They find non-constructive existence proofs unacceptable.
The relationship between constructive mathematics to computation was readily discernible.
When opinions vary, he said, constructive conflict is where a team shares a goal that ultimately benefits the company.
Any theory of law, he claimed, needs to be a ‘constructive interpretation’ of legal practice.
Rathjen (2006) has also considered various constructive choice principles and their mutual relations.
With his doctrine of constructive empiricism, van Fraassen is widely credited with rehabilitating scientific anti-realism.
In working in constructive and intuitionistic ZF we can thus to some extent rely on our familiarity with ZF and its heuristics.
Constructive type theory is usually considered the most satisfactory foundation for constructive mathematics Bishop-style.
The constructive empiricist argues that constructive empiricism “makes better sense of science, and of scientific activity, than realism does” (van Fraassen 1980, 73).
Intuitionistic type theory (also constructive type theory or Martin-Löf type theory) is a formal logical system and philosophical foundation for constructive mathematics.
The former theories are adaptations of Zermelo-Fraenkel set theory to a constructive setting, while in type theory the constructions implicit in constructive statements are made explicit in the system.
Revisionary intuitionists who object to non-constructive existence proofs in mathematics are not just expressing a preference for constructive methods: they find the notion of non-constructive existence unintelligible not just unappealing:
epsilon calculus | Hilbert, David: program in the foundations of mathematics | logic: linear | mathematics: constructive | proof theory: development of | reasoning: automated | semantics: proof-theoretic | set theory: constructive and intuitionistic ZF | type theory
While the constructive empiricist view is a view about the aims of science and not a normative theory in epistemology, the constructive empiricist is an individual who values the sort of epistemic modesty which might motivate one to harbor anti-realist sympathies in general.
It is also worth noting that the present convention on the use of the words “constructive” and “intuitionistic” differs from that made in the context of constructive mathematics (see, for example, the entry on constructive mathematics and also Bridges and Richman 1987).
A sufficiently unreflective constructive empiricist might adopt this construal of empirical adequacy for her theory, but a more sophisticated constructive empiricist would probably embrace an account of empirical adequacy akin to that which van Fraassen develops later in The Scientific Image.
The relatively permissive epistemological view of a constructive empiricist who is also an epistemic voluntarist helps explain why such a constructive empiricist would be prudent not to take constructive empiricism to be a normative theory concerning the deliverances of science.
Acceptance of a theory, according to constructive empiricism, correspondingly differs from acceptance of a theory on the scientific realist view: the constructive empiricist holds that as far as belief is concerned, acceptance of a scientific theory involves only the belief that the theory is empirically adequate.
logic, history of: intuitionistic logic | logic: intuitionistic | mathematics, philosophy of: intuitionism | mathematics: constructive | proof theory: development of | semantics: Montague | semantics: proof-theoretic | set theory | set theory: constructive and intuitionistic ZF | set theory: Zermelo’s axiomatization of | type theory | type theory: Church’s type theory
(f) Constructive set theory (as do the theories under (d) and (e)) sets out to develop a framework for the style of constructive mathematics of Bishop’s 1967 Foundations of constructive analysis in which he carried out a development of constructive analysis, based on informal notions of constructive function and set, which went substantially further mathematically than anything done before by constructivists.
constructive
adj all
- emphasizing what is laudable or hopeful or to the good
Example: constructive criticism
adj all
- constructing or tending to construct or improve or promote development
Example: constructive criticism
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f Constructive set theory as do the theories under d and e sets out to develop a framework for the style of constructive mathematics of Bishops 1967 Foundations of constructive analysis in which he carried out a development of constructive analysis based on informal notions of constructive function and set which went substantially further mathematically than anything done before by constructivists