When does ideology trump common sense? This question is very relevant to the sad situation with modern pure mathematics, which is in a dire logical mess. All manner of dubious concepts and arguments are floating around out there, sustained by our fervent desire that the limiting operations underlying modern analysis actually make sense. We must believe — we will believe!

And there is hardly a more obviously suspicious case than that of *space-filling curves*. These are purportedly one-dimensional continuous curves that pass through every (real) point in the interior of a unit square.

But this contradicts ordinary common sense. It imbues mathematics with an air of disconnection with reality that lay people find disconcerting, just like the Banach-Tarski Paradox nonsense that I talked about a few blogs back.

In mathematics, dimension is an important concept; a line, or more generally a curve, is one-dimensional, while a square in the plane, or a surface is two-dimensional, and of course we appear to live in a three-dimensional physical space. But ever since the 17th century, mathematicians had started to realize that the correct definitions of “curve” and “surface” were in fact much more subtle and logically problematic than at first appeared, and that “dimension” was not so easy to pin down either.

In 1890 a new kind of phenomenon was introduced which cast additional doubt on our understanding of these concepts. This was the **space-filling curve** **of Peano**, which ostensibly fills up all of a square, without crossing itself. This was a contentious “construction” at the time, resting on the hotly debated new ideas of George Cantor on infinite sets and processes. But the influential German mathematician David Hilbert rose to defend it, and so generally 20th-century pure mathematicians fell into line, and today these curves are considered unremarkable, and just another curious aspect of the modern mathematical landscape.

But do these curves really exist? More fundamentally, are they even well defined? Or are we talking about some kind of mathematical nonsense here?

While Peano’s original article did not contain a diagram, Hilbert in the following year published a version with a picture, essentially the one produced below, so we will discuss this so-called **space-filling curve of Hilbert**. It turns out that the curve is created by iterating a certain process indefinitely. Along the way, we get explicit, finitely prescribed, discrete curves that twist and turn around the square in a predictable pattern. Then “in the limit”, as the analysts like to say—as we “go to infinity”—these concrete zig-zag paths turn into a continuous path that supposedly passes through every real point in the interior of the square exactly once. Does this argument really work??

The pattern can be discerned from the sequence of pictures below. Consider the square as being divided into 4 equal squares. At the first stage we join the centres of these four squares with line segments, moving say from the bottom left to the top left, then to the top right, and then to the bottom right. This gives us a U shape, opening down. Now at the next stage, we join four such U shapes, each in one of the smaller sub-squares of the original. The first opens to the left, the next two open down, and the last opens to the right, and they are linked with segments to form a new shape, which we call U_2 as shown in the second diagram. In the third diagram, we put four smaller U_2 shapes together, also oriented in a similar way to the previous stage, to create a new U_3 curve. And then we carry on doing the same: shrink whatever curve U_n we have just produced, and arrange four copies in the sub-squares oriented in the same way, and linked by segments to get the next curve U_{n+1}.

“Hilbert curve”. Licensed under CC BY-SA 3.0 via Wikipedia – https://en.wikipedia.org/wiki/File:Hilbert_curve.svg#/media/File:Hilbert_curve.svg

These are what we might call **Hilbert curves**, and they are pleasant and useful objects. Computer scientists sometimes use them to store date in a two dimensional array in a non-obvious way, and they are also used in image processing. Notice that at this point all these curves are purely rational objects. No real number shenanigans are necessary to either define or construct them. Peano and Hilbert made a real contribution to mathematics in introducing these finite curves!

And now we get to the critical point, where Hilbert, following Peano and ultimately Cantor, went *beyond the bounds of reasonableness*. He postulated that we could carry on this inductive process of producing ever more and more refined and convoluted curves *to infinity*. Once we have arrived at this infinity, we are supposedly in possession of a “curve” U_{infinity} with remarkable (read unbelievable) properties. [Naturally all of this requires the usual belief system of “real numbers”, which I suppose you know by now is a chimera.]

The “infinite Hilbert curve” U_{infinity} is supposedly continuous, but differentiable nowhere. It supposedly passes through every point of the interior of the square. By this, we mean that every point [x,y], where x and y are “real numbers”, is on this curve somewhere. Supposedly the curve U_{infinity} is “parametrized” by a “real number”, say t in the interval [0,1]. So given a real number such as

t=0.52897750910859340798571569247120345759873492374566519237492742938775…

we get a point U_{infinity}(t)=

[0.68909814147239785423401979874234…,0.36799574952335879124312358098423435…]

in the unit square [0,1] x [0,1].

(Legal Disclaimer: these real numbers are for illustration purposes only and do not necessarily correspond to reality in any fashion whatsoever. In particular we make no comment on the meaning of the three dot sequences that appear. Perhaps there are oracles or slimy galactic super-octopuses responsible for their generation, perhaps computer programs. You may interpret as you like.)

The infinite Hilbert curve U_{infinity} cannot be drawn. Its “construction” amounts to an imaginary thought process akin to an uncountably infinite army of pointillist painters, each spending an eternity creating their own individual minute point contributions as infinite limits of sequences of rational dots. Unlike those actual, computable and constructable curves U_n, the fantasy curve U_{infinity} has no practical application. How could it, since it does not exist.

