Moreover, in Markov's school, Markov's Principle of unbounded search is accepted:. MP: For any binary sequence, if it is impossible that all the terms are zero, then there exists a term equal to 1. This reflects the idea that if it is contradictory for all the terms to be zero, then although we do not have an a priori bound on how far in the sequence we must seek eventually any machine that is looking for a 1 will find one in finite possibly large time.

In part because of this, the constructive status of this principle is not so clear. While the omniscience principles are not accepted by any school of constructivism, there is at least pragmatic that is, computer-implementable reason to admit MP: the algorithm which will compute the 1 in a binary sequence for which it is impossible that all terms are 0 is simply to carry on the process of writing the sequence down until you come across a 1.

Of course, MP does not provide a guarantee that a 1 will be found in a sequence before, say, the extinction of the human race, or the heat death of the universe. Recursive function theory with intuitionistic logic and Markov's principle is known as Russian recursive mathematics. If Markov's principle is omitted, this leaves constructive recursive mathematics more generally—however, there appear to be few current practitioners of this style of mathematics Beeson, , p. A central tenet of recursive function theory, is the following axiom of Computable Partial Functions :.

CPF: There is an enumeration of the set of all computable partial functions from N to N with countable domains. Much may be deduced using this seemingly innocuous axiom. For example, PEM and weakenings thereof, such as LPO may be shown to be perhaps surprisingly simply false within the theory. Constructive recursive mathematics presents other surprises as well. However, some amazing and classically contradictory results can be obtained, such as Specker's theorem Specker, :.

While it seems like a contradictory result, the theorem becomes clear when one considers the objects of study: sequences and functions and real numbers, are recursive sequences and real numbers; so Specker sequences are sequences which classically converge to non-recursive numbers. Another interesting result is the constructive existence, within this theory, of a recursive function from [ 0 , 1 ] to the reals which is everywhere pointwise continuous yet not uniformly continuous as such a function would be both classically and intuitionistically, as a consequence of Brouwer's fan theorem.

Not only did this assuage the worries expressed by some leading mathematicians such as Weyl about the feasibility of constructive proofs in analysis, but it helped lay the foundation for a programme of mathematical research that has flourished since. It captures the numerical content of a large part of the classical research programme and has become the standard for constructive mathematics in the broader sense, as we will shortly see. Bishop refused to formally define the notion of algorithm in the BHK interpretation. It is in part this quality which lends Bishop-style constructive mathematics BISH the following property: every other model of constructive mathematics, philosophical background notwithstanding, can be seen as an interpretation of Bishop's constructive mathematics where some further assumptions have been added.

Even classical mathematics—or mathematics with classical two-valued logic—can be seen as a model of Bishop-style constructive mathematics where the principle of excluded middle is added as axiom. Another factor that contributes to this versatility is the fact that Bishop did not add extra philosophical commitments to the programme of his constructive mathematics, beyond the commitment of ensuring that every theorem has numerical meaning Bishop, The idea of preserving numerical meaning is intuitive: the numerical content of any fact given in the conclusion of a theorem must somehow be algorithmically linked to or preserved from the numerical content of the hypotheses.

Thus BISH may be used to study the same objects as the classical mathematician or the recursive function theorist, or the intuitionist. As a result, a proof in Bishop-style mathematics can be read, understood, and accepted as correct, by everyone Beeson, , p. If one reads a Bishop-style proof, the classical analyst will recognize it as mathematics, even if some of the moves made within the proof seem slightly strange to those unaccustomed to preserving numerical meaning.

The axiom is in fact provable in some constructive theories see Section 5. And as we have seen, for some constructivists it would actually be inconsistent to accept PEM. It thus seems that AC, while very important, presents a significant problem for constructivists. However, as we now discuss, such a conclusion is hasty and unjustified. The key observation to make regards the interpretation of the quantifiers. Thus only if there is no extra work to be done beyond the construction of x in showing that x is a member of A that is, if this can be done uniformly will this be a genuine function from A to B.

We will revisit this in Section 5. Let us return to Bishop's statement at the start of this section, regarding the existence of a choice function being implied by the meaning of existence. The context in which his claim was made illuminates the situation. Some constructive mathematicians adopt various weakenings of AC which are perhaps less contentious; Brouwer, as we saw in Section 2 , adopts Continuous Choice.

Leaving the question of foundations open is, as mentioned, partly responsible for the portability of Bishop-style proofs. Some work has been done on founding BISH. A set-theoretic approach was proposed in Myhill and Friedman ; Feferman suggested an approach based on classes and operations. It is interesting to observe that, as a rule of thumb, the more ingenious a proof looks, the less explicit numerical content it tends to contain.

