Is university math all about proofs? [closed]

Solution 1:

Yes, mathematicians do a lot more besides writing proofs! While some already pointed out that “they also have lunch” I assume that you are more interested into the intellectual processes involved in the mathematical activity.

When I¹ examine my intellectual processes, I can see that some exercise my intuition and some exercise my rationality. My intuition tells me “there is something there” and with my rationality I can organise intuitions, see causal relations between facts and all the like. I am not, by far, versed in that area of philosophy, and I am confident that many people investigated this and wrote beautiful books on that topic. But for the purpose of this answer, I kindly bid the right of being superficial and approximative.

Mathematics is all about the interaction of intuition and rationality. As I study mathematics, my intuition leads me to genuine astonishment and makes me wonder about mathematical phenomenons I experience:

• “could the same obstruction be at work in these two seemingly unrelated problems?”
• “what is the geometric meaning of that algorithm?”
• “where do the solutions of that problem live?”
• “why do this topologist and this algebraist run into the same problem?”
• “is there a formalism turning this gibberish into a straight computation?”

Producing all these questions is the work of my intuition – and this is a pretty common experience for mathematicians. Sometimes, my intuition is on a big day and can even give me a glimpse at the answer of the question! This is where my rationality comes into play, and helps me to reconnect the new intuitive idea to the body of knowledge which I am already familiar with. And this reconnection is what we call a proof, and this is how I can tell other mathematicians how they can approach and tame the intuition I had.

Think of mathematics as a journey in a place where you cannot take pictures of what you see (your intuitions) but you still can draw a “treasure map” to teach others how they can go the same place where you had been (rationality at work).

As a conclusion, proofs are the most visible aspects of mathematics, but as in many other activities, there is much more that one can experience than one can see. If mathematics were all about proofs, then mathematicians would be like computers which randomly explore the space of proved or provable statements by combining known facts together, but this is not the case, as this method would hardly produce anything interesting.

¹ I write this at the first person to provide a vivid and entertaining picture, with the implicit assumption that the case I describe is the common case.

Solution 2:

Literally all math you've ever done are "proofs". You just get more rigorous about it.

"Solve the equation $2x+5=0$" really means "prove that there exists a solution to $2x+5=0$ and give an expression for it". Doing math is just doing logical reasoning with certain rules. In higher math, it is more explicitly logical and rigorous but honestly IMO the difference is kind of overstated. Once you get past junior/senior/first year of grad school you stop being so persnickety as well. I recommend https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

Solution 3:

Mathematics is about understanding things. Proofs are one part of this. The bulk of the time is spent asking questions, working out examples or doing calculations, figuring out what others have already done, positing conjectures/hypotheses, and thinking about ideas for why something should or should not be true. There are many famous unsolved problems that mathematicians have spent a lot of time thinking about, so no, we do not spend all of our time writing down proofs. That is just the last, optimal stage of a project (at least for pure mathematicians).

Why you may have this impression: the classes you're taking are likely on well studied topics, and, for efficiency's sake, you're presented with a streamlined version of the material, rather than how it was actually developed. Proofs teach you a way to think about things so you can learn to reason out and make confident deductions on your own, and justify the material you're learning, so they're definitely very important, but not of sole importance, in mathematical training.

Personally, I usually try to spend a lot of time in class explaining motivation and underlying ideas and working out examples, but as a result go through material slower than many of my colleagues. Quite possibly, there's more going on in the courses than just proofs, but depending on the course, they can take up the largest amount of time.