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On June 11 2017 09:08 thePunGun wrote:Show nested quote +On June 11 2017 08:04 Biolunar wrote:On June 11 2017 04:48 CecilSunkure wrote:On June 11 2017 04:32 Biolunar wrote:On June 10 2017 23:39 Silvanel wrote:
In reality the roll of dice or throw of a coin is perfectly deterministic. Is it? Quatum mechanics say otherwise. You cannot now the position of every particle exactly or you change the outcome of the experiment. That doesn't mean it's not deterministic. It just means we can't know all of the initial state. Hence it is not deterministic. Einstein's special relativity disagrees with you, spacetime and every aspect of our universe are entirely predetermined. Edit: You also might wanna reevaluate your point about quantum mechanics, "every quantum state can be represented as a sum of two or more other distinct states" and "the result will be another valid quantum state". (Source: Quantum Superposition)
https://en.wikipedia.org/wiki/Laplace's_demon
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I'm currently a web dev, who has moved into analytics, data science, and usability. My next stop in my career is eventually aerospace. I'm looking to get back into some general math courses as a refresher. Any thoughts I where I should start? I've been looking into QPL (Quantum Programming Languages), and find that interesting as well. I'm currently in the position where I might be able to go to a University for free, and I would like to eventually pivot my career. I feel like a general refresher from the beginning would help, I know khan academy has a few courses, any thoughts on where to start from there?
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On June 12 2017 03:03 Manit0u wrote:Show nested quote +On June 11 2017 09:08 thePunGun wrote:On June 11 2017 08:04 Biolunar wrote:On June 11 2017 04:48 CecilSunkure wrote:On June 11 2017 04:32 Biolunar wrote:On June 10 2017 23:39 Silvanel wrote:
In reality the roll of dice or throw of a coin is perfectly deterministic. Is it? Quatum mechanics say otherwise. You cannot now the position of every particle exactly or you change the outcome of the experiment. That doesn't mean it's not deterministic. It just means we can't know all of the initial state. Hence it is not deterministic. Einstein's special relativity disagrees with you, spacetime and every aspect of our universe are entirely predetermined. Edit: You also might wanna reevaluate your point about quantum mechanics, "every quantum state can be represented as a sum of two or more other distinct states" and "the result will be another valid quantum state". (Source: Quantum Superposition) https://en.wikipedia.org/wiki/Laplace's_demon
Linked this one earlier:
https://en.wikipedia.org/wiki/Sunrise_problem
The question is whether or not we can induce events we have not seen yet based on past events (by coming up with deterministic laws governing physics). So at best, you can achieve a near 100% probability.
Also related:
https://en.wikipedia.org/wiki/Problem_of_induction
For example, Newton used induction to come up with his laws of motion based on astronomy data collected by tycho brahe. He stated in Principia that the laws that govern these can be generalized to all objects -
"In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies and the laws of motion and law of gravity have been found by this method. And it is enough that gravity should really exist and should act according to the laws that we have set forth and should suffice for all the motions of the heavenly bodies and of our sea."
In other words, he observed the orbits of the planets, extracted out the laws of motion, and said they applied for everything. Its a big jump. So although the physical laws that humans have come up with are deterministic and have been verified empirically, you cannot know with 100% certainty that they apply deterministically to everything in the universe, because we have not observed everything in the universe. For a contrived example, if we are living in a simulation, and some coder changes some variables, then the laws of motion could change tomorrow. Or consider dark matter/ dark energy, from what I understand, our current laws of physics cannot describe the phenomenon we are seeing, so those laws don't apply to a vast majority of the universe.
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When this thread was created, I had high hopes. Now this discussion has been interesting so far, but imho it not so much mathematics, but more physics/general science principles, mixed with some philosophical consequences.
Maybe I'm having a narrow view of mathematics, but maybe I dont and it is also time for a "physics Thread".
edit: Also unfortunately I dont know much about mathematics literature for non-mathematicians.
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Literature for mathematicians is perfectly fine. This is the math thread after all.
E: Also I vote to start banning goons coming in here to talk philosophy. It's annoying and pointless for a math thread.
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On June 13 2017 06:23 CecilSunkure wrote:
Literature for mathematicians is perfectly fine. This is the math thread after all.
E: Also I vote to start banning goons coming in here to talk philosophy. It's annoying and pointless for a math thread.
A large chunk of computer science is based on discrete math, whose underpinnings are the same as philosophy. Famous philosophers such as Bertrand Russel and Descartes had huge contributions to mathematics. The overlap is real.
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Threads existed for 3 days 
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On June 13 2017 07:57 fishjie wrote:Show nested quote +On June 13 2017 06:23 CecilSunkure wrote:
Literature for mathematicians is perfectly fine. This is the math thread after all.
E: Also I vote to start banning goons coming in here to talk philosophy. It's annoying and pointless for a math thread. A large chunk of computer science is based on discrete math, whose underpinnings are the same as philosophy. Famous philosophers such as Bertrand Russel and Descartes had huge contributions to mathematics. The overlap is real.
Sure overlap is real. Real annoying. Just open a new thread
Also I'm sure nobody would mind if the philosophy discussion was actually related to some real math discussion. But like 99% of the time that won't be the case.
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Anyone have suggestions for (introductory/intermediate) books/open courses related to functional analysis?
