ℝolliℵg M∀th Thr∑a∂

i know there are at least a couple mathematicians in the building. do mathematicians have anything cool to talk about? would be fun to hear about what research ppl are doing, or did in past lives, what courses people are taking or teaching, books we're reading, problems we're stuck on etc

i'm in my last year of undergrad program in pure math right now, currently taking general & alebraic topology (with a total madman who looks like this http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Daniel_Wise.jpg/220px-Daniel_Wise.jpg), real/functional analysis, and probability. kind of toying with the idea of pursuing grad school. i've taken a couple grad courses so far & they've gone down pretty smoothly, and there are a lot of opportunities to teach math at CEGEP (2 year mandatory pre-university program) and a masters degree is the only requirement. buuuut idk. currently applying for funding to do some research in geometry & topology with a postdoc this upcoming summer. will be spending this weekend trying to understand automorphism groups of covering spaces ^_^

We could discuss whether there is a satisfying proof for the Nilsson Conjecture.

my own contribution: why does well-being look like this?

xp never heard of it!

viz 1 is the loneliest number

oh lol -_-

exactly

i am a dormant mathematician

xp now you have ammunition for chatting up a mathematrix at your next conference

i teach math

cool! how is it?

high school?

yes. prealgebra through AP calculus. it never fails to surprise me how much easier it is to teach "hard math" than it is to teach "easy math"

i'm teaching logic right now, to the least academically prepared students i've ever taught (it's not a huge difference, but it's significant), and that seems to be otm.

the extra complement of math that people generally take to get into a university, even if it stops right at or just before calculus, does seem to make a huge difference in terms of experience, comfort, confidence, mastery of working with forms and symbols.

How are we supposed to search for this thread title?

Somebody pointed out recently that the Notices of the AMS are online and free which is great for people like me who are not in academia anymore and don't get a paper copy. Been meaning to read through the article on Beethoven's Metronome: http://www.ams.org/notices/201309/rnoti-p1146.pdf

Also recently bought a copy of Best Writing on Mathematics 2012 which is very nice and pitched at a similar level, the article on Math and Music was very interesting, written by a guy who is both a practicing musician and research mathematician.

mathematical dorks should be able to remember 98225

thats an elegant number

Yes!
225 is square of 15. (Which you can remember as fact in itself or as exemplar of (10 + x)^2 = 100 + 20x + x^2.
98 is multiple of 7 and figures in the decimal expansion of 1/7 if you think about it.

you are ridic

Let's see if I can do this:
100 = 2 mod 7 so 100^3 = 8 mod 7, 100^3 =10^6 = 1 mod 7 so 7 should repeat with 6 digits
1/7 = 7/49 = 14/98 = .14 * 100/98 = .14 * 1 /(1.0 - 0.02) = .14 * ( 1.0 + .02 + .0004 + ...) = .14285(6+1)..

currently participating in a seminar on homotopy type theory and learning the rudiments of algebraic topology. been spending time with v/a Theory and Applications of Categories reprints.

Really? Where?

the seminar? at the cuny grad center http://nylogic.org/homotopy-type-theory-reading-group

didn't realize that someone else on the board would have an interest in this!

Are you enrolled in any kind of graduate program at CUNY or elsewhere? What about the other people in the seminar?

the seminar is open to the public, like lots of cuny things actually. i'd say its only about 30% people with any current relation to NYU, grad students included. v/a former students/phds, as well as just a cross-section of ppl currently just employed but with a math background, as well as maybe students at other campuses.

we're meeting in the eves biweekly to make it feasible for people who work

Thanks. By NYU I assume you mean CUNY or CCNY, although there are probably people from NYU there too.

Also how long does it go for: two hours,

oh yes by nyu i meant cuny. goes about 1.5 hrs and then some people go off to the pub afterwards for however long. there's a google group too https://groups.google.com/forum/#!forum/hott-nyc

whats yr maths background and how does it bring you to HoTT?

'how did you first know christ?'

pretty much

constructivism had that flavor

How are we supposed to search for this thread title?

yeah sorry, had that thought while making the thread but couldnt resist lol. bookmark it, i guess?

s clover we should def homotope it up itt, stoked. what books are you reading? my course is based on munkres (<3) but prof is doing some additional topics, so far i've only looked at hatcher... it's all very intense, though. i slayed point-set but algebraic topo makes me feel v dumb

had some halfbaked thoughts while hi the other day about some group theoryish intepretation of (basic) music theory. trying to figure out how to make like, arpeggios/chords subgroups of keys/scales?? i don't know enough group theory tho

Don't have much background in this area per se, but have some background in math in general, was on the math team in junior high and then at the Bronx High School of Science, undergrad degree in math at Yale and graduate degree from NYU. First professor I had Freshman year was topologist William S. Massey, at graduate school I took the topology course from Sylvain C, but it was kind of easy and I didn't really learnd too much and most of NYU is about applied math anyway. Actually guy who wrote that music and math paper did a post-doc in topology at NYU, now he is at Lehman College. Sort of curious about this topic but more importantly like the idea that it is free and of being on an equal footing with the other students, other people not currently in academia but presumably mathematically literate - have toyed with idea of going back to finish (start?) thesis and get PhD now that my advisor is the chair but don't really want to deal with have to crank out research-level material, plus I didn't always like the Eloi vs. Morlocks setup of formal grad school, as I recently discussed with former ilxor Casuistry.

I don't know how much higher math you really use to understand music or music theory, the article I was talking about basically says that he thinks a lot of mathematical research on music is sort of bogus, drawing a picture or creating blinking lights that sort of looks like what music is doing. To him the most interesting thing seemed to be the way the similar processes of collaboration jazz improvisation and mathematical research.

i teach high school math but im an econ undergraduate and yall are simply adding fuel to the fire that rages inside me that i shd not be teaching math

plus I didn't always like the Eloi vs. Morlocks setup of formal grad school,

what does this mean?

xp would u rather be teaching econ? my first major was in econ, then i was like ah fuck it and turned my math minor into a full major

The few mathematical ideas I use to think about music are:
*Thinking about the twelve semitones as the Abelian group Z12. (supposed to be a subscript)
*Looking for least common multiples when counting out polyrhythms
*Realizing that the names of the intervals are Ordinal Numbers and
*Realizing that sometimes there are Fencepost Problems in either counting out beats or even in that the octave is the same as the one, which is why some people talk about a heptatonic scale

*Thinking about the twelve semitones as the Abelian group Z12. (supposed to be a subscript)

yeah this was basically the extent of my ruminations lol

well hey you know

http://en.wikipedia.org/wiki/Set_theory_%28music%29

Fully funded PhD student with good background (perhaps trained abroad) and famous advisor, who can sail through qualifyings, has good chance of helping advisor push out papers and securing post-doc and just generally burnishing his legacy = Eloi
Rest = Morlocks

hmm, in the humanities the 'fully funded' part of the distinction doesn't really apply anymore at lots of places, programs looking to trim, improve graduation rates/times etc. figured maybe they should only admit those who they could fund

the other part, though…

Someone told me Hindemith was into group or set theory as it applies to music but I couldn't find a reference. Maybe that Wiki page will have.

Telling you, that Fencepost thing saved me some headaches.

Oh yeah, one more thing
*Thinking about musical entities as partitions- triads as partitioning into three (with the fourth being a "generalized, augmented third*), seventh chords as partitioned into four (with the second between the seventh and the octave as a *generalized, diminished third*)

homotopy type theory isn't exactly homotopy theory bear in mind -- its this wild new realization that homotopy theory is precisely isomorphic to that branch of logic known as martin-lof type theory, and this all works only in a constructivist setting. so you have math/logics ppl without much topology background (hi!) and topology people trying to understand type theoretic notation all sort of in the mix together. i have a coworker with some topology so he's good at explaining things like fibrations and fundamental groups etc. oh right and the other part of the hott project is now you have a setting in which the claim is you can do the foundation of _all maths_ as an alternative to set theory, so eventually (tho it may be 6 mos before we get there, if we can keep up the momentum) we'll stop just building up the theory and do applications and use it to do classic results in e.g. actual homotopy theory, set theory, category theory, analysis etc.

the hatcher book is very approachable but it does sort of require someone with a bit of background to explain what he means by certain things, since the geometrical intuitions aren't totally obvious without someone drawing things or waving their hands or etc. to illustrate movement.

sounds cool xp

xp would u rather be teaching econ? my first major was in econ, then i was like ah fuck it and turned my math minor into a full major

― flopson, Saturday, November 9, 2013 4:53 PM (17 minutes ago) Bookmark Flag Post Permalink

there are many things for which i am better qualified to teach. don't get me wrong, i know the hs curricula v well, i just didnt take much advanced math and sometimes feel like i am training my kids for a sport ive never seen or played before. being the good economist, tho, i know my comparative advantage is in the high-need HS math field rather than the social studies or journalism positions i'd have an absolute advantage in.

Now I remember the problem with math, you have to beef up all this apparatus, before you can really understand or prove anything- except for trivial variations on the proofs in the book with slightly different initial conditions- let alone do a calculation, for what seems like an eternity.

Assuming you are an applied mathematician and want to do a calculation.

calculations! what, like with numbers? i thought we were talking about math!

Ha, exactly. I never forget in Group Theory class when Jonathan Rogawski told us "The idea that mathematics is about numbers is false. Mathematics is about the relationship between mathematically interesting objects and other mathematically interesting objects." RIP, Jon.

lmao. had a group theory prof tell me "math is just patterns. that's why group theorists are like the high priests among mathematicians"

I was a chem major bison

Chemists I've known have always been good at math.

For instance, my friend's dad is a retired chemistry professor and he was writing papers on things like statistical mechanics and Brownian Motion and co-wrote an undergraduate math textbook with a super-famous mathematician, well famous in the field, anyway.

plus I didn't always like the Eloi vs. Morlocks setup of formal grad school,

what does this mean?

― flopson, Saturday, November 9, 2013 5:52 PM (1 hour ago)

http://en.wikipedia.org/wiki/The_Time_Machine

i'm good at math but i've never taken like hardcore college classes, i can just do stuff in my head pretty well and am good at algebra, that's basically the math i need or will ever need

I was trying to teach a student synthetic division today and I kept fucking up! So frustrating! Somehow stressing that you don't actually use that in real life isn't a satisfactory explanation!

