Wrong on the Internet II: Norman J. Wildberger

[jestingrabbit]

Humans, from an extremely young age, think of the number line as an order-complete field. They don't realize they think about it this way, but they do. And personally, I think the definition is very straight forward. Not at all arbitrary.

Humans also, from a young age, think that the product of two negative numbers should be negative. They think that every number should have a next bigger number and a next smaller number. Then think that 1-.999...=.000...1. They think that (a+b)²=a²+b², and √(a+b)=√a+√b. I'm not impressed with what humans think from a young age, and I don't think it makes a very good argument for teaching things a specific way.

Yeah. I was expecting appeals to authority in this thread, but this is ridiculous.

[Mindworm]

But the topology argument is a strong one. Independent of children, I like having a line through the centre of a circle bisect the perimeter. And without the reals about half of all topological terminology becomes useless. And it's the important half ("line", "manifold" etc). The rationals or even algebraic numbers are no replacement for the reals, with their being totally disconnected.

[Eebster the Great]

But the topology argument is a strong one. Independent of children, I like having a line through the centre of a circle bisect the perimeter. And without the reals about half of all topological terminology becomes useless. And it's the important half ("line", "manifold" etc). The rationals or even algebraic numbers are no replacement for the reals, with their being totally disconnected.

The rationals are only totally disconnected in the subspace topology of the reals. If instead you take the (usual) order topology on the rationals, then that topology is connected and has many of the desirable properties of the reals in their usual topology.

Similarly, if you define "paths" based on the rationals rather than the reals, this topology is path-connected.


Note that the disconnectedness of the rationals is not particularly profound. Every totally ordered subset whose complement is dense is totally disconnected.

[skeptical scientist]

The rationals are only totally disconnected in the subspace topology of the reals. If instead you take the (usual) order topology on the rationals, then that topology is connected and has many of the desirable properties of the reals in their usual topology.

No. There are lots of disconnections, i.e. partitions into disjoint open sets, and the connected components are the singletons. That's what it means to be totally disconnected, and is an intrinsic topological property of the rationals. Also, those are the same topology.

Similarly, if you define "paths" based on the rationals rather than the reals, this topology is path-connected.

You can't do that, because then "path connected" doesn't imply "connected".

Note that the disconnectedness of the rationals is not particularly profound. Every totally ordered subset whose complement is dense is totally disconnected.

Complement? What is this complement of which you speak? Being connected, disconnected, or totally disconnected is an intrinsic property, not an extrinsic one.

[Qaanol]

The rationals are only totally disconnected in the subspace topology of the reals. If instead you take the (usual) order topology on the rationals, then that topology is connected and has many of the desirable properties of the reals in their usual topology.

No. There are lots of disconnections, i.e. partitions into disjoint open sets, and the connected components are the singletons. That's what it means to be totally disconnected, and is an intrinsic topological property of the rationals. Also, those are the same topology.

I don’t quite understand what you’re saying. What is a partition of ℚ into disjoint open sets? I don’t see it as being possible except for the trivial partition into one open set (and perhaps some copies of the empty set). A singleton subset of ℚ is not open in ℚ, since every neighborhood of a rational contains infinitely many rationals.

[Eebster the Great]

The rationals are only totally disconnected in the subspace topology of the reals. If instead you take the (usual) order topology on the rationals, then that topology is connected and has many of the desirable properties of the reals in their usual topology.

No. There are lots of disconnections, i.e. partitions into disjoint open sets, and the connected components are the singletons. That's what it means to be totally disconnected, and is an intrinsic topological property of the rationals. Also, those are the same topology.

I don’t quite understand what you’re saying. What is a partition of ℚ into disjoint open sets? I don’t see it as being possible except for the trivial partition into one open set (and perhaps some copies of the empty set). A singleton subset of ℚ is not open in ℚ, since every neighborhood of a rational contains infinitely many rationals.

I think he is right, actually. The sets (-∞,√2) and (√2,∞) are still open in ℚ; the left interval contains all negative rationals and all nonnegative rationals whose square is less than two, and the right interval contains all positive rationals whose square is greater than two. I was thinking it was necessary to define the real numbers to do that, but clearly it isn't, given all unions of open sets are open (and (-∞,√2) for instance is the union of all left-rays (-∞,x) where x is as I said above).

