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3:39 AM
No one talks about the Irish mathematician Richard Hayes and his contributions to the war effort.
copper, given that it was ireland, and before i google it, did he make his contributions to the allied side? :)
2 hours later…
5:48 AM
@copper.hat wikipedia says he had degree in languages
@Jakobian I am using the word mathematician generously en.wikipedia.org/wiki/Richard_J._Hayes
6:15 AM
Is the complexity class ALL the same as the arithmetic hierarchy $\Delta^0_{\omega_1}$?
Or under ZFC, $\Delta^0_\Omega$, where $\Omega$ is the smallest ordinal with cardinality $\beth_1$?
7:00 AM
Nevermind. The halting problem prevents that.
7:13 AM
Q: A paradox about cardinality of ALL and arithmetic hierarchies ― Did I just prove that ZFC is inconsistent?

Dannyu NDosThis problem arose when I tried to find the arithmetic hierarchy that $\mathsf{ALL}$, the class of all formal languages over a finite alphabet, corresponds to (like how $\mathsf{R} = \Delta^0_1$ and $\mathsf{RE} = \Sigma^0_1$). Since the cardinality of $\mathsf{ALL}$ is $\beth_1$, I thought that,...

5 hours later…
12:22 PM
@Jakobian That surprises me. I read Hodges' biography of Turing when it was new, so my memory of it is a little fuzzy. But I certainly remember him mentioning the original Polish work on the Enigma, and how important it was. Hodges still maintains a Turing website. Here are a couple of links from it that mention the Polish work on the Enigma. turing.org.uk/sources/nov39.html turing.org.uk/publications/mathswar1.html
@s.harp You can improve that approximation of the central binomial coefficient by multiplying it by (16n-1)/(16n+1). sagecell.sagemath.org/…
Sep 5, 2022 at 12:39, by robjohn
@Ajay One nice series to remember is $$\sum_{k=0}^\infty\binom{2k}{k}x^k=(1-4x)^{-1/2}$$
Sep 5, 2022 at 12:40, by robjohn
The generating function for the central binomial coefficients, which gives $$\frac{x}{\sqrt{4-x}}=\frac{x}2\sum_{k=0}^\infty\binom{2k}{k}\left(\frac{x}{16}\right)^k$$
12:58 PM
A couple of weeks ago I was playing with the Kolakoski sequence, an infinite sequence which encodes its own run-length. 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2, ... oeis.org/A000002 Wikipedia claims it's a fractal.
I was curious to see what it looks like, wrapped into a rectilinear spiral. Start at the origin, facing the +X axis. Move forward 1 step to (1, 0). Turn left. Move 2 steps to (1, 2). Turn left. Etc. The first few points are (0, 0), (1, 0), (1, 2), (-1, 2), (-1, 1), (0, 1), (0, 3), (-1, 3), (-1, 1). You execute a pseudo-random walk, revisiting many lattice points, so keep a count of how many times you visit each point, and use that count to assign a colour to each point.
Here's what it looks like after 100,000 steps, using a logarithmic function to convert visit counts to palette indices.
And here's 9,000,000 steps.
This could be a fractal. But isn't the original sequence 1-dimensional (only ones and twos) ? How did you convert this into two dimensions ?
@Peter Yes, the Kolakoski sequence is a sequence, so it's 1 dimensional. ;) As I said above, I wrapped it into a rectilinear spiral.
The Kolakoski sequence tells you how far to move forward at each stage. After moving forward one or two units, turn left.
Anyway , a fascinating picture.
Thanks. :)
My script can go a bit higher than 9 million, but if you go too high, the server will drop the job. That's unfortunately necessary to stop stuff like people using the Sage server to mine crypto.
The thing to remember is that a "generic" set is fractal. Being non-fractal is weird.
It is similar with irrational numbers: a generic real number is irrational. Rational numbers are weird.
So the question "is this fractal?" is not, in-and-of itself, terribly interesting or enlightening.
1:12 PM
My favourite fractal is the real number line. :D
@PM2Ring I know of no definition of "fractal" which would include the real number line. X(
Isn't it self-similar?
@PM2Ring I know of no mathematician who uses self-similarity as a defining feature of a fractal. :D
Ok. It doesn't have a fractional dimension.
@PM2Ring That's a more common approach (though it depends on what you mean by "dimension"). It is still imperfect, but as a first-order approximation to the question "Is this fractal?", I'd say that's a good place to start.
1:15 PM
Personally, I like my definition: a set is fractal at a point $x$ if it has non-real local complex dimensions at $x$.
Fractals are those sets with "interesting" local geometric zeta functions.
FWIW, I used to have a copy of Mandelbrot's Fractals: Form, Chance and Dimension. So I knew a bit about fractals before the Mandelbrot set became famous. The Mandelbrot set isn't in that book (at least, not in the edition I had), but it might have had Julia sets.
That being said, the Kolakoski sequence, as you have drawn it, looks very fractal-y (for what it's worth). It would be interesting to compute the empirical box dimension to confirm.
@PM2Ring Yeah, I have that book.
My advisor and Mandelbrot were pretty close friends.
It seems to grow at the speed you'd expect for a random 2D walk.
