Mathematics - 2024-07-30

Akiva Weinberger

00:00

Here's an example. Let $\zeta(n)=1+\frac1{2^n}+\frac1{3^n}+\dotsb$. Very little is known about the irrationality or transcendentality of $\zeta(n)$ for most values of $n$. However, it's known that $\sum_{n=2}^\infty\big(\zeta(n)-1\big)=1$.

(See if you can prove this!)

zeta space

I will definitely try that

zeta space

00:20

(a) show invariance of $L_{\rho}$ for general mappings $\rho:\Bbb N \to \Bbb R_{\ge 0}$

$$L_{\rho}= \lim_{k \to \infty} \frac{\int_0^{e^{-\sqrt{\rho(n)}}} \sum_{n=1}^k e^{\frac{\rho(n)}{\ln(t)}} \, dt}{\int_0^1 \sum_{n=1}^k e^{\frac{\rho(n)}{\ln(t)}} \, dt}$$

(b) show that $L_{\rho} \in \Bbb Q.$

(see if you can prove these!)

btw each term in these series' is unknown to be rational irrational or transcendental

except the numero uno i guess

Jakobian

Jakobian

00:40

oh I forgot to type

How should I in general interpret $X\cong Y\text{ rel }A$ in the context of homotopy equivalence of spaces

in this case I could give it an interpretation, but what about more generally

porridgemathematics

11:50

I don’t think the accepted answer or the answer with 2 upvotes here works: math.stackexchange.com/questions/489098/…, e.g because the answer with 2 upvotes involves assuming the deck transformation group is non-trivial (but is right in spirit), and the accepted uses a theorem I think is false, but I’m not completely sure if I just don’t get their solutions. What do you guys think?

noballpointpen

13:32

Hello. I was working on a basic problem in linear algebra, which is easily solved by considering eigenvalues, but when I first solved it, my argument was using "same linear transformations" which opened doors for me that my understanding of matrix similarity was somewhat incorrect.
The problem is that I left an answer on MSE, and then tried to amend it with some further considerations, then I've got a downvote. This prompted me to rethink this problem again. Could someone explain to me why this argument fails in the basic case of matrices over $R$ https://math.stackexchange.com/a/4950603/11…

Thorgott

14:01

@Jakobian ah, right, but then the homotopy just shifts $Z$ one to the right, but I don't really see visually that this extends continuously to a homotopy of the entire space

@Jakobian both spaces are spaces under $A$, i.e. equipped with a map $A\rightarrow X$ resp. $Y$. homotopy equivalence means there are maps $X\rightarrow Y$ and $Y\rightarrow X$ under $A$, i.e. the composite $A\rightarrow X\rightarrow Y$ is the given map (a triangle commutes) resp. other the way round, and these are homotopy inverse, i.e. there are homotopies under $A$, meaning they are maps under $A$ at each time $t$, from either composite to the respective identity

Jakobian

@Thorgott I think I see it visually, but its the formal justification that's giving me trouble

this is precisely the map that these two answers talk about, but written formally

Thorgott

oh wait, I do see it

pretty neat

Jakobian

no point of this space is a strong-deformation retract, so one would think that one has to carefully prove that this space is actually contractible, no proof that I've seen does this

in fact nothing I've seen I would call an actual proof of this

Thorgott

14:22

you have just linked me one

Jakobian

No its not

It only says what one should do to prove this

And I've defined $F(g(z, s), t)$ already i.e. a supposed homotopy $F:Y\times I\to Y$. There is a problem to prove its continuous

Thorgott

@Jakobian no, it tells you how the homotopy is defined. it does not explicitly check the continuity, but that is not required for an "actual proof" by any reasonable standard.

unless you're claiming the continuity is not a very routine thing to verify, in which case I would disagree

leslie townes

@noballpointpen i didn't downvote, but i can't tell what the argument even is in the case of R. the paragraph beginning "if A + I was similar to A, you would expect" is entirely unclear to me. it might become clearer if you wrote it in the language of matrix algebra instead of prose, but my guess is that if you did try to do that, you'd get stuck somehow.

Jakobian

@Thorgott then your definition of "actual proof" isn't very good. Its lazy hand-waving

leslie townes

as a commenter notes, your answer seems awfully close to suggesting that similar transformations must be literally the same, or at least do literally the same thing to some unspecified vectors. maybe it doesn't do that, but it almost sounds like it does.

