ℝolliℵg M∀th Thr∑a∂

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.