It is this fact about reals that allows us to define the reals, actually.

[skeptical scientist]

Aesthetically, I prefer {x : x²>2} and {x : x²<2}.

[Qaanol]

Ah, right. So the point is that, for any two rationals p and q, there exists a partition of ℚ into open sets such that p and q are in different sets?

[skeptical scientist]

Ah, right. So the point is that, for any two rationals p and q, there exists a partition of ℚ into open sets such that p and q are in different sets?

Exactly. Hence ℚ is totally disconnected.

[Eebster the Great]

Aesthetically, I prefer {x : x²>2} and {x : x²<2}.

Using interval notation does make it clearer that these sets are open, though.

[Yakk]

Aesthetically, I prefer {x : x²>2} and {x : x²<2}.

Using interval notation does make it clearer that these sets are open, though.

But using elements from not-the-set to describe subsets of the set seems impolite.

And > is tied enough to open sets as it is.

[skeptical scientist]

Aesthetically, I prefer {x : x²>2} and {x : x²<2}.

Using interval notation does make it clearer that these sets are open, though.

Well, yes and no. Remember that (-∞,√2) is not a basic open set in the order topology on the rationals, so showing that it is open requires much the same argument as showing that {x : x²>2} is open (or else a proof that the order topology is the same as the subspace topology viewing Q as a subspace of R).

[Eebster the Great]

Aesthetically, I prefer {x : x²>2} and {x : x²<2}.

Using interval notation does make it clearer that these sets are open, though.

But using elements from not-the-set to describe subsets of the set seems impolite.

I don't know, that seems pretty standard to me, given how the rationals work as a subset of the reals.

[silverhammermba]

Well... anyway.

I'm sorry that my rhetoric offended you, jesting. I said that Wildberger and Gabriel would be "best friends" because they both want to abolish irrational numbers. That's it. I have already pointed out that I am aware that their approaches are fundamentally different. You seem to be consistently missing my point.

I agree that Wildberger's all rational approach would be very useful for applied mathematicians - especially if they could start off with it at square one and never have to worry about the nuances of real numbers that almost never come into play in applications. However imagine if Wildberger had it his way and we ditched real numbers entirely. Vast swathes of mathematics would have to be completely rebuilt from the ground up with a reemphasis on explicit calculation, error bars, and precise decimal expansions. Other fruitful fields would need to be abandoned entirely!

I'll even agree that pre-college math education focuses too much on real numbers - especially since most high school math students will never see a rigorous treatment of the subject. But in this lies a deeper, constant struggle of math education. On the one hand there are the people who will only ever use math for strict applications and don't require a deep understanding of pure mathematical theory. On the other hand there is a definite need for pure mathematicians and it would be remiss to not even give high schoolers a taste of pure mathematics to acclimate them to the subject.

Wildberger seems to have decided (arbitrarily) that the only numbers that are worthwhile are those that can be explicitly written as fractions or finite decimals. But why does this matter? Using our extensive knowledge of the real numbers, it is entirely within our capability to create real world approximations of irrational numbers with arbitrarily small error bars. By Wildberger's approach, isn't that good enough? Indeed, isn't this the entire stance taken by those who accept the reals? Leave everything in symbols and abstractions and then - if you really, really need a decimal expansion - apply a rigorous method of approximation to find the number with sufficiently small error.

Wildberger's math is 100% correct. However his agenda and his argument supporting it are fundamentally wrong. On the internet.

[skeptical scientist]

Well... anyway.

I'm sorry that my rhetoric offended you, jesting. I said that Wildberger and Gabriel would be "best friends" because they both want to abolish irrational numbers. That's it.

That's not what you were doing at all. In the context of your original post, you were saying that they would be best friends, and inviting the reader to form a mental image of Wildberger based on that piece of information (and only that piece). In other words, you were intentionally painting a picture of a similarity between the two of them, which, it turns out, was tremendously misleading.

[silverhammermba]

I resent words being put in my mouth. I know what I meant, and I have edited the OP to make clear the distinction.

[capefeather]

I don't have the time to look at this stuff, but just from reading the thread, I'm confused as to what this man's goal is, and what we're talking about exactly.

