Why's math more abstruse, perplexing than philosophy?

  1. This feels true, as intelligent laypeople can understand unsolved philosophy problems and competing possible solutions, but can't understand or how to begin unsolved math problems.

  2. Yet math and philosophy use logic, which feels mathematical enough to fall under math.

    Optional Information

    A friend apprised me of a leading fee-paying UK high school's experiment for Sixth Formers (i.e. seniors) to substantiate 2.

  3. Teachers notified all students that they'd be tested in 4 weeks, in 4.5 hours in class without notes, on 3 IMO questions from the Algebra section randomly selected from 2011-2016 papers whose solutions are posted.

    They were advised to study the solutions, if they remained baffled after trying. Yet on the test, nobody solved 1 problem or finished early. When asked why, students said that IMO problems needed too clever tricks or inventions or "machinated ploys" (as one student complained).

  4. After this IMO test, all students were instructed to read this list of unsolved philosophy problems for homework. In 2 weeks, they'd write a précis $\ge$ 300 words outlining the problems, on 4 randomly chosen topics from that list, in 2 hours in class without notes. Everyone aced this. Some finished before the time limit.

The snag: 4 weeks after this test, the teacher dispensed a pop test with the same format, but on another 4 randomly chosen topics. Yet everyone still aced it.

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    $\begingroup$ My point is, they were expected to solve the math problems but they were not expected to solve the philosophy problems. Isn't that unfair? $\endgroup$ May 8 '18 at 6:40
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    $\begingroup$ Could you provide a link to the study? $\endgroup$ May 8 '18 at 6:41
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    $\begingroup$ This is like the ten gazillionth time you've asked this. $\endgroup$
    – guest
    May 8 '18 at 8:34
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    $\begingroup$ Just to add: There are a lot of easy to understand math problems (that are still unsolved), e.g. Goldbach's conjecture that every even number (except two) is a sum of two primes. On the other hand, there surely are problems in philosophy that are really hard to understand for a non-expert. $\endgroup$
    – Dirk
    May 8 '18 at 11:00
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    $\begingroup$ The premise of the question is at least dubious, and, probably, nonsense. Sorry... $\endgroup$ May 9 '18 at 2:22

If you solve a mathematical problem the wrong way, you get a wrong result, which can be checked. If you "solve" a philosophical problem there is no way to check the result in any decent timeframe. May be you made no sense, may be you invented a new branch of philosophy.

In the hard (read: checkable) parts of philosophic thought, the tools are mathematical ones, but vastly reduced - the logic components used by philosophers usually don't rise above the introductory courses for logic in maths. Philosophy needs to inject natural language at some point (otherwise it stays math), which is where checkability goes out the window.

The rigor needed to perform good philosophy may be as high as the rigor needed to do good math, but the ability of tutors to check for rigor is hampered by the impossibility of said task in philosophy. Thus, more students will seem to pass philosophic hurdles than mathematical ones.

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    $\begingroup$ To do mathematics, all you need is a pen, paper, and a wastebasket. To do philosophy, all you need is a pen and paper. $\endgroup$ May 8 '18 at 13:11
  • $\begingroup$ @JoelReyesNoche Sorry. I don't understand? I know that you can type philosophy, but math one ought write on paper. Is this what you intended to say? $\endgroup$
    – NNOX Apps
    May 10 '18 at 6:55
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    $\begingroup$ I was adding to what bukwyrm was saying. It seems that mathematics is falsifiable (hence the need for a wastebasket) whereas philosophy is not. I am only joking, of course. $\endgroup$ May 10 '18 at 8:29

I'm apologizing in advance for the subjective answer but I think a short story is the best way I can organize my thoughts around this.

I started out in Medicine in the mid 1990's. But I was unhappy. Medicine was too concrete. While we learned plenty theory in medical school, that theory wasn't very helpful in making accurate predictions. You never knew that a treatment would work for a patient until you tried it. Often, when a treatment failed, you had no idea why. So you fell back to statistical methods that would give you the best probability of success. Medicine had too much rote memorization because you were too often unable to make accurate predictions from first principles. Medicine wasn't abstract enough for me.

After med school, I specialized in Psychiatry. I thought psychiatry would be more abstract. And it was. Yet psychiatry gave me even less job satisfaction than I had before. And that was the first time I think I really confronted and understood the difference between abstraction and ambiguity.

Both math and philosophy require comfort with abstract ideas. But math has virtually zero ambiguity to it. For example, the natural logarithm of pi is several layers of abstraction deep, yet it has absolutely zero ambiguity. Philosophy has more ambiguity. Definitions aren't as precise. There are shades of gray. Priorities come in to play.

I think that's why math is more difficult than Philosophy. Math requires that you need to walk a tightrope of abstraction on one side, and formal rules and definitions (to avoid ambiguity) on the other side. Philosophy is more forgiving of ambiguity.

  • $\begingroup$ Actually, I have to disagree. There is endless ambiguity in mathematics, though not of the sort in everyday discourse. Those elementary seemingly non-ambiguous mathematical "facts of nature" are not what mathematics is. After all, "definitions" are human constructs... $\endgroup$ May 9 '18 at 2:24

This is too long for a comment. This question reminds me of a story that Feynman told in Surely You're Joking, Mr. Feynman:

There was a Princess Somebody of Denmark sitting at a table with a number of people around her, and I saw an empty chair at their table and sat down.

She turned to me and said, "Oh! You're one of the Nobel Prize winners. In what field did you do your work?"

"In physics," I said.

"Oh. Well, nobody knows anything about that, so I guess we can't talk about it."

"On the contrary," I answered. "It's because somebody knows something about it that we can't talk about physics. It's the things that nobody knows anything about that we can discuss. We can talk about the weather; we can talk about social problems; we can talk about psychology; we can talk about international finance--gold transfers we can't talk about, because those are understood--so it's the subject that nobody knows anything about that we can all talk about!"

Edit: As @KCd suggested, I should include Feynman's conclusion to this story:

I don't know how they do it. There's a way of forming ice on the surface of the face, and she did it!

  • $\begingroup$ Your citation had to end somewhere, but still I want to note that you ended it right before Feynman's funny description of the reaction of that Princess to Feynman's remark. $\endgroup$
    – KCd
    May 12 '18 at 2:00

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