/sci/ - Science & Math

>>6914477
Shit was easy to follow even as a layman. I like this guy.

>>6914520
>Shit was easy to follow even as a layman. I like this guy.
He's the greatest math prof ever... and a genius. His channel is fucking goldmine!

>>6914477
Another non-argument from the Don Quixote of math. No one cares about this false dichotomy between "choice and algorithm" he's invented besides his naive followers. It's pure sophistry. What his argument is going to boil down to again and again is "infinity not real". This is childish and unmathematical and he should know better.

>>6914590
this.
His argument boils down to the assumption that defining sets by choice is bad... newsflash, it's not.

>>6914595
Why?

>>6914761
Because it produces unique results and no paradoxes.

>>6914590
>>6914595
ITT plebs who've never done any mathematics outside of classical logic.

>>6914595
Axiom of choice has always been regarded as kind of an inelegant tool.
Of course, still all the mathematicians (outside set theory) have no problem with using it and probably this is the healthy way.

It should be a rule that any mathematician who wants to believe in the reals should be forced to do all arithmetic—by hand or by computer—using their preferred construction. Then we'll see how much they think infinity is practical.

Of course they don't, because it isn't.

>integers are just pairs of naturals!
>rationals are just pairs of integers!
>reals are just infinite sets and/or sequences of rationals!
>so simple!

Thankfully we finally have computers so people can start holding people accountable for their unworkable bullshit.

>>6915262
The axiom of choice is cosidered non-constructive. It implies the law of the excluded middle and both in general just produce bad proofs. It's possible to do mathematics without the axiom of choice and the law of the excluded middle (along with double negation and other similar things) but many things have to be redefined and one has to be extremely careful about how they proceed. This is formalized with constructive logic.

Due to the way constructive logic works it has several close connections to formal languages and computability theory. In particular, something is constructible if there exists an algorithm that can compute it. This is why Wildberger always goes on about algorithms vs choice, the only problem is that he dumbs it down so much that dumb undergraduates who've never heard of constructive mathematics just think he's a nutjob.

In constructive logic one is able to create some really powerful axiomatic systems like Synthetic Differential Geometry or work in a version of analysis with an axiom stating that every function is continuous (note that your functions have to be constructible and defined <span class="math">\forall x\in\mathbb{R}[/spoiler] while taking the lack of the law of the excluded middle into account).

>>6915274
I've been studying logic and categories for some time now and all the professors I've met doing stuff in intuitionistic logic don't go "Cantor was wrong" or "real numbers are inconsistent because I can prove they're all the same".
So yeah, constructivism is an active and interesting area of mathematics, but that doesn't mean Wildberger isn't a nutjob.

>>6915317
It's a combination of two things. In intuitionist logic one typically describes the reals using an axiomatic system rather than attempting to construct them out of the naturals. Wildberger's rants only claim that the reals cannot be constructed out of the naturals. Whenever faced with an axiomatic definition of the reals he mentions that it makes no attempt to construct the real numbers and he's not interested in it.

It's not that he's a nutjob, it's that he's only focusing on one very specific issue.

Oh look, it's this crackpot again.

>(..)Thankfully we finally have computers so people can start holding people accountable for their unworkable bullshit.

>(..)This is why Wildberger always goes on about algorithms vs choice, the only problem is that he dumbs it down so much that dumb undergraduates who've never heard of constructive mathematics just think he's a nutjob.

Thanks so much. Im hugging the screen right now. The people who disagree with wildberger dont want to, or cant understand what hes talking about.

The math that comp scientist have to learn these days is a pain in the ass to use as a tool for shaping and analyzing algorithms, its just an inconsistent clusterfuck. The idea that matematicians doesnt need to care about implementation because "muh platonic realm" is insane. An idea is implemented in our brain just like an algorythm is implemented in a computer and both are following some physical laws.

>"B-but I can fly in my mind but I cant fly in reality and this proves mind > reality"
If you think like this then gtfo, you understood nothing.

>>6915338
>Wildberger's rants only claim that the reals cannot be constructed out of the naturals.
That's not all of it. He frequently tries to prove the inconsistency of infinitary mathematics through finitary methods, he's hostile to non-constructivism and still claims Cantor was wrong.

https://www.youtube.com/watch?v=XKy_VTBq0yk
>Here I start the uphill battle to convince you that talking about`infinite sets' is just that---talk, not mathematics.

>>6915342
Math was not invented just so that you could make calculators, stop being retarded.

>>6915387
There's nothing new about this. Many other prominent mathematicians in the past have held this view. Wittgenstein and Poincare come to mind.
http://plato.stanford.edu/entries/wittgenstein-mathematics/#WitLatFinCon

Here are some quotes.

> Wittgenstein's finitism, constructivism, and conception of mathematical decidability are interestingly connected at (RFM VII, §41, par. 2–5).

>> What harm is done e.g. by saying that God knows all irrational numbers? Or: that they are already there, even though we only know certain of them? Why are these pictures not harmless?

>> For one thing, they hide certain problems.— (MS 124, p. 139; March 16, 1944)

>> Suppose that people go on and on calculating the expansion of π. So God, who knows everything, knows whether they will have reached ‘777’ by the end of the world. But can his omniscience decide whether they would have reached it after the end of the world? It cannot. I want to say: Even God can determine something mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not decide for us.

>> We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is a ‘2’ at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want to say: No. (MS 124, pp. 175–176; March 23–24, 1944)

There are several sections on Cantor and infinite sets.

>Wittgenstein denies this, saying that the diagonal proof does not prove non-denumerability, for “[i]t means nothing to say: “Therefore the X numbers are not denumerable” (RFM II, §10). When the diagonal is construed as a proof of greater and lesser infinite sets it is a “puffed-up proof,” which, as Poincaré argued (1913b, 61–62), purports to prove or show more than “its means allow it” (RFM II, §21).

It's all very much in line with Wildberger.

>>6915399
Wittgenstein is just such a boss.

>A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and… he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification!
shed dat dogma bro

ITT: passionate sucking of NJW cock

>>6915399
Yeah, but Wildberger is no Wittgenstein, or Poincaré, or Errett Bishop, or Kronecker, or Brower, or Weyl...
But even Bishop and Kronecker are kinda nutty and Brower used what he thought was inconsistent in his own work.

Anyway, serious mathematicians doing research in different kinds of logic are much more open minded than Poincaré and Wittgenstein and they understand formal systems a lot better than people did 70 years ago. You just don't have to pick a side.

>>6915465
I agree with everything you've said. I was only pointing out that Wildberger is no more of a nutjob than anyone who always assumes the axiom of choice and refuses to consider different systems of logic. In other words, anons like these (or really a large majority of working mathematicians).
>>6914595
>>6914590

>>6915465
>serious mathematicians doing research in different kinds of logic are much more open minded
The mental contortions necessary to say this sentence without a shred of irony are truly staggering.

He pretty much destroys theory after theory! Genius.

>>6914477
This guy is the '10thdim' guy of Mathematics...

>>6915511
I'm not the guy you're responding to but you are very obviously wrong.

>>6915523
A more appropriate comparison would be a relativistic theorey of gravity vs quantum gravity. Both are consistent within their own systems.

>>6915523
except Wildberger is an accomplished mathematician with a great pedigree and his criticism are justified and well-grounded in logic, mathematics and philosophy.

other mathematicians, as he's shown in so many videos, just sweep the foundational issues under the rug and don't bother with them. some, who are a bit more honest, also mention that there are issues with real construction but then decide not to bring them up.

>>6915546
>accomplished
What exactly has he accomplished besides confusing undergraduates with youtube videos? Mathematicians don't sweep the construction of reals under the rug. If Wildberger actually looked for proper sources rather than trolling through general textbooks not focused on set theory or such constructions, he would find several rigorous constructions. And he knows they're rigorous. He can't find a contradiction in them. The only thing he can do is complain that they employ an axiom he doesn't like, and he acts as if this invalidates them or makes them dogmatic. That's not how mathematics works and it's very childish.