I started to follow along with a course from Coursera a few years ago, but didn't really stick with it. I've done a bunch of undergraduate level calculus/algebra courses, but never really got into analysis. Seems like it would be interesting to learn 
RE: The recent discussion, who volunteers to start the epistemology thread? Seems like that's almost what people want, haha
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On June 13 2017 13:13 Mr. Wiggles wrote:Anyone have suggestions for (introductory/intermediate) books/open courses related to functional analysis? I started to follow along with a course from Coursera a few years ago, but didn't really stick with it. I've done a bunch of undergraduate level calculus/algebra courses, but never really got into analysis. Seems like it would be interesting to learn RE: The recent discussion, who volunteers to start the epistemology thread? Seems like that's almost what people want, haha
Well, functional analysis is a rather advanced topic. Typically, when you study it, you will have under your belt both real analysis and complex analysis. To take the most out of it you will need measure theory. For this reason most of the books are pretty difficult to read.
Now, if your math background is not that strong (i.e., you are not math major) the best book I believe is "Introduction to Functional Analysis" by Kreyszig. Here is the link www.amazon.com
I think this is the only book prior knowledge of any analysis.
I have to say that functional analysis was always at the top of the things to learn, but at this point I think I will never have time to properly study it. It is really fascinating but complex subject.
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On June 13 2017 05:57 Mafe wrote:
When this thread was created, I had high hopes. Now this discussion has been interesting so far, but imho it not so much mathematics, but more physics/general science principles, mixed with some philosophical consequences.
Maybe I'm having a narrow view of mathematics, but maybe I dont and it is also time for a "physics Thread".
edit: Also unfortunately I dont know much about mathematics literature for non-mathematicians.
Please, no!
The ask and answer stupid question already derail often enough, there is no need for a dedicated thread for derailment 
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Russian Federation421 Posts
On June 13 2017 13:13 Mr. Wiggles wrote:Anyone have suggestions for (introductory/intermediate) books/open courses related to functional analysis? I started to follow along with a course from Coursera a few years ago, but didn't really stick with it. I've done a bunch of undergraduate level calculus/algebra courses, but never really got into analysis. Seems like it would be interesting to learn RE: The recent discussion, who volunteers to start the epistemology thread? Seems like that's almost what people want, haha
IMHO, a good introduction book to functional analysis is Rudin's Functional Analysis. I should note that it is a good book to study functional analysis; it is really useful to have at least basic understanding of linear algebra, real and complex analysis, and topology beforehand to understand the motivation behind the theory.
On the other hand, I've heard good things about Introductory Functional Analysis with Applications by Erwin Kreyszig for people who look for applications. I haven't read it myself though.
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On June 14 2017 01:43 JimmyJRaynor wrote:
it is obvious and intuitive that multiplication and division are inverse mathematical operations. does any one have a simple 2 or 3 line intuitive way to summarize why differentiation and integration are inverse operations?
You have a description in en.wikipedia.org :
Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or over some other variable) adds up to the net change in the quantity.
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France12904 Posts
I guess the physics example? Position -> speed -> acceleration
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On June 14 2017 03:03 Ingvar wrote:Show nested quote +On June 13 2017 13:13 Mr. Wiggles wrote:Anyone have suggestions for (introductory/intermediate) books/open courses related to functional analysis? I started to follow along with a course from Coursera a few years ago, but didn't really stick with it. I've done a bunch of undergraduate level calculus/algebra courses, but never really got into analysis. Seems like it would be interesting to learn RE: The recent discussion, who volunteers to start the epistemology thread? Seems like that's almost what people want, haha IMHO, a good introduction book to functional analysis is Rudin's Functional Analysis. I should note that it is a good book to study functional analysis; it is really useful to have at least basic understanding of linear algebra, real and complex analysis, and topology beforehand to understand the motivation behind the theory. On the other hand, I've heard good things about Introductory Functional Analysis with Applications by Erwin Kreyszig for people who look for applications. I haven't read it myself though.
I disagree about Rudin's Functional Analysis. This book targets graduate math audience. Unless you have really strong math background that includes Measure Theory, that's not the book to start. But at the graduate level it is a great book.
Beside Kreyszig that Ingvar and I mentioned in our posts, another introductory books aimed at people with modest backgrounds are "Functional Analyisis for Beginners" by Saxe, and "Elementary functional analysis" by MacCluer. I own the latter and I found it quite easy to follow if you have some background in analysis.
I should say that I self-study quite a bit of math and I often find that it is best to own a couple of books. Because invariably you will get stuck at some points. But since you are doing things on your own there is no professor or classmate to ask or help you clarify your confusion. So I found that the best is to have a couple sources to consult.
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On June 14 2017 01:43 JimmyJRaynor wrote:
it is obvious and intuitive that multiplication and division are inverse mathematical operations. does any one have a simple 2 or 3 line intuitive way to summarize why differentiation and integration are inverse operations?
It's a little difficult without drawing but here's my best attempt.
I'm guessing you're thinking about integrals as area under a curve. (Alternatively, you could think of it as the inverse of differentiation, which would make the question trivial) So the question is, if I'm looking at the area under a curve, why does it change exactly as fast as the height of the original curve?
Well, area is height * width, at least for rectangles. The slope of the area function (derivative of the integral function) is rise over run taken at smaller and smaller intervals. So it's (height * width) / width as long we take a small enough interval where the area under the original function, f(x) looks like a rectangle. Substituting f(x) for height, we get that the derivative of the integral at some value x is:
(height of f(x) * width) / width = f(x), that is the value of the original function. So differentiating the area function gives back the original function.
There's a number of small assumptions there that aren't necessary but omitting them would make the explanation even more complicated.
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Thanks for the recommendations, Lebesgue and Ingvar, I'll check them out!
I've got a bit of a weird math background, as I got three-ish years into a physics degree before switching into computer science and taking a few additional algebra courses that interested me. The end result is a mish-mash of different subjects, along with a bunch of implicit stuff that I learned in my non-math courses.
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Oh, there is a TLer called Lebesgue.
He should be one of the best to talk about integrals then :p
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EPT
