Wrong on the Internet II: Norman J. Wildberger

[silverhammermba]

So here's another mathematician I found on the internet with some pretty strong, interesting views. He has draws the same conclusions as the infamous John Gabriel, however his approach and argumentation is fundamentally different.

Here he explains the basics of "Universal Hyperbolic Geometry"
http://www.youtube.com/watch?v=wHQwJhL5i0E

Spoiler:

it's all just an attempt to get rid of irrational numbers

It starts off a bit slow, helped by his long-winded explanations and dramatic pauses for effect, but it really starts getting good around 20:00 minutes. He has tons of YouTube videos, though I have not had the pleasure of watching most of them yet. He also wrote a book six years ago called "Divine Proportions" subtitled "Rational Trigonometry to Universal Geometry". It's available for sale from Wild Egg Books, which is

a new, small publishing company specializing in high quality, ground-breaking mathematical books.

despite the fact that their only product is Divine Proportions and that they appear to be owned by Wildberger himself.

Fun.

Edited for clarity.

[xkcdfan]

I don't feel like listening to this drivel, anyone have a transcript?

[jestingrabbit]

a new, small publishing company specializing in high quality, ground-breaking mathematical books.

despite the fact that their only product is Divine Proportions and that they appear to be owned by Wildberger himself.

This isn't his only publication though. If you have access to mathscinet you'll see what I mean. He gets published in journals, and he is actually employed by a maths department.

He takes a stand against real numbers. He says that they represent a problematic way of looking at things. He's got his reasons, and he's written them up.

Comparing this guy to an antisemitic moron like Gabriel is completely unwaranted.

[z4lis]

I read one of his papers. He appears to dislike the real numbers, indexing, the fact that mathematicians like to form equivalence classes ("But why should we have to consider infinite sets to talk, for example, about three quarters of a pie?"), anything that might already be established by mathematics, and doing anything but informal exposition and definitions. He finally actually does prove something, which he calls an "Inventory Version of Hall's Theorem" that works for multisets that I'm almost certain I could make equivalent to the plain old vanilla Hall's theorem but don't care enough to do so. Perhaps not.

Meh. I would guess that he's not wrong as much as he's full of himself?

http://web.maths.unsw.edu.au/~norman/pa ... isets5.pdf

Edit: In the video he makes the statement "Nobody has come up with a proper theory for infinite decimals." I mean, I don't know what in the world he wants from a "proper theory" other than it should avoid infinite processes like forming equivalence classes. His reasons for not liking nonrational numbers seem to be:

1. Their decimal expansions are really long! And take an infinite amount of memory and time to compute! How can such a thing exist?
2. If you write out the decimal expansions for two nonrational real numbers, how can you add them up? Oh, my! You have to "carry" an infinite number of digits! We can't do arithmetic anymore! We can try to represent nonrational reals by computer programs, evidently! But how can we figure out how to add computer programs? Nobody knows how. And there are soooo many computer programs that give the same number?

[jestingrabbit]

His reasons for not liking nonrational numbers seem to be:

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".

If you consider the situation with rationals, or even Rat, you have several things that are nice. You know exactly what you're writing down, you have a simple description of the arithmetic of the two usual operations, and you have a simple procedure that decides equality.

By contrast, if you "write an infinite decimal" what is it exactly that you're doing? I might write 3.14... and its reasonable to ask what precisely that "..." is. One could contend that it is some function from the naturals to digits. However, its not an arbitrary function, its a particular function. Its reasonable, imo, to point out that testing whether two such functions are equal isn't something that we can test with an algorithm, and that even if we have a pair of algorithms, that doesn't help us. So, we don't have an easy time deciding equality anymore, even if we restrict to computable numbers.

Regarding the operations, it doens't seem to be as straight forward to make it precise as one might hope. To get into addition in some depth, let a = 0. a_1 a_2 a_3\ldots and b = 0. b_1 b_2 b_3\ldots. The most straightforward definition of addition seems to me to be to define s_i = a_i + b_i \pmod{10} and

c_i = \left\{ \begin{array}{ll} 0 & \text{if}\ a_{i-1} + b_{i-1} < 10 \\ 1 & \text{otherwise.} \end{array}\right.

and then have a + b = s + c, which is pretty clearly problematic. If one keeps making the reduction, forming s^{(i)} and c^{(i)}, is one guaranteed that there is some i such that c^{(i)}=0? If you try to fix it up with limits, you'll notice that the definition requires that you take the difference of numbers ie that you use the calculation that you are trying to prove is well defined. This is reparable, but its perhaps something worth thinking about.

