Douglas Bruce finds himself one day riding the subway in New York without knowing where he is going. In fact, he doesn’t know where he is, he doesn’t recognize any of the buildings. Panic sets in once he realizes he doesn’t know where he took the train, or even his own name. Doug suffers total amnesia, a rare condition in which one forgets everything about one’s life. Well, this is not exactly true: He knows how to talk, and when in the hospital someone asks him to sign, he remembers his signature. This strange and fascinating condition is explored in this documentary that uses footage shot by Doug himself and directed by his (former) friend Rupert Murray.

I found the documentary quite interesting, but I also found it wanting in several respects. Part of it may be due to the simple fact that we still know very little about how memory works, so surprisingly little time is devoted to hard data, to what may be happening—as for why it is happening, nobody knows. Some hypotheses are mentioned, and as the story progresses we get some clues. But part of the problem with the story I think is due to what I perceive as a shortcoming of the director: There are questions that do not get asked, some that beg to be asked, and why they are not seems to be because everybody is so fascinated by what is happening that they assume that filming it is enough. Part of it I think is due to the friendship between Doug and Mr. Murray. Mr. Murray seems to go out of his way to make Doug feel comfortable, while obviously the subject matter may make him uncomfortable. So, at the end of the day, I find the final product a bit awkward. There are a few extras in the DVD that leave me feeling the same (at least there is consistency); I missed (being in Vienna) the controversy on the veracity of the story, so the short section addressing it didn’t mean much to me. There is a long section explaining how a sequence was shot. I found it very curious that the director put so much thought into the visual look of the final product instead of trying to add a bit more substance to it.

I like explorations of memory, and mental problems intrigue me to no end. So this was a good movie overall. It complements well other documentaries in similar subjects, like anterograde amnesia, the disease that Memento popularized. I had the fortune of watching in 2004 an excellent documentary by Koreeda Hirokazu about one such case in Japan, Without memory.

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This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here: Paul Erdős. Some set-theoretical properties of graphs, Univ. Nac. Tucumán. Revista A. 3 (1942), 363-367 MR0009444 (5,151d). (Anyway, it is probably easier to read a more modern pre […]

One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about. As far as I can tell, the question is sometimes called the Erdős-Turán con […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on $X$ such that $|X|

A solution can be obtained as suggested by Keith Conrad in the comments, via Chebotarëv's theorem. Details can be found in $\S3.4$ of Coloring the $n$-Smooth Numbers with $n$ Colors Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach The Electronic Journal of Combinatorics 28 (1) (2021), #P1.34, 79 pp. DOI: https://doi.org/10.37236/8492 Many t […]

No, this is not possible. Dave L. Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two parts (with numerous references). See here: 1 (dated May 9, 2002 6:18 PM), and 2 (dated May 19, 2002 8:29 PM). As indicated in part 1, in Zygmunt Zahorski. Sur l'ensemble des points singuliers d'une fonction d'une variab […]

This is a difficult question in general. Ideally, to show that $f$ is analytic at the origin, you show that in a suitable neighborhood of $0$, the error of the $n$-th Taylor polynomial approaches $0$ as $n\to\infty$. For example, for $f(x)=\sin(x)$, any derivative of $f(x)$ is one of $\sin(x)$, $\cos(x)$, $-\sin(x)$, or $-\cos(x)$, and the error given by the […]

To complement Yann's answer: This is a nice problem, the possible length of Borel hierarchies in different spaces or without assuming the axiom of choice. It has been studied in detail by Arnie Miller. See Arnold W. Miller. On the length of Borel hierarchies, Ann. Math. Logic, 16 (3), (1979), 233–267. MR0548475 (80m:04003), Arnold W. Miller. Long Borel […]

This is a good question, because a priori $\mathsf{PA}$ lacks the flexibility of $\mathsf{ZFC}$ that allows us to deal with consistency problems semantically (by building models) and, anyway, the obvious model of most subtheories of $\mathsf{PA}$ is just the standard model. The way this is done in the context of $\mathsf{ZFC}$ is using the reflection theorem […]

Yes, of course. An example is the statement that all Goodstein sequences terminate. The point is that this sentence is not only independent of $\mathsf{PA}$, but in fact of the theory resulting from adding to $\mathsf{PA}$ all $\Pi^0_1$ statements true in the standard model of arithmetic. Note that $\mathrm{Con}(\mathsf{ZFC})$ is an example of such a $\Pi^0_ […]