>>6915585
As I said the last time it comes up, why don't you let him know which books he should refer to.

>>6915585
>>6915585
>>6915585
winrar

>>6915602
Because it's a waste of time to converse with this guy. There are actually several satisfactory constructions on wikipedia for fucks sake. I assume he has a computer. But again, the real point of these videos is not to show that mathematicians are wrong, it's to convince his naive fans that mathematicians are "avoiding" the issue and trying to fool them. It's essentially conspiracy fearmongering, not math. This is why Wildburger is so popular.

>>6915613
>Because it's a waste of time to converse with this guy.
How convenient!

>>6915619
So convenient it must be a conspiracy to hide the truth!

>>6915622
Never attribute to malice what can be explained by stupidity.

>>6915585
>Mathematicians don't sweep the construction of reals under the rug.

yes NJW goes through dozens of books and only one or two attempt to properly define reals! Some point to other books which don't construct it either.

> If Wildberger actually looked for proper sources rather than trolling through general textbooks not focused on set theory or such constructions, he would find several rigorous constructions.

Oh, so all these textbooks that teach analysis and calculus are not proper sources? Can you list some proper sources then?

>>6915677
>yes NJW goes through dozens of books and only one or two attempt to properly define reals! Some point to other books which don't construct it either.
That's because constructing reals is not relevant to the topics the books discuss. There are several sources that explicitly focus on such constructions and they can easily be found with google. What is the point of ignoring them and going through irrelevant textbooks if Wildburger is actually trying to address the argument and not create a strawman?

>Oh, so all these textbooks that teach analysis and calculus are not proper sources? Can you list some proper sources then?
You just said they don't discuss construction of the reals, therefore they are not sources of constructions of the reals.

Here is a whole wikipedia article devoted to just this topic: http://en.wikipedia.org/wiki/Construction_of_the_real_numbers

It lists several sources as well as having good descriptions of the constructions.

>>6915692
>What is the point of ignoring them and going through irrelevant textbooks if Wildburger is actually trying to address the argument and not create a strawman?
It's not a strawman when his point is that students are indoctrinated into the reals. He knows what the usual constructions are for god's sake.

>>6915697
>It's not a strawman when his point is that students are indoctrinated into the reals. He knows what the usual constructions are for god's sake.
That's like saying students are indoctrinated into the Peano axioms. It's a meaningless and mathematically irrelevant point.

>>6915705
No it isn't, because people actually state the axioms.

>>6915692
>That's because constructing reals is not relevant to the topics the books discuss.
that's bullshit and you know it. <span class="math">\mathbb{R}[/spoiler] is not obvious and you have to be extremely careful about how you define it because operations will not work if you don't define it well. And how the fuck can you reason about <span class="math">\mathbb{R}[/spoiler] if you don't know how it's constructed?!?!

>>6915711
None of the textbooks I have used have discussed the Peano axioms.

>>6915711
People state the construction of the reals. I just gave you an article all about it. Either address the argument or show it doesn't exist. You can't have it both ways.

>>6915720
Were they about Peano arithmetic?

>>6915677
You don't need to construct it. You can go with the axiomatic method, it's the dedekind complete archimedean field, its consistency is already established in the context of ZF.
And to go through the construction of reals you gotta assume some axioms anyway, including the existence of an infinite set.

But if you really want to see it happening you can get Introduction to Set Theory by Thomas Jech.

>>6915712
The same way you reason about natural numbers without constructing it, or groups, or rings, or any kind of mathematical objects presented in an axiomatic manner.

>>6915720
Because they're incomplete and you use the standard model for the natural numbers.

bless you for posting this /sci/

>>6915712
Person A constructs it and proves it has certain properties
Person B proves that from this properties it follows that... etc. etc.

You don't need to know exactly how a car functions to drive it.

>>6915724
>People state the construction of the reals.
Let Norman know which textbook does this.

>>6915712
>that's bullshit and you know it. R is not obvious and you have to be extremely careful about how you define it because operations will not work if you don't define it well. And how the fuck can you reason about R if you don't know how it's constructed?!?!
Most undergraduate math is down without lots of constructions. Constructing the reals is not actually necessary to do math with them. It's just necessary if you want a certain level of rigor. We did it for hundreds of years.

>>6915728
They had arithmetic of natural numbers, yeah.

>>6915734
I already posted the wikipedia link with several sources here. You can tell him if you want to.

>>6915743
>They had arithmetic of natural numbers, yeah.
so, more than just the reals are brushed under the rug. What's so scary about the foundations of mathematics that we don't trust people with the deep knowledge, like some mystery school?

>>6915745
You are literally retarded.

>>6915745
kek. You're an idiot if you think we need to teach little children the Peano axioms before they can add apples. It's not only unnecessary, it's pointless. You don't explain quarks before you explain atoms. If you want more rigor than what's being taught, all you have to do is ask your professor or research it yourself. Nothing is being hidden. But you can't demand that everything be taught in the limited space of a textbook or the limited time of a class.

>>6915244
ITT a CS pleb who doesn't know what math is.

>>6915342
>this much retardation
Oh my god, your fedora is literally erect. News flash, there isn't a single mathematician that gives a fuck about babby's first computibility crisis. We do math for its purity. If you want to do CS, then quit your frivolous mooching and figure out how to make math concepts work for you. You retarded piece of shit.

>>6915762
>You're an idiot if you think we need to teach little children the Peano axioms before they can add apples.
If they don't need Peano axioms to add, then what we mean by addition isn't the Peano axioms.

>>6915786
But is. Natural numbers and addition on them is not constructed before we use them.

<span class="math">I'm \ shocked \ that \ there \ people \ in \ this \ thread \ who \ defend \ ignorance \ and \ think \ it's \ OK \ not \ to \ study \ and \ do \ actual \ proofs \ and \ constructions! \ You \ people \ are \ obviously \ not \ mathematicians \ and \ you \ should \ never \ call \ yourselves \ such. \ You're \ nothing \ but \ applied \ science \ monkeys \ who \ obviously \ don't \ care \ about \ mathematics. \ If \ you're \ a \ pure \ math \ student \ and \ you \ approve \ of \ all \ this \ ignorance, \ you \ should \ be \ ashamed \ and \ switch \ majors \ immediately.[/spoiler]

Is he going to propose an alternative to the real number system or will he just keep uploading his autism onto the internet?

>>6915274
isn't synthetic differential geometry suppose to be some super hardcore shit?

>>6915465
> or Errett Bishop, or Kronecker
>But even Bishop and Kronecker are kinda nutty
sounds like you just set up your strawman and started hacking at it bro

>>6915816
Actually we were explaining why the textbooks Wildburger was looking at do not have such constructions. It's perfectly fine to study such constructions and they are widely available. Wildburger and you idiots are obscuring this fact by making it seem like textbooks are "hiding" these constructions from readers. In fact they simply don't mention them because they are unnecessary for the task at hand, just as the Peano axioms are not necessary to learn how to add natural numbers. So all of this is really a pointless argument. Either address the constructions as they exist or move on.

>>6915858
actually I'm one of those idiots, and I just find it interesting how horribly some people take criticism, and those same people love using words other people don't understand without a second though, and as a primary method of discourse. Maybe that's how upper level mathematics works? Just find out which jargon other people don't understand in order to win debates?

>>6915768
Good for you now write it on a paper and shove it up your ass

To all the "muh axiom" faggots: if you assume shit in the first place you will get shit, its that simple

>>6915872
Assume 6915872 knows what he's talking about.

Then 6915872 knows what he's talking about.

But that is a shit conclusion because 6915872 obviously does not know what he is talking about.

