earlgray's comments

That sounds like a cool project. Do you know of any transcendental number that does admit an intuitive proof of irrationality (or, more generally, of not being algebraic to any specific degree)?

My gut feeling is that this shouldn't typically be the case, but I'm tired and struggling to convey why without descending into downright criminal levels of vagueness. I'd be more hopeful about an algebraic number of degree >1 having a 'nice' reason for specifically being irrational.

If it's even odds then I expect it's a losing bet no matter what anyone writes down - that's why in the imagined scenario it's forced. The pertinent question isn't whether you expect to lose money by playing, it's whether you expect to lose less money by including non-motorised vehicles in your write-up than by excluding them.

Personally my instinct is that I'd lose more money by specifying skateboards and bikes as I've usually seen those addressed by their own signs rather than being included under "vehicle".

I agree 100% with the spirit of your point and I think imagining a forced bet scenario can help to clarify things. There are three main concepts we want to interpret within the context of the phrasing of the rule: (1) the intended referrent of 'vehicle'; (2) the intended meaning of 'in' the park; (3) the actual intention of the rule regarding emergency vehicles.

This is the scenario: imagine you're forced to wager a nontrivial sum of money on the following bet. You have to write down how you interpreted (1), (2), and (3). Then we randomly pick a real park that has this exact rule phrased in this exact way (I'm hopeful there'll be at least one out there), find the person who wrote the rule, give them your written interpretation, and ask if they agree. You lose if they don't. Notice we're not asking them to also write down a longer interpretation and comparing word-for-word. Just whether they think you got the gist of it.

I would write down that 'vehicle' was intended to refer to motorized passenger vehicles, 'in' was intended to mean that the vehicles shouldn't be in/on water or land within park boundaries, and that the rule wasn't intended to restrict passage to emergency vehicles responding to emergency situations. I expect most people would write something similar if they had real money on the line.

The trouble with the horrible website is it's trying to prove that nebulosity makes content moderation difficult by forcing people to disagree, but this disagreement almost entirely pertains to a point that has nothing to do with nebulosity: the park rule would only ever be written within a wider legal framework and doesn't make sense in isolation.

If I take my answers to (1) and (2), I'm forced to conclude that the emergency vehicles were violating the rule within the ridiculously artificial scenario presented. However, I'm also confident that this rule would only have been written verbatim within a wider legal framework that provided exceptions for emergency vehicles.

Consider self-defence in the context of murder or manslaughter. In the UK at least, the first thing the court does is establish whether the defendant would fit the criteria for murder / manslaughter ignoring the self-defence aspect, because otherwise it's a moot point. Once this is done, they would then establish whether the defence of self-defence also applies, which would then negate the conviction. If you wanted to prove that law is complex because it's hard to define words, would you really make a website that says "Ignore everything else you know and suppose that murder is only defined as killing a person" and then think you're being really smart when people disagree on the scenario involving clear self-defence? Hopefully not, because they're really only disagreeing with being forced to invoke your artificially-restricted definition.

That said, the website demonstrates the real reason why online moderation is hard: because it disproportionately attracts the sorts of people who answered 'yes' to the ISS question in this quiz. So you often end up with lots of users sharing a reasonable consensus on what the rules mean being moderated by a tiny group of... we'll say 'non-representative' moderators. It's a common problem with any banal form of authority, and isn't specific to website moderation at all.

Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners this is a perfectly sensible way of approaching it and they wouldn't be interested to know more, yet I found it rather clinical and dry.

As I'm sure you know, group theory arose as a novel yet natural way of investigating the conditions under which polynomials can have closed-form solutions, and when Galois began to sniff this out it allowed him to get to grips with the impossibility of a closed-form solution to the general quintic polynomial: a problem which had beguiled mathematicians for centuries, solved by a theory which crystallised a more structural way of approaching abstract mathematics. Even though Galois theory is a relatively niche topic within the broad context of academia, for me it draws out the true character of group theory in a way that matrix groups don't, and neither do contrived examples formed from the natural numbers under various quotients. And it clearly solves a problem that required a new way of thinking.

There's certainly a balance to be struck here. If you fixate on historical origins then you aren't engaging with the reasons why the topic is still relevant today, so you risk missing the point. It also slows down the pace at which you can absorb the tools needed for applications. But if you only engage with the modern approach you risk building a disconnected, lifeless archipelago of knowledge, unable to see the beautiful links that unify so much of mathematics.

I grew up with the Times crossword over dinner. We're not the most talkative family so a puzzle gives us something to do together while we're eating. A few observations, learned from introducing various other people to the hobby over the years:

1) The number one problem for new people is simply absorbing the rules of how to parse a clue. The linked article does a good job of going through these. Try to avoid the sense of learned hopelessness that often sets in early on.

2) The second most common isue is not managing in practice to get away from a literal reading. You need to take every word in isolation and try to escape the inevitable misdirection. For example, if the word 'rose' is in a clue that also contains the word 'flower', it's very unlikely that you're supposed to read 'flower' to mean something with petals. You need to think of any possible interpretation other than the obvious one. The classic second meaning here would be to read it as 'something that flows', which will mean the name of a river. Which brings us neatly onto the third difficulty:

3) Cryptic crosswords are heavily grounded in old-fashioned English culture. A reference to a river could mean a major international river or an obscure one from the British Isles, but it would be considered unfair to refer to an obscure river from another country. There are also some incredibly dusty references: 'sailor' could mean 'tar' (an archaic english slang that now exists only in crosswords) or 'AB' for 'able-bodied [seaman]'. 'Men' could refer to 'RA' (Royal Artillery) or 'RE' (Royal Engineers) among other things. One of the worst is 'posh' (or synonyms of it) to clue the letter 'u', which comes from high-society slang in the early-mid 20th century. 'Home counties' would be SE for South East [of England]. Cockney rhyming slang also often features, among many other things.

