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Of course.

Anything that can be described well, to the extent that it can be described well, is essentially math.

Math, at its core, is just statements whose statements are carefully defined in their own framework.

Now, whether those constructions can accurately model the world or its parts is a deep question in philosophy. But the question then isn’t whether math can do it, it’s whether it can be done at all. If you can’t do it as math you’re essentially saying it can’t be done. This would be in the area known as epistemology (the study of what can be known).

An example of this is mathematical models of consciousness. Which take, as axioms, some descriptions that philosophers give to “conciseness” and then use the power of mathematical formulation to see what the implications of that are. What ‘things’ in the universe would be described as conscious then, when is a person a dingle consciousness vs many, etc.

The center of that particular space is Tononi’s IIT (integrated information theory) - which has spawned many papers examining the implications, soundness of axioms, and mathematical implications. [an example paper, chosen somewhat at random here: Is Consciousness Computable? Quantifying Integrated Information Using Algorithmic Information Theory

[Note: I am a consciousness skeptic; I tend to think the concept is vacuous chauvinism at heart, but this approach to addressing it — essentially “if true then what” is valuable I think.]

There’s an excellent, incredibly short, and easy to read book on this general idea. One of the best examples of concise, readable, and deep writing imo. It’s Vehicles: Experiments in Synthetic Psychology by Valentino Braitenberg.
Again, tiny volume. It uses simple thought experiments to examine artificial machines “vehicles” that exhibit behavior we would naturally use emotional vocabulary to describe. It challenges the assumption that organic internals like “desire” and “anger” needs be endlessly complex. I highly recommend it. It does not drop many, if any equations, but the controlled nature of the experiments drops them firmly in a mathematical framework as desired.

Thanks for the explanations! For a legal link to this text, here's Amazon's (US) page for Elements of Information Theory, Second Edition by Cover and Thomas.

>First, entropy is always positive, so you are indeed correct that those values should not be negative. Second, those values given are not relative entropy, but conditional entropy.

In the original paper to which OP linked, the last line reads

>Since [; H(C|X_2) > H(C|X_1), ;] the second component is more discriminative.

As I understand it, you're explaining that entropy is always positive, so these values were computed incorrectly with respect to sign. Accordingly, this would mean that [; H(C|X_1) \approx 0.97 > 0.72 \approx H(C|X_2), ;] instead, right? (Or does the FUBARness extend to these computations, too?) And in general, is the claim that [; H(C|X_i) > H(C|X_j) ;] implies that [; X_i ;] is the more discriminative component still true? (Or again, more FUBARness?)

Oh, and OP (/u/hupcapstudios)? This is a response from someone who actually understands this material. By contrast, I was just trying to apply formulas semi-blindly, aided by a little Googling.

If f(x) = x^5 , then...

f'(x) = 5x^4 (first derivative of f(x), with respect to x)

f''(x) = 20x^3 (second derivative of f(x), with respect to x)

f'''(x) = 60x^2

f''''(x) = 120x^1 = 120x

f'''''(x) = 120x^0 = 120

f''''''(x) = 0

In case you didn't see the pattern, if f(x) = ax^b then f'(x) = abx^(b-1) . This is detailed here, under "derivatives of powers" under "derivatives of elementary functions":
http://en.wikipedia.org/wiki/Derivative#Derivatives_of_elementary_functions

This trick only works for polynomials. If you want to take derivatives of more complex functions, like f(x)=x! or f(x)=sin(x) or f(x)=e^x*, that is more difficult.

If you are interested in this stuff, there is a book I would recommend. I, too, learned about derivatives years before I took high school calculus. But I didn't stumble on to it myself, as you did. I wasn't quite that smart, lol. Somebody gave me this book, called "Gravity" by George Gamow. Its a really great little book. It introduces basic calculus, in the context of basic physics (velocity, acceleration, etc). It goes through exactly the sort of stuff you are asking about, and explains it better than I can.

Here's the amazon link. Its not available on Kindle as an ebook, but maybe some other place sells it as an ebook, I don't know. Otherwise, I highly recommend buying a printed copy. It goes through exactly the sort of stuff you are asking about, and explains it better than I can:

http://www.amazon.com/gp/product/0486425630/ref=kinw_rke_tl_1

I liked my textbook. It's not advanced, but it does have proofs. It's accessible to engineers.