Thus such proofs are usually more difficult to translate into constructive proofs. Of course there are exceptions, but the comparison between say Robinson's nonstandard analysis and Bishop's constructive methods appears to be a confusion between elegance and content propagated by some authors; see, for example, Stewart and Richman It encapsulates mathematics based on recursive function theory, with a background of the algorithms of Post However he soon revisited foundations in a different way, which harks back to Russell's type theory, albeit using a more constructive and less logicist approach.

Derivations convince us of the truth of a statement, whereas a proof contains the data necessary for computational that is, mechanical verification of a proposition. Thus what one finds in standard mathematical textbooks are derivations; a proof is a kind of realizability c. Kleene, and links mathematics to implementation at least implicitly. The theory is very reminiscent of a kind of cumulative hierarchy. Propositions can be represented as types a proposition's type is the type of its proofs , and to each type one may associate a proposition that the associated type is not empty.

One then builds further types by construction on already existing types. Recall that this would be inconsistent, if AC were interpreted classically.

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However the axiom is derivable because in the construction of types, the construction of an element of a set type is sufficient to prove its membership. In Bishop-style constructive mathematics, these sets are "completely presented", in that we need to know no more than an object's construction to determine its set membership. It is worth noting that sets construed in this theory are constrained by this realizability criterion.

The program of Reverse Mathematics, instigated by Harvey Friedman , aims to classify theorems according to their equivalence to various set-theoretic principles. The constructive equivalent was begun in a systematic manner by Ishihara and, separately and independently by W. Veldman in The field has since had numerous developments, the interest of which lies mainly in the metamathematical comparison of different branches of constructive mathematics and the logical strengths of various principles endorsed or rejected by the different schools.

This also points to the requirements on computational data for various properties to hold, such as various notions of compactness Diener, It should be noted that arguably the first author dealing with reverse-mathematical ideas was Brouwer, although he certainly would not have seen it as such. The weak counterexamples see Section 2a he introduced were of the form: P implies some non-constructive principle. Though Brouwer may have been aware of the possibility of reversing the implication in many cases, to him non-constructive principles were meaningless and as such the full equivalence result would be of little interest.

The philosophical commitments of constructivist philosophies Section 1b are:. This naturally leads to intuitionistic logic, characterized by the BHK interpretation Section 1c. Omniscience principles, such as the Principle of Excluded Middle PEM , and any mode of argument which validates such principles, are not in general valid under this interpretation, and the classical equivalence between the logical connectives does not hold. Constructive proofs of classical theorems are often enlightening: the computational content of the hypotheses is explicitly seen to produce the conclusion.

While intuitionistic logic places restrictions on inferences, the larger part of classical mathematics or what is classically equivalent can be recovered using only constructive methods Section 4 ; there are often several constructively different versions of the same classical theorem. The programme of constructive reverse mathematics Section 6 connects various principles and theorems such as omniscience principles, versions of the fan theorem, etc.

Furthermore, the use of Brouwerian counterexamples Section 2a often allows the mathematician to distinguish which aspects of classical proof are essentially nonconstructive. Throught the article, several schools of constructivism are outlined. Each is essentially mathematics with intuitionistic logic, philosophical differences notwithstanding. Different schools add different further axioms: for example, Russian constructive recursive mathematics Section 3 is mathematics with intuitionistic logic, the computable partial functions axiom and Markov's principle of unbounded search.

Classical mathematics can be interpreted as Bishop's constructive mathematics, with PEM added. The contrast between classical and constructive mathematicians is clear: in order to obtain the numerical content of a proof, the classical mathematician must be careful at each step of the proof to avoid decisions that cannot be algorithmically made; whereas the constructive mathematician, in adopting intuitionistic logic, has automatically dealt with the computational content carefully enough that an algorithm may be extracted from their proof.

In an age where computers are ubiquitous, the constructivist programme needs even less pragmatic justification, perhaps, than the classical approach. The link between programming and abstract mathematics is stronger than ever, and will only strengthen as new research emerges. For further reading in intuitionism from a philosophical perspective, Dummett's Elements is the prime resource. The Bishop and Bridges book is the cornerstone textbook for the Bishop-style constructivist.

For the finitist version, see Ye , where it is shown that even a strict finitist interpretation allows large tracts of constructive mathematics to be realized; in particular we see application to finite things of Lebesgue integration, and extension of the constructive theory to semi-Riemannian geometry. For an introduction to computable analysis, see, for example, Weihrauch As a technical concept it was developed by Prawitz ; ; , by turning a proof-theoretic validity notion based on ideas by Tait and originally used to prove strong normalization, into a semantical concept.

