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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.

J. M. ain't a mathematician
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Steven Pece
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    "I'm sure most of them have already been proven anyways" For the record, 0% of all true statements have been proven. Formality aside, there are FAR too many known unsolved problems in math to list, all of which require proofs to settle. – Alex Becker Apr 09 '12 at 04:11
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    Also, you seem to be asking two different questions: "How would being able to prove theorems make my life better?" and "Why are proofs an important part of mathematics?" Perhaps you could clarify for us which one you mean. – Alex Becker Apr 09 '12 at 04:13
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    The ability to write proofs in mathematics is one of the foremost skills to have in order to become a mathematician. If you can't write a proof, or put ideas coherently together to form a proof, you can forget about being a mathematician. –  Apr 09 '12 at 04:22
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    Do you want to know how powerful proofs are? In the australian parliament people always squabble, like how the prime minister Julia Gillard squabbles with leader of the opposition Tony Abbott. The squables never end. Each person wants to be right. When you produce a rigorous proof, it ends all arguments. This is the true power of mathematical proofs. –  Apr 09 '12 at 04:25
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    Logic is the tool we use to discover truth, so the more skill in logic one has the easier it is to understand truth. If you don't value truth, then you probably won't care much about logic. – Matt Gregory Apr 09 '12 at 04:25
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    Primary and secondary mathematics education is largely about computation. So is first-year calculus, and even a lot of lineary algebra. Computation is important, but it’s not at the heart of mathematics: the heart is theorems and proofs. And I will be very blunt: I do not think that anyone who hates proofs ought to be teaching secondary school mathematics: the students would not be well served thereby. – Brian M. Scott Apr 09 '12 at 04:30
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    I do have a few qualms with the amount of importance given to the axiomatic method, where most of the rigor, axioms and proofs are usually worked out after the fact. In A typical abstract algebra book the first few chapters are on notation, axioms etc (boring), the fascinating stuff comes later and rigor is built after the initial creative act. That being said, good for you if you instantly intuitively see the correctness of the many mathematical proofs without going through the formal derivation - i surely don't, and without the rigor, one cannot assert validity to another person. – user996522 Apr 09 '12 at 04:31
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    Steven, I think the issue for you may be that you are still at the stage where the problems people ask you already have known solutions. In most fields where one would be expected to apply higher mathematical thinking, the goal is to solve a problem that has never been solved before and in order to do this, you need to be able to show that what you are doing is actually correct. That is where proof comes in. Intuition is well and good (and in fact most mathematicians are highly intuitive in their approaches to problems) but you need to be able to make it rigorous to be sure it is correct. – Chris Janjigian Apr 09 '12 at 04:54
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    I think what you really want to do is engineering. That would suit your needs best. – Raskolnikov Apr 09 '12 at 05:21
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    If we take calculation as a jumping-off point, we can ask some questions ... How is it that we know that the methods we use for calculation always work? Having discovered a neat way of calculating something, can we apply it to the widest possible range of problems? The first question has to do with proof, the second with generalisation - a key feature of mathematics which hasn't yet made these comments. – Mark Bennet Apr 09 '12 at 07:30
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    You don't understand the importance of proofs, think we might as well do without them, but want to become a high school math teacher? The mind boggles. Then you complain about teachers who put students to sleep -- presumably because they don't care about their subject -- all while explicitly planning to become such a teacher yourself? What is this I don't even – hmakholm left over Monica Apr 09 '12 at 11:25
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    A high school math teacher typically has to prepare students for mathematics found in non-math majors and non-math careers. Perhaps it's exactly the focus on proof (irrelevant to high school students) that is soporific. – Ronald Apr 09 '12 at 12:14
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    PLEASE do not become a maths teacher! You have no idea how much harm you can (and by the sound of it will) cause. Believe me, the job market us not in dire need of maths teachers who don't understand what mathematics is about. – Alex B. Apr 09 '12 at 12:24
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    Anyway, if you meant to ask a question, you better ask a question. This is just an (incredibly misguided) rant. – Alex B. Apr 09 '12 at 12:33
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    I voted to close because I feel the question is argumentative and the answer is overly subjective. – Carl Mummert Apr 09 '12 at 12:44
  • regarding the closure decision: what a shame. that this has been favourited 3 times suggests it is a valued question. – Ronald Apr 09 '12 at 12:46
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    @Ronald: I apologize, I realize that my comment above might be read as saying that your answer is subjective. That is not what I intended. I was saying that in principle the collection of answers will include many subjective opinions, because I feel that the question is really a prompt for discussion. – Carl Mummert Apr 09 '12 at 13:30
  • @CarlMummert thanks :) that's understood. I don't take this personally! – Ronald Apr 09 '12 at 13:38
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    I wish to echo Alex's statement: please steer clear of education. There are more than enough lackluster education majors--with their nearly incessant tantrums and whines of "Whaaaaah! When am I ever going to use this abstract algebra/topology/analysis/etc in real life?! Whaaaaaaahhhhh!!"--doing enough damage to K--12 math education. Do something else. –  Aug 24 '12 at 06:31
  • "If you can't write a proof, or put ideas coherently together to form a proof, you can forget about being a mathematician." I am not sure I can agree with that. I mean few people were born being able to write proofs. – Adam Dec 08 '13 at 14:34
  • Reminds me of: Is it unheard of to like math, but dislike numbers? – Adam Dec 08 '13 at 14:35
  • @MattGregory: I dont't care about "truth" and I like logic. – Adam Dec 08 '13 at 14:38
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    Now I read the entirety of Steven's question. I will probably have to calm down by proving some theorems. – Adam Dec 08 '13 at 14:48

