/sci/ - Science & Math

>>12454230

Because it is.

The goal of analysis isn’t to somehow find a perfect pair of straight line segments, the shorter representing a unit, and the greater perfectly measuring [math] \pi [/math], and to put these segments in a glass case for engineers to calibrate their tools.

Perfect calibration can’t practically exist.

The goal is to enable any independent scientist to define their tolerance for error, and to calibrate a version of [math] \pi [/math] that works for them.

In the vast majority of scenarios, you are modeling a non-chaotic system, so coarse representation of your numbers will get you a coarse representation of your result.

>>12454915
>π
Doesn't exist. Take your meds.

>>12454916
Cope and wrong

>>12454916
What is a circumference of a circle with unit diameter?

>>12454918
Holy cope!
>>12454961
That's a meaningless question.

>>12454230
You can draw sqrt(x) in R^2 but not in R.

Therefore, is sqrt(x) a real number?

>>12454230
When numeric density reaches one h-state is achieved and continuum ceases within an infinite range.
finale of infinitism is finitism of infinite.
the count limit is achieved and the set arbitrarily ceases.
hitomi's number limits count, thus is is grand and secure.
also reals don't exist, literally retarded.

>>12455077
yeah I'm thinking based

>>12455077
>That's a meaningless question.
Apparently it's so meaningless you autists won't stop sperging out about it.

>>12454915
>The goal is to enable any independent scientist
I don't think post 1880 investigations relating to the substantive definitions of the set of real numbers has done much for that.
The real numbers as Dedekind cuts have little to do with analysis - at best you just use the completness property

>>12455344
>I don't think post 1880 investigations relating to the substantive definitions of the set of real numbers has done much for that.
We have decidability of the theory of real closed fields (Tarski-Seidenberg), that should count for something at least.

guys what did I do wrong? I was trying to prove pi.

>>12455077
>That's a meaningless question.
Horrifyingly based

Bump. Mathematicians must not get away with this one

>>12457893
>you can't do it on a computer
nobody cares

>>12457912
computer meaning anyone who computes including humans
you ought to care

>>12457893
>>12454230
Stop pretending like you have a problem with just the real numbers. You are being completely disingenuous and just waste time. Admit that you have a problem with set theory so everyone with more than half a brain can dismiss you schizos immediately.

>>12457915
But we don't, and your schizo ramblings won't change anything.

>>12457916
Yes, set theory is a disaster. And of course it is, being conceived as a foundation for analysis...

>>12457915
But I can do it.

1 + 1/2 + 1/4 + 1/8 +... = 2

Don't need to actually compute each operation, it's equivalent.

>>12457934
Are you an engineer? Why are you satisfied with such logically flimsy calculations?

>>12457952
It's not flimsy at all. For any n, 1+1/2+...+1/2^n<2, and for any epsilon>0, we can find an N such that 1+1/2+...+1/2^N>1-epsilon.

>>12457952
>for every eps>0 there exists N such that n>N implies |(2^n-1)/2^n - 2| < eps
how flimsy

>>12457952
If I were an engineer, I wouldn't even take it to infinity, just to the tolerance bound.

But, since I do take it to infinity, it is equal to 2.

>>12457952
1 = 1/2 + 1/4 + 1/8 + ...
just try it
1 = 1/2 + 1/2
= 1/2 + 1/4 + 1/4
= 1/2 + 1/4 + 1/8 + 1/8
and you can keep reducing the right term until 1/h = 0

>>12457965
And let's see how you take it to infinity: the ELLIPSIS, the favorite tool of the analyst. Of course. Can't come up with a sound justification for what you're doing? Just stick an ellipsis!

>>12457960
Yow!

>>12457999
Wasted trips of retardation. The quantors "for all" and "there exist" provide a mechanism for this. But again, your problem has nothing to do with this. Stop pretending like you care about or know anything about math. Your problem is entirely to do with set theory, a subject which you have only studied at the very surface level if at all.