I have heard from Falconer that that Mandelbrot never wanted to moderate a session in which my advisor was speaking---he didn't want to have to tell him to stop talking. (My advisor will generally prepare about 200 slides for a 20 minute talk---most of his 20 minute talks will go an hour or two if you don't tell him that his time is up).
1:21 PM
It's mostly fairly blobby. But it sends out new branches every now & then.
@PM2Ring That is generally to be expected. Random walks are good at locally exploring a space. They sort of meander in a small area, then take big leaps, then meander a bit more.
In the 80s, I attended a public lecture by Mandelbrot here in Sydney. It was packed, as you'd expect. And of course it was very basic. But I'm still glad I went.
Indeed. I am (academically) to young to have ever seen him speak. I kind of regret that.
Here's a prime-based pseudo-random walk I worked on a few years ago. math.stackexchange.com/a/2079346/207316
Here's a fractal fern I did a few years ago, using an iterated function system. I read Barnsley's book. But I could never figure out how to make nice-looking stuff that wasn't some kind of fern. ;)
1:37 PM
I have a basic question. Consider a $2\times 2$ block matrix and a block vector with the appropriate dimension so the matrix multiplication makes sense. Is the multiplication of these two objects simply $$\begin{pmatrix} A&B\\ C&D\end{pmatrix}\begin{pmatrix} u \\ v\end{pmatrix}=\begin{pmatrix} Au+Bv \\ Cu+Dv\end{pmatrix}?$$
@PM2Ring One of the favorite things I've ever done is yozh.org/wp/wp-content/uploads/2011/03/mosaic-5000.jpg .
@XanderHenderson Awesome. I can see that the 1st one is a Mandelbrot set made of its Julia sets. But I'm not quite sure what's going on in the 2nd one. I assume it's a similar construction...
It is a Mandelbrot-set style rendering, but the colors relate to escape speed.
It took about a week to actually render it, and my laptop spent most of that week in the freezer.
Ah, it's a cos Mandelbrot.
I spent many hours generating Mandelbrot images on an Amiga 2000, running at 30 MHz. I still have those images on an ancient hard drive, packed in a box. Somewhere...
A Mandelbulb, rendered using POV-Ray
I think I did that on the Amiga. It was a while ago...
2:31 PM
@psie yep. Multiplication of block matrices is the same as scalar matrices
Of course keeping in mind the order of operations
(Whats on the left goes on the left, whats on the right goes on the right)
3:20 PM
Has anyone here read Dummit and Foote?
Parts of it, yes.
It is on my shelf at home.
Though I learned most of the material in the text from the undergraduate Hungerford and from Hatcher.
I had three questions related thereto as I intend to start it in place of Enderton :)
1) Obviously, is it all and all a good book to learn from? I think yes from reviews but always eager to hear opinions
2) The opening of their preliminary Ch 0.2 makes a lot of statements about elementary number theory which they say they'll use. In what sense do they use it? The claims are without proof so I was wondering if it gets used as examples?
3) What would be a source for proofs of all these elementary number theory claims? I've only seen a proper subset of these claims from my analysis book's intro
D&F is boring. The place it excels is that it's big and contains a lot of exercises.
I wouldn't say it's a feat of pedagogy (it definitely isn't), but it's also the standard for undergrad algebra in many places.
Do you have any particular interests/goals further down the line, @EE18? It might help in finding a better book, though D&F you will probably be fine with.
For 1) I should amend: given that I own it, is it bad enough to throw away
I imagine no
I want to roughly have the background of an undergrad math major eventually
3:35 PM
Is it your first time reading abstract algebra?
I'm familiar with some of it from other sources
i.e. sources not dedicated to abstract algebra like DF
My analysis book had a lengthy section in its intro, I've seen some group theory for physicists, etc.
@EE18 It's a great reference in the least. So don't throw it away.
But if you have certain inclinations to other subjects, you might get a lot more from other books.
There are also just other books that people probably would advocate over D&F for pedagogy.
(usually they are shorter, though, and cover less material---this oddly enough is characteristic of pedagogical materials since making you sick of the topic is not a goal)
most math books aren't mean to be read front to back and D&F certainly isn't
I agree it has some good exercises
@EE18 I think you should start with something easier like Artin
also I don't think anything mentioned in 0.2 is beyond the scope of what you already know
3:45 PM
@Thorgott Yeah there is usually a summer break in the middle. ;)
@Thorgott No I mean I can follow all of the claims and have seen some of them proved, but here there are just the theorems and no proofs. So I'd say more this is a question of where is a decent source of elementary number theory?
Probably just a dedicated set of lecture notes? But I can't find anything decent which just has these things ordered sequentially
instead of going to wikipedia/proof wiki etc. ad hoc
Rosen is pretty standard for elementary number theory.
And not particularly unusual in presentation, either.
cool, this looks great, thanks for the tip anak
maybe ill do this before DF. seems less demanding and probably prudent given I've got other stuff on the go
I think worrying about those is absolutely not what you should be doing
and they're all simple enough to be proven ad hoc
But if you want to do them first from another book, then that's fine, too.
It's still algebra, just out of another book. :DD
3:53 PM
Go with DF instead you say Thorgott?
eh, I'm not sure about that, I'm just recommending to not get lost on an elementary number theory tangent
I think more specifically Thorgott is just saying you shouldn't be worried about those early statements.