Dummett provided much philosophical underpinning to this notion see Dummett, The objects which are primarily valid are proofs as representations of arguments. In a secondary sense, single rules can be valid if they lead from valid proofs to valid proofs. In this sense, validity is a global rather than a local notion. It applies to arbitrary derivations over a given atomic system, which defines derivability for atoms. Calling a proof which uses an introduction rule in the last step canonical , it is based on the following three ideas:.

This self-justifying feature is only used for closed proofs, which are considered primary over open ones. Ad 2: Noncanonical proofs are justified by reducing them to canonical ones. Thus reduction procedures detour reductions as used in normalization proofs play a crucial role. This definition again only applies to closed proofs, corresponding to the introduction form property of closed normal derivations in natural deduction see section 1.

Ad 3: Open proofs are justified by considering their closed instances. These closed instances are obtained by replacing their open assumptions with closed proofs of them, and their open variables with closed terms. For example, a proof of B from A is considered valid, if every closed proof, which is obtained by replacing the open assumption A with a closed proof of A , is valid. In this way, open assumptions are considered to be placeholders for closed proofs, for which reason we may speak of a substitutional interpretation of open proofs.

Formally, this definition has to be relativized to the atomic system considered, and to the set of justifications proof reductions considered. Furthermore, proofs are here understood as candidates of valid proofs, which means that the rules from which they are composed are not fixed. They look like proof trees, but their individual steps can have an arbitrary finite number of premisses and can eliminate arbitrary assumptions. Validity with respect to every choice of an atomic system can be viewed as a generalized notion of logical validity.

In fact, if we consider the standard reductions of intuitionistic logic, then all derivations in intuitionistic logic are valid independent of the atomic system considered. This is semantical correctness. We may ask if the converse holds, viz. However, no satisfactory proof of it has been given.

There are considerable doubts concerning the validity of this conjecture for systems that go beyond implicational logic. In any case it will depend on the precise formulation of the notion of validity, in particular on its handling of atomic systems. Philosophically, it shares with Prawitz the three fundamental assumptions of standard proof-theoretic semantics, mentioned in section 2. This can be formalized in a calculus for type assignment, whose statements are of the form t : A.

A proof of t : A in this system can be read as showing that t is a proof of A. First we have proofs of statements of the form t : A. These statements are called judgements , their proofs are called demonstrations. Within such judgements the term t represents a proof of the proposition A. A proof in the latter sense is also called a proof object. When demonstrating a judgement t : A , we demonstrate that t is a proof object for the proposition A. Within this two-layer system the demonstration layer is the layer of argumentation. Unlike proof objects, demonstrations have epistemic significance; their judgements carry assertoric force.

The proof layer is the layer at which meanings are explained: The meaning of a proposition A is explained by telling what counts as a proof object for A. The distinction made between canonical and non-canonical proofs is a distinction at the propositional and not at the judgement al layer. This implies a certain explicitness requirement. This certainty is guaranteed by a demonstration. Mathematically, this two-fold sense of proof develops its real power only when types may themselves depend on terms.

Rather, propositions and proofs come into play only in the context of demonstrations. Being a proposition is expressed by a specific form of judgement, which is established in the same system of demonstration which is used to establish that a proof of a proposition has been achieved. A recent debate deals with the question of whether proof objects have a purely ontological status or whether they codify knowledge, even if they are not epistemic acts themselves.

For variants of proof-theoretic harmony see Francez and Schroeder-Heister a. Proof-theoretic semantics normally focuses on logical constants. This focus is practically never questioned, apparently because it is considered so obvious. The rise of logic programming has widened this perspective. From the proof-theoretic point of view, logic programming is a theory of atomic reasoning with respect to clausal definitions of atoms.

Definitional reflection is an approach to proof-theoretic semantics that takes up this challenge and attempts to build a theory whose range of application goes beyond logical constants. Such clauses can naturally be interpreted as describing introduction rules for atoms. From the point of view of proof-theoretic semantics the following two points are essential:. Interpreting logic programming proof-theoretically motivates an extension of proof-theoretic semantics to arbitrary atoms, which yields a semantics with a much wider realm of applications.

For example, the head of a clause may occur in its body. Well-founded programs are just a particular sort of programs. The use of arbitrary clauses without further requirements in logic programming is a motivation to pursue the same idea in proof-theoretic semantics, admitting just any sort of introduction rules and not just those of a special form, and in particular not necessarily ones which are well-founded. This carries the idea of definitional freedom, which is a cornerstone of logic programming, over to semantics, again widening the realm of application of proof-theoretic semantics.