Therefore you are a gay nigger faggot.

>>6915858
No one is acting like someone tries to hide something.

If I want to learn real anal ysis I need to know how R is contructed or I cant understand properly what convergence towards pi even means

>>6915868
To criticize something you have to have a point. Wildburger has none.

>>6915868
What is wrong with the constructions then?

>>6915886
That's literally what Wildburger says in his textbook video. He claims that there is no rigorous construction of the reals and that the lack of such a construction in textbooks is evidence that mathematicians are attempting to hide this lack of rigor to protect their "dogma".

>If I want to learn real anal ysis I need to know how R is contructed or I cant understand properly what convergence towards pi even means
But you don't. Many people learn real analysis without knowing the construction of the reals.

>>6915886
>If I want to learn real anal ysis I need to know how R is contructed or I cant understand properly what convergence towards pi even means

That's the position of someone who wants to be a mathematician or mathematical professional. People who don't care about constructions are people who also don't care about proofs and just want to learn applied mathematics. That's fine as well.

But let's not kid ourselves that not knowing how to construct <span class="math">\mathbb{R}[/spoiler] is somehow not important. Of course it is. And if you're a pure math person, this should be of vital importance to your understanding of real numbers and real analysis.

>>6915888
His point is that other's mathematician's method's are invalid. Could you rephrase what you mean?

>>6915893
I'm still trying to keep an open ear as to why. Last time we had a wildberger thread I came down to a complex conclusion relating to the grammar peripherial to the math, but it's not important because it'll just create unnecessary interference to more constructive discussions over these constructive mathematics.

>>6915899
What is the problem with the construction then?
If you can't point that out I don't see why it's bad that people don't know the construction.

>>6915908
This is not true. The only important thing of R is that it satisfies certain axioms (for example: every bounded subset of R has an upper bound in R). There are various ways of constructing R and the important point is that they all end up with something that satisfies these axioms. The particular construction itself is not very interesting, in the same that it doesn't matter what definition you use for ordered pairs, as long as they satisfy certain properties.

>>6915918
But they aren't invalid. And he doesn't show that they are invalid. The argument he actually puts forth is that mathematicians use an axiom he doesn't like and that they hide this by not discussing certain constructions in certain textbooks. Neither of these are relevant towards showing that some method is invalid.

>>6915921
>There are various ways of constructing R and the important point is that they all end up with something that satisfies these axioms

yet NJW shows weaknesses and faults in all of them.

>>6915921
>What is the problem with the construction then?
THERE IS NO PROBLEM WITH THE CONSTRUCTION

>>6915899
>Many people learn real analysis without knowing the construction of the reals.
Because the "construction" of the reals have nothing to do with anything, it's fucking trash. Just like the peano arithmetic has nothing to do with teaching children to add, and therefore cannot be said to be what the kids are "really" doing.

Either you teach them PA or you don't. There's no "don't teach them PA but really it is PA because I scribble garbage on pages."

>>6915927
No he doesn't. He just complains that they use infinite sets.

>>6915932
OK, I see the problem. You don't understand the difference between understanding/defining a concept and constructing a concept. Neither are "trash". You can understand what the natural numbers are in terms of their properties, and you must understand those properties in order to do arithmetic on them. However there is a separate understanding of them that defines them according to a more simple or "fundamental" principle from which you can then derive those properties. But that is unnecessary if you simply want to understand those properties and use them. So why construct things at all instead of just viewing everything as collections of properties? Because it is a basic idea of logic that reducing properties to principles will increase understanding. And in math there is a strong incentive to generalize and simplify concepts, so construction is naturally going to be done. But there is a problem. You can't construct forever. At some point we need to simply admit that mathematical objects are based on intuitional, irreducible axioms. So if you believe that everything must be based on construction then you are not going to be able to do any math.

>>6915924
>that they hide this by not discussing certain constructions in certain textbooks
That point of his seems valid, and factual to me though, despite the intent for doing it.

>>6915868
Just hopped in this thread and am on mobile so can't be arsed to read who you're replying to but, have you ever maybe just thought that you're fucking stupid instead of upper div mathematics being super jargony?
I get really annoyed when someone starts an argument about a subject in which they have no expertese or experience in and then complain when they don't understand what's going on.
The same thing happened to me a few months back on this board when some EE undergrad tried to argue with me about black hole physics and specifically the Schwarzchild solution. I clearly stated where his premises were wrong and then he proceeded to complain in a rant that revieled that he didn't even understand basic tensor calculus.
tl;dr you might just be retarded (or uneducated)

>>6915991
It's patently false and ridiculous. If mathematicians were trying to hide these constructions they are certainly doing a bad job of it considering they can be found so easily.

>>6916004
If it's not in the textbooks then it's hidden.

>>6916035
It is in textbooks, just not the ones Wildburger looked at. It's actually very easy to find, therefore it's not hidden at all.

>>6916043
Ok, could you point out an example to clear up the record and my wrong impressions.

>>6916049
Real Mathematical Analysis by Charles Chapman Pugh

>>6916053
So, just taking your word on it I guess wildberger is just being unfairly difficult pending no other technicalities.

>>6916066
google "Construction of the reals" and wonder if Wildburger is being facetious about the abundance of sources or just lazy.

>>6915883
Q.E.D.

>>6916070
>Google
Just to be clear though, it's purely an issue over whether textbooks actually in institutional use contain the information that's a valid cause of concern for the sake of prosperity.

Saying root2 doesn't exist is saying the unit square doesn't exist... better have some good arguments.

>>6914590
>Don Quixote of math

mysides.jpg

Second best nickname after Wild Hamburger.

>>6916049
You can see Dedekind's construction of the reals very well explained in Rudin's Principles of Mathematical Analysis. The construction does not even use axiom of choice. (That I noticed, I may have missed it.)

>>6916131
Not really. There are many many constructions not included in textbooks. This is not a problem or a cause for concern. Constructions are not necessary to understand the properties of an object they are a waste of space if you are not interested in constructions. If you are interested in constructions then you can find them easily. So where is the concern?

>>6916073
> Q.E.BTFO

>>6914590
infinity isn't real though...

>>6916308
neither are any mathematical objects

>based wildberger bringing up stern brocot tree again
I love the Stern Brocot tree :)

Wildburger talks an awful lot about something which is senseless, i.e. has no meaning.

You constructionfags do realize that a construction is a generation of a model for your axioms? In other words, you need both set theory and axioms to even begin with giving a construction. And the point of a construction is precisely that THE CONSTRUCTION DOES NOT MATTER, i.e. the generated set satisfies the intended axioms, so that everything you deduces from the axioms will automatically be true for your set. Else, by the very fucking definition of the concept, it is not a construction! Thus, you really do not need a construction of the reals of you want to learn some calculus. It's like demanding from a software engineer that he can build his own microprocessor. If you want to code programs, you are only interested in the processor as black box. In fact, you rely on the fact that it's a black box, i.e. interchangeable with any other processor meeting the same requirements.

It is nice to know that there is a construction possible for a given set of axioms, for in that case the consistency of your set theory will imply the consistency of your axioms. And we have an abundance of constructions for the reals, honestly using very modest set-theoretic axioms.

>>6916313
then>>6914590
is getting butthurt for no reason

>>6916341
>It is nice to know that there is a construction possible
possible!

>>6915839
Yes. Here is a really approachable introduction paper on intuitionist mathematics that also talks about Synthetic Differential Geometry amongst other topics.
http://math.andrej.com/wp-content/uploads/2014/03/real-world-realizability.pdf

>>6915768
I'm a pure mathfag. Which is exactly why I'm familiar with constructive mathematics.