As they said: you can just prove them when you need them.
my recommendation is to first settle on a specific subject you want to learn
is it linear algebra, group theory, ring theory, etc.
well, you already know the former
my point is that D&F consists of many chapters that are mostly independent and is too unwieldy to attempt to read at a full book if you wanna go anywhere
so you should first specify a subject within abstract algebra you wanna learn that is more attainable and then you can think about reading the corresponding chapter(s) of D&F and perhaps combining them with another source specific to that area
And unless you have had troubles finding books that suit your learning style in the past, I would not fret about just settling for D&F. If you have it, it's decent enough you will learn what you want.
I've seen many people go down the route of wanting to find the "perfect" book to learn a subject, and they just end up never learning anything.
4:10 PM
@EE18 you can try Basic abstract algebra by Bhattacharya
et al
4:23 PM
Thanks all for the tips
4:57 PM
@EE18 I think it is okay for learning. There is a fair amount of exposition, there are a lot of exercises, and no one section is terribly long (so you get a lot of exercises in digestible chunks). But it is also rather encyclopedic, and doesn't really tell any particular story.
@EE18 This doesn't seem like something that I would worry about... (but I don't have the book in the room with me, so *shrugs*).
@EE18 Again, I really wouldn't worry about this. If you want to study [A] and the author tells you "Here are a bunch of things from [B] which you will need to know, but don't worry about proving them," then don't worry about it. Mathematics is turtles all the way down, and you are never going to make progress if you insist on understanding all of the foundations first.
This is not to say that I don't understand the temptation---I felt the same desire to see the foundations at one point in my career. But part of growing up as a mathematician is coming to terms with the fact that you can't know everything.
@EE18 I wouldn't throw it away. I like Dummit and Foote. It is a good reference to have, if nothing else.
@Thorgott Yes! Exactly this. When I say that D&F doesn't really have a narrative, this is what I mean. There are a lot of topics presented in the text which are independent of what comes directly before, or anything that comes after. It is just a recitation of facts and results in different areas, without a ton of connective tissue. This makes it useful as a reference, but it is kind of inchoate as syllabus for learning.
Like I said above, I learned what little algebra I know from Hungerford's two books (the undergrad text when I was an undergrad, and the graduate text when I was in grad school) and Hatcher's Algebraic Topology. I used D&F as a reference when working through Hatcher---I found D&F's treatment of homological algebra to be a little more accessible.
But I'm an analyst. Algebra is dumb.
My opinions on these books is based only on having been a student, and not from having taught out of them, nor from having experience with deeper topics in the field.
5:16 PM
> Algebra is dumb.
I love algebraic structures in my maths though
5:43 PM
@XanderHenderson I second this, learned it myself only way too late and am still not great at it.
@XanderHenderson neither would be my go-to reference for homological algebra specifically, but that's also not something I think EE18 has to worry about yet
@anak this might be one of the biggest unique advantages of learning in a classroom environment tbh. not so much the quality of the teaching (highly variable) or the ability to ask questions and get individualized feedback (highly variable), but someone saying, this is going to be our foundation, and in a very real sense you are just stuck with it, regardless of whether it's the best, and you have to make do with that. (oddly enough, a very 'real world' experience)
@Thorgott If I were serious about algebra, I doubt that D&F would be my goto for homological algebra, either. But it isn't my field, and I didn't want to buy yet another book. :D
@Jakobian I should be a little careful about that, as I think that it is possible that what I said could be taken too seriously. One of the faculty who taught at UCR while I was there had the somewhat joking attitude that "everything I don't do is CRAP!"
Anything that anyone else was working on was crap.
Any paper he wrote more than a month ago was crap.
Any topic which he might someday work on was crap.
Everything but the one thing he was studying was crap.
When I say that "algebra is dumb", I am very much playing a similar kind of character. I don't do algebra. I don't actually understand a lot of algebra. Therefore, it must be dumb.
"i can always switch which book i'm getting this out of," "this way of presenting the material seems worse than some other way," "ooh i wonder what wikipedia has to say about this" doesn't recreate the real world experience of not having a curated world of resources to choose from, not being able to meaningfully change the difficulty of something by playing with which things are called definitions and which things are called theorems, etc
But to any student who might be in the room right now, please recognize that this is a joke. Algebra is probably the bees knees to someone (just not me (or any sane person)).
algebra can be extremely boring and computational - therefore dumb
5:49 PM
getting results out of a suboptimal set of definitions that you are 'forced' to use, or a pedagogically poor ordering of material might even be a better learning experience, objectively speaking, than reading the "perfect" book
@leslietownes Yeah, I got something like this from a student this morning. I give them a worksheet at the beginning of the semester where I ask them to try to interpret several definitions which have very little to do with the class itself (basically, "convex" and "star convex").
I had a student ask for help, because Google wasn't bringing anything relevant or understandable up.
usually by the time someone has a strong opinion about the "perfect" book, they already know the material, and there's a pretty good chance that they did not actually learn the material from that perfect book
But, like, you aren't supposed to Google. All of the information you need is right there, on the page.
I don't think I'm offending many algebraists since what they're doing is actually interesting
@leslietownes It's a curse of knowledge kind of thing.