>>12458012
Yeah? What about it?

How do you feel about the fact that this series makes a perfect three-point landing at exactly zero after a completed infinity of terms have been added together?
And that each individual term is itself made up of two unrelated and completed infinite additions added together?

{8/(3π)-e, 8/(35π)+e, 8/(99π)-e/2, 8/(195π)+e/6, 8/(323π)-e/24...} = 0

Is that shit cool, or what?

>>12458012
Just to make it sure you get it: I know set theory quite well. You think you've come up with some a-ha discovery here
>A-ha! This guy thinks he has problems with the reals, but little does this idiot know his problems ackshually lie in SET THEORY!
Well, the reals predate set theory, which was conceived as a FOUNDATION TO ANALYSIS!! "Rigorous" definitions of the reals are based on it (there were none before set theory).

>>12458039
I'm not thinking it's some "aha" discovery.
I just want you to stop being disingenuous and wasting time talking about how you don't accept the reals, when the reality is that you just don't accept set theory.
All you do by saying that you don't accept the reals is throw in intermediate steps in the discussion until you reach the point where you just plug your ears to any reasonable explanation because you don't accept anything beyond strokes on a whiteboard as real.

>>12454230
What are your thoughts on the infinite diagonal?

>>12454230
>>12457893
>>12457999
>>12458021
>>12458039
if you already collecting the notes , how about some more rigor ? or money ? if you collect all the notes from all the lectures please drop them. i want to study them. or if you go to wild egg and buy the lecture notes. drop me an eth address. i want to chip in and get the notes.

>>12458071
>infinite diagonal
What's that?

>>12458021
I heard about an infinite hotel. idk
Even if you guys need to compute your stuff, we use formally defined concepts to prove that we can approximate to arbitrary precision everything in analysis. Maybe imagine an idealized universe where you do the computation infinitely may you make understand the definition of the limit.

>>12455077
>>12454916
based schizo

>>12454230
What problem? Dont see any.

>>12457893
What logical problems are there for rational numbers?

Daily reminder that Zeno's dichotomy paradox kills the utility of the analytical continuum as a physical model (Σ 1/2^n converges to 1 but never reaches it)

It's only a question of whether the analytical continuum deserves a place in an axiomatic framework divorced from physical reality. Much of math is divorced from physical reality, so the divorce isn't a deal-breaker per se.

>>12458039
Sets are just (0,0)-categories, what are you on about.
Diagonal arguments work in any cartesian closed category.
Whats more, even in topology (which is not a cartesian closed category) there are notion of fixed points, and that is closely related to continuity.
Sets have nothing to do with it. Its an primordial instrument, sure, but we got same things in other mathematical models.

>>12459654
>(Σ 1/2^n converges to 1 but never reaches it)
it reaches 1 at ∞

>>12459654
>Much of math is divorced from physical reality

>>12459639
For the `infinite set' of `all' rational numbers? Do you need to ask?

>>12459702
Yeah, what is the problem?

>>12459537
The diagonal of a (vertical) list of all infinite two-element (horizontal) strings. Say the two elements are m and w. Inverting every element in the diagonal (m->w, w->m) purports to show that there is always another infinite string which fails to be listed in the list of all infinite strings. And that as such, some infinite sets cannot be counted by the natural numbers.

>>12459657
>Diagonal arguments work in any cartesian closed category.
>>12459776
Here is some abstract nonsense for you
https://bartoszmilewski.com/2019/11/06/fixed-points-and-diagonal-arguments/

>>12459809
I was asking for your thoughts. Are you this blogger?

>>12459820
This is an anonymous board.
Are you this blogger?

>>12459660
Can't decide whether joking or not.

>>12459776
Yeah, that is nonsense. As are `infinite sets' which allegedly can be indexed with the natural numbers. The problem is having `infinite sets' in the first place.

>>12459839
So you deny the natural numbers altogether then.

