E. Bishop. Mathematics as a numerical language. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 53--71. North Holland, Amsterdam, 1970.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Program extraction, Constructive mathematics, Formalized mathematics, Type Theory. 1 Introduction It has long been realized that constructive mathematics has computational content in the sense that proofs of existential statements actually correspond to algorithms to compute a witness, see [3,14]. Also intuitionistic logic, which describes the reasoning in constructive mathematics, is the natural language for type theory based proof assistants. Among these, Coq currently provides a tool that translates proofs of mathematical statements into functional programs which are guaranteed to be ....

Errett Bishop. Mathematics as a numerical language. In Intuitionism and Proof Theory (Proceedings of the summer Conference at Bualo, N.Y., 1968), pages 53{

.... interpretation to analysis by means of bar recursion) 3] a very readable and comprehensive treatment of the whole subject) A variant of functional interpretation which interprets A # A # A in a simpler way so that the decidability of prime formulas is not needed was developed in [31] [13] contains an interesting discussion about the functional interpretation of # from a constructive point of view. Applications of functional interpretation to systems of bounded arithmetic are given in [29] Recently, a functional interpretation for constructive set theories has been developed in ....

Bishop, E., Mathematics as a numerical language. In: Kino, Myhill, Vesley (eds.), Intuitionism and Proof Theory, pp. 53-71, North-Holland, Amsterdam (1970).

....running in Linux on a computer having 256MB RAM and an Intel Celeron 375 MHz as CPU. Most of the Mathematica syntax used here is self explanatory, with a few exceptions which are listed here: Arguments of functions are written inside square brackets (and not parentheses) e.g. f(2, 3) becomes f[2,3] in Mathematica. Variable arguments in function definitions are always followed by an underscore. e.g. f(x, y) x y is written f[x ,y ] x y in Mathematica. All built in functions have names with initial capitals. e.g. sin x is written Sin[x] in Mathematica. If[cond,t,f] returns t ....

....p[#1 2 CompareToZero[CRabs[r[x,p] 0] CRdiv[r[x ,p ] r[y ,q ] CRmult[r[x,p] CRinv[r[y,q] As a sidenote, the problem of separating constructive real numbers equal to 0 # from those apart from 0 # could be done by introducing them as di#erent types in some typed # calculus. Errett Bishop [3] was the first to suggest that constructive mathematics can be formulated in typed # calculus together with intuitionistic logic. This would make it possible to avoid the use of a search procedure such as the one above, since the numbers are marked as being apart from 0 # or as being equal to 0 ....

Errett Bishop. Mathematics as a numerical language. Intuitionsim and Proof Theory, pages 53--71, 1970.

....A) 9y 2 B) P (x; y) constructively means to give a function f that when applied to an element a in A gives an element b in B such P (a; b) holds. For a presentation of the ideas of constructive mathematics we refer to [Bis67, Dum77, TvD88] A constructive proof can thus be seen as a program (cf. [Con82, Bis70, Mar82, Moh86, NS84]) and in type theory execution of the program corresponds to normalisation of the proof. The basic idea behind using type theory for developing proofs and programs is the CurryHoward isomorphism [How80] where propositions (specications) are identied with types and proofs of a proposition ....

E. Bishop. Mathematics as a numerical language, In Myhill, Kino, and Vesley eds., iIntuitionism and Proof Theoryj, pp. 5371, North-Holland, Amsterdam, 1970. 9

.... This paper followed the first expositions of Martin Lof s ideas in [29] and in some lecture notes, made by Sambin during a course in 1980, published as [31] It is interesting to note that Bishop foresaw the possibility of using constructive mathematics as a basis for programming; he suggested in [4] using Godel s theory of computable functionals of finite type. In his series of papers Martin Lof first develops the philosophical and formal basis for his constructive set theory, or constructive type theory, and then points out and exploits the identity between mathematics and programming. In ....

Errett Bishop, "Mathematics as a numerical language", in Intuitionism and Proof Theory (A. Kino, J. Myhill, and R.E. Vesley, eds), 53-71, North-- Holland, Amsterdam, 1970.

....for algorithms. The schools of interest are recursive analysis RA, Martin Lof theories MT, and Bishop constructivism BBB 3 . RA is Kleene s recursive functions: calculus and classical logic. MT is the proposals of Per Martin Lof. MT is close to RA and BBB, differing in the details. Bishop s view[4, 5] is that every theorem should have numerical content and computational meaning. In Beeson s original paper[1] there are lengthy comparisons of the various viewpoints. Bishop s desire was to have a system that could be useful to any and all mathematicians, from his own followers to those with ....

