Primes and Squares

El'endia Starman

03:26

@flawr That was a pretty sweet proof. Also, lol at her answer to the green screen question.

Feeds

04:35

Nathan Merrill has made a change to the feeds posted into this room

The Mathematics of Quantum Computers | Infinite Series

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Splitting Rent with Triangles | Infinite Series

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Infinite Chess | Infinite Series

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5 Unusual Proofs | Infinite Series

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Proving Pick's Theorem | Infinite Series

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flawr

07:43

@El'endiaStarman I agree, the only nitpick I have: She did not show that you can triangulate every such shape by "small triangles".

@StackExchange tempted to flag as spam XD

Strangely in any lecture that I had so far where we used triangulations of some surface, nobody ever proved the existence of a triangulation.

It is probably just too cumbersome to do:)

Martin Ender

10:40

@NathanMerrill 1. yes. 2. by presenting a useful error message and requiring an explicit specification of the interface to be used (by cast or similar). 3. see 1. 4. see 2. 5. in that case you could consider going with the more specific overload.

Nathan Merrill

13:16

@MartinEnder 2. At compile time or run time? 4 Same question?

Martin Ender

@NathanMerrill compile time

Nathan Merrill

14:03

then, for #3, wouldn't it be a compile time error to even write func(val: A){} and func<T>(val: T)?

because the first function would never be callable

Martin Ender

oh right, for that one I had assumed you were intending to interpret that as specialisation anyway

Nathan Merrill

I mean, the other side is that you can determine all available methods at compile time, ensure that there will never be two methods with the same priority at compile time, and then at run-time, choose the method to call

so, for the above example, if you had a T that happened to also implement A, then it would call the A function

because T doesn't actually declare any interfaces, so it has a lower priority than A

Madhuchhanda Mandal

14:25

Can I discuss a math problem?

Nathan Merrill

absolutely

Madhuchhanda Mandal

Ok

We know that a line is a collection of points arranged in a specific direction. Right?

Anyone?

Martin Ender

I guess that's a way to look at it? Just go ahead. :)

Madhuchhanda Mandal

Ok. Now number of points on the line must be proportional to its length

Lets say the constant of proportionality = k

So No of points on a line , P= k L , where L is length of the line

Right?

Martin Ender

What kind of points are we talking about here? Real numbers? If so, then I don't arguments of proportionality apply. Or do you just mean integers along the line? If so, then yes.

Madhuchhanda Mandal

14:40

Ok. I want to know why arguments of proportionality don't apply

Martin Ender

there's an infinite number of points on either line, and it's the same "kind of infinity"

would you say that there are twice as many integers than there are even integers?

Madhuchhanda Mandal

Won't we?

Rather if we limit ourselves to a finite portion of the integer set

?

Martin Ender

of course, if we're looking at a finite set then we get twice as many integers.

but not if we look at all of them.

Madhuchhanda Mandal

But why?

Martin Ender

because there's an obvious one-to-one mapping between the integers and the even integers: if I map each integer N to the integer 2N, then I've assigned each integer to a different even integer. so there can't be more integers than even integers.

you might want to read up on the cardinality of infinite sets and Hilbert's hotel (there's a good youtube video somewhere)

oh, there are two actually:

The Infinite Hotel Paradox - Jeff Dekofsky

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Hilbert's Infinite Hotel - 60-Second Adventures in Thought (4/6)

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Madhuchhanda Mandal

14:45

Isn't it cobclusive that there must be twice integers as to even numbers in any integer set, ofcouse we don't know the exact number.

Ok. I am reading the above messages. Wait.

Nathan Merrill

infinity is weird. Trying to say that 2*infinity is greater than infinity doesn't really work

Madhuchhanda Mandal

(By mapping you mean E=2*K ; where E is the image and K is the integer. Right?)

Martin Ender

yeah, an injective function if that helps

Madhuchhanda Mandal

Ok.. I was thinking that a Line segment of length 1m must contain exactly half the number of points as to a line segment of length 2m

Wait... Let me think.

Martin Ender

not really. you can still find an injective function that maps each point in the first line to exactly one point in the second line, without leaving out any points in the longer line

Madhuchhanda Mandal

14:52

Ok.. See

You agree that in a finite set of integers, the proportionality holds. Right?