5:52 PM
bad textbooks and portions of textbooks definitely exist, and i do think it's useful to call out where people might actually save meaningful time by avoiding something
but i'm less sure that perfect books or portions of textbooks exist
and nobody ever listens to "well just avoid wikipedia for a while" :)
@XanderHenderson i try to avoid this curse by learning/knowing as little as possible
6:06 PM
@leslietownes I hear wikipedia is pretty good for learning set theory
oh yeah. one stop shop
this is because a couple of people that I could say seemed to me to be proficient at set theory learned it mostly from wikipedia
@leslietownes Good strategy. I might have to adopt it.
give me a point of view so neutral that i can't tell which things are used every day and which things have only ever been used by the people whose papers pushed it over a low threshold of notability
jakobian there is sometimes a possibility of a selection effect or bias though. e.g. even if a lot of very proficient users of X learned X out of Y with no other guidance, "Y with no other guidance" might be a really bad resource. it might just be a really available one
@XanderHenderson huh? who are you? which one of us is talking?
6:34 PM
@leslietownes this is also true
the possible cross-referencing becomes much more natural once you've already established a semblance of foundation using one chosen reference text
I've been having such an experience as I'm currently learning $\infty$-categories. The subject seemed somewhat inaccessible till I just picked one book and started reading it without worrying about how it contextualizes relative to other things.
@leslietownes then I would see the same thing happening for other subfields of math
ooh, it's the unsupported assertion of the day!
i'm kidding, but people who self teach are not necessarily representative of all math learners, people who become proficient at set theory definitely aren't
I'd argue the starred message by anak is unsupported
there's no reason i can see why a self taught person might not be more drawn to wikipedia for some subjects than for others (e.g. its linear algebra is horrible)
in particular there is next to no insight towards if a person have learned anything or not
6:45 PM
well, OK
(I was taking issue with it before it got starred)
@leslietownes I've forgotten already.
whatever they said or how you interpret it, i've definitely seen self studying people fall into a trap of way too accustomed to using "oh, i'll just change the resource i'm working from, and find a better one" as a substitute for actual engagement with a subject
if that was the context then sure, I'd agree
and i've also seen more experienced math people (in real life and in forums and in chats like this one) unwittingly encourage this behavior with "oh, you don't like X? well first read Y, Z, and W. T is also good. so is R. and then of course X. and for culture also Y and Z"
6:48 PM
@leslietownes I have to read Y and Z twice?!
which itself is an easier activity than, say, figuring out how to actually understand how rudin wants you to think about measure theory in his horribly written chapter on it in PMA
and being stuck in a classroom that forces you to work with a less than ideal text is sometimes a useful backstop on this kind of infinite descent
irrespective of whether the book is the best, or even very good, it's just the thing you have to deal with. there's no "oh, i didn't like this author's treatment of theorem 2.2 so i'm going somewhere else"
I've figured out how to integrate with respect to arbitrary measures based on that chapter of Rudin
as an undergraduate
I still don't know why its so horrible
in my mind this also connects a little bit with the idea of advice that works well/best for some students (particularly highly motivated students or students with a lot of relevant background) not always being advice that works for everybody, and sometimes being bad advice for some people
@leslietownes Indeed. That is one of my biggest gripes about a lot of the advice given on SE---it is mostly written by people who have been successful in a field for a long time, who have no idea how "normies" learn.
so even purely in the self study context where you do have the option of switching books at leisure, it's possible for some people that the choice of book is not going to make much of any difference, and for those people, really any time at all surveying the landscape is kind of wasted time
that's on both ends of the spectrum, too. someone who is a prof at [elite university] probably would have learned a subject just fine out of any textbook. the choices made in presenting the material wouldn't matter. someone who is in danger of not ever being able to pass a math class is probably going to be in that danger irrespective of textbook
6:57 PM
Why is Rudin's PMA last chapter horrible?
i mean you can go pretty far down this road into absolute nihilism, or some weird kind of theory of predestination
@Jakobian The last chapter of almost every mathematics text is terrible.
(Though I would argue that the last three or four chapter of baby Rudin are... not great.)
but a lot of the search for the "perfect" textbook is maybe just, at a high level, toying with the fantasy that it's easier to change the future than it actually is
Speaking about last chapters, I've learned a very cool theorem which is sort of like an analogy to the Stone-Weierstrass theorem but for different type of rings relating to "connectedness"
Generally, I get the impression that when someone sets about to write a book, they start with what they know best, or what is most basic, and as they get farther along they get farther afield from what they know best, or from what is "settled law". Hence later chapters tend to be less cohesive. On average.
