Mathematics - 2017-07-25 (page 2 of 5)

Secret

13:01

[Random self coined terminology] A set exists in concept space if there exists a map such that all its elements can be mapped to eulidean or noneuclidean space while preserving all the properties of the set

More mathematically speaking, given any concept, if I specify enough number of its properties (each property can be some nth ordered logic, and it does not need to be finite) then I will be able to localise it as a point or higher dimensional construct in some geometric space

THIS, is what I mean something that looks like space to me

So I guess, having such association might be actually suggesting I have a belief that every concept can be uniquely specified as a formula of its properties

user84215

Please explain your claim by some examples.

Perturbative

@Daminark All hail Jacob Lurie, our higher topos saviour

Secret

Well, let's start with the context of abstract algebra

An identity map is defined to be a mathematical object that maps a class such that its image is the same as its domain

Or in symbols: $\text{id} : S \mapsto S, \forall x \in S, \text{id}(x)=x$

So, if we have the countably infinite parameter space (it has to be countable unless we use an infinite alphabet) where each parameter is a proposition like this, I can uniquely locate the object $\text{id}$ in this space as if this is some manifold with chart and I knew the coordinates of a point precisely

Semiclassical

hi chat

Perturbative

Hey hey @Semiclassical

Secret

13:22

in The h Bar, May 4 at 15:42, by Secret

[Philosophy random] My conception of exists = anything that can be mapped into some kind of parameter space

in The h Bar, May 4 at 15:42, by Secret

this is because anything that look like space is somehow more tangible to me in that I can manipulate them mentally

whether all this tendency for me to think of space will make me more receptive to category theory is currently an open question, especially, my own abstract algebra background is haphazard as many pointed out

Mahmoud

Hello.

So.. this is about the two-variable Schwarz inequality:

GFauxPas

mornin', nerds

Mahmoud

$x_1y_1+x_2y_2 \le \sqrt{{x_1}^2+{x_2}^2} \sqrt{{y_1}^2+{y_2}^2}$

Heya

For the case when the equality holds, I managed to get to this point: $x_1y_2=x_2y_1$

Therefore, it follows that: $x_2=\alpha y_2$ and $x_1=\alpha y_1$

But.. the textbook mentions another case, that is when $y_1=y_2=0$

Can we just attribute it to the special case when $\alpha=0$ and simply dismiss it ?

Astyx

No, that would lead to $x=0$

Mahmoud

Because I couldn't deduce it

Oh right you are

Astyx

13:41

It doesn't seem right though

Mahmoud

How could it be done

(I have to deduce this 3 different ways bases on multiple proofs of the inequality)

Astyx

Wait

Semiclassical

Well, the simplest derivation of the inequality I know is to square both sides and move stuff around to get $(x_1^2+x_2^2)(y_1^2+y_2^2)-(x_1 y_1+x_2 y_2)^2\geq 0$

user21820

@Secret I think what you're talking about only covers concepts that you can think of. In that case, indeed, I agree with you that one can consider the 'space' of one's mental concepts. And as you notice, you're a bit stuck with what you can conceive of using whatever internal language you have in your mind.

Semiclassical

which becomes...$$x_2^2y_1^2 - 2x_1y_1x_2y_2+x_1^2 y_2^2 = (x_2 y_1-x_1 y_2)^2\geq 0$$

Which is indeed true, with equality only if $x_2 y_1=x_1 y_2$

Secret

13:45

@user21820 Indeed. Take art as an example, you can never specify all its properties, and good luck finding anything that can describe the life of one specific individual and reality itself

for a more closer to home example which I don't think is like space (or I am unable to map it): all the survival skills in a society

There is no clear reason other than a long history that explain why they took the form they are in the present

user21820

@Secret I see. I thought your claim was more ambitious, but I can see that you're not actually claiming much. I was thinking of various things that lie outside human perception and imagination.