>>12455428
You just need to come up with a constant to add to that to make it equal to pi, then it will be true science. You can call it Moot's Number.

>>12459833
You're the one who replied with a blog to my question asking for an opinion, you stupid fuck. Your mom dies in her sleep tonight asshole.

>>12459839
Okay so you like neither countable nor uncountable infinite sets.

> /sci/ attempts to debate Norm

>>12459913
The real redpill is that arbitrarily large finite sets are logically inconsistent.

>>12459965
What is logically inconsistent about arbitrarily large finite sets? (other than, perhaps, arbitrarily recursive power setting)

>>12459639
There is nothing wrong with rational numbers, just like with natural numbers. The problems come in when you attempt to complete the infinite and start talking about the "set of rational numbers", whatever that means, and its subsets.
You can't write down infinitely many things, so you're forced to consider the most obvious question of what exactly is meant by an infinite subset.
One attempt at an answer is that an infinite subset is represented by some description or an algorithm, and that's what we're talking about when we're considering these infinite sets. This is deeply problematic at the surface because it could be that two different descriptions or algorithms can be proved to be equivalent, so the objects would not be descriptions but rather some kind of equivalence classes of descriptions. But doesn't work either, because it is self-referential: you're using the completed infinite to describe what you mean by completed infinities. You need a prior description of an "equivalence class" which is as problematic as the problem you're using these equivalence classes to solve.
I agree with Wildberger with saying that the best way forward is to consider pi and e not as actual numbers but meta numbers, special objects not part of some completed infinite sets in which they look like any other object.
Another way is to consider, as Wildberger (recalling Wittgenstein) says, "sets" given by infinite choice, not requiring any algorithm or description. But there are no examples of such sequence! By allowing such 'sequences' to enter our mathematics we're necessarily moving to the realm of fairy tales and theology. This forces us to drop any conception of what we're actually talking about and resort to axiomatics (which we don't even know if they're consistent). A game of shuffling arbitrarily chosen strings of symbols that don't refer to anything, don't mean anything and can't be usefully implemented in any way does not deserve to be called mathematics.

>>12459908
>opinion
>math
yeah, sure.

>>12459654
>(Σ 1/2^n converges to 1 but never reaches it)
The sum is not an infinite process in time. Zeno merely divided a physical distance into infinite parts which changes nothing about the movement across it occurring in finite time.

>>12454230
>seriously in logical error
>>12457893
>serious logical problems
>>12457999
>not properly defined
>>12458039
>contradictions
Weird how none of these errors and contradictions ever get explicitly stated, he just strings along his viewers like a con man.

>>12460011
>you need to be
>18 to post here
go fuck yourself loser

>>12459982
>infinite subset is represented by some description or an algorithm
>deeply problematic
> it could be that two different descriptions or algorithms can be proved to be equivalent
>it is self-referential
>infinite choice, not requiring any algorithm or description
>there are no examples of such sequence

It seems like you are entangled in your own shoelaces.

I can write down a symbol of infinitely many things.
Observe, as I describe Natural Numbers with a picture. It comes in a finite amount of bytes, and you can even see the actual number of them near the picture.

But how to give that symbol a meaning? What is a semantic for that picture?

One way I could do that is by recognising Peano construction in this pictorial form.
Other way I could do that is by defining a datatype with accociated mappings.
There are other ways as well.

So then we are inquiring "are those semantics related somehow?"
We associate them with a same symbol, so the relation is evident.
Maybe we can even describe some semantics in terms of the other semantics, and vice versa.

What is the meaning of 2? I can give meaning to that symbol by different symbols, like 1+1, 0+2, 2+0. Somehow they are related.
We can use symbols as meanings for symbols.

What to make of it?
My haskell program with a definition of all natural numbers will compile, and even will produce some result in the finite amount of time.
There it is, in all its glory:
x = [0..] :: [Integer]
Somehow the sequence of instructions for the processor is still finite, and yet there is a symbol for every natural number.
Strange, don't you think? Probably just an elaborate hoax.

