A couple weeks ago I shared a fascinating formula for π. I got it from Carl B Boyer’s The History of Calculus and its Conceptual Development. He got it from Józef Maria Hoëne-Wronski, early 19th-century Polish mathematician. His idea was that an absolute, culturally-independent definition of π would come not from thinking about circles and diameters but rather this formula:
Now, this formula is beautiful, at least to my eyes. It’s also gibberish. At least it’s ungrammatical. Mathematicians don’t like to write stuff like “four times infinity”, at least not as more than a rough draft on the way to a real thought. What does it mean to multiply four by infinity? Is arithmetic even a thing that can be done on infinitely large quantities? Among Wronski’s problems is that they didn’t have a clear answer to this. We’re a little more advanced in our mathematics now. We’ve had a century and a half of rather sound treatment of infinitely large and infinitely small things. Can we save Wronski’s work?
Start with the easiest thing. I’m offended by those bits. Well, no, I’m more unsettled by them. I would rather have in there. The difference? … More taste than anything sound. I prefer, if I can get away with it, using the square root symbol to mean the positive square root of the thing inside. There is no positive square root of -1, so, pfaugh, away with it. Mere style? All right, well, how do you know whether those terms are meant to be or its additive inverse, ? How do you know they’re all meant to be the same one? See? … As with all style preferences, it’s impossible to be perfectly consistent. I’m sure there are times I accept a big square root symbol over a negative or a complex-valued quantity. But I’m not forced to have it here so I’d rather not. First step:
Also dividing by is the same as multiplying by so the second easy step gives me:
Now the hard part. All those infinities. I don’t like multiplying by infinity. I don’t like dividing by infinity. I really, really don’t like raising a quantity to the one-over-infinity power. Most mathematicians don’t. We have a tool for dealing with this sort of thing. It’s called a “limit”.
Mathematicians developed the idea of limits over … well, since they started doing mathematics. In the 19th century limits got sound enough that we still trust the idea. Here’s the rough way it works. Suppose we have a function which I’m going to name ‘f’ because I have better things to do than give functions good names. Its domain is the real numbers. Its range is the real numbers. (We can define functions for other domains and ranges, too. Those definitions look like what they do here.)
I’m going to use ‘x’ for the independent variable. It’s any number in the domain. I’m going to use ‘a’ for some point. We want to know the limit of the function “at a”. ‘a’ might be in the domain. But — and this is genius — it doesn’t have to be. We can talk sensibly about the limit of a function at some point where the function doesn’t exist. We can say “the limit of f at a is the number L”. I hadn’t introduced ‘L’ into evidence before, but … it’s a number. It has some specific set value. Can’t say which one without knowing what ‘f’ is and what its domain is and what ‘a’ is. But I know this about it.
Pick any error margin that you like. Call it ε because mathematicians do. However small this (positive) number is, there’s at least one neighborhood in the domain of ‘f’ that surrounds ‘a’. Check every point in that neighborhood other than ‘a’. The value of ‘f’ at all those points in that neighborhood other than ‘a’ will be larger than L – ε and smaller than L + ε.
Yeah, pause a bit there. It’s a tricky definition. It’s a nice common place to crash hard in freshman calculus. Also again in Intro to Real Analysis. It’s not just you. Perhaps it’ll help to think of it as a kind of mutual challenge game. Try this.
- You draw whatever error bar, as big or as little as you like, around ‘L’.
- But I always respond by drawing some strip around ‘a’.
- You then pick absolutely any ‘x’ inside my strip, other than ‘a’.
- Is f(x) always within the error bar you drew?
Suppose f(x) is. Suppose that you can pick any error bar however tiny, and I can answer with a strip however tiny, and every single ‘x’ inside my strip has an f(x) within your error bar … then, L is the limit of f at a.
Again, yes, tricky. But mathematicians haven’t found a better definition that doesn’t break something mathematicians need.
To write “the limit of f at a is L” we use the notation:
The ‘lim’ part probably makes perfect sense. And you can see where ‘f’ and ‘a’ have to enter into it. ‘x’ here is a “dummy variable”. It’s the falsework of the mathematical expression. We need some name for the independent variable. It’s clumsy to do without. But it doesn’t matter what the name is. It’ll never appear in the answer. If it does then the work went wrong somewhere.
What I want to do, then, is turn all those appearances of ‘∞’ in Wronski’s expression into limits of something at infinity. And having just said what a limit is I have to do a patch job. In that talk about the limit at ‘a’ I talked about a neighborhood containing ‘a’. What’s it mean to have a neighborhood “containing ∞”?
The answer is exactly what you’d think if you got this question and were eight years old. The “neighborhood of infinity” is “all the big enough numbers”. To make it rigorous, it’s “all the numbers bigger than some finite number that let’s just call N”. So you give me an error bar around ‘L’. I’ll give you back some number ‘N’. Every ‘x’ that’s bigger than ‘N’ has f(x) inside your error bars. And note that I don’t have to say what ‘f(∞)’ is or even commit to the idea that such a thing can be meaningful. I only ever have to think directly about values of ‘f(x)’ where ‘x’ is some real number.
