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To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. M

To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions) This reminds me of a short lived stretch in French mathematical education, the "Bourbaki years". For those unaware, Nicolas Bourbaki was "a dozen mathematicians in a trench coat" who, in the early 20th, reformed mathematics from the ground up, their work is notoriously opaque (to some, perhaps a revelation to others), if rigorous. It served as inspiration for ambitious mathematics education reforms in the 60s and 70s. Some of the things my father had to contend with: - naive set theory and non-10 basis arithmetic in primary school (age 7) - set theory in early middle school (ages 10/11), maps over sets, so I imagine things like injectivity and bijections. "[...] a different approach of arithmetic, computations often replaced by a more abstract theoretical approach": I wonder how that went in the classroom - general Algebra introduced at 15yo, groups, rings, fields, vector spaces; 16yo mostly linear algebra (vector spaces, linear mappings etc; in true French fashion I'll bet they didn't see many 2x2 or 3x3 matrices that year) I'm quoting Wikipedia "Mathématiques modernes" here, my father only told me of the last point himself. Though I do have his notebooks from early higher ed, they had general topology before metric space topology, for example, in year 1 or 2 (probably 1, because how can you do anything before you know topology...). It was all like this, a theory that builds on itself and you can't skip any steps. A finite dimensional space is a particular case of an infinite dimensional one, so you start with the latter. Backwards from how things were constructed and are understood by people. The fact these reforms survived about 10 years and have been completely reverted is testament that this approach probably doesn't work.

alexey-salmin 2y ago
> To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions) That's the thing: not only he didn't share this intuition, he actively prohibited people from bringing it up in the class because it can lead to mistakes. The program was to start with most generic cases (Banach spaces, metric spaces) and prove whatever is provable there, then continue adding assumptions one by one and proving stronger and stronger theorems. I think we've reached Hilbert spaces by the end of the year and that was a gotcha moment for many students (wait, these vectors were functions the whole time?), but it was too late. Everyone failed miserably at proving or understanding all the preceding theorems because without the intuition it turns into a game of symbols with no structure or hope. The only recourse were those bootleg analogies from finite spaces and Fourier analysis that a few students knew or came up with. The "New Math" you mention above gives a good frame of reference. A beautiful curriculum that's almost impossible to understand unless you already learned math the normal way -- a formally incorrect and inconsistent but reliable way. I also made similar mistake myself when I tried to teach the C language to 12 y.o kids without saying the dreaded "just write it like that, you'll understand later". That experiment failed completely but I only understood why two years later when I myself was subjected to the functional analysis course. Or I think the complete realization came to me even later, with the C language experiment and the functional analysis story becoming pieces of the same puzzle.
FabHK 2y ago
A small note on Bourbaki: Alan Sokal's famous and brilliant hoax Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity had a hilarious footnote that read (my highlighting): > Miller (1977/78, especially pp. 24-25). This article has become quite influential in film theory: see e.g. Jameson (1982, 27-28) and the references cited there. As Strathausen (1994, 69) indicates, Miller's article is tough going for the reader not well versed in the mathematics of set theory. But it is well worth the effort. For a gentle introduction to set theory, see Bourbaki (1970).

alexey-salmin 2y ago

> To be fair, "functional analysis is infinite dimensional linear algebra" is a saying for a reason, that's an intuition a good professor should have given you. Mine gave me it so well, I still see no difference between R^n and a Hilbert space! (on the dangers of intuitions) That's the thing: not only he didn't share this intuition, he actively prohibited people from bringing it up in the class because it can lead to mistakes. The program was to start with most generic cases (Banach spaces, metric spaces) and prove whatever is provable there, then continue adding assumptions one by one and proving stronger and stronger theorems. I think we've reached Hilbert spaces by the end of the year and that was a gotcha moment for many students (wait, these vectors were functions the whole time?), but it was too late. Everyone failed miserably at proving or understanding all the preceding theorems because without the intuition it turns into a game of symbols with no structure or hope. The only recourse were those bootleg analogies from finite spaces and Fourier analysis that a few students knew or came up with. The "New Math" you mention above gives a good frame of reference. A beautiful curriculum that's almost impossible to understand unless you already learned math the normal way -- a formally incorrect and inconsistent but reliable way. I also made similar mistake myself when I tried to teach the C language to 12 y.o kids without saying the dreaded "just write it like that, you'll understand later". That experiment failed completely but I only understood why two years later when I myself was subjected to the functional analysis course. Or I think the complete realization came to me even later, with the C language experiment and the functional analysis story becoming pieces of the same puzzle.

FabHK 2y ago

A small note on Bourbaki: Alan Sokal's famous and brilliant hoax Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity had a hilarious footnote that read (my highlighting): > Miller (1977/78, especially pp. 24-25). This article has become quite influential in film theory: see e.g. Jameson (1982, 27-28) and the references cited there. As Strathausen (1994, 69) indicates, Miller's article is tough going for the reader not well versed in the mathematics of set theory. But it is well worth the effort. For a gentle introduction to set theory, see Bourbaki (1970).