Mahmoud

I need to develop the $x_2y_1=x_1y_2$ a bit further @Semiclassical

Semiclassical

Well. If $y_1,y_2$ are both nonzero then you can divide both sides by $y_1 y_2$ to get $x_2/y_2=x_1/y_1$.

And you can label that common ratio as $\alpha$.

Mahmoud

Semiclassical

If $y_1=0$ then you need either $x_1=0$ or $y_2=0$; if $y_2=0$ you need either $x_2=0$ or $y_1=0$.

Secret

13:49

@user21820 Theoretically, that framework might be able to account for those, but without a word to even describe it you cannot really illustrate such examples. After all, we humans are ultimately bounded by our hardwiring of the brain (and that is assuming consciousness is 100% physical hence the only reality we have is the one we are currently experiencing, the physical reality)

Semiclassical

So the exceptional cases are $(x_1,y_2)=(0,0)$, $(y_1,y_2)=(0,0)$, and $(y_1,x_2)=(0,0)$. All the other cases are covered by $\alpha=x_2/y_2=x_1/y_1$.

Mahmoud

You can see from (d) what he's aiming at @Semiclassical

Semiclassical

The exceptional cases correspond to either $\alpha=x_2/y_2=x_1/y_1=\infty$ (so that $y_1=y_2=0$ or to one of the ratios being formally indeterminate (0/0)

...or I thought it should. Hrm.

Mahmoud

This is getting too frustrating for a Calculus prologue..

Semiclassical

Oh, derp. Should've been $(x_1,y_1)=(0,0),(y_1,y_2)=(0,0),$ and $(x_2,y_2)=(0,0)$. Typo on my part

The middle case has $x_1/0=x_2/0=\infty$, the others have $x_1/y_1=0/0$ or $x_2/y_2=0/0$.

Where exactly are you stuck right now?

user21820

13:54

@Secret Actually, I don't buy the assumption that we are 100% physical, but even if we are, it doesn't change the fact that there are things beyond our description.

Mahmoud

How to deduce the middle case and cancel the others

Since the author doesn't mention them

Semiclassical

Well, you got down to $x_1y_2=x_2y_1$, right?

Mahmoud

Right

Secret

@user21820 This is one reason I hope within our lifetimes we will met extraterrestial civillisations, so we can learn all those alien worldviews that we can never comprehnd

Semiclassical

Well, either $y_1=0$ or $y_1\neq 0$. Same with $y_2=0$ or $y_2\neq 0$.

If both $y_1,y_2=0$ then the equality is fulfilled regardless of $x_1,x_2$. So that's one possibility.

Mahmoud

13:57

But deducing it is still a problem

Semiclassical

If $y_1=0$ but $y_2\neq 0$, then you need $x_1 y_2=x_2y_1=0\implies x_1=0$.

Why? Whatever $y_1$ is, it's either 0 or not zero.

There are two cases each for $y_1$ and $y_2$, so there's four cases altogether.

Mahmoud

Yes, I meant the possibility when $y_1=y_2=0$

Semiclassical

Yes, and that's one of the four cases.

user21820

@Secret Lol.

If we can never comprehend, we can't learn it. =P

Semiclassical

@Secret I land on the pessimistic side of 'will we ever interact with alien life.'

Mahmoud

14:00

@user21820 Maybe, we as a species are too stupid to figure out the universe we live in

Semiclassical

The universe is big enough that alien life strikes me as both inevitable and irrelevant.

That is: The universe is so big that it's hard to believe that we're really the only sapient being in existence, but also so big that we are unlikely to ever come into causal contact with other civilizations.

LastIronStar

phew i thought you meant all life is irrelevant

Semiclassical

lol

LastIronStar

just noticed the alien bit

Mahmoud

“Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.” ― Arthur C. Clarke

LastIronStar

14:03

by the way, you've heard of Isaac Arthur?

he has done many videos on futurism themes, including the Fermi Paradox and implications as well as solutions to the same.

might be interesting to check it out

Secret

And, regarding about your 100% physical comment, well, if consciousness really does have a nonphysical component, reality will become quite interesting, and sometimes scary.

