Asaf Karagila
I don't have much choice...

Syntactic T-Rex: Irregularized

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One of my huge pet peeves is with people who think that writing \(1+2+3+\ldots=-\frac1{12}\) is a reasonable thing without context. Convention dictates that when no context is set, we interpret infinite summation as the usual convergence of a series, namely the limit of the partial sums, if it exists (and of course that \(1+2+3+\ldots\) does not converge to any real number). However, a lot of people who are [probably] not mathematicians per se, insist that just because you can set up a context in which the above equality holds, e.g., Ramanujan summation or zeta regularization, then it is automatically perfectly fine to write this out of nowhere without context and being treated as wrong.

But those people forget that \(0=1\) is also very true in the ring with a single element; or you know, just in any structure for a language including the two constant symbols \(0\) and \(1\), where both constants are interpreted to be the same object. And hey, who even said that \(0\) and \(1\) have to denote constants? Why not ternary relations, or some other thing?

Well. The short answer is that they are not used for anything other than constants, because the readers are mostly human (sometimes computers, and sometimes cats), and they take strong hinting from the choice of letters as setting up context. If I use \(n\) for some index, it hints to the reader that this is a natural number, or \(\varepsilon\) hints at a very small amount when it comes to analysis. If I use \(\kappa\), at least in set theory, this hints at a cardinal. Whenever I work with people, we run into a joke that as far as large cardinals go \(\delta\) is generally a Woodin cardinal, sometimes an extendible, and rarely a supercompact. And \(\kappa\) is always regular, unless it was a measurable that we singularized somehow.

The point is that \[\lim_{\varepsilon\to\infty}\int_\pi^{\frac1{\omega}}\int_\delta^\kappa\varepsilon\cdot\aleph_0(\omega_3,\Omega,\Bbb R)\operatorname d\Bbb R\operatorname d\Omega=42\] is a valid mathematical statement, which should cause most, if not all, mathematicians to cringe, look away, and possibly burst into tears. Because it feels wrong.

But hey, don't leave the site just yet. I know that you didn't come here to read my tirade against people who misunderstand the whole point of an implicit context. You came here for a Mathematical T-Rex comic!

(Thanks to Matt Inman of The Oatmeal for the template, which can be found here.)

There is one comment on this post.

By Asaf Karagila
(Jun 20 2016, 20:42)

Admittedly, I made this comic a few months ago, after some long debate against several people on math.SE; only towards the end of it, it dawned of me that those people who insist that just stating the sum is not a blatantly false statement will also insist that \(0=1\) is in fact a blatantly false statement.

For one reason or another, I didn't post it back then. But now I was thinking about something related, that didn't come into a full post of its own, where syntax and semantics were also separate, and I ran into this comic. So I figured, eh, why not post this one.

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