6:59 PM
I call it Stone-Weierstrass theorem for analytic (sub)rings for lack of words to describe it
@XanderHenderson that's one of the other main advantages of in person anything is not so much "in person explanation" is "just better," but you have more information about whether someone is a normie or not, and can adjust your advice in ways that might look unrealistic or insulting if you presented them to some rando on the internet
@leslietownes Indeed.
if someone comes to you in office hours and asks about outside recommendations (i.e. "what book is better than the one we're using" or "what's the next thing i should read after this") the real life answer is going to be different based on if they're struggling in the class, or excelling in the class, or have really strong ADD/ADHD such that "here's ten other textbooks plus wikipedia" might be the absolute worst thing to tell them
whereas on a web forum or something you often just get "well, the best next book is X"
@leslietownes I don't know if you're into the whole Stone-Weierstrass shenanigans (given that you have functional analysis background I'd be inclined to say yes)
jakobian: yes i like some of it
7:04 PM
So Stone-Weierstrass theorem says that if $B\subseteq C(X)$ is any subset where $X$ is a compact Hausdorff space, then the closed subring that contains all constant functions generated by $B$ consists of functions constant on all stationary sets of $B$
And now say that a subring $A$ of $C(X)$ is analytic if its closed, contains all constants and $f^2\in A$ implies $f\in A$
"C(X)" being C(X,R) ? (most of my interest concerns departures from full 'self adjointness' hypotheses on B although i'm interested in the real case anyway too)
@leslietownes yeah
then Stone-Weierstrass theorem for analytic ring says that the analytic ring generated by $B$ consists of al functions constant on all stationary connected sets of $B$
this condition: $f^2\in A\implies f\in A$ somehow makes connectedness involved which I find interesting
yeah, it is not at all apparent to me why that would be
how to think about it is that $f^2$ doesn't distinguish between positive and negative values of $f$ and this somehow forces connectedness
its not an obvious theorem, it takes some work
I just thought this might be familiar with what you might do in functional analysis
there is some slightly less trivial topology involved, in particular how quasi-components and components of a compact space are the same
this makes me wonder what other versions of Stone-Weierstrass theorem of this form are there out there
7:35 PM
Is it an exercise in logic or futility to transcribe (to first order logic and set theory) euclid's definitions & postulates in book 1 of euclid's elements? I have to imagine the definitions & postulates aren't arbitrary but the way they're worded makes it seem like it could be? I'm sure it's systematic and rigorous intuitively at least
I guess I'd need to know ancient greek
obliv: making formal sense of euclid's definitions and postulates involves making choices about how you formalize them. they don't just arrive fully formed in a way where it's a question of spelling them out in symbols. under most common choices for how to do this (which involve making "reasonable" choices about how euclid's intent might be expressed in a formal system), you find out that they don't give you "enough" to prove all of his theorems.
although where the trouble starts depends on the formalism
Also @leslietownes I don't understand why non-euclidean geometries seem to shake the foundations of mathematics. According to wiki (Ik, you don't like wiki :P) their very existence "proves" Euclid's geometry to be "wrong" and leads to a paradox. But non-euclidean geometries are formed when you relax the metric/discard the 5th postulate so I don't see how there's anything wrong with his axiomatization
"non-euclidean geometries seem to shake the foundations of mathematics" is vague and a great example of something not worth thinking about even if it is findable on wikipedia.
@Obliv they're not as formal as you would want
getting back to your thing, i have only hazy memories of this, but early in book 1 there are situations where you extend a line into some figure, and it starts to matter (relatively early) whether certain other things in the figure are on one side of that line, or the other. and the case analysis implied by "sidedness" is not something that euclid really thinks about in his definition/postulate setup.
7:50 PM
for something actually formal you want something like Hilbert's axiomatization of geometry
it's implicit in how a lot of the arguments are made that he's thought about it, in that he's dealt with the relevant cases, but he hasn't dealt with it as you would have to in a formalization
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff. == The axioms == Hilbert's axiom system is constructed with six primitive notions: three primitive terms: point; line; plane; and three primitive relations: Betweenness, a ternary relation linking points; Lies on (Containment), three binary relations, one linking points...
Euclid did not work under an axiomatic system like how we understand it, to my knowledge
it only got fully axiomized by the work of Hilbert
yeah, obliv, your question is kind of blending two things. "can this be formalized in the modern sense" and "did euclid formalize it" are different questions. and because they get at whether "euclid's postulates" are "axioms" in any modern sense it is very easy to get confused at the level of what words mean, particularly when vague wikipedians are saying them
as far as this translation goes (page 6-7), it seems he does care about sided-ness but in any case, this is simply one informal way to view geometry. I don't see anything "wrong" or paradoxical about it. I'd reserve those words for their technical meaning like a paradox would be a statement that is both true and false (but his is necessarily true since it's an axiom!)
if you want a good source on this, hartshorne's "geometry: euclid and beyond" discusses this in enough detail that you could, via hartshorne's setup (which is a simplification of hilbert's setup), create a kind of "euclid in symbols" if you wanted to do that.
but to get to my first point, this does involve adding things. it's not just a process of transcription
obliv: the point is that even if his analysis involves sidedness, if you just try to sit down and formalize euclid's postulates and definitions, they don't talk address that. he's assuming you have some implicit understanding of it that is not reflected in his "formalism" (using scare quotes because it is not formalism in the modern sense i thought you were initially asking about)
7:55 PM
I think his geometry was informally logical to begin with, but I can't know for certain since I don't speak ancient greek :P
or to put it another way: place "in his definition/postulate setup" in bold in this remark
5 mins ago, by leslie townes
getting back to your thing, i have only hazy memories of this, but early in book 1 there are situations where you extend a line into some figure, and it starts to matter (relatively early) whether certain other things in the figure are on one side of that line, or the other. and the case analysis implied by "sidedness" is not something that euclid really thinks about in his definition/postulate setup.