Now I would ask you to actually think about how multiplication like this would work, not just hand wave it away. I think he has justifiable reservations about the real numbers.

[Marbas]

My strong and immediate dislike to anyone who wants to throw out irrational numbers aside, here is a slashdot review of his book.

It is not favorable. It is also hilariously off the wall. And it's also kind of bad. But the comments are interesting.

[z4lis]

This is going to a pretty incoherent ramble. I just woke up.

@Jesting. I can convince myself to accept that the nonrationals aren't terribly useful in some practical sense. Since all measurements taken have a pretty small degree of precision, there's no practical need to have numbers with "infinite" precision. However, I don't feel that mathematics really cares about how well the scientist or engineer's measuring tools are, because we can be as precise as we like in our minds. And when doing mathematics, it's convenient to have continuous functions take on their extrema on closed, bounded intervals and have bounded sequences have convergent subsequences and know that there actually is something that you can justifiably write down that represents the length of a hypotenuse of a right triangle with unit leg lengths. It makes the theory precise, which I personally find... although I hate the word, beautiful. It's aesthetic. I know that there's "no such thing" as a "real" right triangle with unit leg lengths. If you draw up a triangle, any triangle you like, I know that all of the side lengths will be unmeasurable to full precision. I can't even tell if the side are truly straight.

And do you want a simple algorithm to add two "infinite decimals"? Here's one. Truncate both terms in the sum to the same number of decimals, say n. Add those truncated decimal expressions. Doing this for each n results in a series of numbers that approaches the actual number. Voila. You can produce truncated sums who are as close to the actual value as you please in this way. The whole argument seems to me to reduce numbers to a particular method of representation, note that performing the basic operations is difficult in this notation, and then conclude that the original numbers were "wrong." It seems analogous to an ancient Roman concluding that noninteger rationals were impractical because writing them down, defining multiplication and addition, and the like, are so difficult to do.

I did give the guy a fair shake. I watched most of the linked video, read one of his papers, and watched one of his other videos. His language is full of remarks about how inelegant the current system of mathematics is, how his changes to the foundations make things more beautiful, practical, etc., that he has figured out how to "do it right", and the like. Your crank alarm should be going off when your read stuff like that in a paper. But once he got all his philosophical background material out of the way, it seemed like his paper progressed in a fairly logical, mathematically sound matter. He made definitions, proved some stuff, great. I'll probably actually watch his lectures on hyperbolic geometry, because I'm really curious to see what in the hell he's doing, and I'm expecting to see some sound mathematics padded with the crazy garble you get from loonies. Is he a loony crank? No, not really. Is he about as close as you can get to that and still do actual mathematics? Seems like he's close, yeah.

He also mentions the fundamental theorem of algebra in his video. Are there even proofs of that that don't involve analysis? I mean, of course there aren't, because your polynomials don't all have roots in the complex rationals, but I guess he'd want to be able to approximate them arbitrarily well. Can one prove that without using standard analysis?

[antonfire]

Can one prove that without using standard analysis?

You can take pretty much any simple analysis proof and turn it into a 'finitistic' proof. Take the usual winding number proof. (If f is a degree n polynomial, the winding number of the image of a small loop around 0 via f is 0, the winding number of the image of a large loop is n, so continuously enlarging such a loop changes the number at some point, which involves crossing the origin.) Replace circles with sufficiently fine regular polygons. The winding number is even easier to define when you restrict yourself to piecewise linear curves; and instead of resizing the loop continuously, you resize it in small steps. The argument proceeds pretty much the same way, only with a lot more mess because we haven't set up the nice terminology.

I don't know if you count this as "without standard analysis", but standard analysis proofs (just like standard analysis statements) have content that can be translated into a framework where we don't talk about stuff like "limits" and "real numbers". In fact, let me tentatively make the claim that they wouldn't be much good otherwise.

[jestingrabbit]

And do you want a simple algorithm to add two "infinite decimals"? Here's one.

And it falls prey to exactly what I mentioned. The usual epsilon delta definition of a limit has within it arithmetic operations. You're trying to prove that arithmetic operations are well defined, and you're using concepts that, in some formalisms, rely on that arithmetic.

[Tirian]

I think he has justifiable reservations about the real numbers.

There are several separate issues here. One could argue that there is value in studying how close we can come to analysis over a metric space that is not complete, just to have a broader understanding. To that degree, Wildberger is performing an act of mathematics, and it's kind of cool. I don't know if there are any applications in the world for a rational geometry, but if so they can install a trigonometry.

