ℝolliℵg M∀th Thr∑a∂

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.