Thorgott

14:36

@Jakobian sorry to burst your bubble, but then almost all research mathematics for the past century has been lazy hand-waving

Jakobian

I am not claiming that continuity is not a routine thing to verify in this particular example, but I don't see how it is a routine thing to verify

I think in this case the weight of the claim is on the people who claim that it is

Thorgott

that is a very different thing then, as I've said

but I don't see why you would object to the continuity here

Jakobian

@Thorgott Its not bursting bubble of any kind. This is not research mathematics, and you did a fallacy of false equivalency

in papers not all math is spoken out loud and obvious details are left to the reader - this is standard

but this is not how math actually is, because a reader verifies those details if they're unclear

if someone is writing that you should do this and that, this is some kind of "proof" but not an actual proof, also in this case this doesn't really contain enough steps for me to be able to say, okay they've done something substantial but left some easy details, no

the substantial part in this case seems to be to say that this map is continuous and all else is not

in fact they haven't actually defined their maps precisely they just said out loud how they would like the maps to be defined, and you could argue this is also something left, although this one I managed and it wasn't that hard

and I think its just another sign of it being lazy hand-waving, that's all

they could have at least defined something but instead they opted for intuition-only

so no its not an actual proof, arguably in research mathematics they usually contain more details than present here as well

besides would I call all the proofs I find in research papers to be actual proofs? No I wouldn't

leslie townes

i think i can guess how you feel about hatcher's topology book from this

Thorgott

@Jakobian yes, and what you're saying (just 2 messages after) is that things published in papers are not "actual proofs"

not a reasonable standard

leslie townes

14:48

he sort of has a point about research papers, the last paper i submitted before i left academia, the referee rejected it citing some paper (probably their own paper) as contradicting it, and it was just wrong. it contradicted my paper because it was wrong and been out there for 20 years

porridgemathematics

sometimes experts write things down that are “easy to check” and then it turns out that even they had obviously not done the easy check on paper, because if they had they would have figured out that after the easy check is done, what they claim to be true isn’t, and the entire proof actually doesn’t go through anymore (and there isn’t even an easy fix , the result is just wrong).

leslie townes

the editor was nice about accepting my paper after i pointed this out as additional justification for publishing it :)

Thorgott

the substantial part in this case is just the right image

porridgemathematics

Although admittedly the author I have in mind is someone who claimed to also have proven a big result decades ago and the proof never gained any traction, and then people stopped really looking at his work… He did do some pretty foundational stuff in the past though (and the errors in his recent papers although substantial could just be because he’s pretty old)

Thorgott

the map in the answer is precisely defined; if a reader cannot understand what it means to move along a line at constant speed 1, the issue is their mathematical expertise and not the writer

Jakobian

14:50

@Thorgott it's not, and that's how errors in mathematics arise, see responses from e.g. Leslie above. But it's what it is

Thorgott

yes, but the issue in that case is not that they left a detail to the reader, but that they simply were wrong in what they were insinuating

the former is a criticism on formal grounds, the latter one on substantial grounds

the former is not reasonable, the latter is

Jakobian

Yes, but that also was a research paper

Thorgott

anyway, to get back on track, you can check continuity on one comb piece at a time and then it's just the restriction of a piecewise-linear map, so I maintain that this is routine

noballpointpen

hello, leslie, thank you for reply. I don't understand why the following doesn't hold:
if $A$ and $A+I$ would be similar, then I expect this equality
$[I]^E_V[T_1]_E[I]^V_E=A+I=[I]^E_W[T_1]_E[I]^W_E+[T_2]_W$,
and I expect that if $x$ is a vector in standard basis, then after multiplications (considering $[x]_W$ and $[x]_V$) and after going back to $E$, the transformed vector is the same, but it doesn't seem to be the case in this case. What is wrong?