Suddenly, you need to consider about someone you don't know that live in the opposite end of the earth on how it affect your life and other things

More generally, as I have found when talking to a neuroscientist x philosopher, in such a scenario, subjective realities and beliefs suddenly become something as physical as electromagnetic fields and you need to extend the notion of social interaction to such things and ho…

Semiclassical

Compare that with the situation of explorers sailing across the globe and finding indigenous settlements: It was not inevitable that a given area would have been discovered and settled by then, and their existence (or lack thereof) was decidedly relevant to how those areas were exploited by their discoverers.

The difference being that the distance scales involved for sailing the Ocean versus travelling the stars are so vastly vastly different.

LegionMammal978

Can anyone here come up with an interesting/unexpected bijection on the natural numbers?

Semiclassical

@LegionMammal978 Euclid: For any positive integer n, there is a unique nth prime. :)

LastIronStar

I guess you could say that we are not even equipped to discover life in different formats given how our definition of life is lacking given the information of our own biochemistry!

@LegionMammal978 the set of rationals to naturals can be put in bijection

Secret

14:07

The set of all formulae writable in a finite alphabet bijects with the naturals

Semiclassical

The Earth feels big to us, but it's small enough that the possibility of 'alien' civilizations (from a Western perspective) could be both surprising and very relevant.

LegionMammal978

@Semiclassical That's just an injection

Semiclassical

woops.

LegionMammal978

@LastIronStar And the domain of that is the set of rationals, not natural numbers

@Secret See ^

LastIronStar

@LegionMammal978 it's a bijection, you can flip it.

Semiclassical

14:08

There's probably a lexicographic ordering of the natural numbers based on their prime factorizations, though.

And that would be a bijection.

LastIronStar

The set of all roots of polynomials with rational co-efficients can be put in bijection to naturals!

Astyx

Here

Secret

2

Q: The set of all finite subsets of the natural numbers is countable

Could someone verify my proofs? Proposition: the set of all finite subsets of $\mathbb{N}$ is countable Proof 1: Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$. We can have a function $g_{n}: \mathbb{N} \rightarrow A_{n} $ for each subset such that that function is surje...

user21820

@Semiclassical I don't even believe there is any life on any another planet, except perhaps those we brought over there.

Astyx

Wait, not what I had in mind

Semiclassical

14:10

Eh. I think the universe is big enough that our being unique is hard to believe.

But "exists" and "can interact with it" are different questions, is my point.

user21820

@LastIronStar @Secret: Obligatory comic:

schlockmercenary.com/2017-03-31

LastIronStar

agreed, what if our physiologies prohibits the very interaction!

Semiclassical

I didn't even mean that. I mean in terms of scale considerations alone.

Astyx

"If a tree falls in a forest and no one is around to hear it, does it make a sound?"

Semiclassical

That alien civilizations are simply too far removed in space/time for there to be a possibility of meaningful interaction.

Mahmoud

14:12

@user21820 The elements of life are everywhere, the time sufficient for evolution is there, our imagined self-importance is tricking us into believing that we are special.

Alessandro Codenotti

@Astyx yes and an hipster buys its album

LastIronStar

@user21820 nice

user21820

@Mahmoud I don't think we are important.

Astyx

@AlessandroCodenotti That's so 2016.

LastIronStar

@AlessandroCodenotti Haha

@Semiclassical You mean something outside the observable universe type of deal?

Semiclassical

14:13

No.

Our being able to observe it simply requires the light to get from there to here.

Our being able to interact with it requires both directions.

Astyx

There are some things in the observable universe we'll never reach - not even close

user21820

@Secret My current guess is that our consciousness is intertwined with our physical self, but extends beyond the material world, and that we are permitted to interact with the physical world through our physical bodies, and that our brain is the primary interface connecting our bodies and our self. But it's just a guess and I won't attempt to justify it.