>>6915921
>>6915613
>>6915677
>>6915692
>>6915724
>>6915858
>>6915921
>>6916043
>>6916053
>>6916070
>>6916163
>>6916177
There's a lot of posts in this thread that aren't familiar with Wildberger's arguments, these posts included. I'll try to clarrify the issues.
1) Typical constructions of the reals begin with the natural numbers (which are typically stated in second order arithmetic, constructed in ZFC, or are just taken for granted, Wildberger takes them for granted). First one uses the integers to construct the rationals and then uses the rationals to construct the reals in one of two ways, either Dedekind Cuts are used or Cauchy Sequences are used. Then given a set of cuts or sequences one defines new operations on it and creates a mathematical object isomorphic to the reals. Every book on analysis, foundatios of analysis, or logic that has a construction will use one of these approaches.
2) Alternatively the reals can be given with a set of axioms (where every model is isomorphic to the reals). Some analysis books will use the axiomatic approach instead of an actual construction.

(cont.)

>>6916468
3) The wikipedia page lists a couple other constructions.
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_using_hyperreal_numbers
a)Stevin's construction, in practice works out very similarly to cauchy sequences and dedekind cuts the way that Wildberger deals with them (you can use the same arguments he uses against cauchy sequences and dedekind cuts against Stevin's construction).
b) The hyperreals are a non-standard analysis field extension of the reals first proposed by Abraham Robinson in the 1960s.They would have the same flaws as the reals, similarly the surreal numbers.
c) I don't believe I've ever heard him mention the Eudoxus reals construction and am not sure if he would actually have a problem with it.
4) Wildberger is ONLY interested in constructions of the reals, this means that he doesn't care about axiomatic descriptions of the reals. His main "thesis", if you will, is that the real numbers cannot be constructed out of the integers.
5) Wildberger's issues are founded in constructive logic, computability theory, and a lot of the same philosophical stuff that Wittgenstein was into. He has more restrictions on his mathematics than a mathematician working in classical logic (this is not uncommon as there are many types of constructive logic and many of them have ideological and philosophical origins). In order for something to be constructible there must exist an (computaility theory) 'algorithm' that can 'decide' on every input in finite time, this is more or less in line with constructive mathematics and computability theory.
6) Because of this he disagrees with the notions of cauchy sequences (because they are infinite sequences of rationals) and dedekind cuts (because given a complicated enough real they can become sets defined with an infinite number of logical statements):
<span class="math">\{q\in\mathbb{Q}\colon P_1 \vee P_2 \vee \cdots \}[/spoiler]
Similarly decimal expansion (Stevin's construction) is ruled out.

(cont.)

>>6916473
7) In his textbook video he disqualifies any textbook that only gives an axiomatic definition and disapproves of any that give the standard cauchy sequence/dedekind cut constructions. He also mentions Landau's Foundations of Analysis which is an entire textbook dedicated to a construction of the real numbers using dedekind cuts (though he disqualifies dedekind cuts for the reason described above).

>>6915729
>The same way you reason about natural numbers without constructing it, or groups, or rings, or any kind of mathematical objects presented in an axiomatic manner.
This is valid, but it's just not what Wildberger is talking about.

>Because they're incomplete and you use the standard model for the natural numbers.
The reals are typically constructed in Second Order Arithmetic. It doesn't really matter whether or not it is incomplete.

>>6915918
This isn't new, refer to >>6915399

>>6916341
Again, the issue is wih a construction of the reals using the natural numbers. Given second order arithmetic the construction of the natural numbers does not matter, obviously. However, the question is, is it enough to have just the natural numbers? In order to show that it is, one must construct the real numbers out of the naturals.

>>6916473
>I don't believe I've ever heard him mention the Eudoxus reals construction and am not sure if he would actually have a problem with it.
It still requires infinite sets, it just dispenses with rationals for defining the reals.

>>6916355
I don't see how that invalidates his point. I think his point is that Wildburger complaining that people can't do things infinitely in real life is not a mathematical criticism.

>>6916518
Can <span class="math">\aleph_1[/spoiler] angels dance on the head of a pin?

>>6916473
Integers can encode provably inexistent entities. Does he deny the existence of integers as well?

>>6916558
>2014
>still believes in cantor's bullshit

kek'd

Did you ever wonder what would happen if an ancient greek came to the present and learned modern maths?
This is the result, a guy who thinks that math exists for some sort of real world, and that everything in math has to be explicitly constructable with finite resources.
Geometry was created to measure the earth, and it grew beyond that.
Also, this is some old shit, I remember first reading about this literal tard a while back when I heard about "Rational Trig"

>>6916558
What's that symbol mean?

>>6916889
Thats called Aleph, we say it represents the "number" of elements in infinite sets.
Aleph 0 is how many natural numbers exist
Aleph 1 is how many real numbers exist
Aleph 2 is how many times shit posters will make this thread

>>6916889
http://en.wikipedia.org/wiki/Aleph_number#Aleph-one

>>6916885
>mfw I tried as a little 10th grader to convince my math teacher that we should be using "rational trig." because it's "better" and he has literally no idea what the fuck I was talking about.
>mfw I had no idea I was literally telling the guy "no, everything is wrong. Sine and cosine are bullshit constructions. Free your mind. You should be squaring everything because this guy online told me so.
>he had no idea what the fuck I was doing, was probably trying not to laugh too hard at me

>>6916885
The fact that math (it's math, not maths you limey bastard) is "unreasonably effective" at describing the real world leads many to believe that universe is a mathematical object (some call it extended platonism).

So if you tend to believe that, it's good to look at how universe or the nature works. One thing to note is that the universe is not continuous but discrete. One other thing to note is that there are no infinities anywhere in nature. Yes, there are lots of large numbers but there's no infinities. No infinities anywhere).

That should give us a clue that maybe we're on the wrong track.

http://edge.org/response-detail/25344

>>6916920
Can you prove the universe is discrete?
And I swear to god if you say "M-Muh plank length" im gonna eat my own hat.

>>6916920
> One thing to note is that the universe is not continuous but discrete.
stop spreading this disinformation

>>6916920
Most numbers don't have any physical representation.
Infinity is just an idea, not a number. It's the idea what would happen if a pattern continued on and on without stopping. Why is this a hard concept to grasp?

>>6916920
>One thing to note is that the universe is not continuous but discrete
Yo what

>>6916925
>And I swear to god if you say "M-Muh plank (SP) length" im gonna eat my own hat.

no need to invoke Plack's length. Quantities such as energy of bound states or the angular momentum can only take "quantized" or discrete values (eigenvalues), for example. You have a ton of other examples of fundamental units taking quantized values.

>>6916942
M8, do you know WHY they take discrete values
Go ahead and show me how we construct these discrete values.
also before you say "hur dur impylin u no" ill tell you it involves the creation-annihilation operator

>>6916928
>Most numbers don't have any physical representation.
But natural numbers of objects do exist! Infinity excluded, of course.

>>6916946
Show me one thing in reality that's continuous. Just one thing. Protip: you can't.

>>6916479
>In order to show that it is, one must construct the real numbers out of the naturals.
Yes it's done a lot. Just Google. I got this shit in my first year.

>>6916920
>They're still not opposites
>how universe or the nature works is still irrelevant for mathematical foundations
>you still lack a God's eye point of view to know there are no infinities anywhere
>still the ontological status of large numbers or any given number is on par with that of a real number.

>>6915585
Why don't you google a bit before spitting dogshit out of your stinky mouth?

https://www.researchgate.net/profile/Norman_Wildberger

He's done quite much in Lie theory.

>>6917171
> He's done quite much in Lie theory.

yes, he is a liar.

>>6916927
Can you justify the opposite?

Are you that retard who I was argueing with the last Wilberger thread? You were saying that physical discretness and continuity can't be opposites.

Well, obviously everyone who believes in physical infinitude is so much mentally-ill. I can't even imagine how hard is it to live with such viewpoints.

Btw, I was asking on why are there so many quantized integer quantities in our world and no one responded. Indeed, why is it so? Why exactly two charges, two kinds of particles, spinds, colors etc.?