Jakobian

@Thorgott I disagree with those points. The map isn't precisely defined, and while I agree that if the reader cannot understand what it means to move along a line at a constant speed 1, then the issue is their mathematical expertise, I disagree that this somehow makes writer less responsible

Its still lazy and doesn't justify the author

leslie townes

14:55

well, could you write out the "i expect that" in algebra? i guess i don't know why you would expect that

i'm not sure [I]^A_B is the best notation for the change of basis matrix, although i've never found a choice of notation for that that i like

Thorgott

but there is no gap

if you know what it means to move along a line at constant speed 1, you know the map

Jakobian

no you don't

you understand what the map is supposed to do, but you don't know the map

Thorgott

no, you understand what the map does, which is the same as knowing the map

or are you gonna give me some "you don't know the map unless you write down an explicit formula" bs

Jakobian

Its not the same

@Thorgott no because that's not true

and it doesn't mean you are right

Thorgott

so what's missing

what's the difference between knowing a map and knowing what it does

Jakobian

15:00

knowing what it does is just knowing some properties of it

knowing a map is to be able to tell precisely what it is

on formal grounds

this doesn't necessarily mean it has to be given by an explicit formula

noballpointpen

are you talking about the expectation about the vector or my equation?
if the first, if $x$ is in the standard basis
$[i]^E_V[T_1]_E[I]^V_E[x]_v=[I]^E_W[T_1]_E[I]^W_E[x]_W+[x]_W$,
and then after multiplying the left side by $[I]^V_E$ and the right by $[I]^W_E$ and reconsidering that $x=[I]^V_E[x]_V$ and same for $W$,
$[T_1]_E[I]=[T_1]_Ex+x$,
absurd.
If matrices are similar then this holds. Just do the same when you dont have this second term in the first place
I am confused

The whole thing about similarity is that when you go back to $E$ you have the same vector, but this case confuses me a lot

(well although clearly it is because $A$ and $A+I$ can't be similar in $R$ and in $Z_2$ you have specific behavior for $+$)

Jakobian

well, ultimately knowing a map entails some kind of ability to be able to give a formal justification of uniqueness and existence of such map

in this case you have some kind of visual interpretation in your brain of what that map is supposed to be, but you don't even have anything besides visual interpretation of the spaces involved

I don't know how I can call any of this a proof of any kind

noballpointpen

Jakobian, sorry, are you answering to me?

Jakobian

I'm not

noballpointpen

ok

Thorgott

15:06

@Jakobian no, knowing what it does means that if I give you a point, you can tell me where it sends that point

which is the same as knowing the map

and it is in fact how the answer describes the map

it describes, for any point in the space, the path along which the homotopy takes this space by giving line segments along which it travels at constant speed 1, which we've already agreed is a precise formulation

leslie townes

noball: i think some of the confusion is just this notation is getting out of hand. if A is similar to A+I then there is an invertible matrix S and another matrix M with S(M+I)S^(-1) = M. so you can deduce e.g. S(M+I) = MS and a bunch of other consequences from that. none of which seem obviously contradictory to me without getting into field specific stuff that is not in this argument.

Jakobian

@Thorgott you can say that it might mean that, but since this is in scope of language and not formal mathematics, this doesn't matter

Thorgott

whatever, the semantics are besides the point

the point is that the answer tells you the path which each point travels under the homotopy, so it tells you what the map is

leslie townes

it's very reminiscent of the other classic exercise, showing that AB-BA = I has no solutions in square matrices over a nice enough field [or indeed in more general objects such as bounded operators on some hilbert space]

Jakobian

the point is that this is only a description and the claim is that this doesn't constitute an actual proof

I've already described why this isn't an actual proof for my standards

If you define "knowing the map" as this then it might be true that this and this holds, but not necessarily that you precisely know the map, since that entails ability of be able to precisely tell what it does. Not even defining what your space is besides visual intuition is not that

the gap between intuition and actual mathematics, which is something you seem to be claiming we can just forget about, is exactly why this is not precisely defined

where actual mathematics means something written that can be formally interpreted

this can be formally interpreted but with a lot of mind-bending

I wouldn't call this formal in this sense

we can give it setting and all that but its not given by anyone

we are inferring that setting - so its informal

Thorgott

15:15

@Jakobian we are able to tell precisely what it does and the space is precisely described

it's just a bunch of line segments and we travel along them at constant speed

Jakobian

No

Thorgott

it really isn't that mysterious

Jakobian

You're not using the word "precisely" as I'm using it

you are using the common speech usage of the word

or not maybe I should retract that

I think we are both using it in the same way

precisely means without vagueness

now we both have two different interpretations of what this vagueness is

if you are to interpret this from my perspective, there is no such thing as mathematics living in my imagination, for something to be actually proved you need to provide a formal proof of it, and this can be somewhat neglected when necessary

but you are saying its not vague for us to be able to imagine and interpret this in our imagination, I don't disagree with that

for the visualization

it isn't mysterious but perhaps you aren't really trying to understand my point

I'm not saying that this is not precise enough for the purpose of imagining this in your brain

