Not only written text is a faster way to communicate information, it is so because it has much bigger context window:
"A moment" in a video is exactly that, a moment of time, either a frame or a couple of seconds that will stay in short term memory.
"A moment" in a text is a page or two facing pages. There can be diagrams or formulas there. It is extremely easy to direct attention to parts of these pages, in any order.
In a video, "moments" in the above sense are generally low information, quickly changing in linear order. In a text, they are fewer and of higher density. It seems that the second type is easier to commit to long-term memory, to understand, etc.
There is a mildly related math puzzle I learned at some point in high school (iirc):
someone comes to a subway station at (uniform) random times between 6p and 8p; he notices that the first train he observes arriving at the same station is 3 times more often inbound than outbound. He also knows that time intervals between the trains going in the opposite directions are fixed and equal — say, always 6 minutes, so the only random event here is when this person arrives at the platform. Explain how this is possible.
The outbound ones are scheduled a minute and a half after the inbound. (For neat integers, should have been twice as often, or 8 minutes between them.)
But the bigger question: What's the difference between "an inbound train" and "an outbound train"? Aren't almost all trains usually both inbound and outbound? First they arrive at a station, then they go on to the next. Or even at a "terminus"-type station; first they arrive at the terminus, then they go back in the other direction. How does he decide whether a train is an inbound or an outbound one, while it is standing still?
Just as the author of this blog, I found Dark Sky UI uniquely concise and informative. I managed to somewhat replace it with not one, but two apps (on Android):
- OpenWeather for a dense, one-page view of short-term upcoming conditions (temperature, precipitation) and the weekly forecast.
- Shadow Weather for extra details: cloud cover, wind directions, etc.
From a quick reading of the footnote, it seems that Turing uses optimal transport distance rather than the Hausdorff distance. Convergence in the latter implies convergence in the former, but not the other way round.
One thing I remember liking in this post was his personal homepage (a little further down). There has been a recent HN discussion of Glance, a personal dashboard, but Wolfram's seemed more effective.
If I could visually persist 3 lines of a terminal in my vision, and then otherwise use the functionality of a cell phone with ear buds with microphones, I think I could do 60% of my job while walking around in the forest behind my house. Most of the time I wouldn't need the terminal.
Do my cohorts, who have drifted into positions of tech lead, and then onwards as technical mentors to younger dev teams, find the same? I know I betray my communication style by admitting I don't need to see the screen; I prefer to see their world through a mental model built by words and sensing body language through tones and pauses. I'm not even sure I'm in the right line of work most days, at this point.
Unless you're working on undergraduate integrals then it's pretty useless for advanced stuff without writing your own library. By that point you may as well write it in a language that's more performant and cheaper.
> Unless you're working on undergraduate integrals then it's pretty useless for advanced stuff without writing your own library. By that point you may as well write it in a language that's more performant and cheaper.
Hard disagree. Mathematica's symbolic dexterity makes abstract reasoning (with equations/expressions) very easy. Think of it as the companion tool for anyone doing pages of algebra that would go into a paper, or form the backend for some code.
The numerical capabilities of Mathematica are passable but nothing fancy. Once you have the math figured out, you might even want to reimplement "in a language that's more performant and cheaper." But I haven't see anything come close to Mathematica for convenience of symbolic reasoning -- not just as a technology (lisp is pretty good) but as a ready-for-use product.
I agree that only some physicists use Mathematica. But I haven't really seen it being used it for calculus. Maybe some differential equations.
But mostly for symbolic algebriac manipulation. I used it during my phd to work with groups. Instead of having to calculate stuff by hand, you can just ask Mathematica to do it. Also lots of stuff with tensors in GR is so easy to do in Mathematica.
I think it's fair to say that most math/physics people use mathematica from time to time, but largely for different things than they use other programming languages for. It's very good as a CAS, but it's a pretty bad programming language for things that don't have analytical solutions.
It isn't. Mathematica is very much a niche product in academia.
The people who would get most out of it are students, but for some god forsaken reason universities don't support them.
I was in a pilot class with Mathematica back in 2006 and the review of the class were _all_ 5 stars and students on average got 10% higher marks in all other subjects they took that year.
They didn't run the course again.
Sagemath is now equally good if a teacher defines a DSL for the students to use in a class.
The only "problem" with sagemath is that it is based on Python. The rationale is that Python is easy to start using and widely known. This is the usual "make it easy for newcomers" trap.
For the mathematical constructs we care about in symbolic programming, I have found Python's syntax and Sage's menagerie of objects awful to use. Initially you feel comfortable, but when you want to do some real work, it gets horribly in the way. The Wolfram language, a LISP variant, is less familiar and harder for a newbie to learn but it is vastly superior for actual work.
The only "problem" with sagemath is that it is based on Python. The rationale is that Python is easy to start using and widely known. This is the usual "make it easy for newcomers" trap.
For the mathematical constructs we care about in symbolic programming, I have found Python's syntax and Sage's menagerie of objects awful to use. Initially you feel comfortable, but when you want to do some real work, it gets horribly in the way. The Wolfram language, not a LISP variant, is less familiar and harder for a newbie to learn but it is vastly superior for actual work.
One has to be a little careful with designing courses like this. They are most likely to be of interest to people who already know, at least to some extent, both (i) calculus and (ii) programming. That is, the (presumable) target audience — those who are learning either of these subjects — is not really ready to take in such a class.
Anecdotally, my personal attempts at incorporating only slightly exotic CASes (Maxima or Sagemath) into calculus courses were met with tepid response at best. Part of the issue was, I believe, that freshmen are rarely interested in setting up software for a non-CS course.
That being said, for slightly higher-level classes it can work quite well as an optional component — I've had really good results with Python projects in an ODE course. Python not being a niche language certainly helped, too.
This is a common challenge in teaching stats where students have to learn statistical concepts and using statistical software simultaneously. In the end, I think it is worth the challenge and beats having students calculate everything by hand & look up values in tables.
> "I've had really good results with Python projects in an ODE course."
Which textbook did you use for this course? What were the reference material? Could you share with me anything about this course? Lecture notes, code, slides, books, anything.
"A moment" in a video is exactly that, a moment of time, either a frame or a couple of seconds that will stay in short term memory.
"A moment" in a text is a page or two facing pages. There can be diagrams or formulas there. It is extremely easy to direct attention to parts of these pages, in any order.
In a video, "moments" in the above sense are generally low information, quickly changing in linear order. In a text, they are fewer and of higher density. It seems that the second type is easier to commit to long-term memory, to understand, etc.
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