If we're talking about high school education, then I'm pretty sure most people would agree that something is wrong with the way in which concepts are introduced, but removing discussion of irrational numbers entirely seems to be a wrong way of fixing it. (I like the idea of not hand-waving infinity or injecting the idea that irrational numbers are infinite, non-repeating decimal strings.) I think that high school students should get some small amount of exposure to how a working mathematician thinks in a math course. You don't have to learn music "properly" to make a living off of making music (as sad as that is), but that doesn't mean that music students shouldn't experience the way of the professional through discussions of concepts like phrasing and counterpoint.

If we're talking about university math, well, my opinion on that is pretty similar. You don't have to have an intimate understanding of how a machine works to use it correctly. Sure, there's always the danger of using it incorrectly, but not using the machine at all seems a bit extreme, and when there's likely a fellow nearby who HAS looked under the hood, it seems almost ridiculous. A computer will look at the rational indicator function and think it's equivalent to f = 1; that doesn't make ME wrong.

[skeptical scientist]

I think a reasonable thing to try for introducing the real numbers (when students are already familiar with fractions) would be something like the following:

• Introduce the number line as a linear representation of all real quantities.
• Plot some rational numbers on the number line.
• Explain that there are some quantities which are not rational. (E.g. the ratio of hypotenuse length to leg length for an isosceles right triangle.)
• This shows that there are gaps between rational numbers on the number line which are not filled by any rational number. This effectively introduces the idea of irrational numbers, such as √2.
• How to deal with such numbers when they arise? Approximate.
• Introduce finite decimal expansions as a way of representing fractions with denominator 10ⁿ, and talk about approximating real numbers with (finite) decimal expansions. Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11). Some numbers are not exactly equal to any (finite) decimal expansion, but can be approximated as closely as you like by finite decimal expansions.
• Never mention the notation of infinite decimal expansions. Pretend as if it doesn't exist, and if a student asks about it, explain that you need calculus to really define it (i.e. "I'll tell you when you're older").

I'm not sure whether this would work better or worse than the existing curriculum, but I think it would be worth a try.

[Tirian]

Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11).

Wait, what? 5*11 = 55, 7*8 = 56, therefore 5/7 < 8/11.

My treatment in GED classes is similar to what you suggest, but even a bit simpler. I don't bring up irrational numbers here, which is just as well since finite decimals aren't the solution. One of my central observations is that when things are observed a lot in the real world, mathematicians tend to create new theories or shorthand notations to address them. Another example is that trigonometry was developed because our ancestors noted that the world is full of right triangles that need to be measured. And there are a lot of fractions whose denominators are powers of 10, so we need decimals (by which I mean finite decimals). Soon enough, we get to the next stage which is that denominators of 100 are so prevalent that we made percentages to deal specifically with them.

On the not mentioning infinite decimals, I would go so far as to say that there we should make a giant jar and every mathematician in the world (not just teachers) should put a quarter into it every time they write "..." in a formula. It's very very often lazy, and that's especially dangerous when it invites us to use our intuition to contemplate notions of infinity.

[skeptical scientist]

Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11).

Wait, what? 5*11 = 55, 7*8 = 56, therefore 5/7 < 8/11.

Yes, you can do it by multiplication, but that's slower than comparing decimal representations (in computer science terms, multiplication of n-digit numbers is not O(n), but comparison of n-digit numbers is).

[Qaanol]

Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11).

Wait, what? 5*11 = 55, 7*8 = 56, therefore 5/7 < 8/11.

Yes, you can do it by multiplication, but that's slower than comparing decimal representations (in computer science terms, multiplication of n-digit numbers is not O(n), but comparison of n-digit numbers is).

Well, for the most part anyway.

[Tirian]

I'll grant that ordering decimals is easier. But hauling out big-O is not a sensible metric for talking about a comparison, because schoolchildren are never asked to differentiate between numbers that have large n. I continue to maintain that they are both easy, certainly much easier than making a claim that might force you to explain big-O to a sixth grader.

[Yakk]

Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11).

Wait, what? 5*11 = 55, 7*8 = 56, therefore 5/7 < 8/11.

Yes, you can do it by multiplication, but that's slower than comparing decimal representations (in computer science terms, multiplication of n-digit numbers is not O(n), but comparison of n-digit numbers is).