On the other hand, the notion that we should modify the high school math curriculum to teach a skepticism on the existence of the real numbers just because there was a survivor from the philosophical defeat of finitism is crazy talk. I'll grant you that the construction of reals from the rationals using 10-adic numbers is unpleasant, but Dedikind cuts and Cauchy sequences are not. If Wildberger's criticism is truly that you shouldn't need infinite sets to talk about 3/4 of a pie (as someone upthread suggested), that is a wholly irrelevant aesthetic argument.

I will go so far as to say that I would embrace our intellectual reliance on the real numbers even if it were somehow shown that our universe is bounded and wholly quantized, so that every quantity we have ever perceived would be rational and that there would be an upper bound on the conceivable "natural numbers". In the same way that naive set theory continues to be both understandable and useful in spite of being inconsistent, our understanding of the reals give us highly sound predictions in science and engineering and every other walk of life, and there would be no need for 99.9% of us to abandon it even if it weren't true. To be more colloquial: "Calculus: it works, bitches".

[Qaanol]

And do you want a simple algorithm to add two "infinite decimals"? Here's one.

And it falls prey to exactly what I mentioned. The usual epsilon delta definition of a limit has within it arithmetic operations. You're trying to prove that arithmetic operations are well defined, and you're using concepts that, in some formalisms, rely on that arithmetic.

The epsilon-delta argument can be rephrased entirely in terms of rational numbers. For the case at hand we have an epsilon-N argument, so that’s what I’ll paraphrase. “The limit of a sequence b_i of rational numbers is equal to the least upper bound of a bounded non-empty set S of rational numbers if, for every rational ε>0, there exists an integer N such that the following is true: For all x in S, if x+ε is greater than all elements of S, then for all positive integers n>N we have x-ε < b_n < x+2ε.”

This relies on the well-definedness of the least upper bound of a bounded non-empty set of rational numbers. However, that itself may be verified, I strongly suspect, using similar rational-ε arguments. “Let S and T be bounded non-empty sets of rationals. If, for every rational ε>0 there exists an x in S such that x+ε is greater than all elements of T, and there exists a y in T such that y+ε is greater than all elements of S, then S and T have the same least upper bound.” Then we just have to show that “If two bounded non-empty sets of rationals have the same least upper bound, then given any sequence with limit in the above-described sense with respect to one, that sequence also has limit in the above-described sense with respect to the other.”

This also relies on the fact that in a bounded non-empty set of rationals S, for every positive rational ε there exists an x in S such that x+ε is greater than all elements of S. I anticipate this should be fairly easy to prove.

[silverhammermba]

Well I'd like to clarify that I was not equating Wildberger and Gabriel. I am aware that Wildberger does get published in journals and that he appears to have a legitimate background in mathematics (unlike Gabriel). However I still maintain that Wildberger is fundamentally wrong regarding mathematics.

His theorems are fine and I'm sure his system for only using rationals has some nice features, but when he jumps from that to saying that real numbers should not be taught nor researched he goes way too far. At least in the video I watched, he offered several "arguments" for the illegitimacy of irrational numbers but each and every one of them was a proven, well understood property of real numbers that he was simply uncomfortable with. Okay, let's accept that there's no generic algorithm for multiplying two numbers with infinite decimal expansions... so what? It sure would be nice to have such an algorithm, but just because we don't have one doesn't mean that there's an inconsistency in the system (this also ignores that some numbers have infinite expansions in some bases and finite expansions in others).

The one point that Gabriel understood - which Wildberger seems to be completely missing - is that in order to make an argument for completely abandoning the real numbers, you would have to show some logical inconsistency resulting from them. Gabriel failed because his arguments were false, but Wildberger doesn't even try to make an argument. He just keeps saying "I don't like that, let's get rid of it!"

[clonus]

He seemed all right until around 21:00 where he starts talking about how analysis is really a conspiracy(his word not mine).
"Real numbers as usually presented are, in fact, a fraud. They don't actually make sense."

[skeptical scientist]

If one keeps making the reduction, forming s^{(i)} and c^{(i)}, is one guaranteed that there is some i such that c^{(i)}=0?

I'm not sure if this was meant as an honest question or a rhetorical one, but the answer is no.

Example:

Spoiler:

.090990999099990999990...+
.010010001000010000010...

[jestingrabbit]

Well I'd like to clarify that I was not equating Wildberger and Gabriel. I am aware that Wildberger does get published in journals and that he appears to have a legitimate background in mathematics (unlike Gabriel). However I still maintain that Wildberger is fundamentally wrong regarding mathematics.

