121–130 of 185 posts
What do people think of this idea. Let's say you want to casually improve your maths knowledge in your spare time. Let's say you find it frustrating how you generally can't just google concepts as you come across them, because the material is usually presented so obtusely and you need to be able to ask questions and have things explained in different ways. Let's say you live in a university town. Let's say you pay a…
This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own. I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll…
I have studied a lot of mathematics on my own. I did study physics as an undergrad. But, I went back and studied Real Analysis, Measure Theory, Combinatorics, Topology, and Stochastic Calculus. I have found, though, that while I have a decent grasp of the concepts my understanding and ability to solve problems isn't as strong as the math grad students who studied these topics deeply. I have found the knowledge useful…
Lots of mathematicians switch to a different subfield within their careers. And they do so by self studying, obviously.
If your compsci or physics undergrad provided you with a decent degree of mathematical maturity, it should be doable. The problem here is that compsci is still young, and there is a lot of variability. So diving into differential forms after attending a Java school sounds like a bad idea.
On the other hand, if you have the time (and ability) to learn some of this material on your own, for a purpose other than competing for a highly paid job as a mathematician, great.
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Without proofs and rigor, aren't you just reducing mathematics to a set of rules to be memorized? Sure, you can memorize the basics of group theory, but when it comes down to constructing the Diffie-Hellman Key-exchange, you'll still the mathematical intuition derived from learning the proofs.
Maybe, but most people condense this by creating mental models of the problem, like visualisations. That's the aim. For example, consider learning vectors, without the spacial/Cartesian visualisation as an aid. Or geometry without the visuals. An "intuition" wrt skill can only come from experience - repeated exercises and practise. But before that another kind of "intuition" can come from a useful mental model. Maybe…
This only works with visuals due to the relative simplicity of the topic, and simple visuals such as this are commonplace in modern textbooks and lectures. This [1], for example, is a visualization describing the one-way functions with hardcore predicates from a lecture.
However, these visualizations fall apart exponentially as you ascend the mathematical ladder of abstraction. Mathematical nomenclature becomes overburdened by many assumptions, and without proper rigor, becomes incredibly difficult and long-winded to explain. This is why newcomers find it impossible to pierce high level mathematics, each rung of the mathematical ladder builds upon the last. How would you suggest a visualization that is useful for the Kelvin-Helmholtz instability [2] for example? You can look at all the visuals and simulations you'd like on Wikipedia, but unless you're a mathematical savant you'll have to dig deep into mathematical rigor, borrowing work done by giants in the past [3]. There's really no easy shortcut to this.
> But before that another kind of "intuition" can come from a useful mental model
This mental model can be just as unhelpful as helpful. It is notoriously hard to fix false preconceived notions, and someone that develops an "intuition" that only applies as at basic level could easily lead them astray, a la the Dunning-Kruger effect. Beginning tabula rasa is often the path of least resistance, since once someone learns something /properly/ the first time, they're more likely to apply it correctly, rather than trying to apply a model that falls apart at higher abstractions. You can't really jump rungs in the math ladder, or even stave it off as a form of debt, telling yourself you'll learn it later.
[1]: https://i.imgur.com/q5KAelG.png
[2]: https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_insta...
[3]: http://www.rsmas.miami.edu/users/isavelyev/GFD-2/KH-I.pdf
This is stupid. The hard part about a Math degree is the number of hours you have to put in. If you cannot go to university full time, go part time. If you cannot go part time, you don't have enough time to actually learn any of these topics on your own. I've done these classes. It's typically 150 hours per class and it's not something you do after coming exhausted home from work either. After those 150 hours you'll…
Its it stupid at all. I taught myself Triginometry from a book for fun . Sure, not everyone's idea of a good time but I don't see the need to take a University or College course for lots of money when in fact I only spent about AU$34.95 on a good book and I didn't need to feel rushed in getting to grips with the concepts. I learn different things at different speeds, a course makes me go at a speed I don't enjoy. Thi…
And trigonometry is taught in high schools? It's something most children pick up, I don't think it's a fair comparison to say, measure theory.
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I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…
I don't mean to be rude by saying this, but the truly-difficult advanced math - the stuff that's really hard to build an understanding of by yourself, because it's fairly distantly separated from any obvious applications or anything you'd readily have experience with, and heavily obfuscated (to newcomers) by the notation and pedantic proof-focused thoroughness (appropriate for academic math, less so for applications)…
Right, but the math tagged as 'advanced' in the article is fairly applied.
> How to Learn Advanced Mathematics Without Heading to University I wonder if it's even possible. Learning maths requires much work, time and dedication. Doing so alone must be very difficult. There are several things universities provide that are hard to replicate alone: a degree, which gives you access to a job, motivation, learning environment, and "peace of mind". What I mean by peace of mind is that, when you're…
I really hope this doesn't come across as brash. I don't mean it to, but I must disagree with your assertions (In my case at least). Learning advanced mathematics without university is completely possible. Because it is at your own learning pace. Not one of a University. I graduated several years ago with a BS in Computer Science, with a Focus on Networking. And during that time, I held 3 part time jobs while also be…
Most unis I know of (I'm in the US) require those courses to be taken as part of your undergrad before you can attain the CS degree. Furthermore, with the prevalence of AP courses at the high school level, many students enter college already having taken some, possibly all of those courses.
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Interesting. Certainly something to consider. I never thought about being a non-degree seeking student.
Eh, if you're just auditing the class, maybe. When I was in college they made me retake ridiculous prereqs for the most trivial of reasons every time I transferred, allowed no exceptions to these, and personal requests to professors to get out of them were completely ignored. The prereqs were enforced by the computerized registration system - good luck getting past them without a waiver. As one anecdote, they once to…
Although I think your situation was pretty special as well, transferring universities is usually incredibly annoying and filled with road bumps. I've found there's a lot more leniency given to students who remain within the same university.
To get the key ideas behind many of these topics, you could try reading Evan Chen's Infinitely Large Napkin: http://www.mit.edu/~evanchen/napkin.html
It’s great that he’s making the effort, but I’m not sure this is the most useful resource for a typical autodidact.
Earlier quoted context omitted.
* the most advanced math I had was Algebra 2* How could you get a BS in CS without taking calculus courses? Which school did you go to?
It's kind of a shame that so many schools push you to study calculus in order to study CS; digital computers are algebraic machines, not analytical ones, pace Babbage. Combinatorics and graph theory would be far more useful. (Although maybe this will change with machine learning.)
By the end of that summer of 1983, Richard had
completed his analysis of the behavior of the
router, and much to our surprise and amusement, he
presented his answer in the form of a set of partial
differential equations. To a physicist this may seem
natural, but to a computer designer, treating a set
of boolean circuits as a continuous, differentiable
system is a bit strange. Feynman's router equations
were in terms of variables representing continuous
quantities such as "the average number of 1 bits in
a message address." I was much more accustomed to
seeing analysis in terms of inductive proof and case
analysis than taking the derivative of "the number
of 1's" with respect to time. Our discrete analysis
said we needed seven buffers per chip; Feynman's
equations suggested that we only needed five. We
decided to play it safe and ignore Feynman.
The decision to ignore Feynman's analysis was made
in September, but by next spring we were up against
a wall. The chips that we had designed were slightly
too big to manufacture and the only way to solve the
problem was to cut the number of buffers per chip
back to five. Since Feynman's equations claimed we
could do this safely, his unconventional methods of
analysis started looking better and better to us. We
decided to go ahead and make the chips with the
smaller number of buffers.
Fortunately, he was right. When we put together the
chips the machine worked.