The idea of considering introduction rules as meaning-giving rules for atoms is closely related to the theory of inductive definitions in its general form, according to which inductive definitions are systems of rules see Aczel, Formally, this approach starts with a list of clauses which is the definition considered. Each clause has the form.

If the definition of A has the form. The elimination rule is called the principle of definitional reflection , as it reflects upon the definition as a whole. If the clausal definition is viewed as an inductive definition, this principle can be viewed as expressing the extremal clause in inductive definitions: Nothing else beyond the clauses given defines A.

Obviously, definitional reflection is a generalized form of the inversion principles discussed. It develops its genuine power in definitional contexts with free variables that go beyond purely propositional reasoning, and in contexts which are not well-founded. An example of a non-wellfounded definition is the definition of an atom R by its own negation:. Some of its authors use a semantical vocabulary and at least implicitly suggest that their topic belongs to proof-theoretic semantics.

Others explicitly deny these connotations, emphasizing that they are interested in a characterization which establishes the logicality of a constant. However, as some of the authors consider their characterization at the same time as a semantics, it is appropriate that we mention some of these approaches here. The most outspoken structuralist with respect to logical constants, who explicitly understands himself as such, is Koslow.

Koslow develops a structural theory in the precise metamathematical sense, which does not specify the domain of objects in any way beyond the axioms given. If a language or any other domain of objects equipped with an implication relation is given, the structural approach can be used to single out logical compounds by checking their implicational properties. In his early papers on the foundations of logic, Popper a; b gives inferential characterizations of logical constants in proof-theoretic terms. He uses a calculus of sequents and characterizes logical constants by certain derivability conditions of such sequents.

Although his presentation is not free from conceptual imprecision and errors, he was the first to consider the sequent-style inferential behaviour of logical constants to characterize them. However, against his own opinion, his work can better be understood as an attempt to define the logicality of constants and to structurally characterize them, than as a proof-theoretic semantics in the genuine sense.

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He nevertheless anticipated many ideas now common in proof-theoretic semantics, such as the characterization of logical constants by means of certain minimality or maximality conditions with respect to introduction or elimination rules. Important contributions to the logicality debate that characterize logical constants inferentially in terms of sequent calculus rules are those by Kneale and Hacking He understands logical constants as being characterized by certain double-line rules for sequents which can be read in both directions.

For example, conjunction and disjunction are in classical logic, with multiple-formulae succedents characterized by the double-line rules. He explicitly considers his work as a contribution to the logicality debate and not to any conception of proof-theoretic semantics. Sambin et al. The double-line rules for conjunction and disjunction are read as implicit definitions of these constants, which by some procedure can be turned into the explicit sequent-style rules we are used to. So Sambin et al. There are several other approaches to a uniform proof-theoretic characterization of logical constants, all of whom at least touch upon issues of proof-theoretic semantics.

Since the rise of linear and, more generally, substructural logics Di Cosmo and Miller, ; Restall, there are various approaches dealing with logics that differ with respect to restrictions on their structural rules. A recent movement away from singling out a particular logic as the true one towards a more pluralist stance see, e. There is a considerable literature on category theory in relation to proof theory, and, following seminal work by Lawvere, Lambek and others see Lambek and Scott, , and the references therein , category itself can be viewed as a kind of abstract proof theory.

For intuitionistic systems, proof-theoretic semantics in categorial form comes probably closest to what denotational semantics is in the classical case. He has not only advanced the application of categorial methods in proofs theory e.

Most important for categorial logic in relation to proof-theoretic semantics is that in categorial logic, arrows always come together with an identity relation, which in proof-theory corresponds to the identity of proofs. Another feature of categorial proof-theory is that it is inherently hypothetical in character, which means that it starts from hypothetical entities. It this way it overcomes a paradigm of standard, in particular validity-based, proof-theoretic semantics see section 3. For logical constants and their logicality see the entry on logical constants.

For categorial approaches see the entry on category theory. Most approaches to proof-theoretic semantics consider introduction rules as basic, meaning giving, or self-justifying, whereas the elimination inferences are justified as valid with respect to the given introduction rules. This conception has at least three roots: The first is a verificationist theory of meaning according to which the assertibility conditions of a sentence constitute its meaning.

The second is the idea that we must distinguish between what gives the meaning and what are the consequences of this meaning, as not all inferential knowledge can consist of applications of definitions.

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The third one is the primacy of assertion over other speech acts such as assuming or denying, which is implicit in all approaches considered so far. One might investigate how far one gets by considering elimination rules rather than introduction rules as a basis of proof-theoretic semantics. Some ideas towards a proof-theoretic semantics based on elimination rather than introduction rules have been sketched by Dummett , Ch. A more precise definition of validity based on elimination inferences is due to Prawitz ; ; see also Schroeder-Heister Its essential idea is that a closed proof is considered valid, if the result of applying an elimination rule to its end formula is a valid proof or reduces to one.

