Some of you may remember a wild discussion we had at TSZ a couple of years ago, spanning eight months, debating whether “3” and “3.0” refer to the same number and whether measurements can be expressed using real numbers. (Yes, really.) One of the questions that arose during that discussion was on the boundary between pure and applied mathematics, and DNA_Jock referenced the mathematician Norman Wildberger’s opinion on that topic.
Wildberger is an odd bird. DNA_Jock said of him, “I’m not entirely sure that he’s sane,” and I tend to agree. He’s a finitist who believes that irrational numbers do not exist. For him, √2 doesn’t exist, and there are no solutions to the equation x² = 2. π doesn’t exist, and there is no such thing as the ratio of a circle’s circumference to its diameter. Strange (and in my opinion goofy) stuff.
Anyway, Wildberger is in the news these days, and the press is enthusiastic: “Mathematician solves algebra’s oldest problem”, “200-year-old ‘algebra wall’ shattered with a bold new approach”, “Math shaken as 200-year-old polynomial rule falls to Geode number discovery”, and other similarly breathless headlines. Newsweek and Popular Science are among the outlets trumpeting the news. The claim, according to an article in Interesting Engineering, is that Wildberger and his co-author, Dean Rubine, have found “a general solution to equations where the variable is raised to the fifth power or higher”, which is something that mathematicians have long considered impossible.
My eyebrows are raised, and the first sign of trouble is that none of the articles I’ve looked at so far quote other mathematicians who have evaluated Wildberger’s work. I’ll keep looking. Also, there are other signs of trouble. In the Interesting Engineering article, it says
By carefully truncating these [power] series, he and Dean Rubine, PhD, a computer scientist, generated accurate, rational approximations of solutions to complex equations, without leaving the bounds of logical, constructible mathematics.
But approximations are not solutions, and this suggests either that the author of the article doesn’t understand what Wildberger is claiming or that Wildberger is making a claim that isn’t supported by the work itself. Or both.
To me, the quoted sentence suggests that all Wildberger has done is to create a numerical method that doesn’t rely on irrational numbers, trig functions, etc., which are not considered constructible. If that’s right, then this is something of a Wildberger nothingburger. Exciting to finitists, perhaps, but not particularly useful to mathematicians who don’t share Wildberger’s irrational fear of irrationals and who already have numerical methods for approximating the solutions to higher-order polynomial equations. In which case the breathless Wildberger headlines aren’t justified.
I’m not a mathematician, of course, and I hope I’m wrong. It would be great if Wildberger really has done something deserving of the “math shaken” claim. If it pans out, it would be an interesting case where an (in my opinion goofy) phobia regarding irrationals actually leads to a useful result that non-finitist mathematicians haven’t been able to achieve.
I’m skeptical, though. I’ll read more on this and post what I find in the comments.
aleta:
Exactly. “Must be expressible as an integer ratio” is a silly requirement that is more concerned with the representation of a number than it is with the number itself. Nothing goes wrong if you drop that requirement, and I have yet to see Wildberger point to something that does go wrong.
Wildberger’s requirement seems as arbitrary to me as including “must be visible to the naked eye” as a requirement for concluding that an astronomical object is a star.
The distinction between numbers and their representations is crucial, and it’s a point that came up repeatedly during the eight-month discussion with Jock, Flint, and others that I mentioned in the OP. As far as I’m concerned, “√2”, “the square root of two”, and “the positive number x such that x^2 = 2” are all acceptable representations of the number in question. To demand a finite decimal expansion or an integer ratio is pointless.
petrushka:
I was surprised to learn that base 13 actually has a famous application: the Conway base 13 function. It’s a marvelously inventive function (as you might expect of John Conway) that demonstrates that while a continuous function f over a finite interval [a,b] is necessarily surjective onto the interval [f(a), f(b)], it isn’t true the other way around: not every surjective funtion is continuous. He specially created his base 13 function to fit that description: surjective but not continuous.
In easier-to-understand language: suppose you graph a function, let’s say x^2.
f(x) is continuous, meaning there are no sudden jumps in its graph. If you feed in every value of x from 3 to 4, you’ll get every value of f(x) from 9 to 16. No numbers between 9 and 16 are skipped. This is known as the Intermediate Value Theorem, and it’s important.
Is there a function g(x) that produces every value between 9 and 16 yet isn’t continuous? The answer is yes, and while this was known well before Conway, he came up with a cool example that meets the requirements and then some. The base 13 function was the one he came up with. It’s really clever, even drawing upon computability theory and the halting problem.
More on that later.
A couple of cool facts about Conway’s base 13 function: not only does it satisfy the requirements above, it also has these two wild properties: it maps every interval, no matter how small, onto the totality of the real numbers. It’s also discontinuous at every point.
In other words, pick a real number. Any real number, with any value. It could be 3, or 0.0001, or √2, or 6.022 × 10^23, you name it. Now pick the endpoints of an interval [a,b]. No matter how close together a and b are, there will be some value x in between them such that when x is fed into the Conway function, your chosen number pops out.
Example: Let’s choose 9688352.0909633 as our target number. It’s guaranteed that there is some number x between 1.00000000000001 and 1.00000000000002 which, when fed into the Conway function, produces 9688352.0909633 as the output. Bizarre.
This is also a good demonstration that there are as many numbers between any two points on the number line as there are on the entire number line. JoeG’s head is exploding somewhere, but it’s true.
“Discontinuous at every point” means that no matter how far you zoom in, you’ll see jumps in the graph. It isn’t smoothly connected. It’s infinitely jumpy. Contrast that with other more familiar functions whose graphs contain jumps but look continuous if you zoom far enough in on some of the intervals.
All of this would give Wildberger the heebie jeebies since he doesn’t even believe in the real number line.
The construction of the Conway function is complicated, so instead of reinventing the wheel and describing it here, I’m going to look for a good explanation somewhere else on the internet.
Here’s a nice 9-minute video:
This Function Maps Any Interval to the Real Line (Conway’s Base-13 Function)
I made a mistake above when I wrote:
That’s true of a variant of Conway’s function, but not of the original function, which involves straightforward interpretation of digit strings with no reference to the halting problem.