

oh is R a functional language? my wife is learning it because she's moving into ~~~ data science ~~~ (she has a doctorate in math) & all the jobs want R. was thinking it would be a week-long project b/c I thought it was more a tool for stat analysis than for developing apps, but I dunno.

it is functional and a tool for stat analysis

And can generate lots of graphs quickly

That SO answer is really weird to me! Putting dfs into a list to apply a function over all of them is natural, but do you really need to do that so often that you're going to keep all your dfs in a list right from the beginning just in case? Reflects a very unorthodox (but not necessarily wrong, I guess) mindset!

It's possible that I'm missing some fundamental advantage of the approach. I just can't remember the last time I needed to do the same thing to an entire list of dfs.

Ah, okay, a comment on a question linked to that question helps me understand: "If the data.frames have a similar structure, it is a good idea to keep them in a list."

that makes sense

Typically dreadful article on Gödel in the New Yorker

surprised, i think roberts is usually pretty good

Promulgated in Vienna in the early nineteen-thirties, the notion of incompleteness threw mathematics into a hall of mirrors, where it reflected upon itself to alluring, if disorienting, effect: the theorem proved, using mathematics, that mathematics could not prove all of mathematics. Of course, it has a proper and technically precise formulation, but the late logician Verena Huber-Dyson paraphrased it for me as follows: “There is more to truth than can be caught by proof.” Or, as the British novelist Zia Haider Rahman put it in his award-winning début, “In the Light of What We Know,” “Within any given system, there are claims which are true but which cannot be proven to be true.”

until the last sentence, that's just heartbreakingly ugh.

& even the last sentence is super wrong.

Whee shimmering metaphors for incompleteness what fun

i had a friend who was always horrified by explanations of incompleteness theorems intended for popular audiences. those quotes don't look that bad to me and the author is explicit about quoting a logician who is himself paraphrasing, and a novelist who why would you even ask a novelist. idk it never bothered me that much but i only ever read 'Godel's Proof' by Ernst Nagel

imo exactingly precise statements of the gödel thing can seem trivial without historical background, like why would anyone care that you can encode statements with arithmetic etc. but people really believed in Hilbert's program, and iirc even kg himself intended to prove a positive result of 2nd problem

would love to read a good history of devastating negative results in math

i read the first 50 pages of Zia Haider Rahman's book due to a favorable James Wood review, but found it insufferable, partly because of how it tossed math around as this complicated thing you couldn't possibly understand except in the most banal metaphors. i just read Cryptonomicon and was wondering if a real logic/cs/cryptography person would be rolling their eyes at some of the technical stuff in it. but contra Rahman Stephenson really takes time to explain stuff; the gears of a bike analogy for prime factors was great

Don't have to be too exacting to point out that eg elementary geometry is immune to incompleteness

I myself have an allergic reaction to popularizations that go overboard with the far flung analogies and "connections" -which reminds me to ask- why does every other book related to computability have to have an intro by Douglas Hofstadter?- so I can only imagine how a professional logician like Euler feels. Would like to read Janna Levin novel about Gödel and Turing, though.

If I had to summarize Gödel's theorem in four words I would type "Formalize. Apply diagonal argument."

Also, guy who inspired me to start this thread Math & Music: The Severed Alliance. Some Recent Academic Approaches (Do Not Read If You Hate Drums) who is a topologist at Lehman College is playing a jazz piano gig tonight at Mezzrow. Would like to go sometime but probably not tonight.