Well why does he set things up with definitions & postulates to begin with?
I guess I don't even know whether he meant postulate in the same way we think of postulates..
you'd have to ask him? i don't know? my best guess at an answer is, his "definitions and postulates" just mean something different from what we would now mean when (after being drenched in formalism) we would use those terms?
It seems somewhat rigorous though
@Obliv exactly
7:57 PM
it seems like his first postulate addresses "flat-ness" in his geometry though
Like there are some deeper ideas maybe..
another good resource would be the dover elements, translated and with extensive commentary by heath
it's not as obsessed with "formalism" and "rigor" as you or wikipedia might be, but it addresses literally all of these issues
sometimes citing very old sources, too (not surprisingly, people have been thinking about this a long time)
@Obliv if i could speak very roughly, i think the consensus is (1) you can't "directly" formalize euclid in a modern sense, you have to add stuff, (2) it's not that difficult to figure out the kinds of stuff you need to add once you look at things in a modern sense, (3) after you do that, euclid's proofs are all fine or at worst, ignore easy-to-handle cases, (4) in retrospect its pretty amazing that, without formalism, he intuited how the parallel postulate was different from his others
in particular, he didn't try to prove it from his others, as dozens of people wasted their lives attempting to do, and claiming to have done, in the middle ages
lots of discussion of that in heath's commentary too
when i say he didn't try to prove it from his others, i mean that he set up his written work in a way that doesn't attempt this. i bet he did try to prove it from his others, and if he did, there's some kind of genius in his realizing that he couldn't
we don't even know who Euclid was
for the record
yeah it could have been like 10 guys in a toga (like bourbaki)
or a large, multi-island conglomerate that used Euclid for branding purposes
@leslietownes This happened to me, I kept changing books over and over and did not progress at all. Now that I am stuck with only one book, i am actually progressing in the subject.
i wonder if in 2500 years people will be writing about the genius of Taco Bell, the presumably great man who invented both Baja Blast and the Enchirito
8:21 PM
@leslietownes I don't know about that, all I'm saying we just don't have info about him. Other than he made Euclid's Elements iirc
@Jakobian wait really?
Yes. There's a reason you don't talk about him as much as Plato or others
@leslietownes Are you referring to modern mathematicians? Were mathematicians even concerned with proving things rigorously back then?
Even Gauß didn't have a formal system in which he worked in afaik
obliv: i mean pre-1900 mathematicians, so maybe not "modern," and yes they were interested in proving things "rigorously," although the way they thought about that would maybe not have resembled early 20th century formalism
until as jakobian mentioned the work of hilbert formalized proofs like a game of logic
How would you describe the proofs of those times? Were they kind of like outlines or an informal argument etc
8:28 PM
i wouldn't equate "rigor" in the sense of paying attention to details and trying to be clear about what you are doing and unable to do, with "formalism" meaning some symbolic thing that only recently caught on
obliv: it depends on the writer and argument obviously, but quite a lot of old math works just fine if you think about it as statements being made about modern definitions (even if those definitions weren't in use at the time)
and maybe because of how we are educated now, it seems like very clever trick for folks to have been able to write such arguments "without knowing what they were even talking about"
many of us today just have a different conception of what it means to know what one is talking about
and yes a lot of old math was also nonsense, or argument wherein key details were hidden in people silently ascribing varying meanings to different words, but it's not like that somehow went away or stopped when people started getting more formal
go on MSE or the arxiv math.gm and find people addressing unsolved problems in number theory with pages of equations, all perfectly symbolic and maybe even formal and yet not adding up to what they think it adds up to
same [thing], different day
from what I researched, we only can say for certain that Euclid wrote the Elements and his other three books, although if Euclid was one person is unknown (but I'd suspect he is shrug). Moreover he lived in Alexandria in 300 BC. The quote with King Ptolemy and "there is not royal road to geometry" might have never happened for all we know.
so yeah, history of Euclid in one paragraph
@leslietownes That's weird because the whole purpose of formalizing mathematics was to be precise in our language to communicate to others. If one understands the language of the formal system then how can they write/read gibberish
especially on arxiv or a journal lol
@leslietownes I was thinking about this in the car ride, What do mathematicians even do? In the past, they had some things they held to be intuitvely true (I guess not much different than now) and from some revelation/incremental train of thought they just create mathematical statements that follow logically?
The nature of the statements themselves boggles my mind. this quote by Borel offers no comfort :(
the criteria for conviction is itself muddled for me
8:51 PM
Hi guys, I'm not sure if this is the best place to post but I wasn't sure where else. I am an undergrad, and I came up with a nice problem (similar to chip firing on a graph) that I am working through, but it would be nice to have someone else to talk this through with. I'm posting here to see if there's any interest, or advice.
You can ask here, this is for anything related to mathematics
Ah it's not a simple question to state, there's a little set up involved, which is why I didn't want to just spam.
If it is like multiple pages long, you can also link to a pdf or something
You can spam in moderation
@XanderHenderson I think
That's it. I can also take suggestions to change the setup, my original setup was much more complicated and I've simplified it to this, which is still not easy.
This is pretty easy for Paths, Cycles, Trees, and complete graphs, but otherwise I'm not sure.