So you made both threads, with near identical titles, linked the other, said they'd be best buddies, but you certainly weren't trying to equate them. I see.

Okay, let's accept that there's no generic algorithm for multiplying two numbers with infinite decimal expansions... so what?

Indeed, the lack of this certainly doesn't demonstrate any inconsistency, but its scarcely something you'd put in the 'pros" column (I don't believe its something that is claimed in the video, either). Likewise the lack of an algorithm for testing equality isn't a great step forward (which is alluded to).

The one point that Gabriel understood - which Wildberger seems to be completely missing - is that in order to make an argument for completely abandoning the real numbers, you would have to show some logical inconsistency resulting from them. Gabriel failed because his arguments were false, but Wildberger doesn't even try to make an argument. He just keeps saying "I don't like that, let's get rid of it!"

Sure, to show that the reals should not be studied at all one would need a contradiction, or to demonstrate some other way in which the theory was entirely vacuous. However, to show that there are better ways to spend curriculum or research time all you need to do is describe a number system that practically outperforms the reals for tasks that we actually do.

So, for instance, the vast majority of calculations that presently occur live inside the system called, for want of something better, IEEE 754-2008. What sort of arithmetic system is that? What would you want from the successor of that system? What does it do well and what are its shortcomings?

Why do mathematicians spend an inordinate amount of time with a system that is not of practical use, and not used practically, and leave the arithmetic system that runs the world to a part of numerical analysis? If error bars were built into our number system rather than a fantasy of infinite precision numbers existing in nature, would it aid or hinder our thought?

To suggest that real numbers occupy too much of our schooling and research time requires a much lower bar than demonstrating a contradiction, regardless of what you watched his lectures expecting to see. To suggest that the questions he puts are not worth asking and merely the domain of people to be derided and ridiculed requires passing a much higher bar imo.

@skep: it was rhetorical. Interestingly, or not, if C_i is the set of digits whose addition necessitates a carry on the i^th iteration of the addition I suggested, then C_i \cap C_j = \emptyset and C_{i+1} \subseteq \{j -1 |\ j\in C_i\}. From there its not too difficult to use an exhaustion argument to see that digit values eventually stay unchanged from step to step, and therefore the sequence of digits converges in the topology on the sequences generated by finite cylinder sets.

edit: that last paragraph for accuracy and clarity.

[mr-mitch]

If you're uncomfortable with manipulating decimal strings, then don't do it. Decimal strings aren't what the real numbers are about. In the first image of the video linked above, he seems to be perfectly happy that the axes intersect with each other and the unit circle; which leads to a counterargument for abandoning the real numbers.

It's perfectly acceptable to draw a ray from each intersection with the unit circle and the axes in the first quadrant to the other (and he's perfectly happy that these points exist) of lengths q in Q greater than sqrt(2).

What is the overlap of this region? You can't specify a rational number for this, yet the two rays clearly overlap a finite amount. Why you wouldn't include sqrt(2) in your number system I've no idea. Just because it's an infinite string in decimal notation, why should you care?

The thing that irks me the most is that he's upset computation is full of holes, but he's content with the rationals that are full of holes.

Also, in his system, what would be the angle between two lines? Angles in general are irrational.

It only seems that his work can only satisfy that approximations are sufficient.

Finally, he talks about circles and lines, but he never defines them. If he's adopting a different definition, he can prove whatever he likes in that framework. It's just a different framework, and a different problem.

Off-topic: What's his accent? It's not exactly Australian. It sounds more Canadian.

[jestingrabbit]

Finally, he talks about circles and lines, but he never defines them. If he's adopting a different definition, he can prove whatever he likes in that framework. It's just a different framework, and a different problem.

He's just working in Q^2, more or less (actually Rat^2), rather than R^2. He does define the circle, and lines are pretty well known.

Its interesting to note that Q^2, equipped with straight lines with rational gradients and intercepts, and circles with rational centres and whose radii squared are rationals, is a model for Euclid's geometry axioms (or at least those that we consider worth mentioning, "a point is that which has no part" has fallen out of vogue). To get a geometry that is modeled by R^2 and not Q^2 you have to include the idea that circles have an inside and an outside, and that rays that start on the inside and end on the outside intersect the circle, or something similar.

Off-topic: What's his accent? It's not exactly Australian. It sounds more Canadian.

He's a Canadian who's lived in oz for a couple of decades.

[mr-mitch]

I've no doubt there's a system on the rationals that works, but the point is, the reals works too. And it works better. Any solution you have in the reals can be converted to the rationals, as an approximation, or if the solution is rational, exact. In fact in that video he correctly states that we work with approximations in decimal strings. This is because we can't really use infinite strings, they aren't necessary to practical applications; only a sufficient precision.

