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I've been studying undergraduate philosophy for the last two years. It seemed to me that philosophers could (and I suspect that many do) use mathematical methods, other than logic, to investigate some philosophical matters more efficiently. On that thought, I decided to pursue a double major in mathematics and philosophy, doing so involves completing a remedial elementary-algebra course during the summer.

The Problem

Success in this course seems to involve quick and correct usage of the problem solving techniques taught in class. Accordingly, at the end of each class, the instructor assigns about 70 practice problems whose solutions involve correctly using one of the two to three methods taught in that class. Usually, I can easily solve all but a few of the problems. When I find problems that I can't solve, I contemplate why my thinking produced an incorrect solution, and adjust my understanding accordingly. However, the overall repetitiveness of the practice problems bores me so much that I often don't bother to solve all of them. I'd enjoy the homework more if it comprised questions like 'Using principles A, B, and C, prove and explain why method Y is the most efficient method of solving problems of form X' (which I do when I find a problem that I can't easily answer). Regardless, in order to study more math, I need to do well in this class, and doing well in the class involves doing well on the exams, which involves quickly and correctly using the problem solving techniques taught in class, and being able to do that involves a lot of repetitive problem solving, which I can't motivate myself to do.

The Question

It would motivate me to know that success in future math courses depends more on understanding principles and deducing their implications than on rote calculation. Is that the case?

How else could a person motivate themselves to do these problems?

Thank you.

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As soon as you can, get copies of the Gelfand School Outreach Program books: Algebra (Gelfand/Shen), Functions and Graphs (Gelfand/Glagoleva/Shnol), The Method of Coordinates (Gelfand/Glagoleva/Kirilov), Trigonometry (Gelfand/Shen). For later use, you'll want to get Sheldon Axler's Precalculus: A Prelude to Calculus and Courant/Robbins' What Is Mathematics?. – Dave L Renfro Jun 10 '14 at 18:03
@DaveLRenfro Thank you. Why those books? – Hal Jun 10 '14 at 18:09
The Gelfand books because they are fairly elementary (even when compared with U.S. high school texts), quite short, and they have an extremely high focus on the underlying ideas. In particular, the Gelfand Algebra book is something you could read NOW, easily within two weeks, as an antidote to what you're doing in class (which you should continue, by the way). See the reviews for Axler's book, and Courant/Robbins is such a classic that you can find more online comments about than you'll have time to read. – Dave L Renfro Jun 10 '14 at 18:52
Another book you should read: George Polya: "How to solve it" about heuristics, the art of problem solving! It instills the right attitude to problems! – kjetil b halvorsen Jun 11 '14 at 8:15
Tangential to the actual question asked, but you might be interested in reading some of the work of Colin McLarty or David Corfield, both of whom are philosophers who engage deeply with the mathematical community, outside of logic (They are both interested in category theory). – Steven Gubkin Jun 11 '14 at 11:57
up vote 5 down vote accepted

It would motivate me to know that success in future math courses depends more on understanding principles and deducing their implications than on rote calculation. Is that the case?


Honestly I don't even know what more I can say about this. Rote calculation is the lowest-level (and most maligned) form of mathematics.

Unfortunately if you are starting out with remedial algebra then you have a long, tedious slog ahead of you before you get to the interesting stuff, and you might want to take matters into your own hands and learn some "real" math independently while simultaneously pursuing your entry-level course work.

(I also think you should make sure you speak to a college advisor about your double-major plans... If you are in your third year at a university and just starting out with elementary remedial algebra, it may not be possible to complete the graduation requirements for a mathematics major in fewer than four more years.)

Maybe start by reading Hardy's A Mathematician's Apology to get a feeling for what the subject is all about. You might also enjoy the book Gödel, Escher, Bach by Douglas Hoftstadter.

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How long is the slog? When do most schools start teaching the interesting stuff? – Hal Jun 10 '14 at 17:47
You probably will need to take something like Precalculus, two semesters of single-variable calculus, one semester of multi-variable calculus, and differential equations, before you can start on the "core" mathematics courses (the theory-and-proof-based ones, which are probably what you are interested in). However some universities have an honors calculus track, which might be more suited to your interests, but also probably presumes a greater level of proficiency at the rote stuff. See also my answer at…. – mweiss Jun 10 '14 at 17:51
Yeah, all of those exactly, and linear algebra which I've heard some people describe as theoretically fundamental, and others describe as 'cookbooky'. – Hal Jun 10 '14 at 17:56
There are (at least) two really different species of linear algebra course. One is based mostly on matrix methods and is indeed cookbooky; the other is about the theory of vector spaces and linear transformations, and is for many people the first truly theoretical course they take. (That the two theories are essentially equivalent is actually one of the first really deep things they learn.) Most university math depts in the US offer both versions, distinguished by different course numbers; read the descriptions carefully. – mweiss Jun 10 '14 at 18:49

As you are following a definitely non-standard track, you should schedule an in-depth talk with your advisor, and with somebody in such a capacity at the math department. Be prepared, have a reasoned outline of where you want to go and why. Take a look at the math courses offered, set up a sequence of courses to take and be prepared to defend it (and also to change it if the suggestions to do so convince you). Do it soon, changing direction in mid-study is painful; the more time goes by the worse.

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Most schools have a placement examination, in which you can test out of algebra 1, geometry, algebra 2, pre-calculus, etc. and start in Calculus. Assuming you're fairly mathematically astute and self-motivated, you could probably teach yourself the mathematics up through Calculus and then test out of those prerequisite courses. Perhaps get a few friends who are good at mathematics that you can have on-call (preferably that have different schedules so someone's always around to ask) and come up with a pacing schedule for yourself and see how far you can get on your own... You have the freedom this way to choose your own textbook, set your own pace, and ask people you truly genuinely like and enjoy talking to for help when you need it...

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Unfortunately, in order to get to the heavy stuff, the slog is necessary. I think the worst mistake I have seen students make is rushing through math courses, doing just what they need to in order to get to the upper level stuff they need (which they then struggle with because they just skimmed through the boring stuff). When you get to calculus, you will use lots of your algebra and trig principles. In linear algebra, you will use an immense amount of algebra. In statistics, you will use calculus and algebra. And in real analysis and other proof classes, you will use a ton of algebra, though in more abstract terms (without actual numbers often). I would advise to continue thinking about exactly how the solutions are derived and why, because that will help when you get to your proof-based courses and make you stronger with proving theorems. Proofs are 60% philosophy and 40% mathematical theory.

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What does Proofs are 60% philosophy and 40% mathematical theory mean? (Proof... writing? A final written-up proof? What do you mean by philosophy? How is this different from mathematical theory?) And from where have you derived such an assertion? – Benjamin Dickman Jun 23 '14 at 22:20

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