Fuck you rational root theorem!!!

i actually kinda think hs math should be less about like, finding roots of polynomials or learning division algorithms, and more, like, elementary/discrete probability theory, basic combinatorics & graph theory, just the really easy stuff you can do with elementary methods

ne x86 math topics aside from khaninstitute? trying to set up equasions on x11 in darwin

Isn't that what computer science freshman learn in their Discrete Math course, flopson, iirc?

well how does that help high schoolers?

besides logic, teaching a critical thinking course now that's heavy on inductive logic and probability, for the first time ever, and already (not being super deep into the material) i'm struck by how useful it could be if more people learned this stuff early on. i had a science-heavy math education, and i've spent some time thinking about how science works, but i can see that there's a lot about it that's overly opaque to me because my knowledge of probability/statistics is shallow. especially when it comes to the social sciences.

Yeah, obviously doesn't help high schoolers, you are right.

you know what helps high schoolers DAILY BEATINGS that's what

j/k

the question of what type of math to teach in school is so contentious and political that it makes my brain boil every time the subject comes up, that's why i made the lame joke

isn't the typical college-prep math track basically set up to produce calculus students (then engineers, and the few others who need calculus, like physicists)?

sorry, don't mean to incense you, v, obv. you do the lord's work

There are certain problems about way math is taught-even if you major in it and are good at it!-my old math team coach is still big in education wrote an article about his take on it, maybe I can find it. One thing is overemphasis on proofs- "it was good enough for Euclid"- rather than other kinds of mathematical thinking to develop intuition and visualize or in some other way organize mathematical structures. Another think I've talked to people about, at least as far as applied math, is not enough discussion about the subtleties of units. Little kids learn that if they are given a problem about the perimeter of a triangle with the sides given in feet, they should give the answer in feet and they figure units must be trivial but in fact a better understanding of units can really help you solve a problem more quickly and, more importantly, accurately, even if it ultimately requires calculus. See the book Street Fighting Math, freely downloadable, for a good presentation.

xpost

you are correct j, which is sad because the 95% of kids who don't want to be engineers and physicists just end up getting turned off of the subject by the end of high school, if they're not already turned off by 9th grade (i would estimate about half are by that age)

http://mitpress.mit.edu/sites/default/files/titles/content/9780262514293_Creative_Commons_Edition.pdf

^ this one?

When I was in high school basically they were leading up to teaching Calculus senior year, when you could take it only if you were in the AP class, in which it was gingerly taught at an extremely leisurely pace, as if we had to slooow ourselves down time-lapse style in order to observe the delta-epsilon proofs.

Yeah, that one.

my thinking matured a lot, esp. my proofs, once i took a logic course, which suggests that maybe mathematicians in my hood were not as plain about what we were doing as they could have been.

but abstract algebra was always more elusive for me (even though really interesting), despite having lots of proof-theoretic niceties in its usual u.g. presentation, because i didn't have so many of those intuitions and had trouble learning how to visualize it or play with the structures. never had much of a feel for numbers, compared to many math majors, when younger, which i think would have made a difference. number theory as a gentleman's pastime, say, still has mostly zero attraction for me.

sometimes i wish i could just chill and enjoy math instead of stressing about mathematically illiteracy

let's just like, do some problems

"A farmer has some rabbits and some cages. When he puts 2 rabbits in each cage, there are 2 rabbits left over. When he puts 3 rabbits in each cage, there are 16 cages (but no rabbits) left over. How many rabbits and how many cages are there?"

lol i swear the cover of that book practically reads like 'the art of fucking shit up and guessing about things' to me

number theory is ridic ... as far as algebra goes i learned groups rings and fields to pass a test and promptly forgot everything. i know nothing about topology or real analysis.

this course was about as far as i got in math before i gave up (i passed!)

http://www.amazon.com/Foundations-Higher-Mathematics-Peter-Fletcher/dp/053495166X

i just tried to solve that rabbits and cages problem and got six cages and eighteen rabbits

hm

ahem

http://math.arizona.edu/~savitt/GTM.html

okay, fixed

apparently i am

You are William S. Massey's A Basic Course in Algebraic Topology.

You are intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized.

so this test has really revealed something about my self to me

Awesome. I have a copy of that book somewhere that I can give if you want, if I can find it and you are in NYC.

Tried to do that caged rabbit problem in my head but suspect it's broken meaning non-integral solutions. But maybe it's my brane that's broken will solve properly when I get home.

nah there's a simple integer answer

you can do it with just a system of linear equations but wording is so funny it took me a few tries to get the right expressions

The sides of triangle ABC have lengths 6, 8 and 10. A circle with center P and radius 1 rolls clockwise around the inside of triangle ABC, always remaining tangent to at least one side of the triangle. When P first returns to its original position, what distance has P traveled?

^^ this one is grate

and everyone saw this right?

http://sciencevsmagic.net/geo/

OK, took out a pen and the answer is obvious. Now to review why my mental meanderings failed and see if I learned something.

See the error of my ways too, which I may explain to y'all in a bit.

Here's where I went RONG
Wanted to do it without pencil and paper and without algebra, so I decided to
1) figure out what number rabbits had to be multiple of
2) what number cages had to be a multiple of
3) try various values of 1 and 2 until I was close and then
4) perturb to get exact answer

Got to 3,but then 4 wouldn't quite work. Thought it might be because I perturbed in wrong direction.bIt turned out I had mixed up 1 and 2. When I wrote down the two equations I was able to solve it immediately.

Now I remember the potential problem with word problems. For the "smart" kids, they see the same word problems over and over and just know immediately how to translate them into mathematical terms and solve it, it is too easy for them. For the other kids or the old geezers like some of us, the mathematical content is elusive. In my experiment I went into the transporter beam and was split into both smart kid and dumb kid, and hopefully learned something from it.

Can you get to that?

vahd33m's last problem reminds me of PSAT (or SAT?) controversy in the 80s in which one of the Jungreis brothers, must have been Doug, challenged the testing authorities on their wrong answer to a similar problem.

Somehow that is one of those rare bits of information from Ye Olden Times that has not been caught up in the meshes of the undiscriminating internet.

Here is a book of high school contest problems from way back when: http://www.math.nyu.edu/cmt/pdfs/NYCIML.ProblemBook.pdf

Another, related book: http://www.mathpropress.com/books/ARML/ You could also look at some of it in google books.

http://www.mathpropress.com/books/index.html

Looks like you can't really get it nowadays.

Isn't that what computer science freshman learn in their Discrete Math course, flopson, iirc?

― I Wanna Be Blecch (James Redd and the Blecchs), Saturday, November 9, 2013 9:21 PM (Yesterday) Bookmark Flag Post Permalink

well how does that help high schoolers?

besides logic, teaching a critical thinking course now that's heavy on inductive logic and probability, for the first time ever, and already (not being super deep into the material) i'm struck by how useful it could be if more people learned this stuff early on. i had a science-heavy math education, and i've spent some time thinking about how science works, but i can see that there's a lot about it that's overly opaque to me because my knowledge of probability/statistics is shallow. especially when it comes to the social sciences.

― j., Saturday, November 9, 2013 9:41 PM (Yesterday) Bookmark Flag Post Permalink

isn't the typical college-prep math track basically set up to produce calculus students (then engineers, and the few others who need calculus, like physicists)?

― j., Saturday, November 9, 2013 9:51 PM (Yesterday) Bookmark Flag Post Permalink

yeah i guess what i meant was that you spend most of high school math doing pre-cal + geometry, but don't even get to the punchline of the former unless you take calculus in university, by which point your relationship to math is already largely determined. i enjoyed geometry but that's not most people's experience, particularly due to the focus on tedious things like trigonometry. i think those topics i mentioned have some fun, big results that come very quickly and can be derived and made intuitive with elementary methods accessible to hs students, and broadening the amount of topics you see might make more people likely to realize that they like math, and would give people a better idea of what it's all about. i don't have very strong feelings about pedagogical approaches to math and i know that it's very hard to get people to learn even the simplest stuff. but i don't think my suggestion necessarily means students should learn more or harder material, if anything it's more about giving a practical layman's set of tools for people not continuing calculus

I’m fascinated by this debate. Excuse my rambling:

Calculus is an opportunity to survey mathematics. It’s possible that a student’s calculus sequence will be their first, and only exposure to foundations (e.g. proof by Riemann sums), abstract algebra (e.g. vectors and vector spaces), combinatorics (e.g. infinite series and sequences), geometry (e.g. differential and topology), and a variety of philosophical and applied areas of mathematics. If you care about mathematics as a subject of humanity, there’s no better introduction to the themes, relationships, and problems within mathematics. In a perfect world, we’d rename the courses “An Introduction to Mathematics.”

Nonetheless, I don’t think the typical calculus sequence is useful for non-hobbyists (e.g. “engineers”)—it’s far too comprehensive and not enough time is spent on useful knowledge (e.g. dynamics, differential equations, and numerical approximation). I also don’t think it’s useful for young mathematicians—too much time is spent on sharpening basic knowledge (e.g. algebra, counting, geometry, sentential calculus) rather than rigor. For example, my undergraduate track:

* Introduction to Calculus
* Intermediate Calculus
* Multivariable Calculus
* Functions of a Complex Variable (i.e. complex analysis)
* Introduction to Analysis
* Functions of a Real Variable (i.e. elementary real analysis)

I also took a differential equations and Fourier analysis course. I suspect most people had a similar experience.

In retrospect, I feel like the calculus sequence could’ve been reduced into one course—using Spivak—better preparing me for the jump into analysis and freeing my schedule to study non-analysis subjects.

Incidentally, I feel like algebra has the reverse problem. In my experience, most students enter Ph.D programs with two courses in algebra: linear and an abstract algebra course (covering groups, rings, fields). If you replaced a calculus course from the standard calculus sequence with an algebra course, you could then require an upper-level algebra course in the junior or senior year (e.g. commutative algebra) that’d better prepare everyone.

Also: I’ll join the interested in Homotopy Type Theory choir. While I’ve spent the past while working in compilers, I’ve been studying PLT topics wherever necessary (mostly semantics and types). I enjoyed Types and Programming Languages, but I’ve been looking for something a bit wilder. If there’s enough interest, maybe we can use this tread as an informal book club/Agda help desk.

My interest in mathematics is very much related to computer science. I didn't post here earlier, because I thought this thread would only be mathematics.

But I see some may appreciate these drawings that use Turing machines: http://maximecb.github.io/Turing-Drawings/

Also recently bought a copy of Best Writing on Mathematics 2012 which is very nice and pitched at a similar level, the article on Math and Music was very interesting, written by a guy who is both a practicing musician and research mathematician.