This conception keeps two of the three basic ingredients of Prawitz-style proof-theoretic semantics see section 2. Only the canonicity of proofs ending with introductions is changed into the canonicity of proofs ending with eliminations. Standard proof-theoretic semantics is assertion-centred in that assertibility conditions determine the meaning of logical constants. By that one understands a one-place primitive operator of negation, which cannot be, or at least is not, reduced to implying absurdity. It is not classical negation either. It rather obeys rules which dualize the usual rules for the logical constants.

Essentially, the denial rules for an operator correspond to the assertion rules for the dual operator. The main focus has been on his systems later called N3 and N4 which differ with respect to the treatment of contradiction N4 is N3 without ex contradictione quodlibet. Using denial any approach to proof-theoretic semantics can be dualized by just exchanging assertion and denial and turning from logical constants to their duals.

It can be understood as applying a Popperian view to proof-theoretic semantics. Another approach would be to not just dualize assertion-centered proof-theoretic semantics in favour of a denial-centered refutation-theoretic semantics, but to see the relation between rules for assertion and for denial as governed by an inversion principle or principle of definitional reflection of its own. Whereas in standard proof-theoretic semantics, inversion principles control the relationship between assertions and assumptions or consequences , such a principle would now govern the relationship between assertion and denial.

Given certain defining conditions of A , it would say that the denial of every defining condition of A leads to the denial of A itself. For conjunction and disjunction it leads to the common pairs of assertion and denial rules. This idea can easily be generalized to definitional reflection, yielding a reasoning system in which assertion and denial are intertwined. It has parallels to the deductive relations between the forms of judgement studied in the traditional square of opposition Schroeder-Heister, a; Zeilberger, Dauben, see Dauben, , p.

In the midst of the epic tale, Bishop inserts58 a vitriolic paragraph against Robinson-style infinitesimal calculus. The remarks he added to the galley proofs 5 8 of Bishop, indicate that he was already aware of infinitesimals in the classroom, at least two years prior to writing his book review Bishop, Meanwhile, K. Could it be because Bishop was not interested in field studies? Could his opposition to the non-standard approach have been based, not on any of its perceived shortcomings in imparting knowledge of rates of change, derivatives, areas, integrals, etc. That a certain amount of foundational material should be expected to be taken on trust in a first calculus course, is a feature that infinitesimalbased calculus shares with the standard approach.

Very rare indeed is the first calculus course that deals with equivalence classes of Cauchy sequences or other constructions of the real number field. In the decade of his life that Bishop seems to have become interested in calculus methodologies, he is not known to have written critical book reviews of any methodologies other than the hyperreal one, which— curious coincidence! His fear seems to have been that, unless he personally intervened to prevent it, the continuum would turn out to be discrete.

Could his interest have been foundationally motivated, as indeed it was. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line. His presentation suggests that Bishop himself favors the classical real line. Yet Bishop states clearly in his Manifesto. Bishop, that from his constructivist viewpoint, it is LEM, and not the axiom of choice, that is the source of non-constructivity.

To summarize, Bishop seeks to convince his classical reader of the idealistic. The similar foundational status of the classical continuum, on the one hand, and the hyperreal number, on the other, constitutes an astonishing point of convergence between Bishop and Robinson. They are mere conveniences for generating proofs, whose intuitive content will certainly [ emphasis added— authors] escape the students Bishop, , p. The lack of numerical meaning, to Bishop, corresponds to a reliance on LPO or more generally, the law of excluded middle see glossary in Subsection VI. Bishop expresses the sentiment that the notorious ; definition of limit is common sense Bishop, , p.

Dauben speculates that there may be a very good reason for their disbelief, namely,. The vast education literature on the student reception of quantifiers see a useful bibliography in Weller et al. However, our main goal here is not to gauge the difficulty ofWeierstrassian definition of the limit concept based on epsilontics.

In traditional LEM-dominated mathematics courses, students become accustomed to going through the numerically meaningless motions of idealistic mathematics see glossary in Section VI. De-basing in constructivism. A number of familiar terms in the mathematical lexicon are displaced from their base, or basic, meaning when they are rendered constructive.

We will analyze several such phenomena in this section. Feferman writes that Bishop deals with. Continuity differs from uniform continuity in at least the following three ways: 1. This is possible because ordinary continuity is a local pointwise property, whereas to define uniform continuity, one needs to work with pairs of points.