Complex Variables by Stephen D. Fisher


If you don't mind learning some history too, then An Imaginary Tale is excellent. My professor recommended it when I took Complex Analysis. Also, there is a nice appendix that covers the basics of Complex Analysis at the end of Gamma by Havil. The Road to Reality is good.

could you elaborate the context for why you want to learn Stochastic Calc? The book that is recommended is for people who are more mathy and maybe into modern interpretation: modern is not always clearer or easier. Given your background, I would recommend something less ambitious but easily accessible like this book . You should be able to read it given your current background in the sciences.

When I first started learning math on my own, I started learning calculus from something like this. Though I enjoyed it, it didn't really show me what 'real math' was like. For learning something closer to higher math, a more rigorous version would be something like this. It's all preference, though.

If you don't know much about calculus at all, start with the first one, and then work your way up to Spivak.

Dive in, number theory doesn't need any real prerequisites beyond being able to count and an eagerness to learn. As for reading, you can probably find lecture notes from any university on a first year course on number theory on the web, e.g.

http://www.pancratz.org/notes/Numbers.pdf

https://dec41.user.srcf.net/h/IA_M/numbers_and_sets/full

If you want a book, I recall that I liked Numbers & Proofs by RBJT Allenby.

I would start with something simple. Open University uses this textbook and it's a very nice gentle introduction to analysis https://www.amazon.com/First-Course-Mathematical-Analysis/dp/0521684242

I like this one: Field and Galois Theory by Patrick Morandi (Amazon).

This is also a classic: Galois Theory by Emil Artin (cheap Dover book on Amazon, legally free on Project Euclid).

I took a History of Mathematical Thought class in college where we used this book. I actually enjoyed reading through it, which is weird for a text book. I recommend it if you want a brief history, and some neat practice problems from all types of math throughout history.

Piaget has a good theory for cognitive development which proves to be quite valid when applied to learning. Visible learning and their further publications ground that up nicely in science. People who are 'good at math' often have developed up to their "formal operational stage" earlier than others, quite simply, and teaching maths that are at the appropriate level of cognitive development will yield the best results. Not everyone develops the same, so just like walking and talking, ease in doing higher maths might set in later on.

This means that some people, who only get to that stage in their late teens and truly develop it in their early 20's, will have been a bit worse at math than others - often identifying themselves with being bad at it, feeling sorry for themselves over it, and overall despising the stuff. This in turn leads to less interest in developing this part of their mind, making them even worse off. To a similar extent, this is true at every step of Piaget's cognitive ladder, so those who skipped early, well... even if only one stage was late, the rest might feel sluggish and forced, like an old injury nagging at your pride.

TL;DR Good news/bad news: in their 20's, everyone is on equal footing, very capable of learning it all, but few are still in school and most have willingly locked themselves out of learning math.

Anything you want to know about symmetric functions and characters is probably in:

https://www.amazon.com/Symmetric-Functions-Polynomials-Physical-Sciences/dp/0198739125


This is also a popular text on the subject that starts from more of the basics of representation theory:

https://www.amazon.com/Representation-Symmetric-Encyclopedia-mathematics-applications/dp/0201135159/ref=sr_1_1?s=books&ie=UTF8&qid=1520448586&sr=1-1&keywords=James+and++Kerber+group

Maybe this book?

Or a standard Riemannian geometry textbook like do Carmo might suit your needs.

James Stewart: 5th Edition

Oh yea, interesting that they teach number theory even in CS. I guess CS is most mathematical field if you compare it other fields except math?

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I haven't gotten into this stuff very deep, I am studying through this. I am very sure I wanna pursue math but there are only limited amount of areas to have time to study and I am not quite sure how 'active' the field is on that area (foundations of mathematics).

I have no idea if its relevant to your job interviews, but there's a book by Grimmett and a Stirzaker called "1000 exercises in probability" https://www.amazon.com/Thousand-Exercises-Probability-Geoffrey-Grimmett/dp/0198572212