Meta-level revolutions in mathematics - 11.7. CONCLUSION

So, to summarize, in contradiction of the traditional view that the development of mathematics is purely cumulative, I have argued that revolutions do occur in mathematics, but are confined entirely to the metamathematical component of the community's shared background. It is this that gives mathematics its appearance of having developed cumulatively: its evolution is conservative on the object-lecel but revolutionary on the meta-level.

Several example of revolutions in mathematics have been adduced. Two occurred in ancient Greek times: the reformulation of existing mathematics as axiomatic-deductive systems, and the Pythagorean discovery of incommensurable line segments. The others, no doubt interdependent, all took place during the eventful nineteenth century: the full acceptance of negative and imaginary numbers, the discovery of the possibility of non-commutative algebras and non-Euclidean geometries, the arithmetization of analysis, and the invention of transfinite set theory. All these constituted advances that were inclusive on the object-level but demanded the replacement of meta mathematics beliefs.

If we rephrase the above analysis in the language of Lakatos's model of the development of mathematics, we can say that the hard-core of a research programme is a set of elements drawn from all six components of the mathematical world, the community's shared background, some explicit and some only implicit, that the community considers to be consistent and fruitful, that it chooses to select and work with unchallenged for the time being. These elements are chosen rationally, and the stricture not to violate them is an implicit one that can be broken if there is a sufficiently strong indication that it will be useful to do so. So what is happening is that the hard-core of the research programme within which mathematicians are working at any given time (which, in particular of course, incorporates a selection of values from the metamathematical component of the mathematical world) induces in them certain expectations about the results they are to produce. Any anomalous result that violates these expectations is resisted and musst struggle to the surface, the strength of the community's reaction to an anomaly being in direct proportion to the strength of the beliefs its violates. Indeed, as Dauben points out, the community's resistance to change is a good indication of revolutionary developments. So, research programmes are sometimes worked through quite smoothly with no hiccups; but sometimes unexpected results accrue, forcing the mathematical community to change its point of view. Occasionally, such a change will entail jettisoning a major elements of the hard-core of the programme and diverting to an alternative programme with a different hard-core. But this is the point: the element whose abandonment causes a revolutionary shift of research programme is not from the theorem component, but rather the metamathematical one, of the community's shared background.

Thus it is indeed true that mathematics, unlike the natural sciences, appears to grow very largely by accumulation of results, with no radical overthrowing of theories by alternatives. But what do change in revolutionary ways are the implicit metamathematical views of the community that generate and guide their research programmes. It should be noted that, while it may be true to say that the metamathematical value whose abandonment and replacement causes a revolution is wrong (it is simply not true that Euclidean geometry is the only one possible, nor that all numbers must be less than or equal to or greater than zero and have non-negative squares), yet it would be inaccurate to say that the community was mistaken in believing in them for as long as they did. On the contrary, these metamathematical elements of the hard-cores of research programmes have been quite rationally adopted and worked with on the basis of previous experience; equally, they are quite rationally rejected when it is recognized that more fruitful results could be achieved without them.

So, whereas in the empirical sciences, where there are two different types of revolution, those in which some concepts are totally discarded and those in which they are retained as special cases of some more general concept, in mathematics there is only one kind, but they all exhibit both characteristics: exclusiveness on the meta-level and inclusiveness on the object-level. In mathematical revolutions, concepts are always conserved, but what are irrevocably discarded are metamathematical principles; so that, while concepts are not actually discarded, their scope and meaning can be altered by reinterpretation in metamathematical component of the mathematical world.

--Revolutions in Mathematics, DONALD GILLIES, Page 223-225