I dunno -- is turning a pair of numbers into decimals with enough precision to distinguish them fast enough?

The worry I'd have is that the number of decimals you'd need to calculate would grow. But I guess we might be able to guarantee that to be bound by some constant times the number of digits in the denominator?

[Eebster the Great]

Observe that there is an easy algorithm for comparing numbers in this representation to see what is bigger (unlike, say, 5/7 vs. 8/11).

Wait, what? 5*11 = 55, 7*8 = 56, therefore 5/7 < 8/11.

Yes, you can do it by multiplication, but that's slower than comparing decimal representations (in computer science terms, multiplication of n-digit numbers is not O(n), but comparison of n-digit numbers is).

I dunno -- is turning a pair of numbers into decimals with enough precision to distinguish them fast enough?

Well of course if you have to actually carry out the division you don't save any time. I think the assumption was that numbers could be stored as decimals in the first place.

[mr-mitch]

Turning a rational number into a decimal string is O(n) for n decimal places. All you need is a times table. Then each new digit requires a look up (binary tree would be good for this), and a calculation of the remainder. So if you do them in parallel (compute next digit(s) and compare) then it's still O(n).

(The tree only has 10 entries so it doesn't grow)

[Yakk]

Ah yes, you take the denominator, build a 0-9 multiplication table.

Then for each digit in the numerator, multiply the accumulator by 10 and add the digit (as the accumulator is bounded by the denominator, this is O(lg(denominator)). Search the table for the value that is just less, then subtract (O(lg(denom)) again).

As there are lg(numerator) digits, this is O(lg(numerator)*lg(denom)) time.

In terms of length of input, this (naive) example is O(nm), where the numerator has n digits and the denominator has m.

Assuming the digits before repeat are bounded by a linear function in n and m (which I think they are), this gives you O(nm + m^2).

In comparison, FFT multiplication can be done in time linear in the length of the values (admittedly with a large constant). Comparison for equality can also be done in time linear in the number of digits.

So I don't see a O-level domination of decimal digits over rational equality or comparison.

[skeptical scientist]

I'll grant that ordering decimals is easier. But hauling out big-O is not a sensible metric for talking about a comparison, because schoolchildren are never asked to differentiate between numbers that have large n. I continue to maintain that they are both easy, certainly much easier than making a claim that might force you to explain big-O to a sixth grader.

Big O notation was only one way of going about things, and certainly I wasn't suggesting explaining it to a middle-schooler. In practice for schoolchildren comparing decimals is a lot faster than comparing fractions, and I can't imagine you would have to go to any extraordinary lengths to convince them of this. If you wanted to compare, say, 61/87 and 68/97 you would have to perform a pair of two-digit multiplications, which are hard to do in one's head, so one would either have to juggle a lot of numbers in one's head, get pencil and paper, or use a calculator. On the other hand, it is quite easy to compare 0.70103 and 0.70115. The point is that comparing two decimals continues to be easy even when numbers have many digits, whereas comparing fractions can be rather tricky.

Moreover, this sort of problem comes up all the time in real life. Say you're in a grocery store and you want to get rice as cheaply as possible, but different bags have different prices and different sizes. In such a situation both the quantity and price will routinely be two or three digits. Do you compute which is cheaper by mental arithmetic, or do you just look for the little place on the price tag where it shows the price per unit?

FFT multiplication can be done in time linear in the length of the values (admittedly with a large constant). Comparison for equality can also be done in time linear in the number of digits.

So I don't see a O-level domination of decimal digits over rational equality or comparison.

Well, I'm talking about the ease for schoolchildren, rather than computers. And schoolchildren rarely compute products using Fourier transforms.

However, even with computers, comparing decimals is faster than comparing rationals. FFT multiplication is not quite linear, and has a much larger constant, and even the linear time algorithms someone linked (which rely on certain models of computation) will probably be at least an order of magnitude slower. But in practice, with a computer, both are so fast that you rarely care.

[Eebster the Great]

But in practice, with a computer, both are so fast that you rarely care.

The speed at which an APU multiplies floating point numbers is a standard metric of its performance and the limiting factor in the speed of quite a bit of research in many fields (particularly scientific research).

[Yakk]

Well, I'm talking about the ease for schoolchildren, rather than computers. And schoolchildren rarely compute products using Fourier transforms.