Problem 3 is the most insurmountable. It's also largely unfixable. If the range of acceptable references were broadened, it would become almost impossible for anyone to finish any given puzzle. But the references were fixed at a time that is no longer relevant and provides a huge barrier to entry for new people, which is why I expect these puzzles will largely die out over the next generation.

The bottom line is that if you feel like you struggle with cryptic crosswords, it's probably not because you're being stupid. There's a surprising amount of domain-specific knowledge you have to absorb, and in the best of cases any given puzzle will typically contain one or two absolute stinkers. A few references to help:

1) A list of common abbreviations: https://www.dummies.com/games/crossword-puzzles/cryptic-cros...

2) A website that solves clues and tries to explain: https://www.crosswordgenius.com/

3) A blog where people solve puzzles and explain them so you can learn how it works: https://www.fifteensquared.net/

4) I haven't seen squarepursuit before (linked by tclancy) but it looks like an excellent resource.

A few random tips:

1) Try to get the clues from the first row and the first column early on, as these give you starting letters for other clues

2) Get used to looking for anagram indicators. 'drunk', 'rotten', 'altered', anything like that. The other anagram indicator is always from adding up letters: if the answer is nine letters, look for combinations of words that add up to nine letters. Once you identify them, anagrams are a solid place to start once you've looked at the first row/column.

3) If possible, crosswords are best done with company. Everyone thinks in different ways.

4) Be wary of fish references. These can be incredibly obscure, and often indicate that setter was struggling to clue the last few letters so just googled them and found some vietnamese river fish that fit the bill.

5) If you're truly fed up, you can use a thesaurus on the word that you think is the definition. This is a bad habit, but if it lets you open up the puzzle a bit then it might be the right course of action.

My favourite clue ever: 'geg' (9-3)

This answer matches my experiences very closely. When you feel you brain begging you to gloss over the equation and move on - that's a red flag that you need to slow down, exercise discipline over your concentration, and figure out what's going on. It might take a few days for the intuition to settle. Personally I've known it to take a couple of years.

edit: patience, self-forgiveness, and a willingness to accept frustration are important traits. You might spend a whole week banging your head against the wall, feeling like you're making no progress, and then one day everything falls beautifully into place. That doesn't mean you did something correctly on the final day - it means you did everything correctly for the whole week before. Don't view a difficult and unrewarding day as wasted time. You're building something very difficult and that takes a bewildering amount of time.

Could you give some examples of notation that you struggled to understand, and explain what about it you find confusing?

One difficulty of notation is that the hierarchy of abstraction builds dizzyingly quickly, and soon you're manipulating symbols that generalise a whole classes of structures that were themselves originally defined in terms of lower-level abstractions. When this becomes overwhelming, it usually means that I didn't give my understanding of the lower levels long enough to settle and mature.

Concise notation and terminology is only useful if the underlying ideas are organised neatly in your mind, and the best way I've found of achieving this is to study a subject obsessively for some time, then put it away for a few weeks, and then go back and try to see the big picture and find out where it doesn't fit together by trying to derive the main results from scratch. Then I start again and fill in the blanks. After a few years things begin to make sense, but this process takes time and it's difficult and tiring (or at least that has been my experience of it).

In order to read research papers fruitfully it's crucial that you understand the basics well, and the best way to do that is to work through books aimed at undergraduates or young graduates. People don't read foreign literature by jumping straight in and looking up every word and every grammatical construction as they go. They become familiar enough with the language by reading easier texts until the language is no longer an obstruction - then they're free to appreciate what's happening at a higher level. The same for driving: you wait until you're comfortable operating the car mechanically before you drive on busy roads. The same, also, for mathematics.

It is not at all unusual to to find notation and technical terminology tiring. Everyone does to some extent. I hate it. But it's necessary.

Some resources I found useful:

Naive Set Theory by Paul Halmos. One of the great mathematical expositors, Paul Halmos here describes the fundamental language of mathematics: set theory. This is a book for people who want to understand enough set theory to do other parts of mathematics without obstacle.

How to Prove It by Daniel Vellemen. A nice introduction to logical notation and common proof structures, aimed at helping incoming maths students to become comfortable with the basics of formal language and notation.

Anything by John Stillwell. Stillwell is an inspiring teacher who insists on including the practical and historical motivations for the abstractions we use (this is, sadly, rare for modern teachers of mathematics). If you find yourself wondering why people cared about a problem enough to solve it, Stillwell might be able to help.

I suspect you'll also need resources on linear algebra (Halmos has 'Finite Dimensional Vector Spaces') and analysis but I'm not sure as I don't work in machine learning. I just sort of learnt linear algebra as I went and I avoid analysis as much as a supposed mathematician can. (context: my undergraduate degree was in economics and didn't carry much mathematical content other than some basic linear algebra - now I'm a graduate student in mathematics and computer science who uses a lot of category theory and abstract algebra. The transition was painful. Really painful.)