I'm saying it's not precise enough for the purpose of translating it into language of mathematics

where even I agree to compromises on that part

2 mins ago, by Jakobian

if you are to interpret this from my perspective, there is no such thing as mathematics living in my imagination, for something to be actually proved you need to provide a formal proof of it, and this can be somewhat neglected when necessary

> there is no such thing as mathematics living in my imagination

I mean there is but that's not what it should really be about to me

in this sense

8 mins ago, by Jakobian

I think we are both using it in the same way

using the same definition but not in the same way

when someone says that a map is precisely defined, I would think they'd use what I meant by saying "precise"

and not what you were saying

I think you are bending the language a little bit there

noballpointpen

@leslie, another point of confusion, stemming on the same grounds (and maybe altogether same)
given matrices $A$ and $B$ that represent $T_1$ in $V$ and $T_2$ in $V$, $A+B$ would automatically represent $T_1+T_2$ in $V$, but it doesn't preclude $A+B$ from representing $T_1$ in some another $W$;
I can see it in examples (in the nice field of $R$), but equations in this notation seem to go nuts:
you have $A=[I]^E_V[T_1]_E[I]^V_E$
and $A+B=[I]^E_V[T_1]_E[I]^V_E+[I]^E_V[T_2]_E[I]^V_E$
and assume you also have $A+B=[I]^E_W[T_1]_E[I]^W_E$,

Jakobian

It seems to me like you are using language in this way to be able to justify your beliefs that maybe visual proofs of this sort are enough for proofs

noballpointpen

15:30

I am probably missing something in this case since it is not contradictory like with $A$ and $A+I$ in $R$, because I am already fed up with this
rewriting this long notation on the paper again and again is tiresome

Jakobian

But this is not how things work. And just because a proof isn't necessarily wrong, that doesn't mean it does its job - which is to remove any reason of doubt

This is different on this site as it is different in a research paper. On this site I expect people to be able to clarify things in a precise way

noballpointpen

...gibberish and/or this notation and problems it entails

Jakobian

In fact I think that visual proofs of this sort and lazy hand-waving, both in research and outside of it, is what mathematics should actively fight

saying things like "this is precisely defined and the space is precisely defined" is maybe not intentionally malicious, but it contains some kind of dangerous for mathematics thinking

besides, the idea of leaving the details to the reader, perhaps this is somewhat of an outdated thing

in a research paper, of course there are some things so obvious that they're not worth mentioning

this isn't an excuse for lazyness

if everyone was more careful when writing proofs and so on, then we'd have less errors overall, and mathematics would be more accessible to people

noballpointpen

likely it is just that $T_1+T_2$ is a different transformation so I am doing incorrect multiplications expecting equality in the first place
what would be correct then, in this notation?

leslie townes

noball: it should be possible to frame the matrix algebra in terms of a single invertible matrix (representing one basis change) and its inverse. up above we have both [I]^E_V and [I]^E_W and their inverses which already suggests maybe a degree of freedom that does not actually exist and maybe is adding to the confusion.

i do like proofs of 'no solutions in [set] to AB-BA = I,' it would be worth seeing if your argument, whatever it is, provides something like that for a larger class than 'real matrices.'

my officemate and i in grad school had a list of different arguments for that. we both hoped to add to it someday and i don't think either of us did

Jakobian

15:41

for example, Terrence Tao, I believe he makes pretty good, clear papers, not tainted by laziness. Why can't more people be like that?

noballpointpen

@leslietownes why?

leslie townes

noball: we just found other stuff to do? :)

Thorgott

18:04

@Jakobian you say this, yet you already have succeeded in translating it into the language of mathematics, even into explicit formulae

I mean, I think it was in that language to begin with, but I don't wanna go in circles discussing semantics

@Jakobian you have just called more Field medalists lazy than you have fingers on your hand

user 20458579510081670432

18:25

To be "fair" most Field medalists are not considered to be as outstanding as educationalists.

