David Corfield
Shaperean Philosophy of Mathematics

Were we to bring over Shapere’s ideas on science to mathematics, what would appear?

Key thought

It is truly all aspects of science, not only what are considered its substantive beliefs about nature, but also its methods and aims, that are subject to change in ways that have continued to surprise us. The problems we face in our inquiry about nature, and the methods with which we attempt to deal with those problems, co-evolve with our beliefs about nature. (Shapere 1987, p.5.)

Dudley Shapere (1987). Method in the Philosophy of Science and Epistemology. In Nancy J. Nersessian (ed.), The Process of Science: Contemporary Philosophical Approaches to Understanding Scientific Practice.

Principle of the Rejection of Anticipations of Nature

The results of scientific investigation could not have been anticipated by common sense, by the suggestions of everyday experience, or by pure reason. (Shapere 1987, p. 1.)

Principle of Internalization

Every aspect of our beliefs ought, whenever possible, to be formulated, and to be brought into relation to well-founded beliefs, in such a way that it will be possible to test that aspect. (Shapere 1987, pp. 3–4.)

In mathematics

Rather than separate scientific concepts – force, electron, species, etc. – from ‘metascientific’ ones – observation, explanation, etc., we find that the latter cannot be timelessly captured but rather undergo modification by internalization. We might expect the same for – space, symmetry, algebra – and – proof, definition, explanation.

Rational continuity of science via chains of reasons

Methods, rules of reasoning, criteria (e.g., of what can count as an explanation) go hand-in-hand with the beliefs arrived at by their employment, and are on occasion altered in the light of the knowledge or beliefs arrived at by their means. Constraints on scientific reasoning develop, being sometimes tightened and sometimes broadened, as science proceeds. And thus, although at one stage of science, what (for example) counts as a legitimate scientific theory or problem or explanation or consideration might differ, even radically, from what counts as such at another stage, there is often a chain of developments connecting the two different set of criteria, a chain through which a “rational evolution” can be traced between the two. We can then recognize that, given the knowledge and criteria available at a particular time, certain beliefs about possibilities and truth were reasonable, even though alteration and improvement were later possible, with the emergence of new knowledge and new criteria. (Shapere 1984, p. 212.)

Functions of the philosophy of science

  1. The critical function
  2. The overview function
  3. The detailing function

Learning how to learn

(1) What considerations (or, better, types of considerations, if such types can be found) lead scientists to regard a body of information as a body of information - that is, as constituting a unified subject matter or domain to be examined or dealt with?

(2) How is description of the items of the domain achieved and modified at sophisticated stages of scientific development?

(3) What sort of inadequacies, leading to the need for further work, are found in the bodies of information, and what are the grounds for considering these to be inadequacies or problems requiring further research? (Included here are questions not only regarding the generation of scientific problems about domains, but also scientific priorities - the questions of importance of the problems and of the “readiness” of science to deal with them.)

(4) What considerations lead to the generation of specific lines of research, and what are the reasons (or types of reasons) for considering some lines of research to be more promising than others in the attempt to resolve problems about the domain?

(5) What are the reasons for expecting (sometimes to the extent of demanding) that answers of certain sorts, having certain characteristics, be sought for those problems?

(6) What are the reasons (or types of reasons) for accepting a certain solution of a scientific problem regarding a domain as adequate. (Shapere 1984, 277-8).

Last revised on December 16, 2014 at 09:16:46. See the history of this page for a list of all contributions to it.