Courtesy of Scientific American, we have the “debate” over whether 0.999… equals 1.
This is really really stupid. Although, to be fair, SciAm‘s attempt to redefine numbers is much less stupid than their attempts to redefine sex.
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Thanks Marty. I read the article (albeit not that closely). I found it moderately interesting. It starts by saying that there are “countless debates” about the truth that 0.999…=1, and the goes on to show different ways of establishing that it’s true. I’m not sophisticated enough to question the evidence they present, but it doesn’t rub me the wrong way. Is it that they are building a straw man by asserting that there is a debate at all that irks? Or is it something else?
What is the purpose of the article?
I presume it serves the same purpose that any pop-sci article does, to inform and entertain. It was a little informative, but my knowledge of mathematics is limited.
In what manner were you informed?
I guess I hadn’t considered different approaches to establishing the truth of the matter. I knew how to establish it algebraically by letting x=.999 recurring and going from there.
The article seems harmless enough to me. I know you have issues with “pop math” and have written on it extensively. I have never been able to distill what it is that you see is wrong, however. Is it that articles like this don’t really teach anything? Are they shallow or flat out wrong? Do they distract people from more substantive learning, or something else? I’d value a short explanation, as I think a summary would help me improve my critical radar.
The article is very badly written. But the main problem is that it pretends there is a debate where none exists. 0.9999… equals 1 and that’s it. It is madness to suggest that “mathematicians” are somehow arguing about this.
It is not simply that the article is bad, and wrong. It is that the article, if sufficient numbers were to read it, would do real damage, by undermining students’ and teachers’ already weak understanding of mathematical reality.
I really really hate this stuff. As I replied to Franz, I don’t care about the author. But I hate it when Numberphile does it. And I hate it when Eugenia Cheng does it. And I hate it when Eddie Woo does it. And, similarly, Scientific American should have *much* higher standards.
Thanks Marty, that makes sense. That’s what I meant when I asked whether the article was building a straw man. I agree it risks muddying what are very clear waters. It’s similar to when reputable outlets publish pieces about the “climate change debate” or “the debate around evolution by natural selection.” There is no debate. Only the uninformed, ideologically committed, or science deniers treat it as one.
I don’t know Eddie Woo beyond his reputation as a YouTube maths personality. My assumption is that he presents settled ideas as more complex or contested than they are, and positions himself as someone who can cut through that noise and should be followed.
A fair assumption, which turns out to be wrong.
A couple of questions (I do not read SA):
1. Is this a reputable source of quality journalism (as in, would people who work in the science industry read it?)
2. Does the article (in anyone’s opinion) make it clear what they mean by “mathematician”, “bitter” or “countless”.
If neither of these questions can be given an immediate YES answer, I think the conclusions are obvious enough to remain unspoken.
The third question I have though, and perhaps the most important is WHY?
Why did anyone think this was an article worth writing? (perhaps also WHO is the intended audience, but I will start with why).
Scientific American was once *the* popularisation magazine. Not expressly for scientists, but for people with enough background and curiosity in science to want a quick overview of topics, and with a bit of meat. I was never that fond of it, but of course it was also the home of Martin’s Gardner’s columns, which were great.
This debate keeps coming up. The point is that we know what a sum is when we are adding finitely many digits (to be more formal, the Peano Axioms set it up; bit it practice, we just keeping until we stop). If we want to find an infinite sum, we first have to define what that means, and the standard limit definition does it for us. There is nothing to fuss about; just follow the definition.
In the eighteenth century, it was not clear what an infinite sum was, and there was an interesting paper by Euler who gave a number of reasons that consistently assigned a sum to 1 – 1! + 2! – 3! + 4! – … ; several perspectives led to the same value. Basically, the idea was that an infinite series represented something, that you can manipulate them as oversized polynomials and get something meaningful (e.g. Newton) and the job of the mathematician was to find out what it connoted.
A rough analogy is the use of complex numbers before the nineteenth century.
Except that there is no longer any debate. And, to pretend otherwise is just screwing people up.
It’s just another version of what those Numberphile twats did.
The author of the article studied mathematics in Germany in the 2010s. Experts texperts everywhere.
“In other words, you have what’s called a geometric series, something mathematicians have known how to solve for several hundred years”. In the name of what used to be German mathematics, I would like to apologize to Euclid and Archimedes. And don’t get met started on “solve”.
Yes, independent of the fundamental stupidity and pointlessness of the article, line by line it is also very very bad.
I avoided naming the author (but I’m fine with your comment), for the reason I usually do so. It’s not primarily the author who is at fault; it’s the association/authority/publisher who provided the platform and then took no proper editorial responsibility.
Plenty of mathematicians, and semi-mathematicians, try their hand at this stuff, and (hopefully) find out it is a *lot* harder than they imagined. In my early popularising days I wrote some very clumsy stuff and presented some very clumsy stuff. I still write some clumsy stuff, and I’m large enough and combative enough that I’m fair game. Smiling Eddie and the insufferable Eugenia are fair game. Smaller guys, I try to avoid putting in the spotlight.
I’m sure the author was doing the best she could. I put all the blame on the education system she had to put up with, including what passes as a university these days.
“Mathematicians” ?
Broadly construed.