Jakobian

19:32

@Thorgott I don't see how I did

@Thorgott yes but the point is that its not precise enough to be able to somewhat straightforwardly translate it. I've came up with a formula but it wasn't straightforward as for how this is supposed to look like

If I were to write such answer then including such details is a must. It doesn't take much to write down an idea about how proof should look like

I don't know, I feel like we've exhausted the topic and it's probably not really worth to talk about

I mean yes, it's important to say proofs should be decently precise

other than that, why are we arguing anyway

Thorgott

20:34

@Jakobian cause a lot of proofs in algebraic topology and the majority of proofs in geometric topology are of a similar visual and/or geometric nature and you have called all of them lazy and/or hand-wavy by proxy, which I don't take kindly to as I like those subjects

@Jakobian your formula is, in my opinion, convoluted. if you find it convenient to work with, then all the power to you, but I don't think the author of the post is to blame for this formula not being straightforward.

it is straightforward to define the zig-zag and it is also straightforward to add a perpendicular horizontal line segment of length 1 at each rational point to that

Jakobian

@Thorgott "and you have called all of them lazy and/or hand-wavy by proxy", I don't see how I did

Thorgott

you have called the answer lazy and/or hand-wavy based on general criteria that equally well apply to many if not the majority proofs in these subjects

Jakobian

I don't really think so, but if they are then they deserved to be called that

the decision to call this particular case lazy hand-waving, I don't think I really gave you enough insight for you to be able to claim that I've based it on some general criteria that would cover a lot of the cases you are talking about

@Thorgott I think you're exaggerating. Its very clear what the formula represents once you think about it for a second

@Thorgott you are talking about things that never happened in any type of formal setting and then calling them straightforward

Thorgott

21:01

@Jakobian I don't disagree that it's clear what it represents, but my personal opinion on it also doesn't matter

the clearer you are saying it is, the less sense it makes for you to simultaneously call it not straightforward

@Jakobian this is the formal setting

Jakobian

@Thorgott I don't know what you're trying to say here

Thorgott

that's ok, it's just a tangential at best remark

Semiclassical

non-analysis brain: If $f(x)\to 0$ as $x\to \infty$, must $f'(x)\to 0$ as well? (assuming $f'(x)$ exists for all $x>0$ for simplicity)

Jakobian

I think that remark might be based on some kind of misinterpretation of something I said

Thorgott

nah, it just isn't really salient

Jakobian

21:10

I doubt that you are in position to say this with such confidence

@Semiclassical $x^{3/2}\sin(1/x)$

no sorry, wrong example

Semiclassical

found a relevant Q&A: math.stackexchange.com/a/2996542/137524

so $1/x \sin(x^2)$ does the trick

Sine of the Time

the standard example is $\frac 1x\sin(x^2)$

Semiclassical

for context, i'm trying to see whether the problem posed here is valid regardless of how badly-behaved $f$ is: math.stackexchange.com/questions/4952542/…

Sine of the Time

if both $\lim f'(x)$ and $\lim f(x)$ exist and are finite, then $\lim f'=0$

Thorgott

@Jakobian I am calling my own remark not salient

user10478

21:18

Is this expression (i.imgur.com/6usYwhW.png) supposed to be the law of total probability, or something else? It looks mistaken to me, like x and D should both be in both probability "denominators" in the integrand or something.

Thorgott

the salient point is that it is in fact reasonably straightforward to translate concrete geometric descriptions like a line segment and moving along it at constant speed 1 "into a formal setting", no matter what exactly the latter exactly means to you

Jakobian

21:39

but it doesn't make it clearer why is this continuous

Xander Henderson

22:00

@Jakobian Alternatively, it turns out that the skills needed to do mathematics are not exactly the same skills which are required to communicate mathematics, and Tao is one of those rare people with both. There are many reasons why a paper might be unclear (maybe the reader isn't a specialist in the field; maybe the field is actually hard; maybe the author just isn't a great writer); attributing a lack of clarity to laziness seems... I dunno... kind of like a jerk move.

Semiclassical

@XanderHenderson i'd add to this that you're ultimately writing for a certain audience, and it's not always easy to gauge what the appropriate amount of detail is

"who you're trying to communicate with" is as much a part of communication as "what you're trying to communicate"

leslie townes

yeah, if anything, ability to do mathematics maybe correlates minorly but negatively with ability to communicate it

Thorgott

@Jakobian I addressed that earlier

Semiclassical

@leslietownes it can also be medium-dependent, i.e., people who can communicate on a blackboard better than in a paper

leslie townes

haha i'm reminded of a guy who had the opposite problem, too good at communicating, not so good at the actual results

he'd write these papers where you'd be like "wow, that's lovely, i get it" and then come back the next day and it all falls apart

not a pure mathematician

Xander Henderson

22:08

@Semiclassical Indeed. Or there may be many different ways of trying to convey an idea, and the author simply chooses one which doesn't click with a particular reader.