Continuum-fags have to justify why the nature restricts everyone from accessing beyond a certain finite scale of everything -- time, space, energy etc.?

Why is our world quantum and not, say, Newtonian?

>>6917175
kek

>>6916896
So it's just equal to powers of omega?

>>6916473
>>6916479
>Wildberger is ONLY interested in constructions of the reals, this means that he doesn't care about axiomatic descriptions of the reals. His main "thesis", if you will, is that the real numbers cannot be constructed out of the integers.

But the reals are defined as the unique set satisfying certain axioms. It's more obvious the reals cannot be effectively enumerated, if that's what he means by constructible. On the other hand, the set of reals is Godel-constructible. On the third hand, the set of reals is not absolute among models of set theory.

And to address any and all concerns by those who doubt the axiom of choice (AC), remember this: it has been proved for a long time that Con(ZF) entails Con(ZFC), so even if you reject choice, you must believe it is consistent.

FOURTH: The definition of the reals doesn't even require the AC! Why are there so many people talking about AC in this thread? The most important part of ZF needed in the construction of the reals is Powerset.

>>6917199
>implying people don't doubt the consistency of ZF
some people don't even think exponentiation is a total function brah

>>6917213
Exponentiation is primitive-recursive. There are really people who think a primitive-recursive function can be non-total? I..

>>6917220
https://web.math.princeton.edu/~nelson/papers/warn.pdf

>>6917223
I heard of this. I've heard that he's since retracted it, after careful discussion with Terrence Tao. It was interesting, slightly, to read it. I still scarcely know what he was thinking; his metaphors seem incorrect. Indeed, whatever he was thinking *was* incorrect; Terrance Tao was able to pinpoint his fault and convince him that P is consistent.

>>6917240
No, that was another thing where he thought he could find a consistency problem due to Chaitin's incompleteness theorem. But he did retract that.

>>6917184
>Continuum-fags have to justify why the nature restricts everyone from accessing beyond a certain finite scale of everything -- time, space, energy etc.?
[citation needed]

Let me remind you where we left off. I repeatedly asked you for a definition of discrete and continuous such that the following holds:
-A space is either discrete or continuous, but never both
-The definition only uses finite mathematics, no ZF, choice or reals allowed
-The universe is 'discrete', without a doubt, so not based on empirical (thus falsifiable) hypotheses
-From the latter, the natural numbers 'magically' arise
-Somehow, this must be relevant for the foundation of modern mathematics.

Good luck

>>6917248
Oh, thanks.

Well, this is interesting. While it's "obvious" that in the "true" model of arithmetic, exponentiation is total, I do indeed admit after reading this that it currently seems to me possible that a non-standard model of PA might not be exponential-closed.

This will stick with me. I sleep on it.

>>6917223
>>6917248
Ok, am still >>6917240 >>6917264.

This is seriously just so exciting. I haven't felt this excited about math in months, let alone excited almost at all. Thanks for knocking me out of my recent depressive slump.

>>6916942
>says no need to invoke Plack's length
>invokes Planck's constant

>>6917272
And honestly he didn't say he now believed in consistently, TT just saw what Nelson was probably trying to do and pointed out why it wouldn't directly work. As I recall it was a problem with some additive constant that was a function of the language, Nelson had some misunderstanding on that point.

It may still end up being a fruitful attack in the future. I don't know how much we can say about the additive constant in Chaitin's theorem.

>>6917223
one already uses induction to get formal languages and logic of the ground, so far before PA or any other theory. Dismissed.

>>6917278
wut

>>6916483
I see, that makes sense.

>>6916586
I believe he has issues with the existence of infinite sets, which again isn't a new idea (see Wittgenstein and others). I'm not sure though. The thing about Wildberger is that he's talking about what are actually really technical logical and philosophical issues but he's dumbing them down a whole lot to reach a wider audience. I find this makes his arguments unclear at times.

>>6917103
>>In order to show that it is, one must construct the real numbers out of the naturals.
>Yes it's done a lot. Just Google. I got this shit in my first year.
No shit, anon. You obviously did it through one of the methods I listed. I was only pointing out that Wildberger disagrees that those constructions are constructable. It's not that he's never heard of it. How the fuck did you miss the entire point of the post.

>>6917199
>But the reals are defined as the unique set satisfying certain axioms.
Close, technically the axioms define lots of different models, EXCEPT that one can prove that every model they define must be isomorphic to the reals. He is not talking about constructing the reals as a model of THAT axiomatic system. He's talking about constructing the reals inside a model of second order arithmetic (Peano axioms expressed in a second order logic). You can find a formal definition in the first pages of this book.
http://www.personal.psu.edu/t20/sosoa/chapter1.pdf
He is not trying to enumerate them which should be obvious to everyone since any formal language will at most contain a countable number of sentences and thus it's impossible to define a semantic (bijective) map to the reals.

(cont.)

>>6917291

>so even if you reject choice, you must believe it is consistent.
No shit, but the system is consistent without the axiom of choice as well. It just means that it's independent of ZF. Many mathematicians do constructive mathematics without the axiom of choice. In many cases mathematicians will even assume anti-classical axioms that contradict the axiom of choice, law of excluded middle, double negation, etc.. In THOSE systems the axiom of choice is not consistent. As an example look at Synthetic Differential Geometry, I posted a link above or view this talk by the author of that paper.
https://video.ias.edu/members/1213/0318-AndrejBauer

>The definition of the reals doesn't even require the AC! Why are there so many people talking about AC in this thread? The most important part of ZF needed in the construction of the reals is Powerset.
Just watch the videos you asshat.

>>6917291
>I find this makes his arguments unclear at times.
Yes. I'd like him to make a specific, clear series aimed at people with the requisite understanding of the issues.

>>6917296
Me too.

>>6917291
>Again, the issue is wih a construction of the reals using the natural numbers. Given second order arithmetic the construction of the natural numbers does not matter, obviously. However, the question is, is it enough to have just the natural numbers? In order to show that it is, one must construct the real numbers out of the naturals.
And the latter is precisely what is done, using cuts or Cauchy sequences. So what's the point then?
Did you observe that nowhere in your prose did you mention an actual problem with the construction of R out of N.
Neither does Wildburger. His whole argument comes down to: we need some axioms of ZF to construct N out of R and the result is non-computable in finite time. No shit, who cares.

>>6917296
Or perhaps a mathematical proposition? Just one? For example the inconsistency of R? Without handwaving? That would be great, thnx.

>>6917262
troll/10. Do you even distinguish between mathematical and physical discretness? Physical discrete space is ismorphic to a finite subset of naturals of rationals if you want.

>>6917330
>is non-computable in finite time. No shit, who cares.

As I understand, that is exactly his problem with it. Again, this isn't my argument. I'm only attempting to clarify what Wildberger is arguing about. If you're fine with defining a set of things that's impossible to compute in finite time and then using it to define your reals then go right ahead. I imagine most people don't have an issue with it.

>>6917296
While looking for a more formal approach to computable real numbers I found this. It's not really the same thing as what Wildberger is talking about but is interesting nonetheless.
http://math.andrej.com/data/c2c.pdf

>>6917343
>Physical discrete space is ismorphic to a finite subset of naturals of rationals if you want.
isomorphic in what category? Clearly not as topological spaces, because N and Q are not homeomorphic

>>6917387
Either you're an idiot or didn't spot the typo:
Physical discrete space is ismorphic to a finite subset of naturals OR rationals if you want

Both have discrete topology. And yes, dude, finite subsets of N and Q are not just homeomorphic, they're perfectly isomorphic.

