mixedmath

A flurry of activity at Math.Stackexchange was just enough to rouse me from my blogging slumber. Last week, the following question was posted.

I have enjoyed math throughout my years of education (now a first year math student in a post-secondary institute) and have done well–relative to the amount of work I put in–and concepts learned were applicable and straight-to-the-point. I understand that to major in any subject past high school means to dive deeper into the unknown void of knowledge and learn the "in's and out's" of said major, but I really hate proofs–to say the least.

I can do and understand Calculus, for one reason is because I find it interesting how you can take a physics problem and solve it mathematically. It's applications to real life are unlimited and the simplicity of that idea strikes a burning curiosity inside, so I have come to realize that I will take my Calculus knowledge to it's extent. Additionally, I find Linear Algebra to be a little more tedious and "Alien-like", contrary to popular belief, but still do-able nonetheless. Computer Programming and Statistics are also interesting enough to enjoy the work and succeed to my own desire. Finally, Problems, Proofs and Conjectures–that class is absolutely dreadful.

Before I touch upon my struggle in this course, let me briefly establish my understanding of life thus far in my journey and my future plans: not everything in life is sought after, sometimes you come across small sections in your current chapter in which you must conquer in order to accomplish the greater goal. I intend to complete my undergraduate degree and become a math teacher at a high school. This career path is a smart choice, I think, seeing as how math teachers are in demand, and all the elder math teachers just put the students to sleep (might as well bring warm milk and cookies too). Now on that notion and humour aside, let us return to Problems, Proofs and Conjectures class.

Believe me, I am not trying to butcher pure math in any way, because it definitely requires a skill to be successful without ripping your hair out. Maybe my brain is wired to see things differently (most likely the case), but I just do not understand the importance of learning these tools and techniques for proving theorems, and propositions or lemmas, or whatever they are formally labelled as, and how they will be beneficial to us in real life. For example, when will I ever need to break out a white board and formally write the proof to show the N x N is countable? I mean, let's face it, I doubt the job market is in dire need for pure mathematicians to sit down and prove more theorems (I'm sure most of them have already been proven anyways). The only real aspiring career path of a pure mathematician, in my opinion, is to obtain a PHd and earn title of Professor (which would be mighty cool), but you really have to want it to get it–not for me.

Before I get caught up in this rant, to sum everything up, I find it very difficult to study and understand proofs because I do not understand it's importance. It would really bring peace and definitely decrease my stress levels if one much more wise than myself would elaborate on the importance of proofs in mathematics as a post-secondary education. More simply, do we really need to learn it? Should my decision to pursue math be revised? Perhaps the answer will motivate me to embrace this struggle.

I happened to be the first to respond (the original question and answer can be found here, and I'm a bit fond of my answer. So I reproduce it below.

I mean, let's face it, I doubt the job market is in dire need for pure mathematicians to sit down and prove more theorems (I'm sure some of them have already been proven anyways).

The idea that most of 'math' has already been solved, discovered, found, founded, or whatnot is nonsense, a misconception arising from a system of education focusing on mastering arithmetic skills rather than performing mathematical thought.

It seems to me that what you like to do is arithmetic. I don't mean this diminutively - I mean that it sounds to me that you like coming across problems where a prescribed solution or apparent solution method exists, and then you carry out that solution. In my undergraduate days, when I was surrounded by the engineers at Georgia Tech, this was a common attitude. A stunning characteristic of many of the multivariable calculus classes and differential equations classes at Tech, which were almost exclusively aimed at the overwhelming majority of engineers at the school, was that many concepts were presented without any proof. And many of the students, whose interest in understanding why what they were taught was true was dulled by years of largely mindless arithmetic studies or whose trust is so great that they willingly give up the responsibility of verification to others (for better, or for worse), this was fine. And in this way, 2 of the four semesters of calculus at Tech are largely arithmetic as well.

But as a mathematician (rather, as a sprouting mathematician), I draw a distinction between mathematics and arithmetic. To approach a problem and come up with a solution that you do not understand is not mathematics, nor is the act of regurgitating formulae on patterned questions that come off a template. While arithmetic skills and computation are important (and despite their emphasis in primary and secondary schooling, still largely weak enough for a vague innumeracy to be prevalent and, unfortunately enough, sometimes even acceptable), they are not at the heart of mathematics. The single most important question in math is why?

I agree with one of the comments above, identifying two questions here: Why are proofs important to mathematics? and How would being able to prove theorems make my life better?

For the former, I can only say that a mathematics without proofs isn't really mathematics at all. What is it that you think mathematicians do all the time? I assure you, we are not constantly computing larger and larger sums. Nor are we coming up with more formulae, eagerly awaiting numeric inputs. A mathematician finds an intriguing problem and then tries to answer it. The funny thing about most things is that they're really complicated, and so most mathematicians must go through some sort of process of repeated modelling and approximation. It is in our nature to make as few assumptions as necessary to answer the problem at hand, and this sometimes leads to more complications. And sometimes, we fail. Other times, we don't.

But then, any self-respecting scholar (let alone a mathematician) who is also interested in the problem gets to ask why the answer is, in fact, an answer. Not everything is so simple as verifying arithmetic details. So everything must be proven. And then some other mathematician might come around, assume more or less, and come up with a different proof, or a different model, or a different approximation, or an entirely different way of viewing the problem. And this is exciting to a mathematician - it leads to connections in an increasingly unwieldy field.

In response to your assumption that most theorems have been done by now, I mention the quick fact that no mathematician alive could ever hope to learn a respectable fraction of the amount of math that has been done. This is a very big deal, and is strange. Just a few hundred years ago, men like Gauss or Leibniz were familiar with the vast majority of the then-modern mathematics. It's hard to express how vast mathematics is to someone who isn't familiar with any of the content of mathematics, but ask around and you'll find that it's, well, huge.

Finally, for the second question: in all honesty, the ability to prove the theorems of calculus or linear algebra might not be fundamentally important to the you live your life. But to lack any concept of any proof is to allow yourself to be completely consumed by not only innumeracy, but also irresponsibility. In particular, I would find it completely inappropriate for someone who disdains proofs from teaching math in secondary school. This would place yet another cog in the machine that creates generations of students who think that math is just a big ocean of formulas and mathematicians are the fisherman, so that when someone needs a particularly big formula they ask a mathematician to go and fish it out. More concretely, it is in secondary school where most people develop their abilities to synthesize information and make evaluative decisions. It is a fact of our society that numbers play an important role in conveying information, and understanding their manipulation is just as important as having the technical skills to undertake the manipulation itself when confronted with the task of interpreting their meaning. And this means that when a student asks their math teacher what something means, that teacher had better have a good idea.

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