What have been some of the most revolutionary philosophical shifts in perspective in mathematics?
Often times, great revolutions in mathematics come from shifts in philosophical perspective. The shift from extrinsic to intrinsic geometry yields manifolds (and much else). The shift in focus from objects to the maps between them is the shift from set theory to category theory. What other shifts in perspective have there been which have led to great advances and revolutions throughout mathematics? I apologize if this question is vague.
The shift in perspective (championed by Hilbert) which arises when we start thinking of mathematical theories as themselves formal structures which we can treat mathematically is surely profound and had has wide-ranging consequences (leading to the possibility of proof theory, model theory, and much more).
The Greek shift from the applied mathematics required e.g. to build pyramids to an axiomatic approach as personified by Euclid deserves mention.
I have two candidates:
Analytical treatment of geometry using coordinate systems by René Descartes. The tools which modern differential geometry first used to address the problem rooted from coordinate geometry. Not to mention its huge impact on how to describe motions of heavenly bodies, thus paving the ground for the invention of calculus.
Modeling physical phenomena using differential equations/calculus of variations during 18th century. Some of the major contributors to this machinery are the Bernoulli family, Lagrange and Euler. The mathematical form of classical mechanics somehow "institutionalized" mathematician and physicists's way to establish a theory with certain explanatory power. Thanks to this, later on in 19th and 20th century, in nearly every field of physics, somehow the great results are all in the form of PDEs. Thus functional analysis, $C^*$-algebra, modern geometry, etc benefited from this viewpoints a lot. Though back then their fishing tools might not be as delicate as we have today, but they got lots of big fishes in the pond for them to catch! :)