>>6917364
>While looking for a more formal approach to computable real numbers I found this.
Check out On Feasible numbers (sorry, I can only find a ps file):
ftp://ftp.botik.ru/rented/logic/papers/SAZONOV/lcc.ps

GUIES
IF INFINITY ISN'T REAL
WHAT AM I SUPPOSED TO DO WITH THIS DELTA DIRAC FUNCTION!?

>>6917443
literal child detected

>>6917443
That is not how the Dirac delta is defined.
It isn't even a function; it's a distribution and therefore lacks any sort of value at any particular point.

>>6917477
It's hilarious how wrong you are.

>>6916341
>you really do not need a construction of the reals of you want to learn some calculus.
You don't need the reals at all either because all you're going to be calculating with a rational numbers + some extension fields (quadratic fields for square roots, etc) and maybe 'algorithmic' numbers like e and pi.
You are *never* gonna do any actual calculations that require axiom of choice for example.

>>6917543
The reals have the convenient property that when you look for an answer on the number line it's magically there. Reals are like a persistent hallucination.

But there are proofs in calculus that depend on this magical property without giving a means of explicit calculation, such as the extreme value theorem. First you define the reals to have the magic property of being there when you look. Then you assume that if you were to be able to look, then they'd be there too, even if you aren't already looking.

Persistent hallucination.

Do you guys realize that we will never be able to represent real numbers perfectly in a computer and compute with them. So they're fucking useless.

We should invent new set of numbers that's actually computable.

Maybe NJW can create one.

And it doesn't matter if the new set doesn't have infinity. In fact, it would be better if we just erased all the Cantor shit.

>>6917564
Computers can do symbolic computation, too. Checkmate.

>>6917567
Any set of symbols on a comp boils down to 0 and 1

Therefore these are the only truly available symbols to work with

Everythig else is just abstracted from that

>>6917567
Symbolic computation ≠ computation with <span class="math">\mathbb{R}[/spoiler]

>>6917521
He is perfectly right you prick.

>>6917543
This. I'm not even gonna mention how analysis is carried out in constructive and finitistic systems. The troll ITT will explode.

>>6917579
Just because you don't comprehend words does not mean reality has to comply with your idea of what it should mean.

>>6917567
Only with a finite set of symbols.

>>6917596
You are stupid

>>6917596
the platonists are getting upset, quick, everyone conform

>>6917400
>And yes, dude, finite subsets of N and Q are not just homeomorphic, they're perfectly isomorphic.
Again, isomorphic in what sense, dude? Still waiting for that awesome definition of yours btw

>>6917543
>any actual calculations
what are 'actual calculations', and why should I care?

>>6917622
>what are 'actual calculations', and why should I care?
The things you are doing when learning and applying calculus.
So lets say you wanna calculate the area under a graph, you are only ever gonna do this with truncated finite decimals (generated from rational numbers, or some algorithm that spits out e to n decimal places) - never needing to reach for 'real numbers'.

For anything applied, real numbers are useless.

As for proving theorems, all you can do is appeal to axioms of course, but you can do that just fine without real numbers, you'll just be proving different theorems, and maybe have to throw out some other ones, but that shouldn't be a problem, all that matters is which axioms you pick, and why not pick axioms that are more inline and useful to applications?
What use are axioms that will never generate a concrete example?

>>6917607
ur a dumdum >:(

>>6917634
You have to be the most ignorant person on /sci/, holy fuck!

ITT

what are alternatives to R? has anyone done any work to replace R with something else?

>>6917443
>function

>>6917649
Q

Can you give a single example of where anyone did any calculation that wasn't just with numbers in Q (and not just tautologies like 2*pi = 2*pi)?

>>6917659
>Can you give a single example of where anyone did any calculation that wasn't just with numbers in Q (and not just tautologies like 2*pi = 2*pi)?

No, of course not. We can't even enumerate R and no one has ever written down the exact value of Pi anyway.

Q does seem like a good alternative and a way to stop wasting time on R.

>>6917649
I think I would start by using Pi, e, sqrt(2) etc. as unique items instead of putting them all into a set. Then I would enumerate iterative steps to give those numbers a "depth".

F.e. Calcuate sqrt(2) recursively up to step 3 != sqrt(2) calculated up to step 4

Then one have to look at what happens when you use sqrt(2) depth 3 + sqrt(2) depth 10 or pi depth 11 + e depth 4 etc. I think this might improve the understanding of our calculations

>>6917622
> Again, isomorphic in what sense, dude?
As sets, dude. There is a bijection between any finite subsets of N and finite subsets of Q with the same cardinality. Omg so simple. What are you even trying to catch me on?

>>6917665
I like the way you think!

ITT ignorant kids, who've never heard of constructive math, trying to elaborate that math is impossibru without the Axiom of Infinity and the Axiom of Choice.

/sci/ has changed for the better, a year ago I was banned for being an ultrafinitist. Now we can actually have some decent discussions, post papers, etc. Good stuff.

>>6917675
>constructive math
Philosophtard detected.

>>6917690
u got banned for that? damn.

>>6917670
It suprises me how one can think this makes any sense at all.

>I-It is! Its j-just c-counter intuitive!

There are obv more rationals between 0 and 1 than there are numbers between 0 and 1.

Yet N and Q should have the same size.

What you are actually saying is: I can enumerate N and Q but I cant enumerate R because I defined R / Q as being a set of infinite sets themselve, for me it doesnt matter if those sets have nothing in common except irrationality and infinity.

Inf is not a special property, even N has it, but calling "numbers" irrational and talking about them is like talking about different kinds of cars by only using the word "car".

>>6917696
> Philosophtard detected.
Bullshit

>>6917708
You bored me, buddy. Just say do you understand the difference between continuous and discrete physical space?

>>6917714
http://www.mnei.nl/schopenhauer/38-stratagems.htm

Nr. 34

"When you state a question or an argument, and your opponent gives you no direct answer, or evades it with a counter question, or tries to change the subject, it is a sure sign you have touched a weak spot,"

I know Schopenhauers ways to win an Argument, can we please get to the topic please? Im not in the mood of getting trolled by plebs.

>>6917740
>1. motte and bailey argument
Schopenhauer is pretty much one of the top western philosophers. Frankly anyone besides Schopenhauer, Hume, and Wittgenstein are just navel gazing faggots.

>>6917564
You can do real analysis where all numbers are computable (just like some constructivist theories prove that all functions are continuous). However, those systems are inherently non-classical and you have to give up some "intuitive notions" about the reals.

The problem lies in the existence of sequences of computable numbers, which have a non-computable number as limit.
If your theory is able to express an decimal expansion, like 1/9 = 0.1111, it will also be able to express stuff like <span class="math">\sum_{n=1}^\omega10^{-n}[/spoiler] and then, if you can encode universal computation, you find this shit

http://en.wikipedia.org/wiki/Specker_sequence

A guy putting "divine" in the title of one of his books is suspicious to me, but his educational stuff looks pretty good and I support that.
I once ask Wildberger why he's wasting his time with this quest of convincing people of his views what he thinks about type theoretic approaches to foundation. I can't remember a reasonable answer to the former and he said he didn't follow the latter.

I consider undefinable numbers to be one of the ugliest beasts, i.e. the continuum of numbers given by
"(all real numbers R) - (the small subset of R for which there is a description in our enumberable language)"
But for doing relevant physics, I must see it form a practical perspective and here we need to realize that differentiable manifolds or probability distributions will not find a workable formalization in constructive world, one were I need to communicate with people who don't want to worry about this stuff.

PS: I think foundations need universal properties form the get go, while at the same time ECTS and friends are unworkable and type constructors too formal. Atm. I take my formal notes over many-sorted FOL with functions, ad hoc subtyping etc. as primitive concept, kek. Come at me, foundational purists.

>>6917740
Your statement: discreteness and continuity are equivalent.

What is hear to discuss?

>>6917802
> A common way to resolve this difficulty is to consider only sequences that are accompanied by a modulus of convergence; no Specker sequence has a computable modulus of convergence.
D-dude constructive analysis is typically carried out via witnesses, soooo ...

