I'd quip:
"Science is the study of concrete (material) patterns;
Math is the study of abstract patterns"
...and since concrete patterns are instances of abstract
patterns, we see why science is fundamentally (but not only)
mathematical (it also has the content of "how these patterns
express themselves in the world": the connection between
world and pattern).
[Which of course is a sort of Platonism, but backwards:
rather than the world being shadows of ideals,
the world exists, and one sees ideals in it
...though of course the ideals have a life of their own.]
the *structure* of science is mathematical,
and the study of the structure of a science is math:
meta-science is math.
"Scholastic" approach to science:
apparently Einstein's relativity was very scholastic,
not data-driven.
"Science is all physics or stamp-collecting"
...is unfair.
some fields are more data-driven, some are more pattern-driven
Learning pattern-driven ones is better for the brain:
data is data.
[NB: this is partly a technological statement:
to the extent that memorization is a valuable skill,
memorizing vast quantities of something makes you better at it;
these days thinking is more prized than knowing.
Knowing isn't useless (and is undervalued), but relatively easier.]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Enjoying math and science
Experimental Science: Math is computations, math as tool
For some, science is the bottom line:
math is only useful insomuch as it helps you do necessary
computations.
Some think of math as just a tool, some mildly enjoy it,
others actively dislike it.
Simon DeDeo
"Curved space-time helps us understand black holes."
%%%%%%%%%%%%
Theoretical Science:
Mathematics in science is cool
Bottom line is science, but really enjoy the math (& heavy-duty math).
Don't care about the math in isolation, but if it's involved in
some scientific theory, relish it.
Anindya Sen
"Singularities in pseudo-Riemannian manifolds model black holes!"
(dually would be applied math: primarily interested in math,
but care deeply about applications)
%%%%%%%%%%%%
Pure math, with applications
For me, the bottom line is math, but I enjoy applications,
both within math and outside of it
(and do use them for some justification of math).
Enjoy math in isolation (from physical world),
but enjoy connections.
EG, I enjoy that phone numbers are self-delimiting codes,
and that this is related to prefix-free computation.
...but I don't really care about phone numbers:
I'm glad that math helps me understand them,
and I'm glad that phone numbers help illustrate self-delimiting codes.
"Singularities in pseudo-Riemannian manifolds are cool!
They can be interpreted physically as black holes,
which give good intuition."
%%%%%%%%%%%%
Pure math, unapplied (Hardy)
Most purely, some people like math for its internal structure,
with no eye for applications (either outside of math or even to other
areas of math).
...or indeed prefer that there not be any applications.