Semiclassical

the benefit of the doubt i'm willing to give does depend on the topic tho. with math i'm prepared to forgive a fair bit

but in more "philosophical" contexts i do take issue with the kind of "academia-speak" that people fall into

writing should ideally be as complicated as it needs to be, but no more than that

Jakobian

@XanderHenderson I don't think I've attributed lack of clarity to laziness

I was just listing properties of what I think Tao's papers have

In this case I think its lazy because you need to verify continuity, and to verify it you need the map, and the people answering didn't even write the map

they didn't do the important step, and I attribute that to laziness

Semiclassical

to me there's also a bit of a coastline paradox here

Xander Henderson

@Jakobian Then perhaps you would like to clarify what you meant by:

7 hours ago, by Jakobian

for example, Terrence Tao, I believe he makes pretty good, clear papers, not tainted by laziness. Why can't more people be like that?

Semiclassical

namely, that the amount of detail you need in a proof can depend on how far you're willing to zoom in

Xander Henderson

22:15

It certainly appears that you are contrasting the clear writing of Tao with the unclear writing of other, lazier authors.

Jakobian

@XanderHenderson I thought I just clarified what you misinterpreted

Xander Henderson

Not really, but I don't care enough to demand further clarification.

Jakobian

I certainly didn't try to imply that lack of clarity implies laziness

but I guess I can agree with you that demanding clarity is too much

my main issue was still laziness

Semiclassical

a more narrow point to focus on is probably this:

7 hours ago, by Jakobian

In fact I think that visual proofs of this sort and lazy hand-waving, both in research and outside of it, is what mathematics should actively fight

Xander Henderson

@Semiclassical Yes, that is what I had in mind when I wrote

14 mins ago, by Xander Henderson

@Semiclassical Indeed. Or there may be many different ways of trying to convey an idea, and the author simply chooses one which doesn't click with a particular reader.

Jakobian

22:23

@Semiclassical do you disagree with it?

Semiclassical

i don't have a particular opinion on it, but i thought it was more productive to narrow in on that as a critique of "visual proofs' rather than of math writing in general

i am fond of visual proofs but i can also recognize how possible it is to lie with them

Jakobian

I do think its dangerous to work with visual proofs, but they can be meaningful

I was thinking more about laziness

laziness is a problem in every type of research because it creates crucial errors

sometimes ones that can't be fixed

but in this context this isn't a problem of course, since its pretty obvious this map is ought to be continuous

but it is bad mathematics

if I had the confidence that at least one of these people did a calculation and was able to finish it, no problem then

I don't have such confidence. It looks to me like no one actually tried this

that's why lazy

you can't finish a proof on claiming that something works and then just leaving, without even trying this yourself

this is precisely the laziness that errors emerge from

I don't see how Thorgott can claim this is an actual proof

this is some kind of recipe on how to begin such proof, but not actually finished, and doesn't contain necessary work

In the same vein all other false proofs where someone claims something and then just stops at not verifying it are actual proofs

I don't get the logic here

Or by the same logic, to extreme, we can claim Fermat proved Fermat's last theorem

or someone else that attempted it

Semiclassical

22:43

@Jakobian eh. if we're going to talk history the example i'd pose is whether Jordan had a proof of the Jordan curve theorem

Jakobian

and then you claim its not substantial because the result is still true

even though there were multiple times people were "proving" Fermat's last theorem

Semiclassical

c.f. the back-and-forth here: en.wikipedia.org/wiki/…

the Fermat example isn't a useful one b/c 1) he never published a claim that he proved it, and 2) the methods required to prove it are so far beyond what Fermat had that it's not a reasonable supposition

Jakobian

sure, its a bad example. This is just about not picking up what you've started, like how you cite Jordan

Semiclassical

whereas the debate about Jordan is whether his published argument is sufficiently precise as to grant him priority

Obliv

what's the structure of cantor's proof?

Semiclassical

22:53

what?

