Paradigm Shifts in Mathematics

Question

in physics there were several clear revolutions or paradigm shifts which fundamentally changed the field. One example is the copernican revolution and the encompassing shift from the ptolemaic to heliocentric view.

Given that mathematics works from axioms, I figured it is unlikely for wrong assumptions to creep into the canon of the field. Additionally, during my math education (as a physicist) I had the feeling, that mathematics evolved rather continuously from the greeks to today, always adding new knowledge on top of the old.

Therefore my question is, whether this is wrong and there were certain paradigm shifts or radical reinterpretations of previous results in the history of mathematics, or was it a continuous growth of knowledge?

Addendum

There has been this question already, which is asking for philosophical shifts in mathematics. However, I figured it is different from this one, since I try to understand whether the body of mathematical knowledge grew linearly or was discontinuous at certain points.

Answer 1

I suppose we might distinguish "revolutions" which bury their dead (so to speak) from "paradigm shifts" (where the game moves on, and work done in the old style is not expunged but no longer looks interesting or important to pursue).

I suppose it was once thought that the 19th century reworking of analysis without infinitesimals was a revolution that displaced falsehood/incoherence (which is why varieties of non-standard analysis which rehabilitated infinitesimals -- sort of! -- came as an intriguing surprise a hundred and something years later). The development of set theory was a revolution, in showing that it was possible to have a coherent theory (of "completed infinities") where previously it was thought that there could only be falsehood/incoherence.

But these sort of cases are surely the exception (in maths at any rate). A paradigm shift need not involve supposing that what has gone before is wrong. Rather, novel concepts are introduced, new problems can be raised, new approaches come to be seen as particularly interesting/rewarding; new exemplars come to be regarded as paradigms to be emulated, and as setting the standards by which problem-solutions are judged. The development of abstract algebra in the last century, for example, would seem to be a paradigm example of this sort of paradigm shift ...!

Answer 2

Mathematics is not an axiomatic discipline. One way a new field is opened is by generally by uncovering examples which have something in common and which appears to point to a new theory.

Take for example homology. This was axiomatised by Eilenberg & Steenrod. But had not people discovered the Betti numbers, had not Poincare discovered homology and had not Noether pointed out that the Betti numbers wrre better thought of as groups there would not have been something to axiomatise.

Hilbert says more or less the same in his Geometry & the Imagination where he classes deductive thinking, that is the thinkng that comes from the axiomatic form of a lower order than that of inductive thinking which he classifies as the true form of scientific thinking.

Personally, a key paradigm shift for me has been the introduction of category-theoretic thinking into mathematics and it also demonstrates the continuity of thought too. For example, the triangle was discovered early, by adding directions to the sides we have the law of vector addition and then by allowing the sides to be curved we can think of them as category-theoretic arrows. This is also revealing: we can think of them as non-Euclidean vectors and in a length space where between any two points there is a unique geodesic we can lift the directed geodesics into such a vector.