There's no point developing the system in the rationals when the reals complete it, and when the reals provide solutions that the rationals cannot.

Why doesn't he try fix the problems he sees with the reals?

[skeptical scientist]

Its interesting to note that Q^2, equipped with straight lines with rational gradients and intercepts, and circles with rational centres and whose radii squared are rationals, is a model for Euclid's geometry axioms (or at least those that we consider worth mentioning, "a point is that which has no part" has fallen out of vogue). To get a geometry that is modeled by R^2 and not Q^2 you have to include the idea that circles have an inside and an outside, and that rays that start on the inside and end on the outside intersect the circle, or something similar.

Erm, you basically sacrifice the entire notion of length as a useful concept, since 1) lengths aren't numbers, and 2) given two different lengths, how do you know which is bigger? In R², the bigger one would be the one with a subsegment of the same length as the smaller one, but in Q² this is no longer true.

[jestingrabbit]

Not really. Squares of distances are numbers and are comparable.

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

[mr-mitch]

Except any polygon in a rational geometry would have to have vertices, and so any shape in which you would require the square of a length in order to express 'length' wouldn't be a shape.

Also, you couldn't add lengths together.

[jestingrabbit]

Except any polygon in a rational geometry would have to have vertices, and so any shape in which you would require the square of a length in order to express 'length' wouldn't be a shape.

What? The triangle on A=(0,0), B=(1,0) and C=(0,1) is a shape. The quadrances (ie square of the lengths) of the sides are 1, 1 and 2.

Also, you couldn't add lengths together.

Yes, some things are easier in rational geometry, others aren't. Such is life.

[mr-mitch]

What's easier in rational geometry?

And that shape can't exist, because it means the geometry is variant under rotations, making it useless. You can't rotate it 45degrees or a "spread of 1/2".

[MartianInvader]

I haven't watched/read it yet, but I like the fact that I can connect any two points on the real line by a path, and when I remove a point I no longer can. I also like that I can connect any two points in R^2, even if I remove a point, but not if I remove a line. Does this new formulation break that stuff? It seems like kind of an ugly patch if it does.

[antonfire]

I haven't watched/read it yet, but I like the fact that I can connect any two points on the real line by a path, and when I remove a point I no longer can.

You can connect any two points in the rationals by a piecewise linear path, but when you remove a point you no longer can.

What's easier in rational geometry?

Telling if two points are different.

[jestingrabbit]

What's easier in rational geometry?

For a right triangle the pythagorean identity looks a lot nicer with Q_1 + Q_2 = Q_3.

For any triangle the analogue of the sine rule is more elegant with \frac{s_1}{Q_1} = \frac{s_2}{Q_2} = \frac{s_3}{Q_3}.

And that shape can't exist, because it means the geometry is variant under rotations, making it useless. You can't rotate it 45degrees or a "spread of 1/2".

Does the lack of line-point duality in Euclidean geometry render it useless? Different geometries have different strengths. To perform simple harmonic motion you do genuinely need the usual transcendental trigonometric functions and squareroots, and transcendentals etc. To say that that renders the whole work useless seems a bit odd to me. You can certainly rotate by some angles, there are still congruent triangles and other figures.

[mr-mitch]

I don't think 'formulae looking better' is a good reason. If you're that concerned, just define Q_side as the square of the length of the side (which it is in the rational geometry) and define s_side as the square of the sine of the angle opposite side, and everything is the same.

I still think that since R² can do everything that Q² can, and that Q² can't everything that R² does, and Q² offers nothing more, the study of Q² is pointless. If you're concerned with the definition of the reals, then study them!

[supremum]

I read one of the articles on this website, and he made a lot of good points and a lot of bad points. I more or less agreed with his point that the foundations of math are really messy and most students of mathematics (and probably some professional mathematicians as well) don't really think about them so much. But then he goes into this ridiculous nonsense about how you don't really "need" axioms for mathematics and mathematics shouldn't be about what you "believe" without offering any foundational alternative.

EDIT: I started reading his second paper, and his way of not using axioms is to just make assertions buried in definitions. For example, he defines a function as a "rule" and then somehow goes to talking about constant functions on the naturals. Why is there a "rule" sending every natural number to 1? He never bothers to explain what he means by "rule." Apparently his alternative to using "axioms" is to make assertions without proof.