Most textbooks adopt the standardWeierstrassian approach using epsilontics. When it comes to continuity and uniform continuity, both definitions involve alternations of quantifiers. However, pedagogically speaking, one can present the definition of ordinary continuity with only three quantifiers, once the point has been fixed.

There is no such device that would permit a definition of uniform continuity with fewer than four quantifiers. Such a definition is almost inaccessible for an average undergraduate. It emerges that due to the intrinsic difficulty of the subject matter, the semester was spent explaining ; for linear functions.

The author concludes that [ a] lthough the ideas concerning quadratics were pursued outside of class with only a limited number of students, it is important to. Many teachers, based on classroom experience, feel that uniform continuity is a concept that is even more difficult for students to grasp than ordinary continuity.

In fact the author stresses the equal hypothesis interpretations of classical theorems, which is misleading since the hypotheses are for the greater part only verbally equal van Rootselaar, What van Rootselaar is referring to is the fact that some of the theorems are worded in such a way as to sound identical to the classical ones, but, of course, the definitions have been tampered with, e.

What does the term finite mean? In this section we will comment on the constructive distinction between finite and subfinite sets, in the context of a result relating the axiom of choice and LEM. There is an intriguing point here that Bishop actually claimed that he has no quarrel with the axiom of choice. In his Constructive manifesto, eight years prior to the publication of Diaconescu-Goodman-Myhill, he wrote:. His remark has apparently been a source of discomfort for some of his followers. The theorem is proved by means of such an application of the axiom of choice see Subsection II.

Note that the law of excluded middle is a constructive consequence of much more elementary statements than that of the full axiom of choice. We will examine the question of such compatibility in this subsection, in the context of the extreme value theorem. Note that the extreme value theorem does not hold constructively. Namely, the existence of a maximum of a continuous function f on [ 0 ; 1 ] , in the sense of the constructive formula. Indeed, let a be any real number such that. Next, define f on [ 0 ; 1 ] by. A weaker principle is the LPO limited principle of omniscience.

The LPO is equivalent to the law of trichotomy:. This property is false intuitionistically. After discussing real numbers. See D. Bridges for details; a useful summary may be found in Taylor The extreme value theorem illustrates well the fact that a verificational interpretation of the quantifiers in Bishopian mathematics necessarily results in a clash with classical mathematics; it is merely a matter of tactical emphasis that the clash is minimized in Bishopian constructivism.

Are there two constructivisms? Hellman has distinguished between liberal and radical versions of constructivism in Hellman, The dichotomy75 was picked up both by by H. Billinge and by E. Davies Rather than borrowing terms from political science, we will exploit terms that are somewhat more selfexplanatory in a mathematical context.

What are the goals of constructive mathematics? Bishop in his Crisis. It seems reasonable to assume that a mainstream mathematician can relate favorably to the methodological approach that seeks greater numerical meaning, as enunciated by Bishop in Bishop, Such an approach could be described as numerical constructivism, or rather, numerically meaningful constructivism. A methodological approach, complementing other approaches, could be a companion to classical mathematics, while at the same time recognizing the coherence of post-LEM numerical meaning see Subsection V.

On the other hand, when the emphasis shifts to the extirpation of the law of excluded middle from the mathematical toolkit, whether there is numerical benefit or loss76 in such a quest, one arrives at an approach that could be termed anti-LEM constructivism, conceived of as an alternative to classical mathematics, and viewing LEM as a variety of mathematical phlogiston.

## Constructive Mathematics | Internet Encyclopedia of Philosophy

In short, constructive mathematics is posited, not as a companion, but rather as a replacement for and alternative to, classical mathematics. Should the gentle reader feel that our description of an anti-LEM species of constructivism is exaggerated, she should ponder H. Elsewhere, Billinge has claimed that E. Bishop is not a radical constructivist: We can safely say that Bishop was a liberal constructivist Billinge, , p.

To elaborate further on the difference between the two approaches to constructivism discussed in Subsection V. The infinitesimal definition of the derivative was already envisioned by the founders of the calculus, and was justified by A. Robinson see Appendix A. Heyting is able to appreciate such an accomplishment see Subsection VII.

An infinitesimal definition certainly leads to computationally meaningful formulas for the derivatives of the standard functions, even though classical. To emphasize the difference between the two types of numerical meaning, we introduce the following distinction.

We will refer to the content of mathematics that is fully constructive from the bottom i. Pre-LEM numerical meaning is, of course, exemplified by Foundations of constructive analysis Bishop, Example V. The volume calculations in Archimedes appear to rely on proofs by contradiction. Thus, a contradiction is derived from the assumption that the volume x of the unit sphere is smaller than two-thirds of the volume of the circumscribed cylinder, and similarly for greater than. The contradiction proves that the volume is exactly two-thirds.

