my wife's field (sociology) is far, far worse, in terms of writing quality. people just write horses--t. you're almost penalized for not writing horses--t.
terrible prose, sentences that go on forever. you can never pin down a meaning. math does better than that, most of the time.
So now if you try to explain it to someone there are lots of preciseness involved in the correctness of the algorithm. We just need to show 3 things - the two major ideas in FT which is periodicity and its behavior across periodic functions + how prime finding can be mapped to period finding and then how we can finally get the result upon certain input.
If I explain all this, still it would not makes sense, why this algorithm would be faster, because then we need to explain the physical charactersitics of so called quantum states - which is irrelevant. So we should just consider it as a black box and lie that "well there is a system which would basically have multiple states instead of 0-1" and then go on about it.
@leslietownes The social sciences got taken over by the post-modernists 20 or 30 years ago. It was sad, and largely why I am a mathematician, and not an archaeologist.
@user27286 The example I like to give here is in dealing with the solutions to the equation $x^2 + c = 0$, where $c$ is a real number.
When students first see such a thing for positive $c$, there are two approaches: (1) we can say that the equation has no solutions, or (2) we can say that the equation has no solutions in the real numbers.
The first is a lie, and is wrong, and should never be told to a student.
@user27286 in my wife's field i think there is a kind of premium placed upon being vague. you never want to say anything outright for fear of giving offense. so this style develops of just, haze and nothingness throughout everything.
That is, (2) is completely correct. The point is that the description given to a student needs to be pitched at the right level. High school algebra students, encountering solutions to polynomial equations for the first time, are likely not mathematically mature enough to handle complex numbers. So we don't work with complex numbers.
But we also don't pretend that they don't exist---they are just out of the current scope.
I know. But this one neither refutes nor confirms what i said regarding partial correctness. I didnt gave the complex number example, I just said it could be something that is close to that. And my example was what I gave.
I believe that everything I tell my students should be completely correct, and that if something is beyond the scope of the current class, I should adopt language which clearly explains that we don't currently have all of the tools needed, and so we are going to focus on a "nicer" special case.
I wish I could give you a better example. But most of the time in computer science and physics these things are the way to go. Building the intuition with less precise thing and then correct it to the true form.
@user27286 Okay... but you were complaining about mathematicians writing badly, not physicists or computer scientists. I don't work in those fields, nor do I read what they write.
I get your point, I guess your point is the less preciseness is not always showing partially correct things because you are mentioning the framework accordingly. And I am saying thaat not showing the full framework which exists is kind of not the truest + correct form.
You keep switching back and forth between "we should teach things which are incorrect" and "mathematicians write badly". I don't even know what I am supposed to be responding to anymore.
@user27286 Okay. I haven't read those books. And you haven't answered my other questions. What is the context in which those books are meant to be read? Who is the intended audience?
Moreover, can you back up your assertion that mathematicians are worse, on average, than writers in other fields?
@Avra Depends. But in the US curriculum, measure theory is usually taught either in the last year of an undergraduate program, or early in a graduate program.
@XanderHenderson I didn't even check it. I expected anyone above 10 years old should able to read and understand it. Because no matter how difficult a topic is it is explainable to the simplest form.
@Avra I've never taken a class in statistical inference. I would imagine that is taught in stats programs, not math programs (and many, many institutions in the US divide the two departments).
@user27286 Why would you expect a 10 year old to understand graduate level mathematics?!
i don't think it's possible to give a simple explanation of everything. i wrestle with this in my day job (attorney). people want to leave details out because they think it will be more persuasive. i find it less persuasive.
we had a sad moment a week ago. we used a glue trap to catch crickets in our garage. they were running rampant. a lizard went after the crickets and died.
@user27286 Have you really taught the Fourier transform to a 10 year old? Really? Which version of the transform? Are you only defining it on $L^1(\mathbb{R})$? or are you extending it to $L^2(\mathbb{R})$? or are you working in higher dimensions? Or are you thinking about locally compact topological groups?
But that was the point. We don't need to show the whole stuff. Just basic explanation of things can sometimes give a better passage way to learn the deep things on our own.
@user27286 It depends. I can explain the Fourier transform to a high school student to a level where they might feel like they have some idea of what is going on. But if I expect a student to actually use the FT to do mathematics, they need to learn quite a bit of theory. That theory is hard. It will take time and perseverance to get them there.
@XanderHenderson But do you understand where from it started? I started reading this book. The author says the field is new and anybody entering this field should use this book as a basic building block but then he fails to clarify where from these things come intuitively.
i actually read a contract the other week that referred to the party of the first part and the party of the second part. someone was using a very very old form.
i had another contract where the contractee had to warrant that he was not a member of the communist party and could be fired if he was. the blacklist clause.
@XanderHenderson Why are you comparing with anyone else? Mathematicians are bad at writing. Period. Well to be honest, if we need to compare, I would have to say whenever people udnerstand something they forget how did they come to understand that. And they never mention the journey of understanding clearly.
Everyone is a set that include mathematicians as well so this statement is also true "Mathematicians are bad at writing". Why are you feeling bad about it?
Define $\Phi: D^n \rightarrow \mathbb{R}P^n$ by $\Phi(x_1,........,x_n) = \left[x_1,.......,x_n,\sqrt{1-||x||^2}\right]$ .