>>12460054
there is the pic

>>12460034
So, will you respond to the diagonal argument, or will you just implode?
Right now it seems that you are angry about totat annigilation of your belief system. Man up.

>>12460071
I know the diagonal argument, schizo. I was curious about OP's take on it, not the canonical.

>>12460085
Oh, 4chan doesn't have an indicator for posts of OP author. What a pity. Certainly a source of confusion.

>>12460054
>Haskell
Prelude> data Nat = Zero | Succ Nat
Prelude> n = Succ n
Prelude> :t n
n :: Nat
Prelude> print "LOL"
"LOL"

>>12454230
>representing the continuum
You can't "represent the continuum" with numbers, that's nonsense. Even in standard model, there's always more continuum in between every represented number.

>>12459838
why would I be joking?

>>12460028
I dropped this guy after he said there was no way to express the length of the hypotenuse of a right triangle with arm lengths 1.
>no no no, you need to use my original metric, Quadrance, not this flawed concept of "length"
fucking idiocy

>>12460054
>>12460058
>>12460153
Bump

>invent math
>use math to invent a very specific tool to help us with math
>disregard all math which isn't compatible with this tool
is finitism a conspiracy of IT corporations? is it just a massive cope?

>>12461384
If you really think we reject real numbers just because computers can't handle them then you're a fucking retard.

>>12460956
That pic is so good lmao

>>12461391
That's literally what every argument you have boils down to. If you have any actual arguments, then feel free to write them down.

>>12461391
it seems to be your favorite argument tho

>>12461403
>>12461406
Read the thread.

>>12461417
Yep, I did. Not a single argument that doesn't rest on
>Muh computer can't do it.

>>12461419
Read better.

>>12454230
I have said it before, I will say it again.

PRAISE THE HOLY BURGER!

Amen.

>>12460956
Listen here, you GOD CURSED SODOMITE, your meme is stupid and illogical, just like your mother when she walked down that dark alley in Detroit whacked out of her mind on the low quality crack her pimp sold her.

>>12460054
Wildberger is perfectly fine with things going on in an unending fashion.
As an example, he takes a CS approach for a function's inputs and outputs by using types (Nat, Int, Frac (Q+), Rat).
He is also okay with some infinite sequences (on going sequences in his lingo). As an example, [n> = 1, 2, 3, ... (a_n = n in regular math), the sequence of natural numbers is perfectly okay.
However, what is not okay is considering [1, 2, 3, ...] (the entire list of numbers from that sequence). Or Nat = N (the set of all natural numbers).
So in other words, there is a finite formula which can produce any finite amount of information that you want it to, so basically this:
>even will produce some result in the finite amount of time.

Now your computer program will never produce all integers because it can never do an infinite amount of work. You can never do an infinite amount of work either, so does it make sense for you to even imagine that you can (even in a platonic sense)? Here is where it gets philosophical. Wildberger says no, but most mathematicians say yes. Fundamentally, this comes from the axiom of infinity, which is meta-mathematical. As I've told many other people, if someone rejects the axiom of infinity, it will likely limit the math one can do, but that does not mean their math is incorrect.

>Somehow the sequence of instructions for the processor is still finite, and yet there is a symbol for every natural number.
No there is not a symbol for every natural number lmao. You have given a formula for producing as many natural numbers as you want. You haven't actually done the work of producing all natural numbers.
>Strange, don't you think? Probably just an elaborate hoax.
No, I think it's just because you fail to understand either the difference between a formula and actually doing the work to compute that formula, or what Wildberger's views on infinity actually are.

>>12461825
if sequences are allowed, are limits are allowed?

>>12461854
Yes, he talks about limits of rational polyfractions in his videos.

>>12461856
can sqrt(2) be defined as a limit or not?

>>12461872
What do you mean by sqrt(2)?

>>12461881
what do you mean by "what do you mean?"