Obliv

for uncountability of $\{0,1\}^{\mathbb{N}}$

what is the starting point & formal argument structure

Jakobian

By diagonal argument

First assume that $\{0, 1\}^\mathbb{N}$ has countable amount of elements, list all of them, and construct new one that isn't on the list

the same proof applies to show that $|X| < |2^X|$ for any set $X$

Obliv

so basically he's saying let $f: \mathbb{N}\to \{0,1\}^{\mathbb{N}}$ be the map where $f(n) = a_n$ for some infinite binary digit decimal expansion

Semiclassical

if all you're looking at is $\{0,1\}^\mathbb{N}$ then there's no real need to think about these as "base-2 expansions" of real numers

Obliv

i thought that's what that set denoted

Semiclassical

22:57

e.g. there's no reason to think of (0,1,1,1,1,...) as being the same as (1,0,0,...)

not really. they've got the same cardinality, sure, and in that sense they're "the same set"

but if you're trying to characterize real numbers in [0,1] then you do have to deal with issues of "some infinite strings of binary digits encode the same real number"

leslie townes

as an aside, i wonder if it's even possible to go online and read "cantor's" argument in the form he gave it

and how closely it resembles what we refer to by that name

Semiclassical

@leslietownes presumably his published paper is available somewhere

leslie townes

was it about {0,1}^N for example or some other equivalent-for-this-purpose object

Jakobian

【数学】無限より大きい無限が無限に存在することの証明【ずんだもん解説】

►

leslie townes

semi - yeah i mean like possible for me (no academic connections, paywalled out of stuff)

due to the shortness of copyright terms there's no legal reason why it couldn't be everywhere i just wonder where it is

nvm it's this anime thing jakobian just posted

Jakobian

23:01

anime is too powerful

Semiclassical

@leslietownes also there's an English translation of his original paper here which doesn't seem paywalled

leslie townes

okay so he basically does use {0,1}^N as his model

that's kind of interesting

Obliv

I haven't read the proof so I can't assess it properly but the "proofs" i've found online aren't rigorous and I can't see the actual logic behind it. I guess I'll watch that video.. lol

@Semiclassical oh thanks

Semiclassical

the funny point with the translation is how obviously the terminology used is different from what it is now

Jakobian

@Obliv the video is good

I wouldn't recommend something bad that's a video

leslie townes

23:05

yeah, although that's somewhat an artifact of machine and not human translation

"The further development of this field is the task of the future." lol you said it buddy

Semiclassical

true

Jakobian

it contains an explanation of the proof and how would one arrive at the proof, step by step

its kind of like a gem, don't mind the anime theme

leslie townes

i do like the idea of machine translation in this context however, the problem with translation that goes too far from what the author actually wrote is that the more you know about the math, the more you are tempted to update things in accordance with a future understanding

so seeing it in kind of clunky old words forces you to mentally grasp it differently

Semiclassical

there's something to it, yeah

leslie townes

even when not translating into a different language, the original wording is sometimes more informative

Semiclassical

23:07

i do like looking at the original versions of well-known formulas occasionally

leslie townes

like reading what newton actually wrote

Semiclassical

for instance, in a recent answer i went and looked up/worked through Rodrigues's derivation of his formula for Legendre polynomials

zeta space

One can seem to be very lazy and still do exceptional mathematical research

leslie townes

its often considerably different from some modern summary of it

euler in translation is pretty close to how he wrote it which is just his genius

Jakobian

@zetaspace I feel like you're just trying to justify being lazy

Semiclassical

23:09

(or at least as much of it as I was willing to---I stuck to the case of m=0, i.e., not associated Legendre polynomials)

it is funny looking at those and thinking "okay, i get what you're saying, but why did you organize it like this?"

Jakobian

I am for laziness in math, but not the laziness of the type "I claim this to be true leaves" but of the type of arriving at arguments in a lazy way, simple way

paradoxically, sometimes you need to think a lot about how to be laziest possible

leslie townes

jp serre, the genius of lazy

zeta space

@Jakobian Yeah I mean mathematicians are often "lazy" in the sense that they like to take shortcuts. In this way it can be more efficient for finding a solution that the person who restricts themselves to rigorously verifying every minutia, especially when working with very likely constraints like time

Jakobian

You seem to be conflicting the two with each other

taking shortcuts doesn't mean you don't verify things rigorously

the two types of laziness aren't the same

also, I'm not claiming its always good to verify everything rigorously

just that blatant laziness in proofs isn't good

Xander Henderson

@Jakobian I don't think that's quite right---rather, you take any map from $\{0,1\}^{\mathbb{N}}$, and show that the map is not surjective. It isn't a proof by contradiction (though that is how it is often presented to students, because it is typically presented at a time when the idea of a "surjective map" is already a little hard to understand).