This is the problem with kids these days. School is too easy.

[Beaver SM]

I've read through the thread and seen part of Norman's video. It's pretty interesting. I can see his point on there are difficulties with the reals and I think many mathematicians have thought of that too. However I disagree on abolishing the real on curriculums or research. I work in topology so there haven't been any problems with the reals. And there is nothing wrong with teaching highschooler the real numbers, they are not ready to see the problems with the real anyway - I think most people don't have a problem. For the applied mathematicians want to deal with that problem, so be it.

Unlike Gabriel. Yesterday I bump into him on some forum and found that he was trying to convince me that complex numbers are not well-defined since i^2 = -1 would imply that i=sqrt(-1) is real just like 2^2 = 4 implies that sqrt(4) is real >.< He then continued to write that Euler's identity e^(i*pi) = -1 is rubbish because that would implies ln(-1) = i*pi >_<
I tried to help him to realize that the real are not closed under square root and that the natural log definition he's using e^a = b <=> a = ln(b) only applies when b > 0 in a really nice way. He ended up deleting my comment (since for some reason on ResearchGate the OP has full power on censoring comments) and then says that "it seems that Beaver seems to think that i is real..." putting words into my mouth after deleting my comment so I didn't have any evidence...

And about his countability of the real... oh gosh. I wanted to be nice and help the guy out but seem like I failed

[Eebster the Great]

And about his countability of the real... oh gosh. I wanted to be nice and help the guy out but seem like I failed

Well it sounds like you tried pretty hard. There are some people out there who will never be convinced of a given argument, even if it is literally mathematically proven. (Though I'm sure he wouldn't accept most of your definitions anyway.)

[lightvector]

Hmmm... I wonder if John Gabriel would accept the following construction of "pair-numbers" as well defined:

Every pair number is simply a pair (a,b) of real numbers a and b. We define two operations we can do to pair numbers. We have "pair addition" defined as (a,b) + (c,d) = (a+c, b+d), and "pair multiplication" defined as (a,b) * (c,d) = (ac-bd, bc+ad).

This also works just as well if a and b are restricted to be rational, rather than real.

[capefeather]

You know, I've thought about the question of why we need infinite sets to describe 3/4 of a pie. The infinite sets actually reflect intuition very well. If I consider the letter "a", would I think of it as the thing that I just bolded, in that specific font, in that specific bolded state? Of course not; I would think of it as an abstract concept that can be given visible form ("mapped") through font/handwriting/etc. This thinking wouldn't change if there were infinitely many fonts or infinitely many people with different handwriting. This can be applied to any letter or word in the English language, or any language, even.

[smectymnuus1]

I think many people who are denigrating Wildberger are missing some really big points.

He has at least the following 4 strands to his work:

(1) Traditional non-controversial Mathematics research published in reputable peer reviewed journals (most of which is >= 5 years old).
(2) A new approach to Trigonometry, "Rational Trigonometry" which doesn't invalidate the traditional approach, but contends it is much clearer, natural, and intuitive done the "rational" way.
(3) Further to (2), a new way of looking at geometry in general (Universal Geometry he calls it), which among other things outlines a new and purportedly better way to do Hyperbolic Geometry
(4) Following on from (2) and (3) but independent of them, he has come to doubt the traditional approach to some aspects of mathematical foundations (notable the Real Numbers as conventionally defined), and has some suggestions as to how it could be better done, but by his own admission does not have anything approaching a viable alternative at this point. What he contends is that there are serious problems currently that are not properly addressed or even widely appreciated, and he hopes to provoke a re-think.

I have met Norman, and he is a completely sane, orthodox and very clever character. Even if you would disagree with various of his contentions, you would find him courteous and thoughtful. He has pondered very hard about what he is proposing and is very knowledgeable about the history and practice of Mathematics, and is 100% aware of the subject matter he is dealing with - ie he knows all about how the real numbers are defined and infinite set theory etc. There are many others who have had similar misgivings eg Brouwer, Kronecker etc

My own view on his work is the following:

(2) is a wonderful new approach to the subject - I wouldn't imagine it is going to replace traditional Trig any time in the foreseeable future, but if you examine what he has done, it is impressive. He does trig without transcendental functions, and reinterprets old results, and generates novel ones. Here I am thinking of the theory of the subject rather than the practise of it in High School.

