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Comment by lambdaphage | Hacker News Reader

I'm always surprised when reading the notes of scientists and mathematicians working in previous centuries to see just how steeped they were in synthetic geometry. This was taken to an extreme in the case of the Principia, but one can't read Gibbs or Maxwell either without realizing that they felt Euclid in their bones in a way that few people do today, with possible exceptions for mathematicians trained under the Soviet system.

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I was born in Eastern Europe, under the influence of Soviets, but a different country, Romania.

We learned planar classical geometry (Euclidean) in the 6th grade.

If someone wants to take a pick (the manual is in Romanian, but you can see the figures and the mathematical notations): http://manualul.info/Geom_VI/

Why are the mathematicians trained under the Soviet regime an exception?

I don't know-- usually the answer to such questions about academic priorities is "because it was cheaper", but they just seemed to emphasize geometry much more at the K-12 level. It's a generalization, of course, but a pretty robust one. One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it.

EDIT: this interview with Izaac Wirzsup comparing the Soviet and US systems confirms my prejudice:

  Another extremely harmful feature
  of [the US] school mathematics programs
  is that only about half of our students
  take geometry, and for only one
  year, generally in a concentrated high
  school course. Students cannot be
  expected to master the material taught
  in this way. Moreover, they are not
  being taught solid geometry, and they
  rarely have a workable perception of
  three-dimensional space, which is so
  essential for studying science,
  technical drawing, or engineering.
  Soviet children study geometry
  extensively for ten years, including
  two years of solid geometry.

Jut from my own common sense it seems like geometry is important for understanding the relationship between abstract things and concrete things. It's easily understandable that shapes are described by geometry and it seems obviously useful. The square footage of a house. The volume of a bath.

If you try to describe what Calculus is or does, it's abstractions of abstractions. Rates of change or 'angle of a curve for a certain values. I think it's hard for students to see this as something useful or even see how it's a description of the world that opens up ways of understanding it.

I don't see calculus as an abstraction of abstractions. The fundamental idea is completely geometric: "break the domain of a problem into a bunch of pieces that can be easily described and related (e.g. by physics) then put the pieces back together." Time is a first-class dimension. Abstractions only enter the picture when you want to separate the problem of picking a mesh from the problem of representing mesh elements.

Differential operators perform the task of "breaking into pieces" in a mesh-invariant way. Differential forms are mesh-invariant pieces. Integration is the mesh-invariant description of putting the pieces back together.

It's convenient that differential forms can be interpreted physically (by normalizing, associating with geometric elements, etc) but I'd hesitate to associate them with any single physical interpretation (e.g. rates of change) because doing so de-emphasizes the generality of the approach; you can have a rate with respect to distance, area, or volume just as easily as a rate with respect to time.

Leibniz notation makes the hop from the geometric approach to the "operator that maps a function to a function" approach seamless, and since the latter description isn't nearly so intuitive, I've always suspected that the geometric approach could profitably be taught first.

> One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it.

Do you remember what was the problem, by any chance?

I don't remember-- it must have been either differential geometry or topology, but I think the theorem in question was the inscribed angle theorem: http://www.proofwiki.org/wiki/Inscribed_Angle_Theorem. Not a difficult theorem, but you had to know it well to be able to see the application immediately.

I've also noticed that the Chinese have a pretty high focus on geometry.

Thanks for the extensive resp.

My mother learned math in the French system in the 50s and it also stressed geometry.

There is an excellent chapter called 'on teaching of geometry in Russia' [1] discussing how Geometry stayed important even after west moved away from euclid.

1. http://www.amazon.com/Russian-Mathematics-Education-Programs...

I've already spent my quota on $100+ books for the year. Is there a free version for this book chapter anywhere? :-)

Well, in 1665 at least it could hardly have been otherwise: modern co-ordinate geometry was only about 15-30 years old.

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