Almost all fields of pure math use a much stronger set of axioms called ZF and essentially everyone also accepts the axiom of choice (making it ZFC). The axioms in ZF are reasonable but the axiom of choice is surprisingly controversial for an axiom. There are some unintuitive consequences of the axiom but even more unintuitive consequences without it or with the negation of it.
Number theory uses a much smaller set of axioms.
It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.
I have no php experience so I have absolutely zero conception of what the language used to be like, just what it looks like now. I can't help but see it as some kind of frankenlanguage put together with pieces of java/python/javascript and a ton of dollar signs.
In other words, it seems too similar to languages that I already know to be worth learning. Lisp was fun to learn because it is an alien language. C was worth learning because of its usability and centrality to modern computing. The only benefit I can see from learning php is being able to understand more of software history. Cool but I'd rather learn another lisp.
Does anyone disagree with my analysis? I wish the language well but I don't see what it offers me. Obviously it offers a lot to the people who see JS as franken-php and so forth.
My belief is that people believe whatever they want to believe. If you are liberal it is because you want to be liberal. Alternatively you might just not want to be conservative. Same for the other direction. Naturally formed beliefs are just an afterthought. It goes without saying that this might not apply to everyone or to every belief.
For the record, I recognize that perhaps I believe this belief strictly because I want to believe it. I also think many people don't want to believe it.
I'm glad the label worked for you but it did not work at all for me. It kind of shoehorned me in to a personality type and limited my social options.
In all honesty I look back at my "smartness" as a kind of attention-seeking behavior where all I really wanted was the attention you got from getting the highest grade in the class or being the only one from your school to go to the ivy league.
To me, the smart label just means you are willing to sacrifice more than others to study/learn/build. Everything I have learned about people since starting college has told me that this is a net negative.
What is everyone’s favorite translation of Meditations? I bought a cheap copy and the translation is unsurprisingly hard to read and distracts from the text’s content.
I've noticed several side effects such as decreased attention span, decreased sociability, decreased desire to do anything other than your preferred method of attention capturing.
Frankly I just think that many of these things are simply addictive and, with an appeal to naturalism, I don't think it is what humans were meant to do. Increased smartphone use or TV watching or video games probably almost never results in increased happiness over the long term.
Not to mention the ability to use attention capturing as a way to hide from one's own emotions.
No denominators for zero divisors in the localization of any ring!!! [English: No denominators where you can multiply the denominator by anything nonzero to get zero.]
Technically, many definitions of localization do allow for zero/zero divisors to be included (we can use any multiplicatively closed set) however by the definition of localization all of the elements in our localized ring become equal to each other.
The mathematical meaning of a/b = c/d in a general ring is that there exists some value s in the set of legal denominators where s(ad - c*b) = 0. If 0 is an allowable denominator then a/b = c/d for every a,b,c,d.
Smartphones are what I think of as "attention capturers" which are able to completely capture almost anyone's attention span for a significant amount of time. Other attention capturers include computers[0], video games, TV, books, chess. Each of these has varying capability as attention capturers but it is safe to say that smartphones are the strongest attention capturer for most people.
I believe that using any attention capturer for long periods is not healthy and should be avoided. The hard part for me is actually finding anything else to do in a new city without family/friends. Especially on a Tuesday night after work where it is less possible to go to a bar and talk to strangers.
I simply don't understand what is going to happen to all of the PhD students that I saw when I was in university. I managed to escape the PhD trap and find software engineering but I was extremely lucky. Had I gone to any other school I probably would not be anywhere near as fortunate.
That being said, I worry about the people who go to PhD programs just because that's what they are "supposed to do" or because they just don't know what else to do. Especially the ones who lack social skills or lack "connections". Many of my friends from college fall in to this category.
I don't even want to think about the people in the above category who get a PhD at a "non-target" school.
Its not really "what we perceive as mental illness", it's just a mental illness. It doesn't have to have an innate biological component to be a mental illness.
I am a big fan of the "humans are social creatures" mantra. In that regard, if something is more stimulating than social interaction it is likely to be harmful but if something is less stimulating than social interaction it is probably fine. [1]
In this simple model, I would probably say books are definitely less stimulation, TV is probably about equal (I have no idea), and modern videogames and social media are definitely much more stimulating. So, books would only become a problem in the presence of other factors but social media can become a problem for anyone.
In highschool I would play high stimulation PC games while browsing the internet on another monitor while listening to music while using voice chat. I also had almost no desire to socialize while in highschool.
[1] It goes without saying that we probably can't reduce activities to a single metric like "amount of stimulation" but I think this is a useful thought experiment.
I can't help but think I would be much happier if I had never seen a computer or cell phone in my life. Maybe its just my personality type. Maybe its because I started at a very young age. It could be anything but I am fairly certain that the constant stream of stimulation to my brain has done plenty of damage.
This article seems analogous [1] to an article that states "I use hard drugs and I'm not ashamed" authored by someone who manages to keep their use under control. Sure, it works for them. Great, but they are not the failure mode of drug use.
Number theory uses a much smaller set of axioms.
It should be stated that most mathematicians don't really mind the logical foundations of their work when they are actually working in the same way that most programmers don't worry about assembly language or transistors.