>>6917834
*here

>>6917802
>The problem lies in the existence of sequences of computable numbers, which have a non-computable number as limit.
Computable reals aren't a good answer for a host of reasons, like Rice's theorem. Also every implementation of computable reals has weird representation problems (roughly 0.999... = 1 problems, but they're peculiar to the representation so it's hard to be more general).

The best we can really do as far as I can tell is
1) use exact rationals, and if you have to,
2) use field extensions of rationals

Anything outside of this will necessarily entail some dark corners of computation. Honestly I can hardly fault people for using floats though they're pretty terrible in a lot of ways and I wish they were never invented.

>>6917865
>use field extensions of rationals
Do you mean like vectors <span class="math">q_1 · 1 + q_2 · a + q_3 · b + [/spoiler] with a,b,c etc. arbitrarily chosen?

>>6917880
well, up to choice of representation, yes

>>6917865
>Honestly I can hardly fault people for using floats though they're pretty terrible in a lot of ways and I wish they were never invented.

I feel like you're the first person I've met who is capable of truly understanding me.

>>6917884
n-no homo

>>6917834
>Your statement: discreteness and continuity are equivalent.

>>6917865
> The best we can really do as far as I can tell is
> 1) use exact rationals, and if you have to,
> 2) use field extensions of rationals
That is the approach of Wilberger

>>6917903
Yeah, everyone I know who is right always agrees with me.

>>6917895
The old thread floated away so ... But that was your point. You said discretnees and continuity cannot be oposites

<3 NJW threads!

>>6917802
interesting post!

I think this subject of constructing is logic and not math(s) which is applied logic. The difference between logic and applied logic being formal acceptance. Really, the English have got it right calling it maths and not math.

What we're left with here is unresolvable childish persuasive argumentation on both sides each of which can just accuse the other of sophistry instead of proving the other wrong, and the team with the most people wins. Nobody's going to want to admit that their argument is a philosophic construct because that means they lose and the other guys win by default, rinse, repeat. But Wildberger seems to hint at the fact that they can't prove their argument in the first place because of the nature of infinity which people are just forcefully shoving into math with a perpetual prayer of sanity that any potential problems will always be worked out or never happen just because they say so and "just look at all this other stuff, it's so great!"

So yeah, this shit is philosophy.
>my philosophy is better than your philosophy
need more hints?
If your shit begins with "suppose", that shit is philosophy

Analysis is fucking disgusting. I'm going to puke if I have to read another epsilon-delta-all-over-the-place proof.

>>6917922
your arguing with someone else now. I'm the saying that they are not necessarily opposites. I asked you for a definition etc. etc. When pressed, you managed to produce the statement
>As sets, dude. There is a bijection between any finite subsets of N and finite subsets of Q with the same cardinality. Omg so simple. What are you even trying to catch me on?
i.e. there is a bijection between a given pair of sets between which there exist a bijection. What a marvelous insight. Furthermore, not considering the topology, order or even the algebraic structure on N or Q, but just as sets, leaves me wonder why N or Q should be preferable over R.

Also I don't have the slightest clue what this has to do with discreteness and continuity in mathematics nor in the physical world. and I'm still wondering what the latter has to do with foundation of mathematics.

But I'm sure I'm a dum-dum because I don't understand your crystal clear distinction between discrete and continuous.

>>6915522
efficentlychopoffyourfoot.webm

>>6919521
This is ridiculous my friend. You're clearly wasting my and your time.

Continuous physical space has the natural topology of R. Finite nature (read: discrete physical space provided that it's bounded) has the discrete topology. Enough.

And there were several arguments to argue for the finite nature. Please read the thread above.

>>6919402
You're an idiot.

I know synthetic differential geometry is sophisticated, but would Wilderger be satisfied with it?

>>6917477
>>6917657
How is that not a function?
<span class="math"> \delta : \mathbb{R} \to \mathbb{R}^* [/spoiler]

>>6919661
<span class="math">\infty \notin \mathbb{R}[/spoiler]

>>6919616
OK buddy, you win. Discrete + bounded is clearly equivalent with finite, and evidently our universe is a-priori finite. And of course, the well-known continuous alternative, the only alternative, is modelling the universe as R with its natural topology. I see the error of my ways now, because it's obvious how this implies our mathematical foundations must be ultra-finitistic and intuitionistic. Above all, everything but finitistic mathematics is senseless, so actually we cannot even consider any alternative models of the universe but your splendid discrete one.

>>6919682
You can consider it. You can pretend god exists—but he doesn't, either. You can perform magic rituals (aummm... for every epsiloooooaaunnnnnnhmmmm) all you like, too. I will still point and laugh.

>>6919674
Lol retard
<span class="math">\infty \in \mathbb{R}^*[/spoiler]
See the difference? Never heard of extended real numbers?

>>6919661
Its integral is still 0 because {0} has measure 0.

In non-standard analysis it is a function, though.
w an infinite (non-standard) number,
f(x) = 0, if |x| > 1/w
f(x) = w, if |x| <= 1/w

>>6919688
>not even a field
son I don't got time to deal with your bullshit

>>6919674
Didn't say it was.
Are you saying it's in no set whatsoever?

>>6919691
Functions don't have to be mappings of fields?

>>6919682
Good.

Gentlemen, it is a very bad style to define distributions by extended reals. Thank you.

>>6919687
existence outside of mathematics is irrelevant for the discussion of mathematical foundations.

You really don't see the mess you're in? Either you accept that ordinary mathematics about infinity and the continuum are meaningful, in which case you can ask the question whether or not the universe, for example, is a locally Euclidean space. Or whether or not it is infinite or not. Or you reject all talk about infinity and the continuum, in which case questions like 'is our universe finite? discrete?' etc. are meaningless. It's like asking 'Does this table have a color or does it have a flopniop'?

Furthermore, if you want properties of the universe to be a justification for favoring one mathematical foundation over another, you better make sure these properties are known apriori. Else, your foundations will become an empirical fact, which is falsifiable by its very nature and hence can change over time.

In short, if you reject all mathematics except finitistic intuitionism on the basis of the 'finite and discrete nature of our universe', you throw away precisely those tools that are needed to empirically verify this hypothesis about the universe.

>>6919713
I don't know if that >>6919682 is you, but I >>6919706 have finished the discussion there. So, I do not throw any insults, just in case.

>>6919697
It doesn't matter because the property of interest is that its integral is 1. There is no function with this property. I urge you to take the time and enjoy proving this yourself.

>>6919786
The function with that property is the dirac delta function.

>>6919786
> property of interest is that its integral is 1. There is no function with this property

>>6919818
nice proof

>>6919825
nice proof

>>6919836
>There is no function with this property.
Nice proof

>>6919853
http://philtynan.wordpress.com/2010/01/08/the-delta-function-is-not-a-function/
please, do your best to enjoy learning

>>6919874
Generalized function is still a function. Nice semantics.

>>6919892
Nice try! Shame you can't fucking read anything.

>generalized function
It's not a fucking function. If you don't want to learn you don't have to, but you can quit shitting up /sci/ with your willful ignorance.

>>6917443
>>6919661
>>6919688
This doesn't make sense if you're not working in non-standard analysis. Furthermore, there are an infinite number of infinitesimals in the hyper-reals and the way infinitely large numbers is defined is by taking 1/d where d is an infinitesimal. So there are an infinite number of infinitely large numbers. So, then the question is: to which infinitesimal is your function mapping? It can only map to one if it is in fact a function. If you say any infinitely large number then it is not a unique function as you claim here >>6919786
>>6919691
In nonstandard analysis, the Hyper-reals satisfy the field axioms. They are actually a field extension. For more information refer to this Monograph or to Abraham Robinson's original book on nonstandard analysis.
https://www.math.wisc.edu/~keisler/foundations.html
Keisler also has a high school level introductory calculus book on the same subject.