Hello everyone, Is there a specific reason why the eigenvalue/eigenvector definition is not written this way $Ax=\lambda I x$?
9:12 PM
I wouldn't write it like that
but some people may probably write it like that because the eigenvectors of eigenvalue lambda are the kernel of A - I lambda
@Obliv it's helpful to separate the goal of formalizing things with how that goal works out in practice. whenever i taught real analysis, some people, usually weaker students, would want to write everything in symbols, i.e. no english words whatsoever. they thought using symbols would be a kind of magic trick, somehow making it impossible for them to read or write gibberish
in reality, it's actually pretty hard to read and write purely symbolic math, and it's not something people are particularly good at, so even if you have this system in which it's possible to check if some symbolic argument is valid or not, people will not be very good at producing inputs to that system, or interpreting outputs of that system
@CroCo is there a specific reason why you would write it this way
it's a bit like handing a kid a calculator and they think "wow, i can never make a mistake again" because the calculator is wired to do arithmetic correctly. they'll still make mistakes in what they submit to the calculator, and in interpreting its output
@leslietownes Coming from the person who never renders chatjax.. and actually I find formalized statements to first order predicate calculus easier to understand than vague statements like euclid's postulates
for me, the odd one among $Ax = \lambda I x$ and $Ax = \lambda x$ is the former rather than the latter
9:15 PM
@leslietownes I think "sovereign citizens" have a similar attitude towards laws / courts as what you describe with the symbol pushing --- if I say the magic phrase everything works out how I want it
@s.harp is the phrase in question "bingus"?
eg "im not driving without a license, im travelling!"
@Jakobian I'm asking this because when we rearrange the equation to find the determinant to find the eigenvalues, we explicitly write the difference to be between two matrices.
@CroCo what came first?
$\det(A-\lambda I)$ or the eigenvalues
is the former not just a computation, with the goal to find the latter?
@Obliv this is certainly what people tell themselves, but in comparing the statements used as illustrative examples in an introductory treatment of logic, to the statements you see in book 1 of euclid's elements (or a calculus book or anything) is apples and oranges
it's like saying "i actually find two and three letter words easier to spell and memorize than paragraphs of text"
9:19 PM
@Jakobian this comes first $\det(A-\lambda I)$
well, of course you do
@Jakobian is that a refernece? it sounds vaguely familiar
@CroCo in the perfect world of good education it doesn't come first
@s.harp just first word that came to my mind
But it makes sense.
how so
explain how it makes sense
9:21 PM
@s.harp yeah it's a combination of (1) looking for a cheat code that makes anything possible, plus (2) retaining enough of the form of the thing you want to cheat at so that it may not be transparent (even to yourself) that you're just trying to make anything possible
@Jakobian how do you move from $Ax=\lambda x$ to $Ax-\lambda x=0$ and then to $(A-\lambda I)x=0$?
@CroCo that's not an explanation
if you're claiming that something makes sense then the burden of the proof is on you
how can someone substract a matrix from a scalar or a vector?
invalid I guess
that's not an explanation
@Obliv there's to truth this in the same way that it's true that you can hold the understanding of the behavior of very simple circuits "in your head" in a way that you can't, say, hold a complex circuit "in your head" except at the cost of abstracting away at least some of the inner workings (which sometimes matter)
9:24 PM
you can't explain something by asking a question that's unrelated to the discussion
but it's not, like, easier to understand the intel 8088 by writing out what it is in individual transistors
@Jakobian but you said it is odd to write it $Ax=\lambda I x$. Is it correct though?
its correct, of course
but I still don't see any justification from you how it makes sense how using $\det(A-\lambda I)$ for definition of an eigenvalue "comes first" rather than $Ax = \lambda x$ for some non-zero $x$
We want matrices to act like multiplication, so we demand $Ax = \lambda x$
@leslietownes yeah but there is no risk of misinterpreting things if you write it down step by step, which should be how everything is done anyway. Take shortcuts when you have to, but I think the sooner we start talking in logic symbols the better (and encrypt the language so the commies don't understand yk)
this should come first, and method with the determinant only to be an afterthought for solving our problem on how to find such eigenvalues
$Ax = \lambda x$ "comes first"
9:29 PM
@Jakobian what next? how do we end up with $(A-\lambda I)$?
@Leslietownes in drafting a question on the main site we should have a drop down menu for users to select the relevant axiomatic formal system (or if they select "other" then have them list the axioms & language)
@CroCo if you put yourself in the place of someone who wants to find solutions to $Ax = \lambda x$, what would you do?
suppose you don't know anything about eigenvalues
the idea comes - write this as simple matrix equation $(A-\lambda I)x = 0$
@Jakobian two ways, by trial and error or $Ax-\lambda x = 0$ as this valid since we substract vectors.
@leslietownes I'm like kevin from the office (the US one) where he condenses his speech to minimize the number of words spoken while roughly keeping the same meaning
“Why waste time say lot word when few word do trick”
@Obliv i can't tell if you're being sarcastic, but either way, would you really expect that teaching people how to use anything like that would be easier than showing them the ropes via comments and questions as is done now? [and it's the same with "teaching real analysis" or anything else, whether to strangers or to yourself]
9:33 PM
@CroCo I'm not talking about a specific problem
@Obliv "write it down step by step" is a good idea, but it's not like that's the choice you make or not based on whether you're writing things in symbols. similarly, "taking shortcuts"- well, in writing things in symbols, will you let a single letter stand for some more complicated set of letters, or won't you? and can you hold several layers of that in your head without messing it up?
