Video summary
In this discussion regarding the pedagogical order of mathematics, Gilbert Strang reflects on the intuitive difficulty students face when visualizing planes in multi-dimensional spaces compared to concepts from calculus. He acknowledges that while mathematically sound, imagining these higher dimensions is not immediately intuitive for learners. Strang notes a historical irony where calculus preceded linear algebra by centuries, with Newton and Leibniz being credited as the great minds who grasped its key ideas early on. Despite this chronological precedence, he argues that linear algebra should conceptually come first because it deals exclusively with flat surfaces, whereas calculus introduces complications arising from curves and bending geometries. Strang emphasizes that the fundamental distinction lies in the nature of the objects studied: linear algebra is defined by everything being "flat," while calculus inherently involves curvature. He suggests that if mathematics were taught starting with these simpler, non-bending principles, it would make logical sense for students to encounter them before tackling curved surfaces. However, he admits that this ideal order has not been followed historically or in standard curricula; instead, high school and college freshmen typically begin with calculus as their first major math course. Strang expresses a desire to move past the basics of calculus quickly so students can engage with what he considers the "good stuff," which includes the broader applications found in linear algebra. The conversation highlights how human perception struggles when moving beyond two dimensions, making higher-dimensional spaces feel scary and dangerous despite their underlying simplicity if they remain flat. Strang points out that while calculus often restricts initial study to one or eventually two dimensions before introducing multivariate concepts, linear algebra effortlessly extends into ten or more dimensions without issue. This capability stems directly from the fact that in a purely linear world where nothing bends, there is no room for error or confusion regarding curvature. Consequently, Strang maintains that while it is acceptable for calculus not to come first historically, pedagogically speaking, starting with flat spaces would provide a safer and clearer foundation for understanding more complex mathematical structures later on.
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so planes in these multi-dimensional
spaces how how difficult of an idea is
that to to come to do you think if you
if you look back in time yeah I think
mathematically it makes sense but I
don't know if it's intuitive for us to
imagine just what we're talking about
feels like calculus is easier to aisi
into it
well calculator I have to admit calculus
came earlier earlier than linear algebra
so Newton and Leibniz were the great men
to understand the key ideas of calculus
but linear algebra to me is like okay
it's the starting point cuz it's all
about flat things calculus has got all
the complications of calculus come from
the curves the bending this is a curved
surfaces linear algebra the surfaces are
all flat
nothing bends in linear algebra so it
should have come first but it didn't and
calculus also comes first in in high
school classes in in college class it'll
be freshman math I'll be calculus and
then I say enough of it like okay get to
get to the good stuff and that you think
linear algebra should come first
well it really yeah I'm okay with it not
coming first but it should yeah it
should it's simpler because everything
is flat yeah everything's flat of course
for that reason your calculus sort of
sticks to one dimension or so or
eventually you do multivariate but that
basically means two dimensions linear
algebra you take off into ten dimensions
no problem
it just feels scary and dangerous to go
beyond two dimensions well that's all if
everything is flat you can't go wrong
you