[jestingrabbit]

@mr-mitch: firstly, I think its important to note that there are two quite separate things going on here. There is the rational geometry aspect of what wildberger is saying, and there is the deprecation of real numbers aspect. I'm happy to argue that both make a certain amount of sense, but I want to make it clear that there are two distinct things going on.

I don't think 'formulae looking better' is a good reason. If you're that concerned, just define Q_side as the square of the length of the side (which it is in the rational geometry) and define s_side as the square of the sine of the angle opposite side, and everything is the same.

The simplicity of the formulae is not just a matter of aesthetics, they are computationally much quicker to work with for things like solving triangles.

http://www.cs.utep.edu/vladik/2008/olg08-01.pdf

Even for finding the spread of an angle between two lines there is a simple algebraic expression. So, computationally and aesthetically, the formulae associated with rational trigonometry are superior.

I still think that since R² can do everything that Q² can, and that Q² can't everything that R² does, and Q² offers nothing more, the study of Q² is pointless.

I think that working with Q² is useful for a few reasons. From a purely theoretical perspective, it clarifies that Euclid's postulates are deficient in specifically delineating the kind of geometry that we typically mean when we talk about Euclidean geometry ie they don't guarantee that all lines through the centre of a circle intersect the circle. From a more practical perspective, studying it led to the gains in computational efficiency that I've already mentioned.

My third reason is a little more subtle and could as easily be grouped with the discussion of the problems with the reals. Real numbers make us lazy and imprecise about approximations. In particular, lets get into solutions and approximate solutions to x^2 + y^2 = 1 and x = y. There are two solutions for real x and y, namely x = y = \pm \frac{1}{\sqrt{2}}. When we talk about approximate solutions, we tend to just be talking about truncating our precise solution (or more precisely, calculating to some precision and stopping) and presenting that as an approximate solution. That is a very vague way of going about things.

An approximation should come with some statement of how good an approximation it is, with some clear explanation of how "goodness" is evaluated. We can talk about some bound on distance between the precise solution and the approximate solution. That is problematic in that the precise solution is generally not something that we can include in calculations, though if we know that some number of digits are correct then we have at least something that we can say here. Another approach is to determine how well an approximation satisfies the equations, with some tradeoff between satisfying one equation over the other. A subclass of that approach is to require that one of the equations be precisely satisfied, and then let the difference between the sides in the other equation be the precision. Truncating the solution in the reals gets us only one class of approximations. Does working only in the reals allow us to determine if there is a point in Q² that is precisely on the unit circle with the difference of the coordinates less than any epsilon?

So saying that, for instance, "Any solution you have in the reals can be converted to the rationals, as an approximation" is I think demonstrative of a really vague way of thinking that working only with the reals tends to inculcate.

If you're concerned with the definition of the reals, then study them!

I don't think that he's concerned with the formal definition, I think he's concerned with the way that they are typically presented at the secondary level, or even at the non-pure tertiary level, and with some fundamental shortcomings that they have.

They're initially presented as infinite strings of digits with a decimal point somewhere, and the operations are glossed over at best, and just assumed to make sense at worst. We have people talking to us about real numbers a long time before anyone talks about limits. Whereas the arithmetic of naturals and rationals is dealt with to a high degree of logical precision, the reals begin as a handwave, and stay there for quite some time. I can't see that as an entirely satisfactory state of affairs, and if there are other ways to work that can be dealt with rigourously at all times, I would prefer those. We might get less "limit of a number", "infinitely small" etc speak if we didn't forcefeed such a weak exposition.

However, even if we have a perfectly logical exposition of real numbers, we don't get around the fact that the arithmetic isn't defined by algorithms, but by supertasks or limits (which are, in general, another kind of supertask), and that there is no meaningful way to test the equality of real numbers, nor, indeed, to precisely describe an arbitrary real without a supertask.

[mr-mitch]

@mr-mitch: firstly, I think its important to note that there are two quite separate things going on here. There is the rational geometry aspect of what wildberger is saying, and there is the deprecation of real numbers aspect. I'm happy to argue that both make a certain amount of sense, but I want to make it clear that there are two distinct things going on.

I recognize this, but they're not very separate topics. In order to abandon the real numbers you'll need to use the rational numbers, and rational geometry. If the reals offer much more than the rationals, and they do, and the geometry in real numbers offers more than the geometry of the rationals, and it does, then I do not see any reason to abandon the real numbers. They work. If someone isn't comfortable with the formality of the definition of the real numbers, and subsequent properties and applications of the real numbers, then this is what should be studied.

The simplicity of the formulae is not just a matter of aesthetics, they are computationally much quicker to work with for things like solving triangles.