>>6919874
Lel it's not a function but an EXTENDED FUNCTION! :^) i totally got you guys hahahaha

>>6919895
>In nonstandard analysis, the Hyper-reals satisfy the field axioms. They are actually a field extension.
They're not doing non-standard analysis in case you didn't actually read the conversation.

>>6919896
Stop being a retard.

Another anon

>>6917802
>I once ask Wildberger why he's wasting his time with this quest of convincing people of his views what he thinks about type theoretic approaches to foundation. I can't remember a reasonable answer to the former and he said he didn't follow the latter.

That's really a shame, though not that surprising. I get the feeling that Wildberger is taking more of the oldschool philosophy/real world approach when arguing against the reals. Kind of like how many mathematicians argue against the hyper-reals.

>>6917865
I agree with this.

>>6919898
The extended reals are a set explicitly created for non-standard analysis. Perhaps the other anons weren't talking about non-standard analysis but the one who introduced the function mapping to the hyper-reals did.

>>6919907
The problems with computable reals can be circumvented. I mentioned that before ITT. Also, there is a strictly finitistic set-up in Bishop style using a bit of type theory and lambda calculus. So, it depends on how you treat the computable functions on naturals.

>>6919910
Extended reals have literally nothing to do with non-standard analysis.

>>6919916
>The problems with computable reals can be circumvented.
>rice's theorem isn't a theorem
cools story bro

>>6919918
> cools story bro
Constructive analysis has been developed for like 60 years or so and you say it's all crap. Nice try

Btw, I see you badly know the subject. You probably have never heard of Markov principle and least principle of omniscience. The problems related to undecidable properties have been addressed even by Bishop.

>>6915342

> being this in love with wildberger

>>6919918
Oh nevermind, I thought you were talking about hyperreals not extended reals.

>>6919924
>Constructive analysis has been developed for like 60 years or so and you say it's all crap.
You heard that. I didn't say it. I said specifically that you can't sidestep Rice's theorem. Neither of the random shit you namedropped has anything to do with it.

>>6919932
No worries.

>>6915399
If you only knew.

>>6919986
Show where Rice's theorem applies for example in here:

http://www.mathematik.uni-muenchen.de/~schwicht/seminars/semss13/constr13.pdf

or in here:

http://www.mathematik.uni-muenchen.de/~schwicht/papers/uppsala05/uppsala05.pdf

to diminish the proofs.

Rice's theorem is about general properties. It does not say anything about particular properties of functions. Cauchy sequences in constructive math are not picked just as is. They are specially constructed and come along with a thing called modulus of Cauchyness.

>>6920138
>diminish the proofs
It's like you literally don't even know what's happening in the conversation. The problem with computable reals in general is in their use; e.g., actual calculations. Because you cannot in general prove any non-trivial property of a procedure you have computable reals that just pop out of algorithms. You don't know what they are.

Now take an arbitrary rational expression with an arithmetic combination of computable irrational reals in the denominator and prove you're not dividing by zero. Take the square root of 2 times the square root of 2 and get 2. (Depending on your system, this may never generate a single term.) These problems abound in the use of computable reals, not your toy scribbles on paper. They are not removable except to abandon the computable reals. You can switch representations to solve one problem but that representation will have its own problems.

And don't tell me "up to arbitrary precision." This is where the problem comes in. If you FIX a precision then you can determine what you need because all reals become rationals. But if you actually have an arbitrary precision—to put it plainly: if you abstract over the k in the precision setting <span class="math">2^{-k}[/spoiler] (assuming base 2 for instance)—then you are in fucked land.

>it's about general properties, it doesn't say anything about particular properties
You're fucking joking mate. Show me the computer representation of the computable reals and the algorithm for division that provably doesn't divide by zero. Protip: your papers always assume that denominators aren't zero in the definitions. That's a cute trick when you're publishing a paper but I demand things that actually work, not fucking navelgazing shit.

Get the fuck out of my goddamn /sci/.

>>6919641
yeah, nah. you're a cunt.
ya'll are so afraid to philosophy it's just dumb

What do you get when you cross an engineer with a philosopher?

>>6920926
a shitposter

>>6915585
/thread
I was appalled watching that video when he just took undergraduate textbooks to try to prove his point. Really?? They are Calculus and intro analysis books.

>>6920926
Tesla conspiracists?

>>6920926
a designer, but isn't that really off-topic?
>>6920933
that costed you a shitpost to say

Anything else to say to reduce this to a ideological battle?

>>6920988

Since when do they engineer anything?

let me clarify my point here >>6919402
Physicists and other use math, because math by itself is worthless unless someone applies it.

How exactly is anyone suppose to apply this super ultra highly valued construction in question of the real number line to anything outside of math?

If you can't it's because it's not math. What's the point of math for the sake of math?

>>6920926
answer
>>6921012
i.e. worst of both

>>6921028
Ok, but that doesn't answer my question and even a child could see that.

>>6921030
I wasn't intending to. Go build a bridge.

>>6921031
that's what I'm doing on this anonymous message board, metaphorically speaking

>>6921031
That's what I'm doing on this anonymous message board, metaphorically speaking. Please don't be so outwardly butthurt that you must resort to nauseatingly obnoxious jokes .

Okay, so how does he define the reals and the irrationals?

>>6921012
It's like art dude, you don't ask an artist what's the point of his work, duh.

New Video:
https://www.youtube.com/watch?v=xYPw2gY_3PI&list=UUXl0Zbk8_rvjyLwAR-Xh9pQ

Can you hear it anon?

Its the sound of the Continuum getting BTFO

>>6921114
18 minutes in and it's a delight

>>6921107
You know, jokes like that can be hard to miss with the distance in some of these topics.

>>6921114
>all this shitty philosophy

>>6921537
>vague comment
>no argument

>>6920959
Tomorrow is always a day away.

>Occam's Razor
>Dedekind Cuts
2edgiee4mee

>>6921559
>I'm so insecure, fight me on the internet

>>6921114
Rarely do I like when people bring up the uncertainty principle, but I have to say this is a brilliant analogy. Inb4 1000 autists say it's a shitty analogy because they don't understand analogies.

this guy it totally disconnected from formal science and math

>>6921849
>HURR DURR THE UNCERTAINTY PRINCIPLE IS ANALOGOUS TO HUMAN ERROR, GREAT JOB!

Kill yourself.

>>6921860
didn't take long

>>6921854
And you fucking cantor-jihadists are disconnected from the real world like theologists.

Your Ignorance wont make the obv problems go away.

>>6921039
Why doesn't Wildburger never ever, not once, prove an inconsistency in the reals or in infinite sets?
>He's afraid he'll run out of words

>>6921969
There is an interesting result in working with ultrafinite mathematics. Suppose your biggest number is two to the power 1000. You can show that any proof of the inconsistency of this theory is longer than two to the power 1000.

So, the system is consistent, just like Gentzen's proof of the consistency via transfinite induction.

This is why ultrafinite math with symbolic "biggest numbers" can develop pretty much most of the results of analysis. Infinity really isn't needed if your system is only inconsistent after you run out of particles in the universe.

A platonist, a formalist and a Wildburger where in the train to London. The saw a species of ornament horses they never saw before/ The platonist says
>I remember this form now! It must be some typical British horse
The formalist answer:
>I understand the symbols you've uttered to me and although I do not agree with some of your ontological background assumptions, the message is clear and reasonable
Where the Wildburger answers:
>I dont believe they exist because I dont see any applications in the real world! You're all delusional! Stop oppressing me!

>>6921980
kek

>>6921982
Why did the Wildburger let the orphanage burn down?
>He didn't believe in the existence of the bastard orphans, for he didn't know how they were constructed.