>>12461889
You obviously know what the expression means as you've just used it yourself. Stop being silly.

>>12461903
let's stop being silly, I agree. so, can sqrt(2) be defined as a limit or not?

>>12461907
>, can sqrt(2) be defined as a limit or not?
Yes, sqrt(2) can be defined as the limit of the constant sequence a_n= (0,1) in the ring of pairs (a,b) with a, b rational and
(a,b)+(c,d)=(a+c, b+d)
(a,b)*(c,d)=(ac+2bd, ad+bc)

>>12461923
can I have the sequence a_n = n-th digit of sqrt(2) in decimal?

>>12461926
> n-th digit of sqrt(2)
What do you mean by this? Give a proper definition.

>>12461928
we back at being silly?

>>12461928
Not that anon, but: the last digit of the largest fraction q = z / 10^n for some integer z such that q * q <= 2.

>>12461872
Limit in what? None of the constructions of real numbers actually work, so if there is a limit it has to be in some other system. What system is it?
You can provide a description for the ongoing sequence of rational numbers whose squares get closer and closer to 2 using an algorithm. However, this will not be able to fit in a number system because if you view numbers as some kinds of sequences given by algorithms then there is no way (and not just using a computer, as some ignorant people criticizing Wildberger have suggested) to tell when two different algorithms will agree on the outputs, which means the best your definition can do is define numbers as algorithms which might actually give the same sequence which is probably not what you want.
If you have any logically sound theory of numbers which allows for sqrt(2) to be defined in some kind of analytic way I'd be really interested in hearing it, however don't get too excited. Many people have tried doing it such as Cauchy, Dedekind, and none of them have succeeded.

>>12461977
>>12461972
here is a proper response
>>12461987

>>12461987
>You can provide a description for the ongoing sequence of rational numbers whose squares get closer and closer to 2 using an algorithm.
can I define sqrt(2) to be this sequence?

>>12461987
>which means the best your definition can do is define numbers as algorithms which might actually give the same sequence which is probably not what you want
Why is it not what I want?

>If you have any logically sound theory of numbers which allows for sqrt(2) to be defined in some kind of analytic way
Cauchy sequences and Dedekind cuts are perfectly logically sound systems. If they don't have some property you would like them to have, please explain what that property is, because "soundness" isn't it.

>>12461998
You can define sqrt(2) to be whatever you want. You can even define it as your mom. The challenge is constructing a logically sound theory of many such numbers that allows you to do arithmetic and roughly corresponds to the thing mathematicians like to call "real numbers". That's the million dollar question, which I'm not sure we're ever going to see an answer to.
>>12462000
>Why is it not what I want?
Because you probably want to identify numbers whose sequences have the same terms. The problem is that there is no way to do this when you're dealing with algorithms.
>Cauchy sequences and Dedekind cuts are perfectly logically sound systems
Have you not watched his videos or actually delved down into the definitions yourself? They're extremely logically flawed. Take, for instance, the notion of a Cauchy sequence
https://www.youtube.com/watch?v=3cI7sFr707s
> If they don't have some property you would like them to have, please explain what that property is, because "soundness" isn't it.
Yes I would like to be able to write them down and do arithmetic with them, like I can do with the rationals or naturals.

>>12462055
>Because you probably want to identify numbers whose sequences have the same terms.
Not particularly. Real numbers are not for doing arithmetic with.

>Have you not watched his videos or actually delved down into the definitions yourself?
I have.

>Yes I would like to be able to write them down and do arithmetic with them, like I can do with the rationals or naturals.
Oh. Yeah, then you'll want to find yourself a different number system, as real numbers are very limited for that. But that doesn't mean there is anything wrong with the real numbers -- it just means they are not useful for your application.

>>12462211
>Real numbers are not for doing arithmetic with.
No? I thought you considered the reals an arithmetical object whose members you can add and multiply. If not, we're actually much closer in out positions that I previously thought.