Even for finding the spread of an angle between two lines there is a simple algebraic expression. So, computationally and aesthetically, the formulae associated with rational trigonometry are superior.

But it's not superior, it's the same. Let's remember that any function you do use to describe an angle must be one-to-one over a reasonable domain. If you're constantly going back and forth between angles and sines then of course you're going to have computational issues. I don't think you can claim rational geometry is the reason for such advances, it exists in the real geometry (intolerant pun not intended) and I do believe such ideas have been around for a long time. In secondary school, inverting the sine would be the last thing I'd do when asked for an angle. Let's also not forget that angles are much more important when you consider proportions. They ratio of the square of sines is not equal to the ratio of the corresponding angles.

To abandon angles all together is a little shortsighted; a lot of problems can be solved considering the angles, but not so with the spread (eg circles).

it clarifies that Euclid's postulates are deficient in specifically delineating the kind of geometry that we typically mean when we talk about Euclidean geometry ie they don't guarantee that all lines through the centre of a circle intersect the circle.

I would say this is more of a limitation of the rationals compared to the reals. The reason is because such non-intersections are irrational.

When we talk about approximate solutions, we tend to just be talking about truncating our precise solution (or more precisely, calculating to some precision and stopping) and presenting that as an approximate solution. That is a very vague way of going about things.

I don't think that's very vague at all. The decimal system is only meant to approximate, it's not ideal for representing any number, even most rational numbers. You could choose a different base but you'll always run into the same problems when you reach the negative powers of the base. The vagueness doesn't come from the real numbers, rather the decimal system. They're not entirely linked.

One of the problems I had with the youtube link above is he states about computation and checking two programs produce the same output. While this particular thought is quite irrelevant to the real numbers (any real number is not necessarily a decimal number, in fact most of them aren't, something which is often overlooked) it's the same for any program that computes a rational number in the decimal system, too.

The problem that real numbers aren't decimal numbers cannot be solved, but that's not a good reason for abandoning the real numbers. You don't need infinite precision, and indescribable numbers are useless.

Does working only in the reals allow us to determine if there is a point in Q² that is precisely on the unit circle with the difference of the coordinates less than any epsilon?

Assuming the appropriate (set theory) axioms, definitely, as Q2 \subset R2.
Algorithmically, perhaps. Any rational number in the decimal system is of two classes, finite or repetitive. Any algorithm that solves a problem in R2 should would output an integer and a decimal string. During computation of that string, in polynomial time, you can check to see if there are cycles. As soon as a cycle is detected, you would then assume this is the solution and use the 999 trick to form a rational number. Then, perform substitution which can easily be checked. If it isn't a valid then you continue computation. If it isn't valid and you run into the same cycle you run into approximations; is it worth continuing the computation or is it sufficient to return the currently computed rational number?

The repeated digits in 1/7 in order are 142857
If it turns out the solution cycles 1428571428571428571428572, then this number is incredibly close to 1/7.

Such precision (based on detecting these cycles) is well defined and is directly related to the length of the cycles. But why, in the practical sense, you would want to detect a rational number with a large amount of decimals (perhaps about 35), I've no idea. Instruments aren't accurate to such a small scale.

So saying that, for instance, "Any solution you have in the reals can be converted to the rationals, as an approximation" is I think demonstrative of a really vague way of thinking that working only with the reals tends to inculcate.

I agree that this does occur with various lies to children. Secondary students are taught to desire decimals instead of working with fractions. I hate working with decimals, but rational numbers aren't the solution.

However, even if we have a perfectly logical exposition of real numbers, we don't get around the fact that the arithmetic isn't defined by algorithms, but by supertasks or limits (which are, in general, another kind of supertask), and that there is no meaningful way to test the equality of real numbers, nor, indeed, to precisely describe an arbitrary real without a supertask.

Definitely, but the question, in a practical scenario, is only limited to precision again. You won't ever need to know if two real numbers are the same, only up to a certain point which is simple.
You cannot describe arbitrary reals without supertasks and limits because most reals are indescribable. The ones that you can describe are either rational or irrational. You can then describe the irrationals by their minimal polynomials, or if they're non algebraic then, I'm not sure. Wouldn't this be more interesting to research?

The thing that I find overlooked the most when we're talking about solutions and the real numbers is that mathematics is at most a model. Why would you settle for "appears to touch" when you can often solve for an algebraic irrational number; in the model. Whether or not this solution applies to the real world is irrelevant. All you really need is an approximation.
Our current system provides all this, much more than the rational system.