Mahmoud

Cough worm holes ? cough cough

LastIronStar

@Semiclassical It's possible, but the lag would make 56k modems look like magic.

Semiclassical

lol

That's assuming you even see it. There's a lot of directions to look.

LastIronStar

14:16

I'm not even sure we are looking in a particular direction completely given again, the "chemistry" could be so very different!

LegionMammal978

@LastIronStar As in, I'm looking from a bijection from $\Bbb N$ to $\Bbb N$

Semiclassical

Sure. My point is again that, even if one is optimistic about there being life in the universe with which we could have meaningful interactions with, the probability of that interaction ever occuring is...not good.

user21820

That I think we can all agree on.

LastIronStar

@Semiclassical Do you have any affection at all for the Hard Science Fiction Genre?

Semiclassical

Not my cup of tea.

LastIronStar

14:19

That explains it!

Semiclassical

About the closest contact I have with it is Schlock Mercenary, and that's more like space opera.

LastIronStar

lol

user21820

@Semiclassical It's a curious concoction of plausible scenarios plus slapstick.

Mahmoud

I recently questioned all my beliefs (especially the faith) and dropped the majority of them, so these kind of conversations make me uncomfortable.. I don't really know why, I used to pretend knowing the answer to all of the problems in the frontiers of our thought, now, I admit my ignorance humbly, it's feels both so humiliating and thrilling.

user21820

@Mahmoud I did the same, and now I only have a few core beliefs that seem to me to be right.

Semiclassical

14:25

Some numbers are useful. The distance traveled by the Victoria (the ship Magellan captained) was over 42,000 miles. The distance traveled by Apollo 1 to/from the moon was about 830,000 miles. The distance traveled by the Pioneer spacecraft so far is beyond 8 billion miles.

Mahmoud

I think Voyager1/2 has traveled the farthest.

Semiclassical

The distance to Alpha Centauri, the next closest star system? 2.6 trillion miles.

LastIronStar

@Semiclassical schlockmercenary.com/2016-12-11

Mahmoud

1 or 2 not 1/2, what a silly mistake to do in a Math chat room.

user21820

@Mahmoud So you mean Voyager(1+2)?

Distributivity, you know.

Semiclassical

14:28

@Mahmoud That's probably right.

Mahmoud

Haha @user21820

Semiclassical

My main objection to scifi is the old trope: Science fiction writers have no sense of scale.

user21820

@Semiclassical So in your opinion does the author of Schlock have a sense of scale?

LastIronStar

I agree, but i'm not sure that applies to some of the writers esp. some of the Hard Science Fiction authors, some are even qualified Astrophyscists and Mathematicians!

Semiclassical

Sometimes. But I don't read Schlock because I'm interested in it as speculative fiction. I like the characters/stories/slapstick.

Mahmoud

14:30

Okay, bye people, and thank you @Semiclassical for the help.

Semiclassical

later @Mahmoud

LastIronStar

ttyl!

Oskar Tegby

I’m going to ask a question that I probably should ask my professors, but I’m on summer break and so are they, and I want to know the answer to the question before the summer break ends.

Where can a person competent in both mathematics and programming find work that combines these skills?

In general, I’m not interested in analysis, stochastics, or related sciences, such as economics. Examples of workplaces within said fields would be much appreciated. If this question is too large, then is there any place that I can turn to besides my professors? My study councillor redirects such questio…

LastIronStar

ttyl @Semiclassical 'twas nice interacting without lag :)

user21820

@OskarTegby Any work that requires algorithms would combine them.

Semiclassical

14:34

lol

user21820

Especially work that asks you to prove your algorithms correct, not blindly contrive them.

