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An undergraduate double major in math and philosophy, my cousin learns philosophy well, but she's failing her abstract proof-based courses (Group Theory, Real Analysis and Linear Algebra). Presume that she studies each subject at least 3 hours daily 6 days of the week.

Why can she learn philosophy, but not abstract math? What can explain her snag: she can calculate or prove something only if it nears something previously solved? On her exam, she failed to calculate, prove, or construct new counterexamples for, anything unseen.

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    $\begingroup$ A wild guess: she lacks the requisite familiarity with the concrete special cases related to the subject at hand. It's much easier to "do group theory" when you have enough experience with actual groups to understand what you're trying to prove and why it should be true. This kind of thing would be easier to pick up by someone who spends all their time doing mathematics. $\endgroup$ – Will R Feb 12 '18 at 9:05
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    $\begingroup$ It's probably because she's memorizing instead of learning but this question is more or less unanswerable without knowing the person. $\endgroup$ – Adam Feb 12 '18 at 14:19
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    $\begingroup$ Perhaps a better question to ask first is, what does it mean for her to "learn philosophy well"? $\endgroup$ – Brendan W. Sullivan Feb 12 '18 at 16:59
  • $\begingroup$ @Adam Please ask if you need details from her? $\endgroup$ – Greek - Area 51 Proposal Feb 13 '18 at 0:53
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    $\begingroup$ These disciplines require very different approaches. Which disciplines do you know that you yourself show very different performance or grades in? What makes the difference? $\endgroup$ – shuhalo Mar 4 '18 at 6:00
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I was also a double major in mathematics and philosophy as an undergraduate (at a state university in the U.S.). I think that both are incredibly important, and I'm happy to have both under my belt. Actually: philosophy was my primary major, mathematics secondary. (This confused a lot of my instructors in higher-level math courses looking at the course registration roster.) Now I'm a lecturer in mathematics.

The truth is, anything in a STEM discipline is simply harder and less forgiving of errors than in courses in the humanities, and this is one example of that. STEM courses like math have problems which are practical and have answers that are clearly correct, or clearly incorrect. In graduate school I once had a four-page proof returned with simply the word "No" in red at the top. The humanities, like philosophy, generally accept a wide variety of answers as long as some background justification is given; and grading is generally much more generous (for a variety of institutional and supply/demand reasons). There's no paper I could possibly write in philosophy for which the response is simply "No".

Now, proof-based math courses are the big leagues. Some people do advanced proof math competitions all through their youth and teenage years before turning professional. Many colleges have a sophomore-level course such as "Introduction to Proof" or "Bridge to Advanced Mathematics" or the like. Fortunately, I had a very good such course where I went to college (and there the professor was delighted to have an apparent philosophy major on the first day), and it at least gave me a fighting chance in later courses like Abstract Algebra and Analysis (love the former, not so much the latter). Maybe such a transition course does not exist at Oxford?

On the other hand: Yes, you can BS philosophy. Not that I'd personally want to. Some might say that's the very essence of philosophy -- but, of course, others would disagree. Hence the point.

Consider also the comments to this question which in turn spawned this question. Everywhere we turn, at every level, no matter how much anyone doesn't want it to be, math winds up being the bottleneck/filter that distinguishes between people who can really do it, and those who can't. I personally refer to this as the brutal honesty of math.

Edit: If it's true that someone is struggling with the transition to proof-based math courses, and lacks a transition course, then my top recommendation would be to look at Richard Hammack's Book of Proof (free, open educational resource) -- ideally before their next course starts.

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    $\begingroup$ Very cool to hear you had that background! I always wanted to take a basic survey class in philosophy. I guess I could go buy the Durant book. Hmm... $\endgroup$ – guest Feb 12 '18 at 19:08
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    $\begingroup$ For what it's worth, I was in a somewhat opposite situation. I took a lot of philosophy courses (upper level and at least one graduate level) as an undergraduate, initially to satisfy non-science course requirements in my last two years but then I found I liked them and also did very well, for a total of 30 semester hours. Sometime in mid August 1980, just before the beginning of my senior (= 4th) year, I noticed that I only needed two courses to get a degree in philosophy --- some kind of writing/research course based on reading papers and writing something akin to an (continued) $\endgroup$ – Dave L Renfro Feb 13 '18 at 10:25
  • $\begingroup$ undergraduate thesis, and something else. I talked to the department chair, who knew me from a graduate logic class, and he was initially confused because he thought I was one of their majors, since I was a fairly familiar face around the department for the past year. Anyway, I wound up not getting the double major, as the writings/research course conflicted with something I was more interested in. (I also toyed with a physics degree, but this would have required more extra work and also one or two upper level labs that I didn't want to take.) $\endgroup$ – Dave L Renfro Feb 13 '18 at 10:36
  • $\begingroup$ @DaveLRenfro: I hear that; likewise, I was accidentally 2 classes away from a religious studies concentration in my philosophy major, but chose not to pursue that. $\endgroup$ – Daniel R. Collins Feb 13 '18 at 15:03
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    $\begingroup$ Anyway, I find it kind of funny that the university bean-counters counted those 10 semester courses in philosophy of mine as "non-science", despite 3 being in logic (one of which was rather advanced taught by this person using his just published book) and one on the philosophy of space and time team-taught by this person and this other person. $\endgroup$ – Dave L Renfro Feb 13 '18 at 17:41
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Obviously it's hard to tell from a distance. Here are some explanations:

  1. She may be more motivated to think about the material in the philosophy course. This may be because she can relate to the topics that they analyze. Like one commenter suggests, if you do not know the basic examples or motivation for group theory, it can be hard to feel motivated to learn the methods. (An intervention that may help here is talking about symmetries of Platonic solids; e.g. show her how to quickly count the number of elements in these groups, how to prove some isomorphisms like $S_4$ isomorphic to group of rotations of the cube, $S_3$ isomorphic to symmetries of the triangle, etc. A fun exercise is to find subgroups of $S_4$ using stabilizers of subsets of the cube.)

  2. She may have some degree of math anxiety. See the paper Mathematics anxiety and stereotype threat: shared mechanisms, negative consequences and promising interventions.

  3. It's possible that she has never learned how to study for a mathematics course at this level. There are books and guides for this. Learning group theory requires doing problems every day for a while; it's a new language. Similarly with analysis.

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