I haven’t read it in its entirety, but I’ve liked everything I’ve read from Paul Hudak’s Haskell School of Music:

http://www.cs.yale.edu/homes/hudak/Papers/HSoM.pdf

(However, I didn’t know anything about music theory. So, the relationship between music and, say, lazy evaluation and typing might be overwrought.)

Sort of agree on calc. Way over done on specific tricks, and not enough emphasis on intuitions and meaning. All the exercises that are just about remembering a zillion tricks and identities for symbolic manipulation and simplification are just really about rote memorization and speed. Calc is a pretty specialized subject and treating it as "genuine advanced math" is pretty maddening compared to what gets left out.

Math is really used due to the history of standards bodies and testing, etc. as much as a screen/weeder as actually taught to teach math, and the emphasis on calc is the biggest symptom of this (but all the trig identities actually similar -- I mean we should really start with the unit circle, euler's identity, etc. and build trig on that.

also hate the way linear algebra is taught, determinants first.

its all derived from teaching things you can test in a certain way.

I'm in the middle of taking a linear algebra course and we haven't got to determinants yet. The text is sort of unorthadox in its presentation, though, according to the instructor.

which one is it?

axler? axler is gr8

Bretscher

wow the amazon reviews for it are beyond vicious. but it all seems to be from frustrated undergrads...

Which text is it? If it was graduate course I might have a guess

(Xp obv)

Dont't Axler was around yet in my time. I liked Valenza, Halmos and, for the more applied approach, that old warhorse Strang.

Don't think

Used copies of third edition of Strang pretty cheap!

Not knowing any alternative, I will say this: Thank god i have a good instructor. The instructional portion of the text compliments the lectures more than anything.

or more than vice versa, i should say

mosly taught myself linear algebra reading wikipedia

man, figures that searching for 'homotopy' to find this thread on ilx would not be a sure bet

...

http://www.theatlantic.com/education/archive/2013/11/the-stereotypes-about-math-that-hold-americans-back/281303/

i like that article but i'm not sure i agree with this

"The U.S. does not need fast procedure executors anymore. We need people who are confident with mathematics ..."

if you are in a college-level math class it is hard to feel #2 without #1 under your belt

i am giving students an 'all proofs' exam next week and they are also of that mind

i love jo boaler though, she fights the good fight

really bummed that she was gone the year i was at stanford

i took half of jo boaler's MOOC this summer, rly enjoyed it and it informed my practice a lot

Famous Mathematician's Daughter Weighs In On Topic Of Interest To Ilxors

assumed that was going to be about tipping tbh

i took half of jo boaler's MOOC this summer, rly enjoyed it and it informed my practice a lot

― shiny trippy people holding bandz (m bison), Wednesday, November 13, 2013 7:16 PM (Yesterday) Bookmark Flag Post Permalink

what was it about?

fixed v growth mindsets, beliefs about innate abilities affecting learning outcomes

a little about stereotype threat as well

hey sterl, can you answer a dumb question for me?

if one of the problems with russell's resort to types is that it is essentially a kludge, what is it about HTT that improves on the kludginess?

Hmm. Apols if this is somewhat rambly. The brief summary is the "point" of type theory is now very different from why Russell was interested in types, and much more about taking types as a foundational setting for logic/computation/topology. However, this doesn't mean that we still don't need a "hack" (in the form of a cumulative hierarchy of universes -- aka a predicative [as opposed to impredicative] logic) to avoid russell-style paradoxes. However, the current approach to dealing with russell-style paradoxes in this setting is arguably not too terrible.

So I guess the argt w/ russell is that types are just a device to prevent your language from saying paradoxical things, but you can see them as imposed from without, semi-arbitrarily.

We don't need to go all the way to HTT, but really should talk about Martin-Löf type theory (HTT is basically just a full model of MLTT in topological terms plus one new axiom [ univalence ] and one new construction [higher inductive types]).

If you haven't read Martin-Löf's three lectures, I'd start there. They're a good motivation of what he's up to in v. philosophical terms http://www.ae-info.org/attach/User/Martin-L%C3%B6f_Per/OtherInformation/article.pdf

The gist is that we don't start with some system and then impose a typing discipline, but instead types can be taken as foundational, via curry-howard.

So in a fully constructivist standpoint we can say types _are_ propositions, terms _are_ proofs, and so we don't have an "object language" and a "meta language" the same way we tack eg first order logic onto set theory. Now types aren't a "kludge" but your foundational syntactic objects, and the fact that your terms have goofy binders like "lambda" becomes the hack. And via Lambek, we don't need that! we have another correspondence/isomorphism that also lets us interpret our language of types into cartesian closed categories, and in a sense MLTT can be (though here my maths falls down a bit) the "internal logic of a topos". (I really need to tackle topos theory at some point but it scary). Here we read types as objects and terms as morphisms.

So this brings us to HoTT that gives us still another isomorphism -- types are propositions are objects in a CCC are also _spaces_. terms are proofs are morphisms are _points_.

But in a sense that doesn't get to what's important about MLTT, which the lectures do. MLTT distinguishes between two notions of equality -- judgmental equality in the meta-language, and which is direct and obvious, and "internal" or propositional equality, which is a proof-theoretic concept.

In this framework you still need to distinguish between levels of universes to prevent paradoxes, and the _hierarchy_ of your universes still needs to come from some external definition, and you still arguably would like "type-in-type" or "universe in universe" to make constructions easier, and doing so still leads to paradoxes. But that's sort of secondary to most of the "important" things type theory does, and universe-bookkeeping is seen as a sort of chore that should be mechanically automated by various tools, just like working in a typed logic is less of a pain if your tools can infer types for you.

Also, arguably, the notion of a "cumulative hierarchy of universes" is itself fairly natural in its own way though, if that helps, in that the construction is iterative and straightforward and can be done "on demand" rather than all at once.

ooh someone put lawvere's "Sets for Mathematics" online http://patryshev.com/books/Sets%20for%20Mathematics.pdf

got a riddle for yall

there are four coins on a turntable, arranged in a square formation. you're blindfolded. your task is to have all the coins facing the same way (that is, all face up or all face down.) you're allowed to make a move of the following sort: you flip any number of coins, and then ask if you're done. if you're done, you're done. otherwise, i randomly spin the turntable (some integer multiple of 90 degrees.)

Interesting.

So, if the base case is verified (i.e. we’re told that the coins aren’t already in a correct position—all H or all T):

   A
B     C
   D

I’ll flip A and D, or B and C. Either:

• I’ve won.
• I have an odd number of Hs.
• I have two bordering Hs.

• I’ve won.
• I have an odd number of Hs.
• I have two opposite Hs (e.g. B and C)

• I’ve won.
• I have an odd number of Hs.

• I’ve won.
• I have two Hs and two Ts.

• I’ve won.
• I have two bordering Hs.

• I’ve won.
• I have two opposite Hs.

• I’ve won.

nice

SPOILER ALERT

y'all wanna see some hardcore bullshit?

http://lmgtfy.com/?q=critical+mathematics

the phrase 'i want to argue...' always a red flag in an academic paper.

http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDMQFjAB&url=http%3A%2F%2Fpeople.exeter.ac.uk%2FPErnest%2Fpome25%2FMarilyn%2520Frankenstein%2520%2520Critical%2520Mathematics%2520Education.doc&ei=E5WOUv79GIiLqAH3yYHYCw&usg=AFQjCNFu31eLETOh7_RmT9x-HgowEtwJPQ&sig2=kyUOwZYChMpCyTHTUff7xw&bvm=bv.56988011,d.aWM&cad=rja

but the idea that math (not just math education) has a socio-ethical dimension is not preposterous. despite how easy it probably is to sound preposterous when you ahem try to 'problematize' that dimension.

Liberatory social change requires an understanding of the technical knowledge that is too often used to obscure economic and social realities. When we develop specific strategies for an emancipatory education, it is vital that we include such mathematical literacy. Statistics is usually abandoned to “experts” because it is thought too difficult for most people to understand. Since this knowledge is also considered value-free, it is rarely questioned.

^ this is otm. imo the thing about applying "knowledge is socially constructed" critical theory to math is, like, you lose the power of math to challenge speak truth to power if you don't believe in its objectivity in the first place. like if someone uses a study drawn from a small sample to justify your oppression and you can find one with a larger sample size and say, "weak law of large numbers" or whatever analogous result for a consistent estimator. but if you're a critical theorist do you just shrug and say, ah it's all subjective?

just because something is socially constructed doesn't mean that it's not objective /spinoza

Mathematics is traditionally seen as the most neutral of disciplines, the furthest removed from the arguments and controversy of politics and social life. However, critical mathematics challenges these assumptions of neutrality and actively attacks the idea that mathematics is pure, objective, and value neutral.

was just going by this

i just think there's a tension bw 'think critically about mathematics' and 'use mathematics as a tool to think critically about other things.' many philosophers seem to have a similar attitude towards language, like they're suspicious of it, they don't trust it, it's unstable. i think mathematicians and people who use math are probably very grateful that they don't have this suspicion.

the same page suggests a pedagogical approach based on "criticality towards received opinion" but indeed mathematics is the field where you can be most critical of received wisdom, because everything is proved rigorously, you don't have to take anything for granted

good mathematicians are very conscious of its limitations

in application, sure. but do you think "good mathematicians" are suspicious of the truth of basic results? a friend of mine became close with an old logic/foundations prof called Pf3nder who has all these crazy sounding theories, like he doesn't believe in infinite sets or the existence quantifier and claims to have a disproof of the incompleteness theorem, but according to friend his views were very far from typical and that he is not taken seriously among the faculty where he used to teach.

what is truth

- Bob Marley

j. btw was my rap on type theory useful/comprehensible at all?

Homotopy reading group was fun tonight. After the rigor and discipline of proof theory, topological arguments are hilariously elegant once you get the intuition. We did eckmann-hilton, and the geometric version is such fun.

also picking up an intuition for path spaces as actual spaces, which works out quite neatly with composition as gluing, etc.

also baker's math essay in sept. harpers is really interesting. i think this is the full text online:

http://democracyweb.com/?p=11417

it was sterl, gonna have to cogitate on it some.

i never took topology :/

skimming the Martin-Löf lectures again, man they're deep. The move he makes on page 18-19 is a long time coming but its really impressive.