The proof appears to depend on the trichotomy for real numbers:. We start with the trichotomy, eliminate the possibilities on the right and on the left, to conclude that the possibility in the middle equality is the correct one. Beeson Indeed, the theoretical entity called the infinitesimal. The equations are thereby placed in the context of a variational problem. Yet the inequality itself possesses indisputable numerical meaning.

The inequality relates a pair of metric invariants of a Riemannian 2-torus T. The first is the least length, denoted S y s , of a loop on T that cannot be contracted to a point on T. The second invariant is the total area, denoted A r e a , of T. The inequality, S y s 2. Burago, who acquainted me with the results of Loewner, Pu, and Besicovitch. These attracted me by the topological purity of their underlying assumptions, and I was naturally tempted to prove similar inequalities in a more general topological framework Gromov, , p.

Our theory of infinitesimal variables [. Connes claims that his infinitesimals provide a computable answer to a probability problem outlined earlier in his text. Avigad of ergodic theory can be viewed as an analysis of post-LEM numerical meaning, in the context of ergodic theory. More generally, the technique of proof mining seeks to extract numerical content from non-constructive proofs, using logical tools, see U. Kohlenbach and P. Oliva That is, one may use all the formal machinery, in particular, nonconstructive but formally valid existence statements such as the Bolzano-Weierstrass theorem , to prove, formally, real propositions, i.

Stolzenberg concludes on the following optimistic note:. Now the constructivisation of classical results may succeed at any rate modulo strengthening the hypotheses or weakening the conclusions, see Subsection IV. But Bishop apparently believed that a Robinson infinitesimal resists constructivisation. If both forms of constructivism discussed in Subsection V. To answer the question, we will start by pointing out that the burden of explanation is upon the constructivists on at least the following two counts: 1.

How is it that one of their own, A. Bishop sometimes seems to identify his notion of numerical meaning, with. One can adopt the methodological goals of a search for greater numerical meaning,. Int merely being polite in conceding a point to Class? The fervor of Bishopian constructivism. Pourciau and published in the Studies in History and Philosophy of Science:. The faith that sustains the classical world view emanates from one belief more than any other: that mathematical assertions are true or false, independently of our knowing which. Every conflict. This is not a belief so much as a hidden cause: it creates the world view— making, for example, proofs by contradiction appear self-evidently correct— but remains transparently in the background, unseen and unquestioned Pourciau, , p.

Such an unquestioning stance naturally leads to calcification, as Pourciau continues:. Once created, the classical world view is sustained and calcified. Talk about sets, functions, real numbers, theorems and so on is taken by the classical mathematician as being literally about mathematical objects that exist independently of us, even though such talk, classically interpreted, has the appearance, and nothing more, of being meaningful Pourciau, , p.

The reader recognizes the pet constructivist term, meaning, already analyzed in Subsection III. In a literal sense, it is shattering. Avigad, while agreeing that [ w] e do not need fairy tales about numbers and triangles prancing about in the realm of the abstracta[,] notes that [ p] roof-theoretic analysis [.

Giorello as follows. To elaborate, we argue as follows. The contradiction proves that Pourciau does in fact think of LEM as mathematical phlogiston. However, we include it here for the benefit of a classically minded reader. Kuhnian revolution. Based on the loss of such assertions in an intuitionistic framework, Pourciau concludes that Intuitionism possessed a potential of developing into a unique mathematical Kuhnian revolution.

Pourciau writes that. But from outside the classical paradigm, [. He argues that both Crowe and Dauben. Pourciau, , p. Pourciau views such cumulativity as a necessary consequence of the classical i. LEM-circumscribed paradigm, suggesting that a Kuhnian revolution in mathematics would in fact be impossible, without first extirpating LEM.

Consider, for example, the largely successful assault on infinitesimals in the aftermath of the rise of Weierstrassian epsilontics. A similar observation applies to infinitesimals themselves see Example V. Thus, Weierstrassian epsilontics constitute a,. Indeed, Robinson , p. Cleave , Cutland et al. This ongoing project appears to be a striking realisation of a reconstruction project enunciated by I.

Grattan-Guinness, in the name of Freudenthal : it is mere feedback-style ahistory to read Cauchy and contemporaries such as Bernard Bolzano as if they had read Weierstrass already. On the contrary, their own pre-Weierstrassian muddles need historical reconstruction Grattan-Guinness, , p. The Brouwer— Hilbert debate captured the popular mathematical imagination in the s. Burgess discusses the debate briefly in his treatment of nominalism in Burgess, , p. Part of the attraction stems from a detailed lexicon developed by Bishop so as to challenge received classical views on the nature of mathematics.