Errett A. Bishop. Mathematics as a numerical language. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 53--71. North Holland, 1967.

....of interest are recursive analysis RA, Martin Lof theories MT, and Bishop constructivism B. RA is recursive functions, calculus, and classical logic oriented. MT is the proposals of Per Martin Lof, also familiar to this readership. MT is close to RA and B, differing in the details. Bishop s view[10, 11] is that every theorem should have numerical content and computational meaning. In Beeson s original paper[5] there are lengthy comparisons of the various viewpoints. Bishop s desire was to have a system that could be useful to any and all mathematicians, from his own followers to the classical. ....

Errett A. Bishop. Mathematics as a numerical language. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 53-- 71. North Holland, 1967.

....us here. It is sufficient to notice that constructive mathematics has some fundamental notions in common with computer science, above all the notion of computation. So, constructive mathematics could be an important source of inspiration for computer science; this was realized already by Bishop [5]. In principle, an implementation of type theory like the system ALF can also be used to express proofs by contra5 diction; in fact AUTOMATH was used to check classical mathematics. For applications in programming, however, we don t know of any example where non constructive reasoning is ....

Errett Bishop. Mathematics as a numerical language. In Myhill, Kino, and Vesley, editors, Intuitionism and Proof Theory, pages 53--71, Amsterdam, 1970. North Holland.

....in practical computer science. 5. Principles of Constructivity 5.1. Received Principles. To set forth a consistent set of principles for constructive programming, there are certain concepts that I inherit from other aspects of constructivity. A succinct list of rules gleaned from Bishop s writings[2, 3, 5, 6, 4, 7] are shown in Figure 6 and expanded below: 1. Objects. Objects in programming are figments of the imagination of humans but representations stored in a computer. The first view is from Brouwer and the second from the reality of computers. 2. Existence. An object cannot be used until it is ....

Errett A. Bishop. Mathematics as a numerical language. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 53--71. North Holland, 1967.

....a different direction from that of Leivant; the induction axiom of S i 2 is retained and the primary focus is definability of functions. At a more general level, the idea of introducing feasibility considerations into logic is a radical extension of the program of constructive mathematics [H] [Bi], TD] Nelson s work itself is in the nominalist philosophy which is highly formalist and, while not finitist, is much stricter than the intuitionistic view. For a general background in logic one can refer to [GG] or [BM] 3 We use the graph of the function here because these systems only have ....

E. Bishop, "Mathematics as a numerical language", in Intuitionism and Proof Theory, eds Myhill et al., North-Holland, 1970.

....specification languages and formalisms for systematically deriving programs from specifications. For constructive mathematics to provide such a methodology, techniques are needed for systematically extracting programs from constructive proofs. Early work in this field includes that of Bishop [Bis70] and Constable [Con71] What distinguished Martin Lof s 82 type theory was that the method it suggested for program syn 5 thesis was exceptionally simple: a direct correspondence was set up between the constructs of mathematical logic, and the constructs of a functional programming language. ....

Errett Bishop. Mathematics as a Numerical Language. In Intuitionism and Proof Theory, pages 53--71. North-Holland, NY, 1970.

....of a constructive proof that type theory can be used as a programming language; and since the program is obtained from a proof of its speci cation, type theory can be used as a programming logic. The relevance of constructive mathematics for computer science was pointed out already by Bishop [4]. Recently, several implementations of type theory have been made which can serve as logical frameworks, that is, dioeerent theories can be directly expressed in the implementations. The formulation of type theory we will describe in this chapter form the basis for such a framework, which we will ....

Errett Bishop. Mathematics as a numerical language. In Myhill, Kino, and Vesley, editors, Intuitionism and Proof Theory, pages 5371, Amsterdam, 1970. North Holland.

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E. Bishop. Mathematics as a numerical language. In A. Kino, J. Myhill, and R. E. Vesley, editors, Intuitionism and Proof Theory, pages 53--71. North Holland, Amsterdam, 1970.

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Bishop, E., Mathematics as a numerical language. In: Kino, A., Myhill, J., and Vesley, R.E. (eds.) Intuitionism and Proof Theory: Proceedings of the Summer Conference at Bu#alo, New York. North--Holland, Amsterdam, pp. 53--71 (1970).

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