"Accepting this, that is, that the proof of a judgement is that which makes it evident, we might just as well say that the proof of a judgement is the evidence for it. Thus proof is the same as evidence. Combining this with the outcome of the previous discussion of the notion of evidence, which was that it is the act of understanding, comprehend- ing, grasping, or seeing a judgement which confers evidence on it, the inevitable conclusion is that the proof of a judgement is the very act of grasping it. Thus a proof is, not an object, but an act. This is what Brouwer wanted to stress by saying that a proof is a mental construction, because what is mental, or psychic, is precisely our acts, and the word construction, as used by Brouwer, is but a synonym for proof. Thus he might just as well have said that the proof of a judgement is the act of proving, or grasping, it. And the act is primarily the act as it is being performed. Only secondarily, and irrevocably, does it become the act that has been performed."

page 27 is extraordinarily funny.

i didn't know you were a mathematician flopz! i had similar interests to you when i was at university but i got a bit distracted with a summer job designing algorithms for data storage hardware. i went on to do a phd in that field ('applied mathematics' let's say) and recently took a job at a large company essentially doing the same kind of research. it's a very nice field for a mathematician actually: the problems are mathematically interesting and extremely 'relevant' from an industrial/commercial point of view. when i get time this weekend i'll try and explain a bit more about it but here are a couple of the areas i'm interested in:

- error correction codes (this area is a particularly interesting mix of probabilistic and algebraic ideas)
- signal processing algorithms (e.g. bayesian inference on graphs)
- statistical modelling and performance analysis (asymptotic analysis,large deviations/study of rare events)

in application, sure. but do you think "good mathematicians" are suspicious of the truth of basic results? a friend of mine became close with an old logic/foundations prof called Pf3nder who has all these crazy sounding theories, like he doesn't believe in infinite sets or the existence quantifier and claims to have a disproof of the incompleteness theorem, but according to friend his views were very far from typical and that he is not taken seriously among the faculty where he used to teach.

― flopson, Thursday, November 21, 2013 8:13 PM Bookmark Flag Post Permalink

Well a disproof of the incompleteness theorem is sorta kooky. But there are well known mathematicians that have produced v. impressive results and also are deeply ultrafinitist or w/e. since their good results don't depend on their left-field views then they're simultaneously respected and listened to on one side and also considered to just have odd philosophical quirks at worst. and also math is a field where you can just make something up, and then study it, as long as you do so consistently.

so we don't need to say "ultrafinite numbers are the only ones that exist" or even accept that this is a meaningful proposition in order to say "i am studying what you can do if you only work with ultrafinite numbers and throw out recursive constructions that let you go beyond them" and that's a valuable thing to do.

back in 2011, famously, Edward Nelson at princeton thought he had a proof that Peano arithmetic is inconsistent. A bunch of ppl took his work seriously and tried to understand his proof. At which point they found some irreparable holes in it. anyway, point being, most ppl aren't going to devote a chunk of their careers to trying to prove something most people consider impossible to prove (like the inconsistency of PA). but ppl that do so, as long as they're doing so in an intelligible way, and not resorting to kook-logic, are appreciated.

also the thing is in 'day-to-day' math its less whether or not you 'believe' in large ordinals or double-negation-elimination or the axiom of choice but ppl just pick a background setting sufficient for the work they're doing (often not explicitly). and then if you can show that you get different results with or without choice, that just makes it more interesting.

yeah this dude apparently started his category theory/logic seminar by claiming to have proved the inconsistency of PA, in response to which my friend laughed out loud in prof's face. thx for your detailed answer, much appreciated

q for all:

is there a simple piece of software / web app out there that does nothing more than support computation w/ bayes' rule when repeatedly updating prior probabilities in light of new evidence?

a first look around has turned up a lot that is way too sophisticated for what i want (basically, a toy for students to play with). or specialized to particular fields or applications, but with a level of parametrization that is unnecessary / distracting.

the mathematicians football league has a different scoring system from regular football. field goals are worth three points and touchdowns are worth five points.

what is the highest impossible score in the mathematician's football league?

what is the general solution to the problem (if we have x points for field goals and y points for touchdowns, what is the highest impossible score?)

what is foot ball

j. -- i asked some twitter folks and they pointed me to this fancy book on probabilistic models of cognition online that has runnable examples including some naive bayes. https://probmods.org/conditioning.html#bayes-rule

also this https://npmjs.org/package/bayes

new riddle

60 immortal impotent chameleons
15 red, 20 green, 25 blue
When two chams of different colors meet, they both change to the third color
When two like-coloreds meet, nothing happens
If all chameleons changed to one color, that configuration would be stable
Question: why does such a configuration never arise?

ok here's a sketch of an answer, which i think is correct.

first recognize that from any 'stable state' we can only get there from any state that include two colors with exactly the same quantity. they can then meet eachother to go all to the remaining color. so now the problem is to equalize any two colors.

if we start equalizing colors, each step we take to move them closer together involves shifting one up by two and the other down by one. this means that they can only approach each other (or run away from each other) by steps of three. since the colors given differ, pairwise, by five, five, and ten, none of which three divides evenly, we can never get to a state where two are equal, and hence never to a state where all are the same color.

ok now for a bit of random further speculation. writing this down actually leads me to an invariant: the difference between any two colors, mod 3, should never vary. furthermore, given the distance between any two colors (mod 3), we know the distance from the first to the third. so if we forget about needing to keep a positive number of chameleons, and if we forget about what they sum to (we can add those conditions in later, somehow?) then we get there are "just" 4 basic quotients going on here:

0 1 1, 0 2 2, 1 1 2, 1 2 0, where the third distance is the sum of the first two, mod 3.

the situation in the above problem is # 3, and the only "unsolvable" one.

i'm sure there's some classic result in group theory or something that this is a very tiny special case of. do you have any reference like that flopson?

wait i fucked up the last a bit. the invariant is weaker

the full list is, unsurprisingly 9 (3 x 3) of which some are symmetric.

we have
0 0 0, 0 1 1, 0 2 2
1 0 1, 1 1 2, 1 2 0
2 0 2, 2 1 0, 2 2 1

of these, the six unique situations (under color permutation) are 0 0 0, 0 1 1, 0 2 2, 1 1 2, 1 2 0, and 2 2 1

we're actually in 2 2 1, which is one of _two_ unsolvable situations (i.e. ones which do not contain a zero).

(or rather the invariant is ok i think its that it just gives more situations than i first listed)

Think I got it. Didn't look at s. clover's solution but maybe I saw it out of the corner of my eye and it inspired me.

1) The next to last step of making them all the same is to have two of the them equal so n of color 1, n of color 2, m of color 3. The n of color 1 meet the n of color 2 and they all become color 3.
2) The simplest operation you can do is have two chamos of different colors meet. What does that do? Decreases the number of each of their colors by 1 and increases the number of the 3rd color by 2.
3) If we think of the differences of the numbers mod 3 the above operation does not change that, since it is just subtracting 1 mod 3 from each population. Obviously if numbers are to be equal they should be equal mod 3. The original numbers 15, 20,25. are 0,2,1 mod 3. Hence all operations will essential permute these moduli, we can't make them equal and we can't get there from here.

Which seems to be exactly what Sterling was going for now that I look.

^_^ nice job fellas

Got thrown off for a while by the fact that they were all divisible by 5 which had nothing to do with anything.

homology is very confusing

feel like it's gonna pay off in a huge way though

Afraid I can't keep up with you guys on that stuff. Did hear that Sylvain Cappell just proved a big theorem though.

OK, that is not quite the case.

usually find these kind of things pretty cheezy but this is a p good lil vid http://vimeo.com/77330591

is there a simple piece of software / web app out there that does nothing more than support computation w/ bayes' rule when repeatedly updating prior probabilities in light of new evidence?

Google Docs, maybe.

also the thing is in 'day-to-day' math its less whether or not you 'believe' in large ordinals or double-negation-elimination or the axiom of choice but ppl just pick a background setting sufficient for the work they're doing (often not explicitly). and then if you can show that you get different results with or without choice, that just makes it more interesting.

Well said.

Separately, when I was in college I tried to envision contemporary algebra or analysis rebuilt with both weak (e.g. Peano) and strong (e.g. New Foundations) programs. My view of modern mathematics was idyllic, but naïve. Later, when I discovered music and poetry, I appreciated the universal adoption of, for example, the axiom of choice, for the same humanitarian reasons I appreciated iambic pentameter and the diatonic scale: universality, expressiveness, simplicity, etc. Nonetheless, the best advice I’ve received on the subject: match the program to the problem and move on.

I should confess that I hated the “match the program to the problem and move on” strategy when I first encountered it, but like the apocryphal Feynman algorithm:

• Write down the problem.
• Think real hard.
• Write down the solution.

from a comp sci perspective these questions become much more worthwhile since "does it compute" is a pretty important question, with "how does it compute" coming a close second.

of course its also worth remembering the famous hamming quip:

"Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane."

ha, that's brutal

altho "would u fly in an airplane that depended on subtle distinction in math" isn't the best way to think about math

also i think there is a physical significance, no? don't you need lebesgue integrals to do, like, integration on manifolds or whatever? (haven't studied this stuff yet)

I have a copy somewhere of this eccentric but interesting book on numerical methods from Dover in which the guys gives an interesting example about why airplane windows are shaped the way the air. At some point way back they were rectangular and, even though the equations describing the stress and strain on the window were smooth, the discontinuity/lack of derivatives at the corners would end up causing cracks. Things that make you go...

The author put the book on line, so here it is, page 38: http://www-personal.umich.edu/~jpboyd/aaabook_9500may00.pdf

Easiest way to think of R vs. L integration is in the former you are thinking of how x->y and using the continuity of the given function to bound the y's in the sums, but in L integration you instead for each y consider the set of x that map to that given y. The latter requires creating a certain apparatus to keep track of how the sets work and ignoring certain sets as "too wild."

Simplest example is a function which is zero almost everywhere (a.e) on the interval 0,1 but takes on some other arbitrary values on some countable subset like, say, the rational numbers. Riemann integration won't work. but Lebesgue integration gives the expected answer zero.

Hamming's actual argument is pretty interesting and subtle i think -- the guy was no slouch. Ultimately the math with regards to an airplane corresponds to real physical things, and these real physical things are what matter, not our models of them, and since R and L agree where they're both defined, and since they're both defined in these types of physical systems, by construction, then the genuine differences between them cease to matter up to a certain point.

Its an argument that we shouldn't think of math in isolation from the reasons we invent it.

I get what he is saying and it's not wrong, I guess, but he seems to be implying that Lebesgue integration is useless, which is a bit of a stretch.

Its an argument that we shouldn't think of math in isolation from the reasons we invent it.