Martin Ender

yes

because there I can't find such an injective function

Madhuchhanda Mandal

Lets consider a infinitesimal length., dl. You agree that it has finite number of points. Right?

Nathan Merrill

is that a thing?

Madhuchhanda Mandal

@NathanMerrill "is that a thing?" ?

Nathan Merrill

like, there's no number next to 0

Madhuchhanda Mandal

14:56

Yes.. It ever tends to 0. Hence it must have finite points.

Nathan Merrill

there's no such thing as 0.00000000....0001

so, by "infinitesimal length" what do you mean?

Madhuchhanda Mandal

*infinitely small

Nathan Merrill

simply a really small length?

infinitely small is 0

Madhuchhanda Mandal

Yes

Martin Ender

if it's infinitely small, then it contains no points (or one point)

there is no length that contains exactly two points

Madhuchhanda Mandal

14:58

It must have undetermined number, yet finite points. Isn't it?

We are never interested in that number of points it contains

Martin Ender

no, if it has more than one point, it has an infinite number of points

Madhuchhanda Mandal

Why?

Martin Ender

because if it contains points X and Y, it also contains (X+Y)/2

Madhuchhanda Mandal

It must not have one point. For sure.

Nathan Merrill

if there is a line of length 0, then it is simply a point

Madhuchhanda Mandal

15:01

And that would be exactly coincident with x and y

Won't it?

Martin Ender

if your line only contains a single point, it contains exactly one point (not an unknown finite number of points, just one). if contains at least two points, it contains an infinite number of points, because you can always add the mean of two points to the set.

there is no way to have anything in between

(not with the usual definitions of real numbers and lines anyway)

Madhuchhanda Mandal

There is a way of having. There must be a.

Because at such dl length, like that infinity, arithmetics don't apply

Same as infinite+infinite=2infinite

I think like that dl+dl=2dl also wont work

Nathan Merrill

2*infinity = infinity

you should really watch the above videos

Madhuchhanda Mandal

Yes. Similarly 2*dl=dp

*dl

Isn't it?

Nathan Merrill

I still don't think that an infinitely small line is a thing.

its just a point.

Madhuchhanda Mandal

15:06

@NathanMerrill but then you would be violating concept of differentials

(Although I never meant inf+inf=2*inf, I should have been more clear with my expression in that statement)

Martin Ender

@MadhuchhandaMandal what concept of differentials?

Madhuchhanda Mandal

That integrating (dl) you would get length l

Would it hold had dl been a point?

Martin Ender

infinitesimals are not actual objects you can treat like that. they are limits.

in particular, an infinitesimal length dl is the limit of ever smaller finite lengths delta-l. but each of those lengths that tend towards the limit have an infinite number of points in them but an ever smaller length that tends to zero.

Madhuchhanda Mandal

Yes that's what I am saying. Won't it contain limited no of points?

Hmm... I understand.

But wgazt

Martin Ender

actually I think what I said isn't even quite right yet. because for infinitesimals you only take the limit after performing arithmetic on your delta-l.

which is why you can't treat them like any other finite length.

Madhuchhanda Mandal

15:14

*But what is causing that contraction in length?

Martin Ender

this is what gives them their special properties but it also means you can't apply the usual reasoning most of the time.

@MadhuchhandaMandal it's just a matter of definition.

Nathan Merrill

no. We talk about approaching a length of 0, and despite the fact that the length of the line gets smaller, the number of points between those two endpoints are the same

Madhuchhanda Mandal

I mean why the length is shortening?

Nathan Merrill

because the endpoints are getting closer

Madhuchhanda Mandal

Why?

Nathan Merrill

15:16

because that's how we define a line

you have two end points

and the distance between them is the line length

consider the coastline paradox:

Coastline paradox

The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. The first recorded observation of this phenomenon was by Lewis Fry Richardson and it was expanded by Benoit Mandelbrot. More concretely, the length of the coastline depends on the method used to measure it. Since a landmass has features at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be measured around...

depending how you measure the coastline, you can get lengths that are absolutely massive

What Is The Coastline Paradox?

