The Two Cultures of Mathematics (2000) [pdf]
It appears that roughly the point of the
OP (original post) is that in math there
are two cultures (1) people who want to
develop new fields of math with
definitions, theorems, and proofs and (2)
people who want to use math, all or nearly
all old, to solve problems usually from
outside math.<p>Apparently the OP is from the UK (United
Kingdom).  Here I attempt to provide a
view and explanation of those two cultures
from and for the US.<p>Math got taken very seriously due heavily
to WWII and The Bomb.  There The Bomb was
seen as heavily from Einstein&#x27;s E = mc^2.
So, suddenly the US (especially Congress)
concluded that for US national security
the US had to lead in science and math.
Over the years after WWII, various events,
e.g., Sputnik, reinforced this conclusion.<p>One result was that the US NSF had funds
for math research for culture (1) in the
US research universities.  The emphasis
was on what was really new; that is, if
there was to be another E = mc^2 result
from math and science, then the US wanted
to be the first to discover that result.<p>But the NSF was not much interested in
funding math in culture (2).<p>Then there was some irony:  For US
national security, the NSF was funding
math in culture (1) while also for US
national security, especially the Cold War
and the Space Race, the US DoD and NASA
were heavily funding applications of math
via culture (2).  In those years, there
were good culture (2) careers, especially
near DC, for people with comparatively
good backgrounds in math and computing.<p>But in the profit seeking, practical,
commercial US, away from the motivation of
the Cold War and the Space Race, math from
either culture was ignored, laughed at, or
in rare but overwhelming cases terribly
feared.<p>The there was and is help for culture (2):
The US research universities also commonly
have schools of engineering, and there are
journals eager to publish applications of
math.<p>E.g., the usual criteria for publication
are &quot;new, correct, and significant&quot;, and
an application of some math that is a new
and correct solution for a significant
practical problem can be seen to satisfy
these criteria and qualify for
publication.<p>So, net, currently, an application of math
-- maybe all or nearly all old math --
that is powerful and valuable in practice
-- i.e., some <i>secret sauce</i> -- can count
on little or no competition.