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While watching this N. Wildberger video, at 12:34 it is mentioned that Modern Mathematics has serious problems with real numbers and that Mathematicians are aware of it.

Can anyone point to what are the problems that he is refering to?

Thank you

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    That is very vague.2011-08-07
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    Sounds like a finitist's talk to me.2011-08-07
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    @Asaf : I don't know much about finitists, but seems to be the answer.2011-08-08
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    It's just Wildberger at it again, not bad ideas, but it would be better if he didn't denigrate the work of others.2011-08-08
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    So I read through the "paper" that comes up with you google Wildberger Set Theory, the one that he mentions in the YouTube comment. I would not recommend it. It is very harsh and punitive. Pejorative. Other p words that mean the same thing.2011-08-08
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    I was going to ask a similar question recently, "Do (non-computable) real numbers exist?" but I felt it was too vague, and too dependent on whether the universe is finite and discrete or not.2011-08-08
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    I may be mistaken, but I believe the Banach-Tarski paradox hinges on equivalence classes of the reals.2011-08-08
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    @mixedmath: You've been right. I read it and regret it now. =)2011-08-08
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    @André: Do you happen to know of a reference containing the basic ideas of this "rational trigonometry" presented in a concise and non-pedagogical way with as little ideological distractions as possible along the way?2011-08-08
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    @Theo: Sorry, I don't. Read Wildberger's book, well, not truly, perhaps the first half.2011-08-08
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    @André: That's what I feared to hear, I hope I'll be able to find the needles in the haystack... Thanks for the quick response!2011-08-08
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    @Theo, the basic idea (as I understand it) is to replace length and angle measure with quadrance and spread, where the quadrance of a line segment is the square of its length, and the spread of two crossing lines is the square of the sine of the angle they make. Wildberger shows that this makes many trig problems easier.2011-08-09
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    @Gerry: Thanks for the summary. Yes, that's about as much as I was able to extract from his texts available on the web. Unfortunately, I'm so thoroughly brain-washed by the malicious guild of logicians that I'm not particularly interested in questions on (non-)existence of infinity and problems of ZF when I'm in the mood of learning some geometry...2011-08-09
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    @Theo, it's quite possible to read the geometry/trigonometry parts of the book without worrying about foundational questions.2011-08-09
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    @Gerry: Thanks, I'll certainly have a look.2011-08-09
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    See also this [question](http://math.stackexchange.com/q/135581/242) on a related Wildberger lecture.2012-04-23
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    I skimmed through [this](http://web.maths.unsw.edu.au/~norman/views2.htm). He writes, for example "Let us have a look at these 'Axioms', these bastions of modern mathematics. `[list of very informally stated axioms of modern set theory]` All completely clear? This sorry list of assertions is, according to the majority of mathematicians, the proper foundation for set theory and modern mathematics! Incredible!" He also complains about "lack of information" or something of the sort: But here is a very important point:2012-10-08
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    we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence In other words, 'arbitrary' sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a 'sequence'2012-10-08
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    which is not generated by such a finite rule? Such an object would contain an `infinite amount' of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics.2012-10-08
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    This is another "interesting" argument: "But... the universe is almost certainly finite. Eventually, you and I may have vaporized and rearranged all the stars, furniture and other creatures in our quest to write down yet bigger numbers, and now we are starting to run out of particles with which to extend our galactic hard drive. (...)2012-10-08
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    At some point, you are going to write down a number so vast that it requires all the particles of the universe (except for some minimal amount of what’s left of you) (...) Now here is a dilemma. Once you have written down and marvelled at $w$ in all its glory, where are you going to find $w + 1$?"2012-10-08
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    @Arjang Did you want to put this link: http://www.youtube.com/watch?v=N23vXA-ai5M&feature=player_detailpage#t=745s (or something similar)?2012-10-08
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    @MartinSleziak : I am at work, can't verify the link (youtube is firewalled), but if you think it is relevant, then please go ahead and edit the post.Thank you2012-10-08
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    > I understand the presentation, and the author (in their works besides this one), to mean that the problem with the $\mathbb{R}$ Numbers is **with the way we handle them**. At the risk of this not being a direct answer to the question: As someone who had a very hard time with the way maths were taught in school, Wildberger's methods were a conceptual breakthrough for me - regardless of his higher theoretical foundations. Most mathematicians don't seem to like him because they seem to think he *'solved a problem that didn't need solving'* by doing it in a different way.2012-10-07
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    @NewAlexandria: From what I can tell, most mathematicians don't like him because he has an unreasonable view of the foundations of mathematics and acts as if clearly developed and useful work has no value just because it's occasionally unintuitive. I can't say I've talked about him extensively, but I've never seen a complaint about the actual *math* in rational trigonometry, only the ideology.2012-10-08
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    @RobertMastragostino this goes exactly to my answer, thank you! I've never been able to access many useful areas of Math because it is unintuitive thus far. Mathematicians (seem to) think that, since they can understand math their way, that Wildberger is doing something wrong by presenting another way. I think it's hubris against him.2012-10-08
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    @NewAlexandria: "Doing something wrong by presenting another way"? Right...2012-10-08
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    @NewAlexandria: I don't think anyone is saying that rational trigonometry is wrong or unwarranted. What they object to is him discrediting all work other than his own. He rejects axioms, but ends up just as reliant on them, simply less precise about it. His issue is that *he* thinks *everyone else* is wrong because *they* present another way. A lot of his complaints are pedagogical, not mathematical, and he doesn't make a good argument for abandoning the reals in professional math. See http://scientopia.org/blogs/goodmath/2007/10/15/dirty-rotten-infinite-sets-and-the-foundations-of-math/2012-10-08
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    @RobertMastragostino thanks. This is getting a bit chatty, though. I'll just close by saying that he wouldn't get anywhere or reach people if he didn't hold a unique position - which fortunately is a position that realizes actual value to some audiences.2012-10-08
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    @Arjang The link in the post took me to a youtube channel, not to a particular video, so I tried to find the original video. You are the author of the post, so you know which video you wanted to have there. Anyway, people trying to find the video can see the link in my comment (and other relevant links in other comments), so I think that no harm is done, if the link is not changed. In case it is relevant, the link I provided goes to video called *Universal Hyperbolic Geometry 0: Introduction* at the time 12:25.2012-10-09
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    Quite the contrary, the new thinking tools may allow access to ideas that were obscured with other methods. ---------- I was always a kid that found complete geometric solutions in my head, very quickly - and the same with complex algebra. Trig. killed me at that age. Then I found Wildberger's work and I was back to realizing complex analyses solutions in my head.2012-10-07