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Madhuchhanda Mandal

I kinda understand

But now I don't understand what's dl

I used to think of it like this : it's an ever decreasing length (which one can never achieve), having exactly undetermined number of finite points

Martin Ender

differentials/infinitesimals are very subtle concepts. the way they are defined, they are just a shorthand notation for inserting a value delta-x, and then after performing some arithmetic (usually a differentiation for derivatives or a sum for integrals) we take the limit of delta-x going to zero.

Madhuchhanda Mandal

But wait... how would you get that x and y in dl?

Martin Ender

my point is, dl itself is not a real "thing"

it's a notation

Madhuchhanda Mandal

I also don't claim it to be real

It's notation of course

Martin Ender

15:25

you still seem to think of it as an object that you can just work with like a finite (or zero) length

Madhuchhanda Mandal

No I never

That's why I thought that it may contain finite points

No I never

Because def. of line don't hold good there

But its also right that if it has finite points, then def of point must hold good and hence it must contain infinite points

Martin Ender

when you write df/dx, there is no infinitely short line dx involved. this is a convenient notation for lim(delta-x -> 0) (f(x + delta-x) - f(x)) / delta-x ... in this equation there is no infinitely short line. there is a finite line delta-x (which contains infinitely many points), and we're looking at a sequence of such expressions with ever-shortening delta-x.

an infinitely short line doesn't really enter into it. that's a simplification that works as a first explanation for infinitesimals, but it doesn't really hold up to any formal properties, because that's not how it's defined.

Madhuchhanda Mandal

@MartinEnder well, I never meant that dx to be infinite small length. I was considering a infinitesimal length on our line of length l , to be dl

That dx is a infinitesimal change in x I suppose

But now I conclude something like this : As point is something like infinity packed into finite (line) , hence though it may seem to be proportional , proportionality don't hold. Right?

But wait. @NathanMerrill if dl is a point, then proportionality must hold.

Hence dl can't be a point

It must be a infinitesimal lengthed line with infinite points

Is anyone there?

Martin Ender

15:43

my point is that dx (or dl) is neither a point nor a line. it's an entirely different concept, that you can in some contexts treat like a very short line but in practice it obeys completely different rules.

I'm not even sure whether the cardinality of it is well-defined, but if I had to guess then I'd say, yes it contains infinitely many points and has length zero.

Madhuchhanda Mandal

Length tending to 0

It must obey rules of line... Otherwise you can't claim it to be composed of infinite points

Rather I think concept of point is not well defined

Martin Ender

@MadhuchhandaMandal I'm pretty sure it is :)

Madhuchhanda Mandal

*concept of point in the context of line

(i.e line is composed of points)

Martin Ender

a line is really a set of points. a (degenerate) line of length zero contains exactly one point. it's like the difference between x and {x}.

Madhuchhanda Mandal

If that degenerate case of line can be considered to be line at all

Because def. of line claims that it must have two endpoints

Martin Ender

15:56

that's right, you can define your line segments to require two different endpoints

Madhuchhanda Mandal

But that again abstracts the concept of point

Martin Ender

hm?

Madhuchhanda Mandal

Because if line is collection of point then at some length it must have finite collection but we just proved that such length is not possible

Martin Ender

"at some point it must have finite collection" <-- I still don't understand how you arrive at that conclusion

Madhuchhanda Mandal

Because removing elements from a collection , we must achieve another collection, and at some point we are bound to get a finite lengthed collection. Aren't we?

But we just proved that such collection don't exist

Martin Ender

16:03

@MadhuchhandaMandal no, why would we? start from the natural numbers. repeatedly remove the smallest integer from the set. you'll never end up with a finite collection. there are always infinitely many values left.

there is no smooth transition from finite to infinite

Madhuchhanda Mandal

Ok so tell : What would happen if we remove 50 points from a collection

?

Does the collection remains same?

Martin Ender

from an infinite collection? we still have an infinite collection.

no it's a different collection, but still an infinite one.

with the same cardinality.

Madhuchhanda Mandal

Has the line changed?

Has it's length changed?

Martin Ender

I guess if you remove exactly 50 points, then you won't have a line any longer, because it's impossible to remove them "from one end"

you'd have to remove 50 individiual points which would break up the line into multiple segments.

