You can take a formal proofwriting class that teaches you to write proofs. It covers topics like propositional calculus, firstorder predicate logic (the forall and there exists quantifiers), set theory, some number theory. Alternatively, Linear Algebra and Number Theory are commonly used to teach proofwriting. Both are really good, but Linear Algebra is light on the proof by induction.
53 Replies  2651 Views  Last Post: 10 July 2013  02:24 PM
#47
Re: Discrete Math and Programming
Posted 10 July 2013  11:51 AM
Interesting. One of these days maybe. "Theoretical" math kind of intrigues me. I think since I grasped the concept of a limit I've kind of been fascinated by the world of math that lies out there that I never knew existed, although for me I always seem to have to bring it back to the real world to really enjoy it. I don't think I could get into chaos math or something that I could not bring back and make it connect with reality in some way. But I think most higher math does come back to reality. Although, I've come to the conclusion that math is a language. And just like you can write a work of fiction in english, you can likewise write a work of fiction in mathematics; I'm not sure everything in the world of theoretical math actually does come back and meet back up with reality.
#48
Re: Discrete Math and Programming
Posted 10 July 2013  11:55 AM
Chaos Theory studies dynamical systems. It applies to physics, engineering, economics, and biology. It very much comes back to reality. Even Abstract Algebra is applicable to Physics.
You should study Axiomatic Set Theory.
Quote
And just like you can write a work of fiction in english, you can likewise write a work of fiction in mathematics;
You should study Axiomatic Set Theory.
#49
Re: Discrete Math and Programming
Posted 10 July 2013  11:56 AM
What is axiomatic set theory, and what is it used for?
#50
Re: Discrete Math and Programming
Posted 10 July 2013  11:59 AM
I'm not overly familiar with it. I found a text with a good introduction to it. It's along the lines of what you were talking about with a Tabula Rasa approach to mathematics.
Source: https://www.dpmms.ca...ofsettheory.pdf
Quote
Its intention is to explain what the axioms say, why we might want to adopt them (in the light of the uses to which
they can be put) say a bit (but only a bit, for this is not a historical document) on how we came to adopt them, and explain their mutual independence
they can be put) say a bit (but only a bit, for this is not a historical document) on how we came to adopt them, and explain their mutual independence
Source: https://www.dpmms.ca...ofsettheory.pdf
#51
Re: Discrete Math and Programming
Posted 10 July 2013  01:25 PM
The interesting thing about math, or one of the interesting things, is that instead of simply descibing the world, it invents worlds to describe and describes them. The fascinating thing is that we then find the reflections of those things in the world, after all. When Galois came up with groups, he was describing an abstraction with no apparent consequences in the world. Turns out they're incredibly handy for describing modern physics and also Rubik's cube. The fascination with primes that took over so many mathematicians over centuries turned out to make a great basis for cryptography, and discrete logs are turning out to be even better (less vulnerable to the advances in quantum computing).
For the programmer, math provides a bag of tools, but they don't come with convenient labels  you don't really know what problems they'll fit until you come across the problem. This means that you have to pursue things that look interesting, and eventually they'll probably also prove useful. Insatiable curiosity and abiding patience are not always virtues, but in this case they seem to come out that way.
For the programmer, math provides a bag of tools, but they don't come with convenient labels  you don't really know what problems they'll fit until you come across the problem. This means that you have to pursue things that look interesting, and eventually they'll probably also prove useful. Insatiable curiosity and abiding patience are not always virtues, but in this case they seem to come out that way.
This post has been edited by jon.kiparsky: 10 July 2013  01:27 PM
#52
Re: Discrete Math and Programming
Posted 10 July 2013  01:27 PM
Quote
When Galois came up with groups, he was describing an abstraction with no apparent consequences in the world. Turns out they're incredibly handy for describing modern physics and also Rubik's cube.
Groups are also incredibly handy for cryptography and graph theory. A lot of Crypto texts talk about schemas over a group. Abstract algebra is the other approach for studying algebraic graph theory, mainly with groups.
#53
Re: Discrete Math and Programming
Posted 10 July 2013  02:09 PM
jon.kiparsky, on 10 July 2013  03:25 PM, said:
The interesting thing about math, or one of the interesting things, is that instead of simply descibing the world, it invents worlds to describe and describes them. The fascinating thing is that we then find the reflections of those things in the world, after all.
That is really cool, but way over my head. I'm still working on Euclid. :)
#54
Re: Discrete Math and Programming
Posted 10 July 2013  02:24 PM
Euclid's a great example! Where do you find an infinite plane, with no curvature, populated by infinitely thin lines and points with no dimension? Well, you find that in your head  and then it turns out you can apply what you learn from that imaginary world to all sorts of neat problems, starting with measuring fields and ultimately giving you insights into physics and even providing a cool proof of the infinity of primes.
Another example is boolean algebra  totally useless, until you extend the logic far enough, and then it turns out to be a great way to construct a propositional calculus. And, wonder of wonders, it turns out that all of the laws of boolean algebra can be modeled in these electronic circuits that were so unimagninable in Boole's time that he couldn't have even imagined being able to imagine them.
To me math is worth a lot of hard work just because it's beautiful, but it consistently shows that it's able to do hard work as well. If you can find a way to see the beauty of the math, it's a lot more likely that you'll find the useful parts as well.
Another example is boolean algebra  totally useless, until you extend the logic far enough, and then it turns out to be a great way to construct a propositional calculus. And, wonder of wonders, it turns out that all of the laws of boolean algebra can be modeled in these electronic circuits that were so unimagninable in Boole's time that he couldn't have even imagined being able to imagine them.
To me math is worth a lot of hard work just because it's beautiful, but it consistently shows that it's able to do hard work as well. If you can find a way to see the beauty of the math, it's a lot more likely that you'll find the useful parts as well.