(3) is an amazing advance that will come to be appreciated as time passes. His model for Hyperbolic Geometry is different to and superior to the conventional Klein-Beltrami or Poincare models. He has obtained completely new stunning results that can be interpreted and validated in the old setup, but come naturally out of his system. I know that is a big claim but I am completely convinced. For example, in his paper at "arXiv:0909.1377v1 [math.MG] 8 Sep 2009", the final "48/64" theorem is profound. I have seen it demonstrated in Geometers Sketchpad and it is not only true but beautiful. He links (2) and (3) together which provides more evidence in support of his approach in (2)

(4) I am less sure about. In fact I am probably not sufficiently equipped with deep knowledge about these foundational issues to pass judgement. What I am sure about is that he is serious about it and is not just superficially taking a swipe at something he does not understand.

There are many people around peddling their new solutions to FLT or the RH or the 4CT etc or disagreeing with conventional ideas without really knowing what they are talking about. Wildberger is the EXACT OPPOSITE of these people ! Look carefully into what he has done, particularly (3), and you be well rewarded.

[Eebster the Great]

^ITT: Norman J. Wilderberger.

[smectymnuus1]

^ITT: Norman J. Wilderberger.

I attended his 3rd/4th Year Course on Rational Trigonometry at the University of NSW, where I learned about his approach to Trig -and also bought the book. I then followed his Algebraic Topology Course at University of NSW, which was later posted on YouTube. Since then I have been watching his various YouTube series when I can, and have looked at some of the papers available from his website. I recommend you watch the first 3 or 4 UnivHypGeom YouTube videos to get an idea of what he is doing there - it is a new approach which doesn't lose what has been achieved by the classical approach, which I studied when much younger, but adds a lot. I have been convinced that this is a superior approach (amazing as that claim may sound, given the giants who formulated the classical approach). One of the reasons why geometry is enjoying a strong renaissance in recent years is the availabilty of Geometry Packages such as Geomters SketchPad which open new avenues for exploration of geometric properties and have seen the discovery of elementary theorems (eg Hoehn's Theorem) which could have been discovered in Ancient Greece but were not, due probably to the inability to experiment and measure which the new software provides. Wildberger use this heavily and has a whole slew of new results.

[isomorphismes]

Silverhammermba, I can't believe you would evaluate a mathematical claim based on rank or how normal it sounds. That's just not the way we're supposed to do it. Since this is about irrationals, how about that fable about the Pythagorean cult drowning the guy who proved &phi; was irrational? They didn't want to hear it, he was weird, he wasn't getting published in journals anyway. We look back on that story and think, How silly! Whether you like the result or the person doesn't matter at all.

I have watched a few of Wildberger's videos and I think he has a few valid points. For example, 32&deg; isn't constructible, which can get under your skin only if you think about it too long.


Por mi parte, I see it this way:

&bull; We know mathematicians were confused about ℝ for quite a long time. Hence the Newton/Berkeley debates, hence Weierstrass making everyone learn calculus the hard way (&epsilon;, &delta; style rather than with infinitesimals). Dedekind eventually figured out that we need a pair of open sets to define a single number. It's not logically wrong or something, but it is rather hairy when all you work with is double-precision floating point numbers.

&bull; But for most of the results mathematicians are just talking about a general field (I prefer the term 'corpus' so as to not confuse a number field with a vector field or scalar field). I only know a little Galois theory but my feeling from studying just that bit was "We can adjoin our way to whatever field we need" (e.g. ℚ adjoin &radic;&minus;1) and thus there's no need to wheel out the Rube Goldberg device of Dedekind Sets just to talk about anything. The algebraic numbers are a very nice field, why don't we just use those?

&bull; Various problems _do_ exist within the reals, Vitali sets and the Banach-Tarski paradox. Either that bothers you or it doesn't, as someone else mentioned most mathematicians do not care about foundational issues because it's too "philosophical" (a four-letter word). But if Norman Wildberger's esthetic is that P is horribly ugly, let him build his own thing without P, what does it harm us? As long as his inferences are valid it can just be "a different mathematics", and there are plenty of people who do such things just because the way other people do it bothers them. Saw a video of Tim somebody who wanted to redo topology from a basis of open lines (something he defined) rather than open sets. Same thing: who are we to stop him from using a different starting point or trying to end up in a different place? Just because it doesn't resemble what everybody else does, as long as it's valid there should be no complaining.