Jakobian

23:20

@XanderHenderson what do you mean

Xander Henderson

I chafe a bit at "Cantor's proof" being presented as a proof by contradiction, because I find direct proofs to be more compelling (and you avoid "but! but! excluded middle!").

Obliv

I don't think there's any other way than contradiction though?

Xander Henderson

Oh, sorry---I'm on my phone---I read "countable" and "contradiction".

Jakobian

You mean to take a map $f:\mathbb{N}\to \{0, 1\}^\mathbb{N}$ and to claim that $f$ isn't surjective

Xander Henderson

@Obliv You take any map, and show it isn't surjective by constructing an element which isn't in its image.

Jakobian

23:22

and this is the exact same as taking a list of such sequences and constructing a sequence not on the list

Xander Henderson

Actually, no @Jakobian, you have presented a proof by contradiction. You start by assuming countability.

I did read that right.

There is no need to do that.

Jakobian

I'm not a constructivist to care about such things. For me the proof is essentially the same

Xander Henderson

@Jakobian Okay, but in Cantor's time the distinction between a direct proof and a proof by contradiction was significant enough (because these foundational issues were not entirely settled at the time) that Cantor did care about such things.

Obliv

I think xander is on the constructivist side of the mathematical philosophical spectrum. I think I lean towards that and intuitionism

zeta space

@Jakobian Yeah definitely, I think there's got to be a balance between the creative (perhaps less rigorous approaches) and rigor

Xander Henderson

23:25

So presenting a proof by contradiction as Cantor's proof isn't quite right, which is all that I am saying.

It is Jakobian's updated version of Cantor's proof (or, really, the version of the proof presented to any second year).

@Obliv I really am not. I am just pointing out the history of the argument.

(Though, when possible, I do tend to prefer direct proofs, as I find them more aesthetically pleasing.)

Obliv

how do you do the diagonalization argument without assuming you have a bijection from the larger set though

Jakobian

You can construct an element not in the image of $f$

Obliv

for example, in the video jakobian linked, they map $n\mapsto \{n\}$ for the map $f:\mathbb{N}\to 2^{\mathbb{N}}$ to show $f$ is surjective and then they do the diagonalization argument to show there is no bijection from the power set back onto $\mathbb{N}$

Jakobian

@Obliv injective

zeta space

Baseball is back!

Semiclassical

23:31

it's the difference between "suppose it worked for some f. show it doesn't work" and "suppose we have some arbitrary f. show it doesn't satisfy, and therefore none could ever work"

Jakobian

If you were to take $\{n : n\notin f(n)\}$ then this should be an element not in image of $f$

Obliv

@Semiclassical so by showing that for an arbitrary $f:\mathbb{N}\to 2^{\mathbb{N}}$, $f$ can't be surjective, that means no function can be defined to be surjective

Semiclassical

that's how i'm reading this, yes

Obliv

yea that makes sense

Jakobian

this boils down to Russel's paradox-like argument

Semiclassical

23:34

bleh

Obliv

u dont have to, but you can

Semiclassical

set-theory paradoxes just kinda tire me out now

zeta space

Is there an "established theorem" involving the anaytic extension of a real analytic function on some compact interval?

Jakobian

If $\{n : n\notin f(n)\} = f(m)$ then either $m\in f(m)$ so $m\notin f(m)$, or $m\notin f(m)$ so $m\in f(m)$

zeta space

as opposed to a non compact one

Jakobian

23:39

@zetaspace I think the problem with this is that analytic functions are defined on open sets

zeta space

@Jakobian sure, you're right I should modify that to some open interval $I=(0,N)$ where $N$ is some finite positive real number

Jakobian

we can have e.g. $\sum x^n/n^2$. Then its real-analytic on $[-1, 1]$. It extends to function on the closed disk $\{z : \|z\| \leq 1\}$

then we'd have to modify the definition of what a complex analytic function means

@zetaspace well then this is clear - you write your function as a power series centered at $N/2$ and then extend that to a complex function

Thorgott

@Jakobian did you read my earlier comment on why the map is continuous? it might've gone under among the other messages.

Jakobian

Yes, and I think this should do it

Just apply gluing lemma

But I didn't try this out yet

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