Or we could just apply the (surely by now well-known) *Law of (Logical) Honesty*, formulated on this blog last year, which states:

Don’t pretend that you can do something that you can’t.

While you are free to create curves U_n even for very large n if you have the patience, resources and time, it is both logically and morally wrong to assert that you can continue to do this for all natural numbers, with a legitimate mathematical curve as the end product. It is just not true! You cannot do this. Stop pretending, analysts!

But in modern pure mathematics, we believe everything we are told. Sure, let’s “go to infinity”, even if what we get is obvious nonsense.

FodeYour argument seems to rest on a single statement being false:

“infinite things exsist”

Yet I don’t see you lay out any specific contradictions that accepting that would lead to in the case of a Hilbert curve. This leaves me feeling like you complained about it rather adding to the discussion. Please help me understand what I missed.

PS: I did notice that you pointed out that the construction of such a object is absurd because it is hard to conceptualize and has counter-intuitive properties (like not being differentiable at any point). While I do happen to agree with you that construction is important, I do not a agree that a construction being unintuitive is proof of it being incorrect.

njwildberger: tangential thoughtsPost authorI am hoping the argument I presented is more than that. What do you feel about the Law of (Logical) Honesty?

ericfodeI feel it is critical to be aware of the limits of any logical system that you engage with. I am not sure that it precludes systems that require undefined or ildefined objects. Especially if that system provides tools to work with and reason about those objects regardless of how well defined they are (and tools to quantify how well defined they are).

Now with that said, I think it hits on my point exactly. I think what you can and can’t do in a system are nesscicarly defined by that system. In the case of real numbers (where this curve claims to be found) I would argue that a choice has been made to trade the ability to meaningfully express precise values for the ability to have meaningful infinite relationships (which may happen to model things like this curve) inside that system. Where as the affine numbers have made the opposite choice and such constructions would be madness.

Also thank you for engaging in conversation.

emperzelosThat you should learn to use it

Michael RosenbaumI saw your MF80 video posted on 4chan, you are the “king of /sci”. i think i see what you are getting at. it reminds me of how calculus is useful for approximation. I am NOT an expert, but i do have a fairly mathematical mind (max high school score on the aptitude test from 1979), and my dad was a rocket scientist (3body problem). your arguments feel right to me. I will try to watch a few more of your vids. thanks for challenging me to think.

Rodney ConnFirst off I just want to say that I am very sympathetic towards your ideas about the foundations of mathematics and I find your approach to mathematics to be quite beautiful! However, having said that, I do have a question to pose to you. If we can’t say anything meaningful about infinite processes (because they don’t “exist”) then how can we say anything at all about geometrical objects like ideal circles, triangles and squares? Those idealizations don’t “exist” either. What we say about these objects is only true, strictly speaking, as the end result of a limiting procedure of more and more accurate constructions and measurements. If you wanted a true line you could only get one by making a line in the real world and slowly, on step at a time, shaving off it’s width until finally it has no width at all! Of course, at that point, it no longer exists!! So, we run into the same issues here that we encounter with infinity and infinitesimals but surely we don’t want to get rid of ideal geometrical objects!! That would result in a very poor mathematics indeed. I would be very curious to see yours thoughts about this. Thank you for all of time you take teaching mathematics! It is much appreciated!

RossI think that you are correct in saying that there are problems in “going to infinity” in the manner of Hilbert and his space filling curves. No matter how many refinements you create for the curve, there will always be gaps – missing pieces that have not been traversed by the curve. We can try by using a hand waving argument to assure others that we have actually completed _all_ iterations, but in fact there will never be a time at which we have already traversed all of the points in the square.

But I think that water is wet, so what do I know 🙂

Arthur OgawaI see your point concerning the error of asserting that the process of creating successively more complex Hilbert curves can be continued to infinity. It seems to me that your arguments are equally applicable to the simpler(?) case of Dedekind cuts. One can, I believe, prove that any rational point in the square can be “hit” by a Hilbert curve U_{N} of an appropriate N.

By way of salvaging some utility from the Hilbert curve(s) (and other such curves), might their properties be defined and investigated in the limit of 1 / N —> 0? Would this not be prey to the very same problems that present when treating differentials and integrals?

Personal notes:

* Thank you for your courageous stand in respect to infinity and “real numbers”. I find that I now have to rethink my own ideas, which were formed while a small child reading George Gamov’s 1 2 3 Infinity (published in my birth year). I found the Dirac delta function just the coolest thing… Such was my benighted upbringing.

* I have viewed all your MathHistory videos with pleasure and look forward to Rational Trigonometry and Foundations. Kudos to you for your generosity in putting your delightful lectures on the Internet!

Peter CottonHi Norman,

This is a bit off-topic. I just wanted to comment because once you said to me that one should think about things hard until they become trivial. That statement, which you may have forgotten you made, has become one of those things that pop up in one’s mind relentlessly. Don’t ask me how it came up today but it did. I came here searching your name and wondering if you had ever put it in print. Then I found your blog. Then my belief system started to turn to dust.

Your blog is an important caveat. Think about things until they are trivial, or not true!

Anyway, to make this comment remotely related to space-filling curves, it might amuse you to know that I use them (the finite ones of course) as a handy but dirty engineering hack on a daily basis. See https://www.microprediction.com/blog/intro if you are interested (scroll down to z2’s, z3’s).

Peter

(UNSW 1995)