Oskar Tegby

Okay. Well, I'll probably end up with about one years worth of university studies and about five years of algebra and related mathematics. Is that going to improve my stance against people studying pure programming? I was initially interested in analysis, stochastics, and computer science, but once I started doing my bachelor thesis in PDE I felt that I was displeased with the actual work of proving such theorems, and the applications within stochastics and computer science.

user21820

If you like algebra, there's coding theory and cryptography.

You'll be surprised at how many percent of people doing programming can't prove a single thing.

Oskar Tegby

I want to have a plan B, and that was my plan B for analysis, but now I've become more interested in algebra. Thus, I want to know more in greater detail what my plan B will be when I specialise in algebra; I need to choose courses to be competent in plan B. Especially now as I'm going to study abroad at University of Melbourne I need to know it farther in advance.

user21820

@Semiclassical Yea I don't think it's meant to be speculative fiction rather than just a comic. But I recall there's plenty of scale-related panels like:

schlockmercenary.com/2006-04-02

Oskar Tegby

14:40

Okay. This is generally the understanding that I've been given; that programming involves lots of advanced data structures, algorithms, and mathematical concepts. I guess that I could email my professors, and look at the handbook for programming courses.

"Coding theory" seems to be my keyword when looking at what beloved Wikipedia has to say about it. :)

user21820

Coding theory has to do with encoding, not with programming, by the way.

Oskar Tegby

14:55

Thanks!

Adeek

15:07

hey @Daminark here

Akiva Weinberger

Scrolling through old MSE posts, I saw this question raised

Adeek

what question @AkivaWeinberger

Akiva Weinberger

$\ln p_i$ provides an infinite $\Bbb Q$-independent set

($p_i$ being primes)

($\Bbb Q$-independent meaning linearly independent over $\Bbb Q$ I guess)

However, $\Bbb R$ is an uncountable vector space over $\Bbb Q$.

So can we write an explicit uncountable $\Bbb Q$-independent set?

From a comment to this answer: math.stackexchange.com/a/6250/166353

Adeek

@AkivaWeinberger It means that we can write every element of Q as a finite linear combination of those elements that is algebraic span

Akiva Weinberger

@Adeek What?

They're not spanning $\Bbb Q$

They're not even in $\Bbb Q$

Adeek

15:12

oh I see I understand the question now

so they are just $\mathbb{Q}$ indepedent set

Akiva Weinberger

That's what I said, yes

user21820

@AkivaWeinberger No we cannot write down an explicit set. It uses the axiom of choice.

Akiva Weinberger

Whoa. From another answer:

> The square roots of the prime numbers are linearly independent over $\mathbb Q$. (Proof: this is immediate given the ability to extend the function "number of powers of $p$ dividing $x$" from the rational numbers to algebraic numbers. $\sqrt{p}$ is "divisible by $p^{1/2}$" while any finite linear combination of square roots of other primes is divisible by an integer power of $p$, i.e., is contained in an extension of $\mathbb Q$ unramified at $p$).

@user21820 Not a complete basis

Just an uncountable linearly independent set

user21820

Yes but you asked for an uncountable set.

Akiva Weinberger

So? It need not be a basis

user21820

15:13

The square roots of primes hardly makes an uncountable set!

Adeek

yeah ofcourse

Akiva Weinberger

Re above thing I quoted: I never saw that before

I don't know how to fill in the gaps but it looks cool

@user21820 Yeah that was the start of an unrelated thought, sorry

Adeek

also ln(pi) is not uncountable

Akiva Weinberger

^Hence the question

user21820

@AkivaWeinberger See citeseerx.ist.psu.edu/viewdoc/….

I think searching Math SE will lead you to some of Asaf's answers about how taking away the axiom of choice prevents you from being able to prove certain things, and I believe it includes the existence of an uncountable Q-independent set of reals.

Of course, you don't need the full axiom of choice. You just need a well-ordering of the reals.

amWhy

15:20

@user21820 Hello! Are you a regular here on this chat? I used to be.

user21820

So that you can go through the reals one by one and add it if it's Q-independent from the previously added ones.