― lollercoaster of rove (s.clover), Monday, December 2, 2013 2:35 PM (2 hours ago) Bookmark Flag Post Permalink

booooooooring :-P

jr -- if you knew his work you'd recognize that he didn't mean it in that way, though yeah, in isolation it can take on that context.

he's actually the author of one of my favorite essays on the relationship of math and physics: http://web.njit.edu/~akansu/PAPERS/The%20Unreasonable%20Effectiveness%20of%20Mathematics%20(RW%20Hamming).pdf

and also "you and your research" is an ur-classic http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

that's great, thanks

OK, looking forward to reading those.

Dudes, you guys are still in academia and have jstor access.

u want me to send you a pdf?

I might, rabbit, I might.

Yes, please.

send me yo email, can't attach a pdf to ilxmail

nm here u go http://www.scribd.com/doc/188781630/2321982

Thanks. Just found it here too: http://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html
Along with :http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Top level is here: http://www.dartmouth.edu/~matc/MathDrama/reading.html The Geometry in Art and Architecture link looks really nice.

OK, that was well worth reading. Thanks, guys.

we should do reading group some time. my greatest regret is not having taken a course in complex analysis.

Took me much longer than I would have liked to grok what was going on in that subject. btw, did you see the thing in Vanity Fair about the guy who reverse engineered device Jan Vermeer used to paint his masterpieces?

like "girl with a pearl earring"?

Yes, only the one that he reproduced was The Music Lesson.

I linked it on some art thread where I don't think anyone looked at it but I'll put it here too: http://www.vanityfair.com/culture/2013/11/vermeer-secret-tool-mirrors-lenses

Anyway most important thing about complex variable

this is the book i wanna read

http://usf.usfca.edu/vca//vca.jpg

Things:
Assumption that the derivative of a function does not depend on the direction you are coming from is a pretty strong one which is why analytic functions have so many things you can say about them.

Function log z can't be defined without branch cut and related fact that integral 1/z on a loop is non-zero

You spend a lot of time switching back and forth from complex variables to rectangular coordinates to polar coordinates.

Have a copy somewhere. It's pretty nice, spends more time trying to generate intuition than most.

Just came across notes on branch cuts by my advisor's advisor: http://math.mit.edu/classes/18.305/Notes/n00Branch_Points_B_Cuts.pdf

heh, a prof in my dept wrote a 1200 page, 2 volume book on the complex logarithm

Really? Who was that?

http://ebooks.cambridge.org/content/978/05/1156/619/6/9780511566196i.jpg

they're enormous

Visual Complex Analysis is terrific (especially if you haven’t been exposed to Michael Spivak’s Calculus or “Baby Rudin”)!

Book club?

http://www.madore.org/~david/math/hyperbolic-maze.html

super down for a book club. would have to start after finals tho

I’d participate!

I started reading the recently published Computability: Turing, Gödel, Church and Beyond edited by Copeland, Posy, and Shagrir. It’s a servant of all. But so far, so good. Especially enjoyed Martin Davis’ essay, “Computability and Arithmetic.” It explores Hilary Putnam and Yuri Matiyasevich’s work on Hilbert’s tenth in a comprehensible way.

super down for a book club. would have to start after finals tho

Good luck!

by servant of all you mean written for a too-general audience?

and thanks!

not sure how much brain i have to tackle another math topic at the moment, but i'm for a reading group as a general notion and i'd try to follow along a bit at least.

i started skimming along the complex analysis stuff and not-incorrectly thought "fibration" so i'm glad i'm building some intuitions.

re. that computability volume, a, uh, friend of mine has to write a review of it pretty soon, so any tips on what worked/didn't from your point of view would be appreciated (my friend hasn't started reading the book yet but the review is overdue, story of his life)

http://i.imgur.com/gxC8u1S.png

why do they schedule exams at 9am? who can even think that early?

http://math.berkeley.edu/~wu/AMS_COE_2011.pdf

Professional development (PD) for in-service math teachers is
generally taken to be \feel-good sessions". Some believe that its
main goal is to give teachers encouragement and sharpen their
pedagogical skills.
Others believe that teachers should be exposed to fun mathematics
(such as the Konigsberg bridge problem or taxicab geometry),
even in the face of their inability to deal with bread-and-butter issues
such as how to teach fractions, why negative times negative
is positive, what similarity means, or why the parallel postulate
is important.

anyone want to take a stab at 'why negative times negative is positive'? seems like a good one.

because negative divided by positive is negative.

i think algebraically it follows from that

"what is division?" is a good problem that i think i've raised on this board before. does 20/4 = 5 mean that if we divide 20 into 4 parts each part is 5 units large, or if we divide 20 into parts that are 4 units large we get 5 parts?

or how about this: if multiplication is defined as repeated addition, then multiplication by a negative is repeated subtraction, and subtracting a negative is obviously positive

"If any single quantity is marked either with the sign + or the sign - without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of -5, or the product of -5 into -5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."

Baron Maseres otm

Use complex numbers. Rotation twice by 180 degrees is the identity

LOL imo if you're using complex numbers to justify arithmetic you've won the battle but lost the war

Well, take out the complex numbers but keep the argument.

or how about this: if multiplication is defined as repeated addition, then multiplication by a negative is repeated subtraction, and subtracting a negative is obviously positive

― the late great, 11. december 2013 00:48 (1 hour ago) Bookmark Flag Post Permalink

This isn't logical, right? Surely -5 - -5 5 times is +20?

that's because what you just described is -5 - (-5) - (-5) - (-5) - (-5) - (-5), no?

should i say subtracting a negative is the same as adding?

tbh i don't completely understand the objection frederick

Seems there are plenty of ways a mathematician could convince himself of why it has to be but not clear what is the most obvious common sense explanation for the layperson.

Actually I might have an idea. But there is not enough room to write it in the margin of this thread.

Oh, I get it. You're right. My fault.

you can just do basic arithmetic on the integers as an additive group, just teach your kids group theory ;-)

for division i guess you either need a euclidean ring or a fullblown division ring, in which case division is just multiplication by inverses

If you believe -1 x a is -a, then -1 x -1 is -(-1), and negative negative 1 is plainly 1.

But once you believe -1 x -1 = 1, I think you believe that a negative times a negative is a positive in general.

elegant

we had to prove all this bullshit in my first real analysis class, to give the impression of "rigour"--but we didn't even construct the real numbers (using dedekind cuts, etc), just stated the Completeness property as an axium--such a waste of time

Think it might be useful to think of multiplication as making a copy or n copies of something to replace the thing and multiplication by -1 as making an inverted copy. So say you have a white disk than multiplying by -1 you replace it with a black disk and vice versa, or better yet you have an Othello token and just flip it over.

is this thread a boys club? where the math ladeez at?

iirc harbl studied math but she said she has forgotten all of it and left it all behind and is a lawyer now

kid i was tutoring deferred his exam :-\

why negative times negative is positive

I feel like I did something like this in discrete math, you start with basic definitions of integers and parity or w/e and then do a formal proof or w/e?

lol n/m i'm drunk and listening to bill withers

how do i shot basic simplification of roots

http://farm8.staticflickr.com/7335/11318231686_aee01101ef_b.jpg

You're asking seriously?

Oh, I see you are making fun of the person who put the question marks.

no i'm asking seriously :((((

defeated by precalc ;_;

Multiply by conjugate?

rotation of axes??

last step looks like some bullshit, no? rationalize the denominator, b then u got sqrt(1 + 2/3sqrt(2)) not sure how much more u can smiplify tho?

it works on a calculator

google it!

sqrt((2+sqrt(2)) / (2 - sqrt(2)))-sqrt(2)

ok got it

Interesting post on zero indexing:

http://exple.tive.org/blarg/2013/10/22/citation-needed/

Author interviewed Martin Richards, author of BCPL and the supposed originator of zero indexing. Conclusion: it was a stylistic decision (i.e. it wasn’t commentary on zero’s inclusion in ℕ or whatever).

interesting to think about how stylistic concerns can be aligned naturally with mathematical principles (vs when they're not aligned). makes me think about what style really means and stuff.

0 is so not a natural number

mathematical principles are always about style

If you believe -1 x a is -a, then -1 x -1 is -(-1), and negative negative 1 is plainly 1.

I once gave this answer but in a much wordier way on this thread:

http://math.stackexchange.com/questions/9933/why-negative-times-negative-positive

considering prepping a talk for an undergrad conference in january, anyone got any topics to suggest?

https://www.simonsfoundation.org/quanta/20121002-getting-into-shapes-from-hyperbolic-geometry-to-cube-complexes-and-back/

great article on the classification of 3-manifolds, written at an extremely accessible level. basically this guy thurston conjectured 23 theorems that, once all proven, would result in classification. my topo prof proved a result that was used to prove the last three conjectures in one sweep, and article goes in some detail into his research. super interesting stuff, to me at least

thanks for the link -- that's very clear!

Yeah

it would be an interesting history of math to classify what programmes have led to the most research -- i suspect classification programmes themselves would probably lead the pack.

classification of surfaces seemed like it didn't take very long once they figured out what they were doing

oh yeah, finite simple groups, too

arguably, figuring out what you're doing is typically the hard part.

great little history of the classification of surfaces on appendix D of this book http://download.springer.com/static/pdf/997/bbm%253A978-3-642-34364-3%252F1.pdf?auth66=1387259121_9a9118105634f100257c6f624c9329f0&ext=.pdf (full pdf)

gah i should be studying for my analysis exam... blegh

which 2 should i take next semester out of these 4

real analysis 4 (measure theory, functional analysis)
differential geometry
topics in geometry & topology course on cube c0mplexes
discrete mathematics of paul erdos (taught by the great vasec chv4tal http://users.encs.concordia.ca/~chvatal/6621/)

real and discrete OR differential and cube complexes

i sort of hated analysis 3 but while studying for it and memorizing all those theorems i became really impressed with it and now have the urge to take the 4th. also i've heard measure theory is one of those things you've just *got* to learn and this guy would teach it properly

interesting, why those 2 diff pairings?

(xp)

i've always had better luck in school when i take courses with some connection to each other rather than courses which have different approaches

although ... is real analysis useful in differential geometry?

yes

pairing of most similar would be cubes + discrete, diff geo + ana

oh okay. that's what i'd do then.