Madhuchhanda Mandal

That's exactly what making me conclude a line can't be collection of points

Martin Ender

16:08

why not?

Madhuchhanda Mandal

Is a collection not contain what it is composed of?

Martin Ender

of course it is. the problem isn't that it doesn't contain those 50 points. the problem is that there are no 50 adjacent points. what are the 50 smallest numbers greater than 1? the question doesn't make sense because there is no smallest number greater than one. whenever you talk about a contiguous range on the real line, you've immediately got infinitely many points.

Madhuchhanda Mandal

Lets insert a,b,c... To a collection. Don't you agree the collection contains a,b,c

Martin Ender

if you talk about a finite number of points, there will always be other points between then.

you can remove 50 points from a line, but they won't be "next to each other", so this is no different from taking the points 1, 3 and 5 out of the line from 0 to 6. what you end up with is not a line, because it has gaps.

(gaps of length 0)

Madhuchhanda Mandal

That's exactly exactly what making me feel that concept of relationship between line and points is not defined.. Atleast not as a collection

I understand what you are implying

But try to think what I am saying

Martin Ender

16:12

since I still don't see how you make that leap, I think the problem is still in the understanding of various concepts of infinity and cardinality

Madhuchhanda Mandal

I think in a collection element 1 and element 2 need not have any relationship

(I don't know much, rather nothing in these concepts, but I am saying what my logic says)

Martin Ender

if by "collection" you mean set then yes that's right. but when we start to talk about lines then we are talking about special sets with geometrical meaning.

Madhuchhanda Mandal

By collection I thought you meant set only

Ok. So you say line is not a normal set of points. (I was told so from the first day of my birth..) Ok then I think there is a long way for me to go before I understand that special set.

Martin Ender

no, a line is just a set of points (although since our objects are points, clearly there's some geometrical relationship between them). however, not every set of points is a line. so if you take a line...and then you add or remove points, you get a different set. and there's no reason a priori why that new set should still be a line.

that is what I meant by "special sets": they are (normal) sets with specific properties (in this case, the property that they contain all the points between two endpoints... if a set does not have this property, it's not a line).

Madhuchhanda Mandal

If its a set, then it must have last element, last to last element (though undetermined), last to last to last element... Wont it??

Martin Ender

16:23

no

Madhuchhanda Mandal

Why?

Martin Ender

the natural numbers are a set. what's their last element?

Madhuchhanda Mandal

Natural number within a finite limit does contain last element

Martin Ender

the natural numbers with a finite limit are not an infinite set

Madhuchhanda Mandal

Natural->Real

Martin Ender

16:27

okay, let's look at the set of real numbers x with 0<x<1

what's the last element of this set?

Madhuchhanda Mandal

Real set of numbers within finite limit does contain last element

Martin Ender

no it doesn't

Madhuchhanda Mandal

[0,1] first element 0, last element 1

?

Martin Ender

okay, fine if you include the endpoints

then what's the second-to-last element?

Madhuchhanda Mandal

And as a line is between two fixed points so it would be (0,1) not [0,1]

Sorry

[0,1] not (0,1)

Ok dinner time... Back within 10 minutes

Martin Ender

16:37

I'm heading out. might be able to talk again later, but I really recommend you try reading up on infinity, cardinality and the real line a bit. I'm sure there are some good introductory examples online somewhere.

Madhuchhanda Mandal

Ok I will

But it seems me to be counterintuitive to think that there cannot be any last to last element in a collection. It might be undermined. But I think it must exist.

*undetermined

By the way thank you very much for your patience and interest

Martin Ender

18:51

@MadhuchhandaMandal no it's not undetermined, it really does not exist. assume there is a largest number x < 1. now consider y = (x+1)/2. we have x < y < 1/2 which is a direct contradiction to "x is the largest number less than 1". so our initial assumption that such an element exists must be wrong.

this is really no different from saying that there is no largest number (in general).

if there was a largest number x < 1, then 1/(1-x) would be "the largest number".

flawr

19:27

@El'endiaStarman I just got a book that you might enjoy too! It is the magic of mc escher (there are various editions). It is a large collection of his prints but it also contains sketches where you can see how he developed some of his works. There is not too much text (a few anectodes and quotations).

Might make a great gift (for someone else or for youself=)

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