One of the ways mathematics has changed me is
that I kind of now expect things to be connected.
So when I first studied algebra and geometry,
they seemed like different subjects. Algebra
was about equations, variables, and symbolic
rules. Geometry was about shapes, angles, and
diagrams. One lived in a world of symbols, and
the other lived in a world of pictures. came
the coordinate plane. With an x -axis and a y
-axis, an equation could suddenly become a shape.
For example, the equation y equals 2x plus 1
was not only a rule for manipulating symbols,
it was also a line. The equation y equals x squared
became a parabola. The equation x squared plus
y squared equals 25 drew a circle. and the connection
worked in both directions. An equation could
produce a picture, and a picture could reveal
information about an equation. This almost feels
magical. Two subjects I had placed in separate
compartments turned out to be different ways
of describing the same thing. Halsberg gave me
one language, and geometry gave another language.
and then the coordinate system allowed them to
speak to each other. The mathematics became richer
because of this connection. An equation might
be difficult to understand symbolically, but
became crystal clear when I could see it in its
graph. A geometric relationship might be easier
to analyze after translating it into algebra.
Neither representation replaced the other. Each
allowed me to notice something that the other
made less visible. And this has influenced the
way I think beyond math. For example, when I
encounter two ideas that seem unrelated, I'm
just more inclined now to ask whether there might
be a bridge between them. Perhaps they are not
competing explanations, but different representation
of the same underlying reality. Perhaps each
perspective preserves something that the other
leaves out. And of course, not everything connects
nearly. And sometimes we force comparisons that
do not belong. But mathematics has taught me
not to assume that the boundaries that I first
see are real or permanent. The categories may
have been created for convenience. A change in
language or viewpoint or representation may reveal
what appeared separate was now connected all
along. And this is one reason why math feels
less like a collection of isolated facts and
more like a landscape. The further I travel,
the more paths begin to cross. Ideas that I learned
years apart suddenly meet. A symbol becomes a
shape, a shape becomes a relationship, and a
relationship opens the way to another question.
Math has taught me that understanding often arrives
when I find the right connection. And once that
connection appears, both sides look different.