My personal predilection is towards the topos idea "Bring your own set theory". You can watch a cool talk by Chris Isham on Oxford or Cambridge's website about how physicists, if you push them, will admit that "points don't really exist" so maybe one wants to rework things point-free and see if at the end of 10 years' labour anything useful came out the other end. Also if you are interested in foundational stuff the ncatlab.org is a better place to go hunting than Wikipedia.



Anyway, these ad hominem attacks are so out of place. Who cares about a Slashdot review etc.

[isomorphismes]

Jestingrabbit,

I'm not sure what his precise reasons are, but I'm not sure that you've given him a very fair hearing on this. I think his clearest reason for rejecting them is that they are unnecessary, that they don't give us anything practically or theoretically that we really need. Practically, I'd say its very hard to argue with that. Theoretically, I think that you do get things from the reals that you don't get from other sets, but I do think his discussion about infinite decimals is a little more complicated than "infinity is big".

I agree with you that Wildberger needs a fairer hearing and I think you are homing in on it.

If, though, it is true that he says we don't need the irrationals, then I disagree with him. &radic;2 is just too frikkin fundamental to throw away. And various algebraic numbers like the ones you use to solve any kind of molecular Schr&odot;dinger, at least as far as physical reality (to say nothing of theory/ abstractions) I would more believe that the number field / corpus for those quantum solutions exists (or whatever the finally correct eq'ns are) than the integers (which I see as a linguistic / human construction).

My perspective on Wildberger: watched some of his algebraic topology lectures, liked those, clicked thru to the Wild Trig videos and thought: "Oh, hey! This is cute. A little weird. But yes I do prefer your geometric constructions to the "Assume 43&deg; exists!" So I can't say I could represent his views well.

[isomorphismes]

Mindworm,

But the topology argument is a strong one. Independent of children, I like having a line through the centre of a circle bisect the perimeter. And without the reals about half of all topological terminology becomes useless.

Topology is about neighbourhood relationships. These do not fall away if the base field is non-real. Eg topology of graphs is quite discrete.

What you're suggesting would be like, "We invented all these definitions with &epsilon;'s in them to deal with the real numbers! And now you're going to tell me we don't need the reals?" Almost like we wrote this bloated code so now we should use it, or we have this buggy whip factory so we're darn well gonna make buggy whips. Not logical IMO.

In the NJ Wildberger videos I saw on Youtube, he goes to extra lengths to make geometrical arguments, thus preserving the meat of the proof in a coordinate-free manner. I think that's better (less dead weight) esthetically.

[j0equ1nn]

I first should confess I didn't read every post on this forum, but I feel I have something to offer on the topic. I've corresponded with the guy and I have a copy of his book.

My opinion is that whether you agree with what Norm Wildberger is trying to do or not, it is a highly rigorous approach to a theory that runs contrary to many ideas that are often accepted without question, and he has a respectable reputation in the mathematical community. He has a PhD from Yale, is a full time faculty member at a University, has published and continues to publish many papers in reputable journals, and based on the devotion he puts into his Youtube stuff, seems to have dedicated his life to his ideas. I also would add that it takes balls to disagree with accepted conventions in math because many (more insecure) people will just ditto what they've heard for fear of being considered stupid. And he definitely backs up his stuff. To read a one-line summary of his work and categorically say "Well I disagree because we need irrational numbers" is an extremely short-sited reaction, not even worth acknowledging considering the amount of rigor he has put into his thoughts.

I contacted him the first time I was involved in some independent research because his methods of avoiding irrational numbers seemed like they could provide some great shortcuts in extending what Joseph Silverman was doing in the early 2000s, about dynamical systems of rational maps on the complex projective line, to the complex projective plane. He was very friendly and provided me with copies of some papers that were not yet published, without even knowing me. It seemed like he was genuinely more interested in the possibility that his ideas could help further the research than he was in getting credit for anything. Unfortunately it didn't go anywhere, but that was my fault not his, as I said it was my first research project. But the guy made a better impression on me than your average person.