2 Answers2

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"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 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.

davidlowryduda
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    A +1 does not seem like enough. Fantastic answer. –  Apr 09 '12 at 12:53
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    +1, "The single most important question in math is why?" I agree with this and everything you wrote. – Américo Tavares Apr 09 '12 at 14:34
  • 'problems where a prescribed solution or apparent solution method exists, and then you carry out that solution' - this sums up 95% of math at 95% of schools. Excellent! – Alex Jun 04 '14 at 15:05
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In Pure Mathematics, proof is crucial. I think this has been discussed.

There are plenty of mathematical fields where proof is not crucial. I would pick out some of the same ones that you did:

  • Statistics (handling data)
  • physics/engineering
  • computer science (software engineering)
  • genetics/systems biology/bioinformatics
  • numerical analysis/optimization
  • finance/risk analysis

I do not claim that proof is not relevant to these fields - but that the majority of the practitioners are not focussing on proof. Essentially - applications of mathematics that require a solution in the real world - these do not necessarily benefit from concentrating on proof at all.

In Computer Science, for example, there is a debate over whether we can prove P=NP. There is a large class of hard problems (NP) which we would like to solve quickly (P), but there is no proof whether that's possible or not possible - it's unknown. In practical reality, computer programmers want to solve their problems in the quickest available way (which may be quickest to write, or quickest to run, depending on the programmers' specific needs), and ultimately have to ignore the lack of proof and concentrate on solving their programming problem.

Once a program is written, it is not even generally possible to prove whether a program will always terminate (or, sometimes run forever). In practice, we can observe easily whether a program will terminate in the realistic cases that we need it to. So, there's no mechanism for a programmer to 'prove' that their program always terminates (although that would not be hugely difficult to implement), because it's simply not necessary in reality.

Even in Cryptography - where proving something is secure is utterly the point of the field - uses systems in practice which rely on things we 'believe' to be true. For example, RSA cryptography (and others) relies on the idea that factoring primes is harder than multiplying primes. This is not, in any sense, proven to be true (we just hope that it is true).

The Millennium Prize Problems of mathematics are all (more-or-less) concepts that ought to be proven: https://en.wikipedia.org/wiki/Millennium_Prize_Problems

This might (correctly) illustrate to you that proof is the primary job of a pure mathematician. However, I would contend that there is plenty of application of mathematics and number that does not require proof.

Having a good sense of society's realistic needs (above proof) will give you access to much larger and more valued career set.

Edited to add: Contrary to some impolite comments, I think that somebody who has a realistic idea of the applications of mathematics would make an excellent high school math teacher.

The job of a K-12 math teacher is absolutely not to prepare a small number of students for university-level mathematics and introduce proof. The job of a K-12 math teacher is to prepare the 99% of students for math they will need in reality.

Ronald
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    OP: I'm an engineer, not a mathematician, but I think being able to prove concepts, however trivial, indicated a higher level understanding of the concepts. Many of the good articles in my field contain proofs that directly relate to real-world applications. – geometrikal Apr 09 '12 at 12:39