@Jakobian I agree on this.
I have no idea what the hell this guy is trying to do math.stackexchange.com/questions/4932621/…
@s.harp what I usually do in these situation is look, think oh differentials, fancy. leaves
I live for differentiability
9:38 PM
@Jakobian we have to put $I$ to make the subtraction valid which bring me back to my question. What is odd about $Ax=\lambda I x$? except it is verbose I guess.
@CroCo yeah so now, we know how decide on amount of solutions to $Bx = 0$ for some matrix $B$. If $B$ is invertible, then $x = B^{-1}0 = 0$, and if $B$ is not invertible then $Bx = 0$ has more than one solution, so at least one is non-zero
that is, if $\det(B) = 0$ then $Bx = 0$ has non-trivial solutions
@leslietownes In the context of set theories, the only primitive notions are classes/sets, elements, and membership. That's it. I was jokingly saying that we could just express everything in first order logic with set theory, so you wouldn't even need definitions like mathematical objects for which single letters/symbols would represent, except to make your writing more terse
@CroCo its redundant writing. Wouldn't call it verbose, though maybe not for me to decide
@Obliv if you look at how people in the automated proving/verification space write up things for various proof assistants, it offers a pretty good illustration of how things begin to look if you start symbol-ifying your arguments, even in terms of "primitives" that aren't primitives. here is some of the fundamental theorem of calculus in Coq, for example: github.com/coq-community/corn/blob/master/ftc/FTC.v
@Jakobian true. I know why $\det(\cdot)=0$. Eigenvectors are basically in the null of $(A-\lambda I)$, if any.
9:41 PM
maybe there's somebody who has ever said "i just find it easier to understand in symbols" who actually thinks of the FTC in those terms (which aren't even primitives), but i doubt it
@Jakobian I'm trying to remember a book I was reading. The author brought up this issue. I was surprised and I said why not.
would you rather write $\text{Id}_\mathbb{R}(x)$ or $x$ (where $x\in\mathbb{R}$)
hell no. I prefer the later.
this is the same
but I believe mathematicians do prefer the former for being absolutely clear.
9:44 PM
I doubt it, but if you can found FTC in some language of primitives and slowly build up the definitions that are to be used (like most math does by default, but doesn't explicitly define in terms of first order logic, primitives, and set theory), then you can kind of understand FTC in terms of primitives using all the usual words used in FTC but in the most technical sense
@CroCo no
mathematicians prefer to write $\text{Id}_\mathbb{R}$ to mean the function (or $x\mapsto x$), but this is its value
@leslietownes that's actually all I've been interested in lately, is tracing all the jargon in math textbooks to the source (if one exists) where the arguments are axiomatic & sound
there's rigor provided by that kind of formalization, in that it can be automatically verified in the way english prose can't, but my own brain is not very equipped to deal with that kind of "rigor," and i think a lot of people who say "i just understand it better in symbols" are just imagining that the impression they get of simple examples extends to anything that might be worth saying
@nnabahi sadly I don't think we have any graph theory enthusiasts hanging around
and in the case of euclid's elements and all other maths developed pre 1900s, there doesn't seem to be a formal system in which the math exists so we've kind of had to adapt the proofs & theorems to the contemporary math foundations (which have had their own issues and which we've not had a thorough consensus on which system to use and for what purpose) like the chinese remainder theorem, galois, cauchy, euclid, etc
9:48 PM
i understand 0, i understand 1, why are there so many letters in the alphabet and why don't we just exchange bit strings. all ambiguity would vanish
a friend made me write down all stable graphs of genus 3 on 4 vertices a couple days ago
thorgott: under a time limit, i hope
@Jakobian see the above picture.
I think the author doesn't refer to $Ax=\lambda I x$. From reading this footnote, I said may be $Ax=\lambda I x$ is a good idea if someone asks me where $I$ comes from. apparently not. :)
one way to convince yourself that $Ax = \lambda x$ means the same as $(A-\lambda I)x = 0$ is to first write $Ax = \lambda I x$ and then $(A-\lambda I)x = 0$, yes
9:52 PM
@leslietownes I haven't even read godel incompleteness theorems but I can already tell the choice of a perfect formal system (where everything we can imagine exists and can be communicated) is impractical.
I haven't really seen decimal expansions yet:
"Determine whether the function defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well defined."
Don't we define decimal expansions so they are unique?
@EE18 huh?
Formally of course
@Jakobian this is what I've said. I believe the author did refer to this in the text. Yes I did remember now.
@EE18 decimal expansions are not unique
9:54 PM
@leslietownes 1s and 0s only make sense to you because you're a robot (clearly from your ability to read unrendered latex) who functions on boolean logic that corresponds to your transistors being on or off
$0.(9) = 1$
I am aware of 1.0000... = 0.999999999... and things of that nature
But don't we generally choose one or the other?
e.g. would you say the real 1 has two decimal expansions?
9:55 PM
oh ok
Thank you
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