The NL Times just translates Dutch articles and editorializes them for a (mostly American) audience. They should be consistently taken with skepticism. In this case, as other commenters have pointed out, this is "just" Google Analytics.
Pretty neat! However, if you wanted to know the _probability_ of a noodle crossing any line in the long noodle case (L/W > 1), the expression is more complex (and I believe would require an integral) :).
It's interesting that the number of crossings is independent of whether L/W is less than or greater than 1, but the probability of crossings is equal to 2pi * L/W only in the short case. This makes sense since in the short case the noodle can at most cross a single line.
This is the crux of the observation. For needles of length less than W, the probability that it crosses a floorboard is equal to the average number of floorboards it crosses. (Exercise for the reader ;))
The point is that the "right" quantity to be considering for the problem is the average number crossings, since that naturally extends to curved noodles, lines of any length, and even circles. The number of crossings is also known as the Euler characteristic of the intersection, and there's a rather deep and beautiful theory of geometric probability that takes this as the jumping off point.
Is the probability actually more interesting though? I find the symmetry of this type of result extremely compelling, beautiful even. Buffon himself restricted his attention to the case where the needle was short enough that "probability" and "expectation" had the same answer. Put simply, math is best when complicated-seeming things suddenly become simple.
Proof irrelevance I don't think is accepted in constructivist situations. Those are, however, not that relevant to the recent wave of AI math which uses Lean, whose type system includes classical mathematics.
I am excited for some alternative syntax to jq's. I haven't given much thought to how I'd write a new JSON query syntax if I were writing things from scratch, but I personally never found the jq syntax intuitive. Perhaps I haven't given it enough effort to learn properly.
You don't learn it properly. It's not supposed to be intuitive, it's supposed to be concise at the cost of it being intuitive. Would be like somebody saying typing words in to Google is more intuitive than writing regex.
jq is supposed to fit in to other bash scripts as a one liner. That's it's super power. I know very few people who write regex on the fly either (unless you were using it everyday) they check the documentation and flesh it out when they need it.
Just use Claude to generate the jq expression you need and test it.
I worked at a family-owned bike shop, at a fancy hotel as a valet/porter, and picked up a few shifts at breweries/events. I’ll be jumping back into software in a few months but it’s been a refreshing year of doing something different.
This is incorrect. The set paradox it’s analogous to is the inability to make the set of all ordinals. Russel’s paradox is the inability to make the set of all sets.
Technically speaking, because it's not a set, we should say it involves the collection of all sets that don't contain themselves. But then, who's asking...
This is the easiest of the paradoxes mentioned in this thread to explain. I want to emphasize that this proof uses the technique of "Assume P, derive contradiction, therefore not P". This kind of proof relies on knowing what the running assumptions are at the time that the contradiction is derived, so I'm going to try to make that explicit.
Here's our first assumption: suppose that there's a set X with the property that for any set Y, Y is a member of X if and only if Y doesn't contain itself as a member. In other words, suppose that the collection of sets that don't contain themselves is a set and call it X.
Here's another assumption: Suppose X contains itself. Then by the premise, X doesn't contain itself, which is contradictory. Since the innermost assumption is that X contains itself, this proves that X doesn't contain itself (under the other assumption).
But if X doesn't contain itself, then by the premise again, X is in X, which is again contradictory. Now the only remaining assumption is that X exists, and so this proves that there cannot be a set with the stated property. In other words, the collection of all sets that don't contain themselves is not a set.
Let R = {X \notin X}, i.e. all sets that do not contain themselves. Now is R \in R? Well this is the case if and only if R \notin R. But this clearly cannot be.
Like the barber that shaves all men not shaving themselves.
The paradox. If you create a set theory in which that set exists, you get a paradox and a contradiction. So the usual "fix" is to disallow that from being a set (because it is "too big"), and then you can form a theory which is consistent as far as we know.
Perhaps I overstated how related the two were. I was pulling mostly from the Lean documentation on Universes [0]
> The formal argument for this is known as Girard's Paradox. It is related to a better-known paradox known as Russell's Paradox, which was used to show that early versions of set theory were inconsistent. In these set theories, a set can be defined by a property.
I came across this when wondering if there were any efforts to give programmers additional information via audio, similar to how colors are used in syntax highlighting.
There are some interested experiments out there in modes of highlighting, such as where colors are assigned to particular variables.
IMO syntax highlighting is still the best default for navigation and general reading, but I can imagine tasks where I might want to temporarily toggle what information I'm getting through that color-of-text route.
