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Apart from books on heuristics by George Polya.

When trying to engage with and understand mathematical concepts and when applying abstract mathematical concepts to model "continuum" or real world problems.

I find it fruitful to engage in Socratic questioning based on the Socratic method and principles of critical thinking such as:

  • Analyzing thought (questioning the components of my thinking, for example questioning goals and purpose, questioning assumptions)

  • Assessing thought (questioning standards of my thinking, for example questioning clarity, accuracy, precision, breadth and depth)

I understand the only way to "learn mathematics is to do mathematics", but if this is approached unthinkingly, the learning in my experience tends to be rather superficial.

Since the quality of our thinking is driven by questions, "doing mathematics" should be approached in the spirit of Socrates who acknowledged “The only true wisdom is in knowing you know nothing”.

I suppose if meaning is context dependent, most of Socrates questions were rigorously focused on definitions. Not sure if Godel's incompleteness theorem regarding logical inconsistency is somehow related to this?

The importance of asking essential questions was fundamentally important to Georg Cantor given that his 1867 Doctoral thesis was entitled "In mathematics the art of proposing a question must be held of higher value than solving it."

Does anyone know of any textbooks which focus on the relationship between the Socratic method and deep mathematical thinking?

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    $\begingroup$ Maybe best for matheducators stack exchange? $\endgroup$ – Jon Bannon Oct 30 at 11:44
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    $\begingroup$ Does so called "inquiry based learning", or "discovery learning" or the "moore method" apply here, or is this more specific? $\endgroup$ – rschwieb Oct 30 at 12:47
  • $\begingroup$ It seems that Cantor's thesis was entitled instead De aequationibus secundi gradus indeterminatis. $\endgroup$ – Denis Serre Nov 12 at 16:19
  • $\begingroup$ On Georg Cantor's wiki page his thesis is given as "De aequationibus secundi gradus indeterminatis." but in Georg Cantor's wiki quotes it reads "In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi." "In mathematics the art of asking questions is more valuable than solving problems." Doctoral thesis (1867); variant translation: "In mathematics the art of proposing a question must be held of higher value than solving it." ??? $\endgroup$ – James Fife Nov 12 at 16:34
  • $\begingroup$ You can read Cantor's thesis here: babel.hathitrust.org/cgi/… The passage in question is the very last line. $\endgroup$ – John Stillwell Nov 12 at 22:14
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An influential book on the teaching of mathematics via the Socratic method is Imre Lakatos, Proofs and Refutations. The full book can be browsed on Google, and individual chapters can be donwloaded from jstor.

The steps in Lakatos' approach are:

  1. Primitive conjecture.
  2. Proof (a rough thought experiment or argument, decomposing the primitive conjecture into subconjectures and lemmas).
  3. Global counterexamples.
  4. Proof re-examined. The guilty lemma is spotted. The guilty lemma may have previously been hidden or misidentified.
  5. Proofs of the other theorems are examined to see if the newly found lemma occurs in them.
  6. Hitherto accepted consequences of the original and now refuted conjecture are checked.
  7. Counterexamples are turned into new examples, and new fields of inquiry open up.
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  • $\begingroup$ Thanks Carlo, I am busy with the Imre Lakatos' "Proof and Refutations" at the moment. $\endgroup$ – James Fife Nov 2 at 15:45
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Socrates 's method, as far a mathematics goes, is described by his student Plato in the dialogue Meno, which can be downloaded here.

As for more modern references, aside the great book by Lakatos mentioned by Carlo Beenakker, I would say that the so-called Moore Method goes a certain way in the same direction (though it does not seem to have much of the interactive Socratic approach) .

Moore was a legendary american topologist (Robert Lee Moore, 1882-1974), who taught topology (and other pieces of mathematics) in a peculiar way: an absolute minimum of knowledge provided (essentially basic definitions and key results) and everything else was to be discovered by doing. Apparently the method works, judging from the long list of Moore's "math children".

You can read it it and find refs here

ADDENDUM If you read French, please download a copy of Recoltes et Semailles, by Alexander Grothendieck. There is an entire chapter on his own view on how to do math research, which is difficult to summarize here, but that has definitely something to do with exploring rather than learning "techniques". I mention just one simple fact: Grothendieck and another great mathematicians, Ennio De Giorgi, share something: both, when still undergraduates, after learning basic real analysis asked themselves whether one could possibly generalize it to measure any set. They independently recreated Lebesgue Measure, without knowing it. Useless? according to modern math education yes, but neither of them thought so....

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    $\begingroup$ Thanks Mirco, "Meno" is one of my preferred Platonic dialogues as it is most pertinent to the method of Elenchus and mathematical discovery. "You can easily judge the character of a man by how he treats those who can do nothing for him." ~Johann Wolfgang Von Goethe. In South Africa where I live, endemic racism and segregation, similar to that of Moore's has been and still is responsible for massive injustice and unnecessary suffering. Socratic questioning seems at it's most powerful when rooting out cognitive biases in our own thinking. $\endgroup$ – James Fife Nov 2 at 21:17
  • $\begingroup$ Are there any possibility of Grothendieck's works being translated in the near future? There seems to be a large number of his works which only exist in French. $\endgroup$ – ArB Nov 12 at 15:58
  • $\begingroup$ A small part of Recoltes et Semailles has been translated: fermentmagazine.org/home5.html $\endgroup$ – Deane Yang Nov 12 at 16:09
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    $\begingroup$ Thanks both to Mirco for initially recommending "Recoltes et Semailles" by Alexander Grothendieck and to Deane for intuitively giving me the above link to the English version of the book, as I err more towards the "polyclot" end of the polyglot spectrum. What little I have read thus far, has been wonderfully inspirational! $\endgroup$ – James Fife Nov 14 at 14:23
  • $\begingroup$ according to Alain Connes (interview with Jean Pierre Serre, in french, available on Youtube, it looks like Recoltes will be finally published (hopefully in both french and english). Recoltes is an incredible piece of work, not always pleasant to read. A lot is (in my modest opinion) not "nice", but it contains pearls beyond belief. If you spend any time reading it here and there, you will nourish your soul, as a mathematician and (most important) as a human being. Best wishes! $\endgroup$ – Mirco A. Mannucci Nov 14 at 18:51

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