(higher math n00b)

cube complexes was developed by algebraic topologists & geometric group theorists, people who exploit an analogy (functor or whatever) between topological spaces, infinite groups, and cayley graphs of infinite groups, to prove results in group theory & 3-manifold theory. so graph theory would come up

the simplest vers of diff geo is, like, in multivariable calculus taking a surface integral

yeah that's about as far as i got in geometry

that article i posted upthread goes into the CC stuff, with some quotes by dude who is teaching the course (and is like world champion of cube complexes)

yeah i read it but my head started to spin halfway through and i definitely got real confused around when the cube complexes came in

word

2 much cool math

How can you not take Erdos class?

i know, right

Students who will make significant progress towards the solution of any open problem on the list posted here will get the grade of A+ regardless of their numerical score.

This is kind of old news, but I enjoyed this article. I never tire of these kinds of stories about unknown mathematicians toiling on hard problems in obscurity for years and then reaching a breakthrough, plus it does a good job of explaining the topic in layman's terms:

https://www.simonsfoundation.org/quanta/20130519-unheralded-mathematician-bridges-the-prime-gap/

yeah that one's great

there was a great slate article where they explained why twin primes conjecture is an obvious conjecture (because primes behave *as if* they are randomly distributed, even though they're not)

this interview with the guy is awesome http://nautil.us/issue/5/fame/the-twin-prime-hero

I am working on a problem related to the Goldbach conjecture.

TIL

flopson: what courses did you end up registering for? Real Analysis 4, I hope. :-)

i have until third week of the semester, probably gonna feel it out. not sure if ana will work out though, it's a pretty heavy workload

you like analysis?

One of the songs on Fade I don't care for, but the math is very good:

https://www.youtube.com/watch?v=4IwA9IimYSc

http://www.cut-the-knot.org/

Reminds me I need to change my screenname.

HI DERE

merry xmas math nerds

^ let xmas be any holiday

man this thread needs some actual words in its title

i cracked open a graph theory book, been doing some problems to relax

i realized that when i tried to 'write math', i automatically write a symbol (kind of a backward epsilon, though it was never quite that before when i used it) for 'such that'—which i/we NEVER do now when writing logic.

what is wrong with logicians?! why don't they write it? ('logically unnecessary', prob.)

do you mean existential quantification? as in ∃ x. x > 5 ?

You need it in higher order logic obv, but it tends to be implicit in first order logics.

yes, i mean after the quantifier and before the statement of the condition involving the bound variable

ah, as opposed to just the dot?

yeah; we certainly never used dots in any of the informal math style/notation i learned. backwards 'epsilons' or words.

maybe it's because logicians tend to be writing formulas after quantifiers which they just intend to be satisfied by values for the relevant quantified variables,

whereas a mathematician writing down an existential quantifier usually intends the subsequent conditions to say something meaningful about the 'x' (etc.) whose existence is being asserted (usually in terms of some conditions stated initially, or a principle or theorem etc.), so that he reaches for a piece of notation that emphasizes the subordination of the condition to the existential quantifier.

(i find that when i'm writing math, i will even put in a comma after a universal quantifier, for somewhat analogous reasons maybe)

do you mean '∍' ? I think that it might have to do with set theoretic roots there. You can read it as there exists an x drawn from the set of 'x > 5' for example. But if you're working without a set-theoretic model in mind then its terribly confusing. or if you have exists x ∍ R, then that's more like giving a 'type' than a condition -- x drawn from the reals.

i do mean that, but in concert with a membership epsilon, and w/in the general background of a naive set theory (the 'jargon of mathematicians' kind)

so it may be e.g. there exists x epsilon R backwards-epsilon x > 5

I wasn’t familiar with ∋ in a set-theoretic context. Nonetheless, Wikipedia’s article on Elements: http://en.wikipedia.org/wiki/Element_(mathematics) mentions it:

Another possible notation for [x ∈ A] is:

A ∋ x

meaning “A contains x”, though it is used less often.

I assumed you were talking about the traditional notation for “such that.” Most contemporary mathematicians, however, use a semicolon.

The relevant Unicode code point: http://www.fileformat.info/info/unicode/char/220b/index.htm

contemporary my ass

contemporary my ass

http://thuginpastels.files.wordpress.com/2012/09/grandpa-simpson.jpg

so they made you chair and you think that means you know about math huh well sonny let me tell you in my department chair was merely an administrative position

giving a talk at a conference on saturday

undergrad conference, and it's just exposition. but im still pretty psyched

http://www.slate.com/blogs/bad_astronomy/2014/01/17/infinite_series_when_the_sum_of_all_positive_integers_is_a_small_negative.html

i didn't realize that the clickbait function was unbounded on some intervals, but this seems to be fairly strong proof

blegh

oh btw this is the best thing ever: http://sciencevsmagic.net/geo/

^ ruler & compass constructions game

wow what a shitty slate article.

anyone have any good suggestions in philosophy of math btw? i'm tempted to get Corfield's Towards a Philosophy of Real Mathematics, but I doubt I'll really get to reading it carefully.

never really understood the point of that stuff tbh, aside from like water cooler chat

He notes that much of the philosophical depth and richness of contemporary philosophy of physics comes from that field’s engagement with cutting-edge topics in physics—gauge theory, string theory, quantum gravity, etc.—and he suggests that philosophical investigation of, e.g., category theory or algebraic topology might prove equally fruitful.

hard to think of something more pretentious than philosophers talking about category theory

Mathematicians are—as Corfield emphasizes in an interesting series of examples and case studies—pretty good at plausible reasoning. They evaluate the plausibility of conjectures; they argue about whether certain proof-strategies are likely to “pan out”; they discuss the likelihood that specific analogies will turn out to be fruitful. Unfortunately, it’s hard to see how they do all this.

lol

reading thurston's geometry & topology of 3-manifolds notes right now, so so good

sterl, the last thing i was read to read/look at in grad school was stewart shapiro's 2006.

by the time i had stopped caring about phil. of math, though, i was really floored by a talk by ken manders about knot theory as a source of examples for philosophy of mathematics. a lot of his older work is on diagrams in geometry, but unfortunately he is apparently a slow one so he hasn't published an inordinate amount - and the book his bio says he's writing is the same one he's been writing for years. still, his stuff seemed so insightful and mathematically realistic, from what i saw.

if you're thinking about reading Corfield, try reading a few reviews of the book first (I have some, er, acquaintance with these).

how gar are you in greek constructions, flopson?

i am stuck on circle packing.

just did the first two on my friend's ipad yesterday. will continue after i finish my analysis assignment

Euler did u write the review i was linking excerpts from above?

that one's not mine, but I know the author very well

corfield took the one you quoted pretty hard though

he was cooler with mine though I think mine's deeper

hard to think of something more pretentious than philosophers talking about category theory

spoken like a person who has never heard category theorists talking about philosophy

thanks for the pointers to the reviews, but they made me want to read corfield more, since i get the sense i'm more sympathetic to the sort of approach he advocates. i've definitely read some mclarty i really like, and have been v. taken by his historical exposition.

any thoughts of/knowledge of john bell btw? i enjoyed his primer on infinitesimal analysis.

also heard him give a good talk, less of a picture of his work, but people who are down with robinson seem good?

the problem with david's approach in the book is that it's too shallow; he just threw quotes out there and expects them to do the philosophical lifting

colin & ken have gone much deeper, but yes, not so much published; and colin is a partisan for cat theory and partisanship of that kind isn't becoming a philosopher

this convo is so weird to me, I come here to talk about rem bootlegs & then there are people talking about my people

what makes you think there's no isomorphism there

april 10 1981 -> the euclidean diagram

terry tao making some interesting points tangential to the whole summing an infinite series debacle https://plus.google.com/114134834346472219368/posts/ZuJDv3daT9n

"colin is a partisan for cat theory and partisanship of that kind isn't becoming a philosopher"

I'm curious about this statement in a number of ways.

A) how is he a partisan for cat theory? partisan in what fashion? to whom? against what?
B) why is partisanship unbecoming for a philosopher? since when? what sorts of partisanship are and are not acceptable? does it depend who you are partisan to, or about what topics?

more generally, i'm not quite sure what constitutes "philosophical lifting"? like what is "deep" philosophy vs not?

i can read mclarty as basically intellectual history with a strong grasp of the concepts under consideration and related debates. in what sense is this or is this not "philosophy"?

intellectual history is important, but ≠ philosophy. philosophy seeks to understand values; in the case of mathematics, mathematical values. you need to do history to bring those values forward, because mathematical practice expresses those values; but describing that history isn't enough

colin is a philosopher though, but rarely in print

yes, being rah rah for category theory isn't philosophical; but articulating what values category-theoretic methods realize could be philosophical. the former ≠ the latter

lol at the idea that sterl is going to accept that intellectual historians do not seek to understand values

describe values, sure, but that leaves the understanding to us. need furthur

still i lol, but then in my experience 'intellectual history' is a label used by boundary-policing philosophers to keep fantasy-land sacrosanct and pure

fantasyland rules, though rip mr toad's wild ride

THERE'S NO PHILOSOPHY IN THE MATH THREAD!

^^ boundary-policing

I liked Terry's post on 1+2+3+4+5+.... by the way

if you don't do your own philosophy they're just going to take it away from you

good understanding of statistics, no understanding of literature

http://blogs.scientificamerican.com/roots-of-unity/2014/01/27/rosencrantz-and-guildenstern-flip-coins/

not very good at paying attention during plays either, show some class

https://www.youtube.com/watch?v=Vsvy8Ko2-YM

http://users-cs.au.dk/danvy/the-ideal-mathematician.pdf

that rang totally hollow for me for some reason maybe because i am a pure math snob?

