Terry Tso's three stages of mathematics

It will hardly come as a surprise that I think computational mathematics
is in Tso's Stage 1.

From Terry Tso's blog

https://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

One can roughly divide mathematical education into three stages:

   1. The “pre-rigorous” stage, in which mathematics is taught in an
   informal, intuitive manner, based on examples, fuzzy notions, and
   hand-waving. (For instance, calculus is usually first introduced in terms
   of slopes, areas, rates of change, and so forth.) The emphasis is more on
   computation than on theory. This stage generally lasts until the early
   undergraduate years.
   2. The “rigorous” stage, in which one is now taught that in order to do
   maths “properly”, one needs to work and think in a much more precise and
   formal manner (e.g. re-doing calculus by using epsilons and deltas all over
   the place). The emphasis is now primarily on theory; and one is expected to
   be able to comfortably manipulate abstract mathematical objects without
   focusing too much on what such objects actually “mean”. This stage usually
   occupies the later undergraduate and early graduate years.
   3. The “post-rigorous” stage, in which one has grown comfortable with
   all the rigorous foundations of one’s chosen field, and is now ready to
   revisit and refine one’s pre-rigorous intuition on the subject, but this
   time with the intuition solidly buttressed by rigorous theory. (For
   instance, in this stage one would be able to quickly and accurately perform
   computations in vector calculus by using analogies with scalar calculus, or
   informal and semi-rigorous use of infinitesimals, big-O notation, and so
   forth, and be able to convert all such calculations into a rigorous
   argument whenever required.) The emphasis is now on applications,
   intuition, and the “big picture”. This stage usually occupies the late
   graduate years and beyond.

_______________________________________________
Axiom-developer mailing list
[email protected]
https://lists.nongnu.org/mailman/listinfo/axiom-developer