Is mathematical physics rigorous with proofs?
Mathematical physics is physics! Mathematical physics is a branch of physics which has solid mathematical foundations and researcher in this area have a high expertise in mathematics. They understand advanced mathematical concepts which they use regularly in their research, but they don't pay much attention to rigor or proofs. The motivation for a mathematical physicist is to study physical systems using advanced mathematics and which sometimes result in predictions about the mathematical objects (underlying the physical problem) under study. These predictions become conjectures in mathematics which are left for mathematicians to study rigorously and prove. Many a times, mathematicians studying the results of physicists come up with more theories and generalizations. Mathematical physicists study the physics implied by these new results. So it's kind of a symbiotic relation. A good example of mathematical physics is the theory of general relativity which was built on the mathematics of Riemann. In modern days string theory, supersymmetry, topological quantum field theories, Ising models, loop quantum gravity etc. are famous areas of mathematical physics. Let me give you another example from the work of Edward Witten (a mathematical physicist who was awarded the Fields medal). Jones polynomial is a polynomial invariant of an oriented knot (Knot theory is an area of topology). Vaughan Jones received a Fields medal for this work. Edward Witten showed that the Jones polynomial can be obtained by considering Chern–Simons gauge theory on the three-sphere. Now Witten didn't prove any mathematical results, but he showed that the same polynomial also arises in a physical system. Later Khovanov defined another invariant of oriented knots called Khovanov homology which is a generalization of Jones polynomial. This opened a questions for physicists to understand the physical meaning of Khovanov homology in the context of gauge theories. Dunfield-Gukov-Rassmusen and Witten independently approached this problem and explained Khovanov theory in physics.
Is foundations math principles or applications ?
I'm not familiar with the exact course descriptions of these two, but they sound like something my high school did. Applications of math will likely be more geared toward practical problems that can be solved using math. Principles of math will cover more proofs, theories, and strategy. The principles of math are indeed the foundations of math, and it is likely to be a more difficult course, focusing more on "why" math is the way it is, than simply how to apply it. In order to be successful in higher math courses, like in a university setting, a fundamental understanding of previous mathematics is important. It's important to know "why" cos^2x + sin^2x = 1, instead of just punching cos on the calculator to find a side length of a triangle. If you know why math works, you'll be more successful in higher math classes, and the applications of math will come very easy. Whether you choose foundations or principles is up to you. If you plan on a math based major (like engineering, computer science, physics, or generally any science major), take the foundations class. If not, you can get by with the applications class (it's likely to be more fun and interesting, but less useful in college.)
Should I study Mathematics or Computer Science?
I am very interested in areas like Algorithms, Recursion Theory(Computability, Complexity), Philosophy of Mathematics, Foundations of Mathematics (Set Theory, Logic, Type Theory, Proof Theory, Category Theory). I am also interested in pursuing research in these topics. I really enjoy doing programming and I love solving problems in the Project Euler Website. I am very bad at Physics and Chemistry and I hate learning about them. I am more interested in subjects that use mathematical maturity and ideas instead of finding patterns and using rote learned formulas to solve problems. I was wondering which education would be a best fit for me.
Why is it that mathematical logic isn't taught in most high schools?
My personal opinion would be for three main reason:Not much motivation for it. Some of the main theories that make us of mathematical logic include set theory, of which at least in the US doesn't seem to be taught often explicitly or rigorously. Although the Peano Axioms provide a good example of a first-order theory, it is ZFC that is perhaps the most instructive (this is debatable, of course) of first order theories. Since logic makes natural use of set theory, this also brings up an issue. For example, the Lowenheim-Skolem theorems wouldn't have much meaning unless students had an understanding of set theory. A fix to this could be teaching both concurrently (as is done at my university).Too abstract for most students to grasp. A proper study of mathematical logic is naturally extremely abstract. Most of the mathematics that is taught in schools is computational in nature (geometry is a nice counter-example to this in some cases), whereas logic is usually not computational. Perhaps just a study of propositional logic would be useful for this because of logic tables, but whether this would provide enough material for a year-long class is unlikely.Teaching-related concerns, by which I mean things associated with the teachers, curriculum, and administration, as opposed to concerns with the actual teaching of the material. In my high school, there was mention of Godel's Incompleteness Theorems within my Theory of Knowledge class. The teacher was no mathematician, and as such the presentation of the result was far from accurate. In general there is the issue of finding teachers with an appropriate background to teach such a course well. It would be dangerous to teach students about logic without being well-versed in the subject enough to be accurate. Other concerns like lack of time and higher difficulty of assessment (one of what I find is a major reason mathematics is taught from a computational standpoint in high schools) are also major reasons why logic would not be taught in schools.In spite of the above problems, I believe that logic should be taught in high schools, especially concurrently with set theory. Understanding some of the foundations of mathematics and things along the lines of "What makes a proof? What is a mathematical theory? What are the extents of mathematics?" are all important questions to answer.
What is the most complicated mathematics possible?
Alright so there's Basic math Algebra Geometry Algebra 2 PreCal Calculus Calculus 2 Calculus 3 Linear Algebra Calculus 4 Differential Calculus Integral Calculus What's after this? Actually I want two answers, What is the most complicated math taught at a university? What is the most complicated type of mathematics possible? Like for the most intelligent people that dedicate their lives to math.