How could we as mathematicians prove to a skeptical outsider that our theorems have meaning in the world outside our own fraternity?

why does it matter if they do? most mathematicians pursue math for little monetary award or prestige and they do it for the love of it. and the comparison to scientology in the next paragraph, ffs -_-

i skipped a large part of that though so maybe that qn was answered, can u sum it up for me sterl?

if mathematicians need to get better at explaining Why it is important to study fucked up geometric spaces that don't make any sense in order to get funding, then yeah whatever it takes to keep letting them do that. but like, idk in terms of beauty + ingenuity it's one of the greatest human achievements AND a lot of that weirdo shit ended up explaining all of 20th c physics (and now physics like quantum field theory is just math out the wazoo so there ARE "applications" in terms of understanding the universe) so like just trust them to keep doing cool shit u know?

a lot of this jsut seems like Why It's Sad and Lonely To Be A Mathematician

He finds it diffcult to establish meaningful conversation with that large portion of humanity that has never heard of a non-Riemannian hypersquare. This creates grave diffculties for him; there are two col- leagues in his department, who know something about non-Riemannian hypersquares, but one of them is on sabbatical, and the other is much more interested in non-Eulerian semirings. He goes to conferences, and on summer visits to colleagues, to meet people who talk his language, who can appreciate his work and whose recognition, approval, and ad- miration are the only meaningful rewards he can ever hope for.

the thing about the writing being undecipherable outside of the community it addresses is interesting. my favourite prof talks a lot about it, how, when you're writing something that will be read by your peers, there's a common base of not just knowledge and terminology that you can assume, but also a familiarity with the same methods, similarly to how you might say "by induction on n" or "by a diagonalization argument," just more convoluted and specific. its true that it would be a shame if the meaning of those papers were lost to future generations, but is that really the case?

ok read the whole thing

is it such a big deal that a vast amount of research math is undecipherable to those outside the hyper-specific community when the amount of math written in an accessible way is more than any average person would ever want to read if they lived to be a million years old?

lol which is like, one book of it, if that

yeah, like given the demand for accessible math expo mathematicians have been more forthcoming w/ exposition than is required

that just shows how concerned they are for our true well-being, rather than with our ignorant conceptions of it

heh

i know it's written by mathematicians but the whole thing felt kinda "ppl who work at record stores are such snobs!"

you'll feel it, eventually

for caek:

http://www.nature.com/news/scientific-method-statistical-errors-1.14700#/b2

P values have always had critics. In their almost nine decades of existence, they have been likened to mosquitoes (annoying and impossible to swat away), the emperor's new clothes (fraught with obvious problems that everyone ignores) and the tool of a “sterile intellectual rake” who ravishes science but leaves it with no progeny3. One researcher suggested rechristening the methodology “statistical hypothesis inference testing”3, presumably for the acronym it would yield.

Guys, come participate on my new thread: Math & Music: The Severed Alliance. Some Recent Academic Approaches (Do Not Read If You Hate Drums)

they're crowdfunding a translation of the grothendieck biography

http://www.gofundme.com/7ldiwo

(i bought vol 1 i think, and skimmed it, and it left off before the math got interesting.)

http://languagelog.ldc.upenn.edu/nll/?p=11326

Snide reviews form part of the folklore of Mathematical Reviews. The most famous one is as sublimely succinct as it is damning: “This paper fills a much needed gap in the literature.”

I review for Math Reviews (as do others ILXors I expect) and I have yet to make a joke, alas. I just say what the paper does and get on with it. My reviews in philo journals are much more critical though I think I've never gone for a lol in one of those either.

on the philo math tip, attended a v. nice lecture by jean-pierre marquis that discussed notions of abstraction in mathematics, and frege's (i guess?) notion of a criterion of identity.

variations on "this paper fill a much-needed gap in the literature" are code for "this is an insignificant paper" in other fields too ime

Remember the relevant bit of verbiage in Lucky Jim?

"In considering this strangely neglected topic," it began. This what neglected topic ? This strangely what topic? This strangely neglected what?

now here's a paper title for the ages! http://arxiv.org/abs/1404.0799

Madame BovaryNed Raggett, c'est moi.

Ha, wrong thread.

so wow i now have an intuition for (co)homology, finally! that took forever.

i still can't crunch out calcs with it, but i get what the chain condition is about, how it relates to "the boundary of the boundary is zero" and i'm now familiar with a few different versions of chain/cochain complexes, how short sequences become long ones, and why "exactness" matters. oh and how this relates in some sense to de rahm's theorem!

thus resolves what is probably my longest period between hearing a word and being able to even describe what it means in vague handwavy terms.

any actuaries in the building? i once did interesting math...

every time i want to find this thread i just search for 'grothendieck'

http://www.nytimes.com/2014/06/30/us/math-under-common-core-has-even-parents-stumbling.html

Laci Maniscalco, a third-grade teacher in Lafayette, La., who said that sometimes her students cried during the past year when working on problems under the new curriculum, said she had seen genuine progress in their understanding — and in her own, as well.

crying, that's how you know it's working right

seven and eight year olds crying during school? this new phenomenon must be investigated!

https://www.youtube.com/watch?v=8wHDn8LDks8

http://cheng.staff.shef.ac.uk/illogic/illogic-web.pdf

Interesting article about the Stanford mathematician who just won the Fields Medal:

http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

This thread is impossible to search for, btw.

So how did you find?

articles on all of the winners are super interesting.

I ended up searching within posts for some mathematical terms such as "topology". Kinda inefficient though. xp

(oddly the R in Rolling has never rendered correctly on my linux laptop, everything else was ok, but the R was just doing the rectangle thing. until tonight that is. it's fine now.)

Rolliag Mallth thrsad

for the layperson obv. but i dug this dynamics-explained bit in slate
http://www.slate.com/articles/life/do_the_math/2014/08/maryam_mirzakhani_fields_medal_first_woman_to_win_math_s_biggest_prize_works.html

go stanford, go Iranian Americans!

i was just thinking about if you knew or had met her

wondering

but yeah, it's really cool

unfortunately not ... i was in school of (math and science) education and my wife in nat'l literatures, so i never met anybody in the math dept proper

would love to though!

also note not only first woman winner, but an indian and a brazilian winner, and also the Nevanlinna prize won by an indian. the simons article on that one particularly deep on the maths: http://www.simonsfoundation.org/quanta/20140812-a-grand-vision-for-the-impossible/

also rumors that they may change fields medal qualifications in terms of age a bit, which is exciting too, since the limit + the quadrennial awards mean lots of ppl slip thru the cracks

in other news, i'm scarily close to putting together a first-pass understanding of sheaves (and maybe even stacks)

Pls to xplain to the rest of us.

This thread is still impossible to search for. I ended up searching for “homotopy.” It wasn’t the first result!

Why not search for "Dedekind"?

Just saw sheaves mentioned in Love & Math: The Heart of Hidden Reality, by Edward Frenkel.

lol i was gonna say, i just search for 'grothendieck' and then

j. wrote this on thread ℝolliℵg M∀th Thr∑a∂ on board I Love Everything on Jun 30, 2014
every time i want to find this thread i just search for 'grothendieck'

O_0 i think this is the start of something

Has the Grothendieck prime come up on this thread yet?

What is relation between sheaf and tangent bundle? Aka How do I shot sheaf?

ack. the general notion of a sheaf i understand only through general abstract nonsense. a tangent bundle involves differential geometry or generalized smooth spaces or something, which gets dangerously close to actual numbers and spaces. i'm of no help there.

You toss it iirc

http://www.heideland-games.de/files/12-06-30_strohsackhochwurf3.jpg

pictured: mirror universe in orbit

Sterling, you got some phase planing to do!

Sorry

http://cdni.wired.co.uk/674x281/g_j/Issey_1.jpg

https://www.youtube.com/watch?v=BJ7_fFABc9s

Sterling, you got some phase planing to do!

― The Wu-Tang Declan (James Redd and the Blecchs), Sunday, August 31, 2014 9:12 PM Bookmark Flag Post Permalink

by this you mean you want me to explain my phrases more clearly?

No, just making a bad joke.

this is 100% a stupid question but still.

I used to be really good at maths in school, didn't carry it on to university, but recently I've got the urge to take it up again. I play a lot of numerical puzzles and things but I really want to do problems and spend time trying to work things out again. Yeah, I know that's embarrassing. Anyway, are there such things as maths books for adults that you can buy that have problems and examples etc or would I just be better off picking up secondary school books?

I used to be really good at maths in school, didn't carry it on to university, but recently I've got the urge to take it up again. I play a lot of numerical puzzles and things but I really want to do problems and spend time trying to work things out again. Yeah, I know that's embarrassing. Anyway, are there such things as maths books for adults that you can buy that have problems and examples etc or would I just be better off picking up secondary school books?

Yes! I recommend Strogatz's books:

http://www.amazon.com/gp/product/0544105850/ref=pd_lpo_sbs_dp_ss_1?pf_rd_p=1535523722&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0691150389&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=17CZ8R5ZTVM1NJ77SRYC

A terrific introduction to mathematics. It starts with arithmetic and ends with calculus and abstract algebra. His writing is terrific, regardless of subject.

http://www.amazon.com/Calculus-Friendship-Teacher-Student-Corresponding/dp/0691150389/ref=sr_1_1?s=books&ie=UTF8&qid=1410292988&sr=1-1&keywords=calculus+of+friendship

Is also very good. The problems within illustrate the usefulness of calculus.

Textbooks are trickier. Are you familiar with algebra or trigonometry?

Yes, both. Trigonometry was actually one of my favourite parts of the course. I'm really more after textbooks rather than books on the topic, but thank you for the recommendations.

Word. I love http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918. Take it slow. :D

Whiplash of nostalgia on seeing the book front on Amazon, though mine must've been an earlier edition (as it was 20+ years ago).

Just this weekend got back into redoing my Masters from that time, which was in algorithmics so a few levels below sterling's. But so satisfying to work out an equation for "What are the points equidistant to these two points and this line", and see an set of clattering clauses snap into a simpler form.

Only have little Spivak.

You know, Calculus on Manifolds

I actually ended up folding and buying an advanced secondary school maths book BUT I have wishlisted that recommendation for later, I don't want to get into it and find I can't do any of it and then get discouraged...so thank you for the rec, much appreciated!

yeah I have discouraged myself on various occasions since graduating trying to sit down and do random math problems from The Art of Computer Programming almost completely cold.

TAOCP is my favorite non-analysis analysis textbook. Concrete Mathematics has some fun problems too.

Song about topology:
https://www.youtube.com/watch?v=chh78JcKfoA

gyac, if you have a good university bookstore or used bookstore in a college area around, you might browse through their dover books selection and see if there's anything to your liking on a given topic. many dover books are old textbooks picked up and reprinted by the publisher, and there are several that try to be inviting in among the fairly hard-nosed coursebooks (especially considering that older textbooks were a lot less hand-holdy).

^^^otm

Think even B&N has some Dover books on the shelves. Used to actually be a Dover bookstore in an office building on Lower Broadway. You could just go to the virtual store: http://store.doverpublications.com/

In addition to the maths book you might also want to pick up http://store.doverpublications.com/0486783405.html

Hello dere

spent the evening perusing the works of shinichi mochizuki

http://cdn.tokyotimes.com/wp-content/uploads/2013/02/abc-conjecture-300x2001.jpg

http://www.heidelberg-laureate-forum.org/de/laureate/alexander-grothendieck/

dudes

Wow.

But what about his prime?

