ginnungagap

As a mathematician this looks like standard usage of "just" in mathematics, but I do try to avoid doing that because it does sound arrogant and patronizing

Where are German users supposed to have received this message? I'm not German but my telegram account is registered with a German number and I haven't received anything

There's also chesstempo for puzzles if you are not aware of it already, the puzzles are great and free, with subscription tiers for more features (unlimited endgame puzzles, custom puzzle sets, opening trainer etc.). The UI is ugly, clunky and old, but as long as it is only for puzzles and not playing I can stand it

> simply because the latter offers superior game analysis experience.

But it does so once per day, unless you pay (the low depth analysis you get for free on chess.com is not great to put it mildly)

I'm Italian, my PhD in Germany in pure maths is paid 40% more than friends working in Italy with engineering degrees. If I stay in Germany after finishing my degree I could easily make twice the money I could get in Italy regardless of whether I stay in academia or go to industry. Not a strong incentive to go back in the immediate future!

This is probably enough information for the folks at the Science Fiction & Fantasy Stackexchange (https://scifi.stackexchange.com/) to identify the story, they get similar story identification requests all the time and are crazy good at it.

If you do ask make sure to include approximately when you read it and please let me know, I'm curious about your story too now!

Your grade reflects how well you learned with respect to your peers.

Getting 80% in an exam in a weak cohort of students might earn you an A, while doing so in a strong one might get you a B instead. Is this fair toward a student getting a B this year with the same score that earned somebody an A the year earlier just because they happened to enroll with different peers?

Also why should the grade distribution of an exam be determined beforehand?

> He hasn't done any harm to anyone.

Reading posts from former employees and plagiarism accusations (look up Matthew Cook) suggests otherwise.

I remember this being discussed on HN before as well, even though the search function is not bringing up anything.

I have aphantasia and I'm doing a PhD in math.

I draw on paper when I'm trying to visualize something, but the area I work in is quite abstract and very often the images are huge oversimplifications that give some intuition, but on which one shouldn't rely too much.

I agree that the suggested method doesn't sound like a good approach at all to me

Makes sense, the French railway metric strikes again...

> Eurostar and (the rest of) the French high speed network really compensate some simple geography.

Wouldn't German travelers also take advantage of French high speed trains when travelling to Spain?

There is no cover-up going on here, it's just that we know very little about preroman and especially preetruscan populations of Italy.

There is a wikipedia page[1] concerning megalithic architecture in Lazio (the region of Italy in which Rome is), which unfortunately is only available in Italian but contains some information about the estimated age of most of the structures from the linked submission, as well as attempted explanations. For example archeologists suppose that some of the most advanced walls where built by local populations helped by wandering Greek builders based on similarities with the temple of Delphi and Greek acropolises. Wikipedia also has individual pages for some of the ruins[2][3] (also not available in English) where other theories are suggested, for example the acropolis of Alatri could have been built by the Hernici[4].

[1] https://it.m.wikipedia.org/wiki/Architettura_megalitica_del_...

[2] https://it.m.wikipedia.org/wiki/Acropoli_di_Arpino

[3] https://it.m.wikipedia.org/wiki/Acropoli_di_Alatri

[4] https://en.m.wikipedia.org/wiki/Hernici

I wasn't aware this is a named phenomenon, thanks!

Definitely this, everyone sounds like an expert to me in the programming/engineering/humanities threads until I see a one about mathematics (I am a mathematician) and I remember that it is just my ignorance of the other topics preventing me from being able to tell whether people actually know what they are talking about!

It also depends where you are in Europe, in Italy gas is still by far the most common way to cook

I'd say it still applies, even when reading a textbook going through the exercises is crucial

Would be nice to have a writeup with some details on how this disproves Connes embedding conjecture understandable for operator algebraists with little (quantum) computing background

In fact it's an open problem (very likely the oldest open problem in set theory) whether in every model of not AC there is a set with such a paradoxical partition! Here's a nice introduction to the problem by Asaf Karagila http://karagila.org/2014/on-the-partition-principle/

When we say that a statement P is independent of ZFC what we're really saying is "if ZFC is consistent, then P is independent of ZFC (hence both ZFC+P and ZFC+not P are consistent)".

This is the only sensible way to interpret claims of independency, since if ZFC is inconsistent it just proves every statement so there's no independent ones

It gets worse than that, by Easton's theorem, except for two mild restrictions provable on ZFC, the function that sends a regular cardinal to the cardinality of its powerset can be anything

Either you show that your statement P is equivalent to another known independent one or you need to produce a model of P and a model of not P.

The two main techniques to produce new models of ZFC are inner models, that is submodels of a model, for example studying the inner model L (Gödel's constructible universe) is how the consistency of the continuum hypothesis, the generalized continuum hypothesis, the existence of Suslin trees etc. was shown. Looking at L also allows one to conclude that the axiom of choice is consistent with ZF. Another commonly study inner model is the so called HOD, and there's a whole area, "inner model theory" that essentially tries to construct canonical inner models for some statements.

The second big way to produce a model of ZFC is by forcing. This is a very versatile tool that allows to extend a given model of ZFC by adding a new set to it (for example by adding a lot of new reals numbers to a model of CH you can make CH false). Forcing is how the consistency of the negation of CH and GCH was proved.

An interisting example is the negation of AC and its consistency with ZF. If you happen to live in a model of AC every forcing extension will still be a model of AC so it seems that our previous techniques are powerless. But actually what can be done is to look at a carefully chosen submodel of a forcing extension that still models ZF but in which AC fails.

If the continuum hypothesis (CH) holds then there is none.

Assume that CH fails and well order the reals. An initial segment of length omega_1 is an example of such a set.

Another famous example of that kind is the existence of outer automorphisms of the Calkin algebra, which is a simple question about a naturally occuring object which turned out to be independent of ZFC by work of Farah and Weaver.

Topology and set theoretic topology is full of those statements, are there Suslin lines? Are there S-spaces? Is the product of two ccc spaces also ccc? The list goes on

Another one from analysis is a very strong form of Fubini's theorem that was shown independent by Friedman.

There's surely more but those are the ones I could remember right now!

Why a 3-category in particular?

This talks about adjoints of operators, which is where the category theory terminology comes from, but a pretty different concept overall

They do different things, ( and ) work just fine, but they are standard sized, while \left( and \right) adapt to the size of what's inside. It's kinda uncomfortable to write, but I guess it is done this way because the \left and \right commands can take a lot of different arguments

The post you linked says that Tao was teaching the class and made notes for it since he didn't like the book

The github link in [1] is dead, where is the repo now? I was curious about that compiler

It is a mnemonic if you're German, which is where the symbol comes from, it's the Z of Zählen!

I'm actually using VS Code for the Haskell project I mentioned in my submission, so I just finished figuring out how git works with VSC, this might be a good solution for me