Where $\mathbb{R}P^n$ is quotient space obtained by $x\sim y$ in $\mathbb{R}^{n+1}\backslash \{0\}$ if and only if $y= \lambda x$ for non zero real $\lambda$
Problem: Show th...
@leslietownes It is very much named after the fruit. At the end of the 19th century, there were HUGE orange plantations all over southern California. Riverside was, for a time, the wealthiest city in the country (per capita) because of the orange industry.
i'll let them have it. you don't see any agriculture down here anymore. a little out near the desert.
i grew up in napa. it's known for grapes, but the first money crop was prunes. you can still find wild plum trees from the old days if you know where to look.
Dates are my favorite, though. I LOVE dates. This is the one disadvantage of my sister moving away from Death Valley---I no longer have an excuse to drive 9 hours to get to the China Date Ranch.
In one of your comments, @robjohn , you write that $u$ is not a function of $h$, yet $u=(t-x)/h$. What is incorrect about treating $u$ as a function of $h$?
we'd send them way out into the hills. it feels like you'll cross over into sonoma, but you can't.
we had a similar phone number to a home goods store and when people would phone us up, i'd announce happily that whatever they wanted was on a 50% off sale and to come down right away.
I think midwesterners do the same thing, but in a more passive aggressive way. "Whelp... Ya just head down the road to where the old Smith farm used to be, and hang a right. Once you get passed the granary they tore down in ought three, turn left and head down the road toward the Johnson's place..."
You know... about five years ago, I was driving through Napa, and got directions from this crotchety old bastard. We ended up way out in the boonies...
i lived on the west edge of town. there was a hill which we called a 'mountain' (it wasn't) but from the top you could see san francisco, 40 miles away.
when i lived in iowa city we were engaged in about a year of combat with a groundhog. other animals kept getting caught in his trap. we never got him. he tunneled under the porch and just lived there.
@TedShifrin I think George Carlin had a comment about this vis-a-vis driving: everyone driving slower than you is an idiot, and everyone driving faster than you is an asshole. Something similar occurs with age, right?
Anyone ever drive up to Mexico City, or down? The city is quite elevated above seal level. I'll never forget the bus I was on going downward, sitting on the passenger side, and not seeing ground to my right, with spirals of constant curves. I shiver thinking about it!
@XanderHenderson I absolutely loved his routine on the difference between cats and dogs! He acted the part of the cat, and the dog, so brilliantly!
i used to bike out there and people would run you over. lots of tight corners and hills where you can't see much. and half of the tourists are half drunk.
It's is interesting to see when I drift out of the room when I sleep my computer, and when I stay here. It may have something to do with someone saying something to me.
@leslietownes A mile less air above you and probably a good distance from a major polluting city.
why anybody wants to host the olympics is beyond me.
i think LA did turn a profit on the 84 olympics because they didn't build anything and just hosted stuff in existing places.
now it seems like they're not doing that.
they do a street race here, it used to be formula one but is something else now. it's loud. you can hear it halfway across town. i like that. it's one day.
Hello, why for upper bound $O$, we can not have $f (n) = O(g(n)) \implies g(n) = O( f (n))$ true? Although if $f \in O(g)$ if $f(n) \le cg(n), ~ n \ge n_0$. Similarity, if $g \in O(f)$ if $g(n) \le cf(n), ~ n \ge n_0$. What if both $f=g$, then it should hold, is not it please?
What is the intuition here please not the solution?
@Avra I guess you are missing the point here. f(n) <= c g(n) for all n>= n_{0} you can't say this if you rotate this definition. The "for all" is in the definition itself.
O-notation gives an upper bound for a function to within a constant factor. We write f(n) = O(g(n)) if there are positive constants n0 and c such that to the right of n0, the value of f(n) always lies on or below cg(n).
btw, just spoke about this and remembered something that happened to me - a russian algebra professor a few years ago allowed the two russian students in class to record him, but didn't allow anybody else (including me, who asked) to do the same thing, showing clear favoritism to the russian students. I just remembered that I didn't do anything about it, and that it wasn't right. Would complaining now to the department head have any effect?
@user27286 There are very few instances when I find myself, as an adult, to not have understood many things I would claim to have understood only a few years ago
Well, there is always a chance that I haven't understood something that I think I have understood. But it's rare. If you are trying to say sth from the previous discussion, I will pass.
Hi can someone please check this: $\forall z=re^{i\theta}\in\mathbb{C}\setminus\{0\}, \exp^{-1}(z)=\{\log(r)+(2k\pi +\theta)i | k\in\mathbb{Z}\}$ and each of these maps homeomorphically to $z$ and all are disjoint so $\exp$ is a covering map from C to C-{0}?
@TedShifrin I need to show that for any point $z$, there exists an open neighborhood s.t. its preimage is the disjoint union of open sets which map homeomorphically to the open neighborhood which I took to be the point itself. I think I did those things; or is it that I need to show that the spaces are path-connected which is part of some definitions?
@schn $u=\frac{t-x_i}h$ is a change of variables. Certainly, if you choose to look at the change of variables $u=\pi x$ in a certain way, then $u$ is a function of $\pi$, but for the integral in which you were computing, $u$ was a function of $t$ and $x_i$ and $h$ are parameters. Move on, look at something else for a while. Come back and look at this fresh later. Maybe things will be clearer then. You are focusing too closely on this and missing the bigger picture.
