In response to PM,
Saying math proofs are like puzzles is quite accurate. Velleman's book is a good start, so it is good you are working through it. Make sure you do a fair amount of exercises too (7 per section seems good), because reading alone is a waste of time. And spread each section over time, ie. do a section every two days. The material you are dealing with is abstract and you will need a second re-reading the day after (or possibly more) to really see the big picture.
What you are dealing with now is symbolic logic, not mathematics. You are manipulating logical statements to get practice with symbolic notation. This is not something you will be doing in math, but the process of proof is very similar - you resolve your questions into the rules and definitions. Also, my "3 months" is quite arbitrary. Proofs do get easier with time, I will stand by that, but there is no set length of time everything just clicks. The more you put in, the faster you see results. It is a skill you will have to practice daily for a few months, and it is a painful process at first. So it is not surprising that many students find they are seemingly "bad" at math when first exposed to it, despite acing high school. This is a common experience among all students exposed to rigorous mathematics for the first time. I know some very very bright students who even gave up on math, while others that persisted and succeeded in it. So you have to be willing to put in a lot of work, more than any other subject imo, to get anywhere with it.
Pretty much all of math uses proofs (or assumptions). Everything you know about math is the result of someone showing the result is true. For instance, did you know the exponent laws can be proved? (that might be a good exercise, then check on the internet for answers). If you take honors math classes in college, they will all be very proof oriented, right from year 1. Non-honors classes will be like high school, just calculations (this is useful to engineers or non-theoretical scientists, who need more time to devote to their main courses). Proofs are especially prevelant in university algebra classes, including Linear algebra, abstract algebra, and number theory. Real analysis and topology, as well as most third year+ math classes will be all proofs.
As for careers that do proofs for a living, I would imagine the vast majority are in academia (ie. teaching). Obviously, the mathematicians will be doing proofs for a living. Physicists don't really do "math proofs", they instead manipulate equations and derive results that correspond to physical phenomena. Theoretical physicists will have good grounding in proofs because of the heavy math courses they took, but it is not something they will be doing in physics. Philosophers and logisticians do proofs as well, although of a different nature. Computer science too has its share of proofs. In fact, most programming languages are written in a very rigorous fashion similar to that of a mathematical proof. This is why a lot of computer scientists make great mathematicians. As you can see, proof is a very general term. All it really means is presenting an argument. English majors and historians do this all the time, as does most of science. But only in math will you find it so precise and accurate.
Since you are in high school, I recommend you take a course on geometry or study Euclidean geometry on your own. A good book is "Geometry, 2nd ed" by Harold Jacobs. Classical geometry is where proofs were born, so it is a good place to begin either concurrently or after Velleman. An introductory to Linear Algebra will also be beneficial, but save that for after you do calculus. Another book I recommend is Courant's "What is Mathematics"?, a book that has some proofs and will surely make you love math.
l). And that's why I'm considering of double major.