Who Is Alexander Grothendieck?

^must read

Also: http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf

http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html

http://projectwordsworth.com/the-paradox-of-the-proof/

^^ more than a year old, sorry

Interesting.

Keep meaning to post this:
https://www.youtube.com/watch?v=chh78JcKfoA

Here are two Dover books I found quite useful and readable:

Advanced Calculus, by David V. Widder
Introduction to Partial Differential Equations with Applications, by E. C. Zachmanoglou, Dale W. Thoe
Mathematics of Classical and Quantum Physics by Frederick W. Byron, Jr. and Robert W. Fuller

And here are some Springer-Verlag Undergraduate Texts in Mathematics that I found readable and useful:
Finite-Dimensional Vector Spaces, by Paul R. Halmos
Linear Algebra: An Introduction to Abstract Mathematics, by Robert J. Valenza
Groups and Symmetry, by M.A. Armstrong
Basic Topology, by M.A. Armstrong

Add to the latter list
Transformation Geometry: An Introduction to Symmetry, by George E. Martin

Wow, I guess these Springer Undergraduate books cost almost twice as much as when I bought them. Guess I shouldn't be surprised.

Much discussion about whether this is actually a 2014 photo of Grothendieck but general feeling seems to be yes:

http://www.heidelberg-laureate-forum.org/de/laureate/alexander-grothendieck/

Is he going to be part of an upcoming Os Mutantes reunion in that getup?

Oh, I get it, those are his monastic vestments.

someone explain the monty hall problem to me please. i just read this

http://mathforum.org/dr.math/faq/faq.monty.hall.html

and it SORT of made sense, but at the bottom the guy gives an example of a 1000-door problem, saying that if monty shows 998 goats, your chance of finding the car out of the final 2 doors is 999/1000 if you switch. this just seems like it cannot be right. doesn't that assume that monty has no knowledge of where the car is?

wait i think i get it now

Monty does know.

it's not a 50/50 at the end because you still have the 1/1000 chance of guessing right originally. since the host shows you 998 goats, the remaining door HAS to have the car if you didn't guess right at the beginning. and the odds of you not guessing right at the beginning are 999/1000

that was fun

You got it

I just tried to write out a game-theoretic interpretation, and then I tried to apply Bayes' theorem, but I got stuck on both of those.

but yeah rly the answer is "because Bayes' theorem is true"

you got it anyway though, hooray!

monty hall is deep and subtle

this, basically http://formalisedthinking.wordpress.com/2010/10/05/bayes-theorem-and-the-monty-hall-problem/

Also, philosophically speaking, the Monty Hall problem concerns an observation selection effect, which gets into all kinds of wooly things like the doomsday problem, I read part of this guy's book. http://www.anthropic-principle.com/

the weirdos over on LessWrong probably eat that stuff up

Really the best way to understand it is to increase the number of doors like you did. Some people use a deck of cards- you pick one, then the dealer flips over fifty of the remaining fifty-one.

Assumptions underlying rules of probability are subtle- recently read an article called "Mistakes in Probability," I think, about how the founding fathers of the subject- mathematical giants!- pretty much all made various errors in one fashion or another.

"Errors of Probability in Historical Context," by Prakash Gorroochurn.
I found it in The Best Writing on Mathematics 2013, which series I recommend to you, any year you can find.

He wrote a book too. Looks nice. Expensive though.

doesn't that assume that monty has no knowledge of where the car is?

I think it's the opposite. It seems to me there's an implicit assumption that Monty does know where the car is, and it's part of his regular routine to open a door and show a goat, regardless of which door you pick. For instance if it was Monty's policy to only show you a goat if you had picked the correct door on your first try, then it would be a bad idea to switch, right? Or if some random guy walks into the studio with no knowledge of anything and opens a random door, and there's a goat behind it, there isn't any advantage of switching either, right?

yeah Monty and his knowledge are a red herring, what matters is that you see a goat behind a door, allowing to update your prior probabilities about which door hid what.

That's not what the link you linked upthread said:

the information is given by the fact that he cannot open the winning door

i.e., he must know which door is the winning one in order to avoid opening it.

practically he has to know which door is the winning one or else the whole point is lost

Oh crap I lied. Yeah this whole thing is not intuitive at all. And not really in my wheelhouse, I should leave it to others.

Yeah, Monty's knowledge is crucial. The thing the problem forces you to realize is that the, um, equipartition of probability one might consider "intuitive" is actually an assumption you make that should be overridden if and when you have new or other relevant data. That's why they always specify a fair die in certain problems, whereas in real life, you might want to consider the probably that the die has been rigged, but then that's too complicated for a homework problem.

A related implicit assumption you might need to be aware of is whether two events are to be considered independent or not. Or to go way, out there, they way in statistical mechanics particles are considered to be indistinguishable in such a way that changes the way one would have thought the distribution should be calculated.

Do u see?

It seems like the Monty Hall problem is a good example of how cultural context gets smuggled into something that is superficially a pure math problem. It shows how there could be cultural bias in something as apparently objective as the math section of a standardized test.

Recent NY Times article about Bayesian statistics featured the Monty Hall problem.

the weirdos over on LessWrong probably eat that stuff up

Yes - Bostrom's big projects – the Oxford future of humanity/Martin school set-up, the book on Global Catastrophic risks he edited – have contributions from Eliezer Yudkowsky who founded LW.

so here's a problem i'm struggling with

a man has three coins. two are made of gold and one is from a foreign land. if we choose one at random, what is the probability that it is a gold coin from a foreign land.

one might say: 2/3 * 1/3 = 2/9

or one might say: either 1/3 or 0 but definitely not 2/9.

who is right?

anyone?

Recent NY Times article about Bayesian statistics featured the Monty Hall problem.

― You Better Go Ahn (James Redd and the Blecchs), Monday, October 6, 2014 6:59 AM (3 days ago)

yeah this is where i got it from

so here's a problem i'm struggling with

a man has three coins. two are made of gold and one is from a foreign land. if we choose one at random, what is the probability that it is a gold coin from a foreign land.

one might say: 2/3 * 1/3 = 2/9

or one might say: either 1/3 or 0 but definitely not 2/9.

who is right?

― the late great, Thursday, October 9, 2014 3:19 AM (9 hours ago)

i guess if you accept that the foreign coin either is or is not gold, then it's either 1/3 or 0

but not knowing that i guess it's 2/9. both could be right depending on how much info you have?

i am probably wrong about this, but here's how i'm thinking about it:

you have two gold coins and one non-gold coin. the chance that the foreign coin is also a gold coin is therefore 2/3. you then pick one of the three coins. so 2/3 * 1/3 = 2/9.

"1/3 or 0" yes but it's not a 50-50 chance between those two, is the thing.

i am leaning strongly toward ⅓ or 0

problem i have is with this

the chance that the foreign coin is also a gold coin is therefore 2/3

not sure about this

i am a realist, and i "know" (?) that if you did this experiment IRL three million times you would get a gold foreign coin either 0 times or approximately a million times, but not approximately 444,000 times

surely it is possible to aggregate those permutations?

enumerate all ways of choosing the two gold coins and the one foreign coin (there are 9) and compute the probability (of gold & foreign) conditional on that configuration (in 6 of them it is 1/3, and in the rest 0). assuming all configurations are equally likely (uniform prior) then the law of total probability says the probability of choosing a gold and foreign coin is 6/9*1/3 = 1/3

would be different for a different prior of course

sorry 6/9*1/3 = 2/9...

6/9 * ⅓ = 2/9

:)

i'm unclear why you would do this:

enumerate all ways of choosing the two gold coins and the one foreign coin

actually i'm unclear what you mean by this in general

I think the answer here is something like, for a frequentist, you'd write "not enough information to tell a priori", whereas a bayesian would be perfectly happy to answer 2/9, and update their priors after each experiment.

If a given coin has the property "is gold" with probability 2/3, and "is foreign" with probability 1/3, and you have no other information, and you assume that those probabilities are totally indpendent, then 2/9 is totally the right answer to the question "What is the probability that you draw a foreign gold coin?" for a single trial. If you draw a non-gold non-foreign coin or a gold foreign coin, you know that the foreign coin is gold, and your probability of drawing a foreign gold coin on a subsequent trial can be updated to 1/3 (the foreign coin is definitely gold). If you draw a gold non-foreign coin, I don't think you can update your priors, b/c you knew that one non-foreign coin was gold before you started (by the pigeonhole principle), so you've not learned anything.

i think the answer might be 2/9 if we imagined an infinite number of people with three coins in their pocket such that ⅔ of them have a gold non-foreign coin and ⅓ have a gold foreign coin

if we summed an infinite number of experiment over all of those people then I have no doubt that 2/9 of the time a foreign gold coin would be drawn

but i think for one person it collapses down to ⅓ or 0

by enumerate i just mean write down all possibilities for the ways the coins can exist (in terms of gold and/or foreign) then assume they are equally likely. it is an assumption though

The thing is that you can't answer a probability question with "x or y" - if there are two possible scenarios like that you have to determine the probability that each scenario will occur. It's not a real world problem so you have to accept that the coins are different every time you pick one. The way the problem is written implies that either you have scenario A, in which there are two non-foreign gold coins and one non-gold foreign coin (I.e the case in which the probability is 0), or scenario B, in which you have one foreign gold coin, one non-foreign gold coin, and one regular coin (I.e the case in which the probability is 1/3). The probability space breaks down so that - given that these coins are assigned their states at random each time you make a selection - scenario A happens 1/3 of the time and scenario B happens 2/3 of the time, as I explained above. So the total probability that you will pick a foreign gold coin is (1/3 * 0) + (2/3 * 1/3) = 2/9.

conditional on the state of the coins it is either 1/3 or zero yeah xp

given that these coins are assigned their states at random each time you make a selection

imo this is the heart of the 2/9 fallacy

coins don't do that

they definitely could have picked something better than coins, like glasses of water that are randomly colored with blue + yellow food coloring & you have to determine the probability of picking a glass with green water

Uh oh is this going to turn into some kind of plane-on-a-treadmill situation

I agree that if you have 9 zillion cups of water, and you randomly add blue food coloring to ⅓ of them and yellow food coloring to ⅔ of them, you will have about two zillion cups of green water and a 2:9 chance of picking a cup of green water if you randomly select a cup from among those nine zillion

but to me that sounds like a fundamentally different problem - equivalent perhaps to my infinite ensemble of men, each with three coins in their pockets, 1:3 of which are foreign and 2:3 of which are gold, evenly / randomly distributed