So! First, let me rewrite Wronski’s formula as a function, defined on the real numbers. Then I can replace each ∞ with the limit of something at infinity and … oh, wait a minute. There’s three ∞ symbols there. Do I need three limits?
Ugh. Yeah. Probably. This can be all right. We can do multiple limits. This can be well-defined. It can also be a right pain. The challenge-and-response game needs a little modifying to work. You still draw error bars. But I have to draw multiple strips. One for each of the variables. And every combination of values inside all those strips has give an ‘f’ that’s inside your error bars. There’s room for great mischief. You can arrange combinations of variables that look likely to break ‘f’ outside the error bars.
So. Three independent variables, all taking a limit at ∞? That’s not guaranteed to be trouble, but I’d expect trouble. At least I’d expect something to keep the limit from existing. That is, we could find there’s no number ‘L’ so that this drawing-neighborhoods thing works for all three variables at once.
Let’s try. One of the ∞ will be a limit of a variable named ‘x’. One of them a variable named ‘y’. One of them a variable named ‘z’. Then:
Without doing the work, my hunch is: this is utter madness. I expect it’s probably possible to make this function take on many wildly different values by the judicious choice of ‘x’, ‘y’, and ‘z’. Particularly ‘y’ and ‘z’. You maybe see it already. If you don’t, you maybe see it now that I’ve said you maybe see it. If you don’t, I’ll get there, but not in this essay. But let’s suppose that it’s possible to make f(x, y, z) take on wildly different values like I’m getting at. This implies that there’s not any limit ‘L’, and therefore Wronski’s work is just wrong.
Thing is, Wronski wouldn’t have thought that. Deep down, I am certain, he thought the three appearances of ∞ were the same “value”. And that to translate him fairly we’d use the same name for all three appearances. So I am going to do that. I shall use ‘x’ as my variable name, and replace all three appearances of ∞ with the same variable and a common limit. So this gives me the single function:
And then I need to take the limit of this at ∞. If Wronski is right, and if I’ve translated him fairly, it’s going to be π. Or something easy to get π from.
I hope to get there next week.
You know, when you first posted about the Wronski Pi, I thought of doing something like this. But then I got distracted by shiny things.
Still, this looks like a fun diversion! I think I’m gonna try my hand at it, too, from a Non-Standard Analysis direction.
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Certainly understand the distractions. I’ve been wrestling with a lot of them myself lately.
I’d be delighted if you did try a nonstandard analysis approach. It would probably be pretty close to what Wronski was trying to get at, if I’m getting from Boyer a fair sense of what Wronski was thinking.
(To those who don’t know what I’m talking about: nonstandard analysis is this approach to analysis that’s grown over the last 50 years. It’s built on ideas like
what if we suppose there's some number that's larger than zero but still less than the reciprocal of all positive integers?'' You might answer there's no such thing. A nonstandard analyst might respond they said the same things about negative numbers and about imaginary numbers and about different sizes of infinity. If the reasoning is sound and the results are fruitful, who's to say there's
no such thing”? It’s still “nonstandard” because, well, it’s not the way we all grow up learning analysis. And we don’t all grow up learning analysis this way because it’s still a relatively new approach and there’s objections to the kinds of abstraction you need to be comfortable with to get interesting results. Maybe check back in another fifty years and see how opinions on it have changed.)LikeLiked by 1 person
This is very interesting! Perhaps Wronski had some method to his madness after all.
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Oh, he certainly had method to this madness at least. I’ve worked through the expressions and I think I know roughly where he was going. It’s just dangerous getting there.
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That is nice to hear. Most biographies I read portray Wronski as a crazy guy.
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I confess I haven’t read a proper biography of Wronski. (Come to it, I’m not sure there is one.) I’ve just had short essays, mostly on the web, that might be factually right but not necessarily fair. It’s hard to judge on just a thousand words about someone.
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Multiplying by infinity seems related to the old joke about asking someone to give more than 100% of effort to something. Something that only exists because you calculated it.
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Well, I’m not sure. I mean, when you’re talking about percentages of something you do get your choice of what the
100 percent'' level is, after all. Like, the Space Shuttle Main Engines routinely ran at 104 percent maximum thrust, and later 109 percent maximum thrust, because the
maximum thrust” level was set early in the design and when refinements came in that made the engine more powerful, it was easier to say normal procedure was running at above 100 percent than to go back and change the value of “designed maximum thrust” everywhere.Multiplying by infinity, that gets tricky because you do have to think carefully what you mean by doing it. And you can easily get ridiculous results if you aren’t thinking carefully. Or if you want to play pranks on people. See every humorous mathematical science fiction story ever that isn’t about Moebius strips. I’m not sure there’s a direct link, although there are clever mappings of the real numbers to finite strips that probably make it possible to get from percentages to infinities quickly enough.
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