2 Answers 2

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If you know about countable and uncountable infinities, consider the following problem:

Is there a subset of the reals whose cardinality is strictly between that of the integers and that of the reals?

Cantor's Continuum Hypothesis says the answer is "No". Godel and Cohen proved that one can neither prove nor disprove the Continuum Hypothesis on the basis of the usual axioms of set theory (ZFC). Some people consider this a serious problem; if we really know what the reals are, we should be able to decide whether or not there's a set bigger than the integers but smaller than the reals. Other people shrug their shoulders and get on with doing mathematics.

If you don't know about countable and uncountable infinities and such, the above won't mean much to you, but then you have some very nice experiences waiting for you.

  • 0
    I'm somewhat sympathetic to him. The set of real numbers has the same cardinality as that of the set of all subsets of the set of natural numbers. But I wonder if an arbitrary infinite subset of the set of natural numbers exists. A concrete subset like the set of even numbers do exist. But an arbitrary infinite subset? You can't construct it in general. You just imagine it. It seems an illusion to me.2016-07-04
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Just realised the same video is also available via a general compilation page.

My question is in first comments, and is answered by the presenter.

Googling "Wildberger set theory" brings up the refrences I was after.

  • 1
    You should re-print the answer here, since we aren't mind-readers. You can then also choose your own answer as the correct answer.2012-10-08