@amWhy: No actually I've rarely come here. =)

I was actually wondering whether or not to ask people here for help with our usual 'chores'. =D

In the end, I saw some interesting conversations here and...

hemant _

Hi there ,I am interested in competitive programming,and it involves a lot of mathematics although I am good at maths yet sometimes questions are a bit high than I am capable of doing.Any general advice to me .Thanks

godskook

@hemant_ have you tried the Euler problems? I enjoyed those.

projecteuler.net

amWhy

@user21820 One place where such "chores" are handled nicely is at the C.R.U.D.E. chatroom. Anyway, there is a lot happening here; I could probably spend all my free time in various chat rooms, even on non.mse. chats!

hemant _

@godskook Yeah but I am a begineer when I start to attempt I possibly can't after 1 or 2 :D

godskook

15:25

@hemant_ projecteuler.net/problem=1 You have a problem with #1?

hemant _

@godskook Yes, some are simple but How to master it

godskook

@hemant_ do it?

@hemant_ that's the best way.

How many have you done?

amWhy

@skullpatrol You 'round these parts?

hemant _

@godskook I started a month ago and tried around 10 to 15 and then I quit as their level reached high

amWhy

@anon Is that still "you" (anon) that I know from way back when?

godskook

15:28

@hemant_ Ok. My advice would be to not quit. If you can do the first 10, you can do at least ~30 with some effort.

user21820

@amWhy I've tried there quite a few times in the past. It never worked.

hemant _

@godskook thanks for motivating me again :)

amWhy

@user21820 Yeah... sort of for me too. But I think it simply needs some new "faces" (users) joining in regularly.

godskook

@hemant_ chat.stackexchange.com/transcript/11?m=38490976#38490976

"Just do it" is the most important thing you could possibly learn. So just go do it :P

hemant _

@godskook Yeah ! Thanks.

Secret

15:34

Why I am more mathematician but not physicists: I rely on the models too much on telling me what the physics are

in The h Bar, 7 mins ago, by Secret

@EmilioPisanty @ACuriousMind ...actually, I think I had all my physics thinking backwards. In general, I have this fixated conception that you can not only use experiments to tell you what model to construct, but you also can start with you model and figuring out what physics it corresponds to by performing known checks, providing the maths obey some known properties that the physical things are supposed to obey

in The h Bar, 5 mins ago, by Secret

Or in short, I thought the model contains all the information that can recover the physics of what each physical based mathematical object (e.g. self adjoint operators) will mean

user21820

@amWhy So who's in? Although I like it better the way we've been doing. It's more fun. =)

Secret

I noticed recently there are many non regulars in this chat in the past 3 weeks. I wonder if that is a sign of something big happening

thought regardless, I interact wit anyone in exactly the same manner, thus it should not be a problem

amWhy

@user21820 I think we've been doing great, myself. I left a comment at the CRUDE, just noting that it seems only two users that maintain it, rather conservatively.

@Secret I used to be a very regular user in this chat. So you can consider me an "alum". I started exploring other sites at a point when the main chat (this chat), mostly derailed from the basis of math; It is absolutely good to have a mix of math and socializing. But when chat users start attacking each other, etc., (as happened when I left), in can get ugly.

Secret

I see, in that case, based on what the chat is like in the past 5 months, I think you will quite enjoy the return. Maths is still the focus of this chat, but ocassionally, other socialising topics and other sciences such as chemistry and physics are common, especially one of the current regulars is a physicist, I am a chemistry PhD, some people sometimes ask chemsitry here and we also have some philosophers

Maths topic itself is not as focused on topology, and more balanced between a range of things

Currently, there is quite a lot of number theory as couple of users found interesting number theoric stuff and posted some MSE about them, they can be quite algebraic

Ted and tobias are stil around and tend to appear during the peak hours, balarka and topology people also appear regularly. The number theory people can appear throughout the day in more or less constant manner

Transcript for

$Mathematics$ Mathematics