The use of computers in mathematics has been somewhat controversial from the very start.
There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understanding. I like this one quote by (allegedly) Wigner that illustrates it well:
"It is nice to know that the computer understands the problem, but I would like to understand the problem, too."
On opening the article, I was somehow expecting a mention of the large deviations formalism, which was (is?) fashionable in late 20th century, and gives a nice information theoretic view of the CLT. Or something like that. There's a ton of deep math there. So having a bio statistician say "look, the CLT is cool" is a bit underwhelming.
Edit: see eg John Baez's write-up What is Entropy? about the entropy maximization principle, where gaussians make an entrance.
While you can certainly argue that some texts have more substance to them than these literary works, you cannot deny that most texts have worse prose than the books.
> abusive language/notation basically everywhere in math
In most cases it is not as much abusing notation as overloading it. If you think of the context of a formula (say, adjacent paragraphs) as its implicit arguments (think lambda captures in c++), then it is natural that curly braces can denote both a set and a sequence, depending on this implicit input.
Such context dependent use of symbols is actually rather convenient with a little practice.
In the context of double precision the article says
> the largest integer value that can be represented exactly is 2^53
— I am confused as to why it not 2^52, given that there are 52 bits of mantissa, so relative accuracy is 2^-52, which translates to absolute accuracy larger than 1 after 2^52. Compare this to the table there saying "Next value after 1 = 1 + 2^-52".
While the idea — of shuffling a societal system a little bit to prevent it from going stale — sounds important, I'm not convinced. Random shuffling leads to good results only when it is combined with a good fitness estimate (see: natural selection). And establishing a fitness test for a societal order seems to be a much harder issue than than that of an organized randomization.
Not necessarily—in 3d there are uncountably many non-lattice packings. They all have the same density as the FCC lattice though. To construct these packings, shift horizontal layers of FCC horizontally with respect to each other.
It is conjectured that in higher dimensions, the densest packing is always non-lattice. The rationale being that there is just not enough symmetry in such spaces.
That's true — time is a big place, and a lot of things are lost in it. I for one am rather more pained over disappearing of physical objects — genomes, books, art.
Plus, it is all the more exciting to think about what caused some languages to exist and thrive for so long, and the information about the past they retained.
If you like this kind of language archeology, check out David Anthony's The Horse, the Wheel, and Language — for how the people that spoke the Proto Indo-European language were located in time and space.
Much of the NYT article can be explained away by the Gell-Mann effect. During most of human history it was hard to maintain multiple strong bonds anyway; long distance communication pre-internet was hard too. There are plenty of modern opportunities for finding friends based on interests: conferences, concerts, sports bars etc. How much of this discussion is a moral panic caused by imprecise notions which by definition cannot be described by hard data?
— of course. In the same way the (standard) Cantor set consists of precisely those numbers from the interval [0,1] that can be represented using only 0 and 2 in their ternary expansion (repeated 2 is allowed, as in 1 = 0.2222...). If self-similar fractals can be conveniently represented in positional number systems, it is because the latter are self-similar.
Vim Startify recently showed me a quip it attributed to Alan Kay. It went something like this:
Modern software is similar to the pyramids: it consists of millions of blocks held together by nothing but sheer weight and the labor of thousands of slaves.
Nice example. You can avoid the issue with these being non-functions (because of the vertical segments) by rotating the picture so that the (0,0)-(1,1) diagonal is horizontal. That way you are talking about a sequence of continuous functions F_n with slopes alternating between ±1 that approach the constant-zero function F_0=0.
The reason your intuition was confused was that you felt that there was a way in which these functions with ±1 slope converge to zero function (it is called uniform convergence), but were unsure about the way in which their derivatives f_n=(F_n)' (which keep flipping between ±1) converge to zero.
The resolution of the seeming difficulty is that there are different modes of convergence. The ±1 slope functions F_n converge uniformly to zero, but their derivatives f_n, which determine curve length, do not converge uniformly or even pointwise to any limit (uniform convergence is stronger — more restrictive — than pointwise). This is why it is reasonable that the F_n curve length stays 2 throughout the exercise.
Note: if the derivatives f_n were converging to f_0=(F_0)'=0 pointwise while staying bounded, then the curve lengths of F_n would converge to sqrt(2), the curve length of F_0. This is called "dominated convergence theorem" for integrals — point being that the curve length of F_n is an integral of sqrt(1+f_n^2). But there is no pointwise convergence, and no such implication.
Finally, you may be curious as to whether there is a sense in which f_n converge to f_0=0 — does all that flipping back and forth amount to some way of converging to zero? Turns out yes! It is called weak* (weak-star) convergence and it applies to f_n if you think of them as measures (or distributions) — that is, if instead of pointwise evaluation you characterize f_n by the way they act on test functions by integration: instead of computing f_n(x) you multiply f_n * phi and integrate. Phi has to be a continuous function. Under this notion of evaluation, f_n do converge to zero.
Yeah, the best part is that a woodpecker's tongue wraps all the way around the back of his head and attaches in the right nostril. The head effectively serves as a spool for the tongue. Awesome technology.
Annnd, this is exactly how humanity figured out astronomy before the telescopes: by taking multi-year observations and keeping detailed records! See Antikythera mechanism, Tycho Brahe, etc. Arguably the entire Greek epistemology was so solid because their astronomy turned out to be an early proving ground for the scientific method.
There are of course all the computer-assisted proofs (see 4 color theorem), as well as the partially-assisted ones (see Viazovska et al on packing problems in dimensions 8, 24). But even finding a solution numerically, then rigorously verifying its properties can leave a lingering sense of incompleteness, of a gap in understanding. I like this one quote by (allegedly) Wigner that illustrates it well:
"It is nice to know that the computer understands the problem, but I would like to understand the problem, too."