[Stig Hemmer]

I should perhaps start by saying that I haven't seen the video or read anything by Wildberger, so I am only reacting to this thread.

If you're uncomfortable with manipulating decimal strings, then don't do it. Decimal strings aren't what the real numbers are about.

Quoted for truth.

It is true that most mathematicians don't think about how reals are defined, but that doesn't matter. What matters is that they know how reals behave, and that has been well mapped by people who DO know how to define them. (Cauchy, Dedekind et al.)

Physicists and engineers will be using calculus regardless of what mathematicians do because it is just so darn useful. That being the case, I think mathematicians should study calculus too. And rational calculus seems rather painful.

By the way, try to formulate the Erdös-Kac Theorem using only rational numbers. I dare you. Mind you, I am not asking about the proof, just the proven theorem. It is a statement about integers, so it should be easy, right? Right? Wrong.

PS: I think most of the problems with reals can be solved by rejecting the Axiom of Choice, but that is another discussion.

[skeptical scientist]

Definitely, but the question, in a practical scenario, is only limited to precision again. You won't ever need to know if two real numbers are the same, only up to a certain point which is simple.
...
The thing that I find overlooked the most when we're talking about solutions and the real numbers is that mathematics is at most a model. Why would you settle for "appears to touch" when you can often solve for an algebraic irrational number; in the model. Whether or not this solution applies to the real world is irrelevant. All you really need is an approximation.
Our current system provides all this, much more than the rational system.

Then perhaps it would be better to introduce decimal notation as a means of approximation, and only deal with finite decimals. You can wait to introduce infinite decimal representations until calculus, when students will have the necessary background (limits, infinite series) to make sense of them properly.

[Eebster the Great]

And do you want a simple algorithm to add two "infinite decimals"? Here's one.

And it falls prey to exactly what I mentioned. The usual epsilon delta definition of a limit has within it arithmetic operations. You're trying to prove that arithmetic operations are well defined, and you're using concepts that, in some formalisms, rely on that arithmetic.

I don't know if I'm missing something here, but couldn't you just take the sum to be the least real number greater than every element of that set? (The order relation just being the dictionary order for {1, ..., 10}^w.)

[Tirian]

Then perhaps it would be better to introduce decimal notation as a means of approximation, and only deal with finite decimals. You can wait to introduce infinite decimal representations until calculus, when students will have the necessary background (limits, infinite series) to make sense of them properly.

This is essentially what happens, although I find it more honest to introduce decimal notation as a specialized format for writing rational numbers whose denominator is a power of ten. There is one place in primary school where infinite decimal representation comes up, which is dealing with repeating decimal notation. You've got to be able to show that 0.083333... = 1/12, and of course there's a sticky thread for this forum that wouldn't have to be there if students were comfortable with the standard technique for solving it.

[skeptical scientist]

You've got to be able to show that 0.083333... = 1/12, and of course there's a sticky thread for this forum that wouldn't have to be there if students were comfortable with the standard technique for solving it.

Why? If we could just satisfy ourselves with 1/12 ≈ .0833 and the approximation becomes better if you add more 3s, at least until students have had calculus, we could avoid a lot of hand-waving and that particular headache.

[Tirian]

Simple long division can assure us of a sufficiently good decimal estimate for 1/12. It's not so elementary to find a sufficiently simple but still good rational estimate for 0.0833333.

Granted, I can guess how your lesson plan would go forward there and I suspect I like it better than what we're teaching now even though re-educating sixth grade teachers would be a total pill. But if you draw up a petition, I'll sign it.

[Nat]

The main problem with people like this or Zeilberger is that yeah, their mathematics is consistent (more or less), but they willfully ignore other ways to think of mathematics for no good reason.

[TwistedBraid]

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.

Also, the reals are uncountable. Computers can't handle that. I don't care.

Humans have a very ingrained understanding of the topology of curves in the plane. Jordan curve is the prime example. If you ask a three year old whether the diagonals of the square intersect, they'll tell you yes. But this is really a pretty non-trivial theorem, and relies strongly on the topology of the reals. On the other
hand, the rationals have a terrible topology.

You can't convince kids that the angle bisector on an equilateral triangle doesn't intersect the opposite side. They aren't dumb enough.

[skeptical scientist]

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.

[mr-mitch]

Off-topic: There's also evidence that people think in terms of ratios rather than sums. People can only recognise units up to about 3-4, and after that people start thinking of the size of the group. There's a tribe which only have words for 1, 2, 3 and after that many and so forth. There's a lot of experiments in Japan with chimps and the like which test this as well.