A constructive lexicon was a sine qua non of his success. It may be helpful to provide a summary of such terms for easy reference, arranged alphabetically, as follows. Debasement of meaning is the cardinal sin of the classical opposition, from Cantor to Keisler, 95 committed with LEM see below. Fundamentalist excluded thirdist is a term that refers to a classicallytrained mathematician who has not yet become sensitized to implicit use of the law of excluded middle i.

Idealistic mathematics is the output of Platonist mathematical sensibilities, abetted by a metaphysical faith in LEM see below , and characterized by the presence of merely a peculiar pragmatic content see below. Integer is the revealed source of all meaning see below , posited as an alternative foundation displacing both formal logic, axiomatic set theory, and recursive function theory. The integers wondrously escape97 the vigilant scrutiny of a constructivist intelligence determined to uproot and nip in the bud each and every Platonist fancy of a concept external to the mathematical mind.

Pourciau in his Education Pourciau, appears to interpret it as an indictment of the ethics of the classical opposition. Yet in his. Bishop, , p. Bishop describes integrity as the opposite of a syndrome he colorfully refers to as schizophrenia, characterized by a number of ills, including a a rejection of common sense in favor of formalism, b the debasement of meaning see above , c as well as by a list of other ills— but excluding dishonesty.

Now the root of integr-ity is identical with that of integer see above , the Bishopian ultimate foundation of analysis. Brouwer sought to incorporate a theory of the continuum as part of intuitionistic mathematics, by means of his free choice sequences. Law of excluded middle LEM is the main source of the non-constructivities of classical mathematics.

Numerical meaning is the content of a theorem admitting a proof based on intuitionistic logic, and expressing computationally meaningful facts about the integers. It connotes an alleged lack of empirical validity of classical mathematics, when classical results are merely inference tickets Billinge, , p. Realistic mathematics. There are two main narratives of the Intuitionist insurrection, one anti-realist and one realist. The issue is discussed in Subsections VI. Since the sensory perceptions of the human body are physics-and chemistry-bound, a claim of such trans-universe invariance amounts to the positing of a disembodied nature of the natural number system transcending the physics and the chemistry.

The anti-realist narrative, mainly following Michael Dummett , traces the original sin of classical mathematics with LEM, all the way back to Aristotle. Generally speaking, it is this narrative that seems to be favored by a number of philosophers of mathematics. The latter requirement, in the context of mathematics, is a restatement of the intuitionistic principle that truth is tantamount to verifiability.

What Dummett proceeds to say at this point, reveals the nature of his interest: but intuitionism represents the only sustained attempt by the opponents of a realist view to work out a coherent embodiment of their philosophical beliefs [ emphasis added— authors]. What interests Dummett here is the fight against the realist view. What endears intuitionists to him, is the fact that they have succeeded where the phenomenalists have not: Phenomenalists might have attained a greater success if they had made a remotely comparable effort to show in detail what conse-. We hereby explicitly sidestep the debate opposing the realist as opposed to the super-realist, see W.

Tait position and the anti-realist position. While Dummett chooses to pin the opposition to intuitionism, to a belief in an interpretation of mathematical statements as referring to an independently existing and objective reality Dummett, , p. Avigad memorably retorts as follows: We do not need fairy tales about numbers and triangles prancing about in the realm of the abstracta Avigad, Turning now to the realist narrative of the intuitionist insurrection, we note that such a narrative appears to be more consistent with what Bishop himself actually wrote.

In his foundational essay, Bishop expresses his position as follows: As pure mathematicians, we must decide whether we are playing a game, or whether our theorems describe an external reality. The right answer, to Bishop, is that they do describe an external reality. In Bishop, , p. Kopell and G. Stolzenberg, close associates of Bishop, published a threepage. Similar views have been expressed by D.

Bridges, see e. Bridges, , and, as we argue in Subsection VII. The Interview with a constructive mathematician was published by leading constructivist F. Richman in In the Interview, Richman seems to reject any alternative to an anti-LEM constructivism, in his very first comment: the constructive mathematician dismisses classical mathematics as an exercise in formal logic, much like investigating the consequences of large cardinal axioms Richman, , p. It should be noted that fellow constructivist D.

This is typical of generalizations: the notion of a normal subgroup is equivalent to that of a subgroup in the context of the commutative law Richman, , p. Yet he quickly retreats to the safety of anti-LEM, observing that the law of excluded middle obliterates [ emphasis added— authors] the notion of positive content [. While little evidence is offered for such a counter-intuitive no pun intended assertion, it sets out the following hope. If a numerical constructivist can, after many years, be led to abandon numerical constructivism and switch to anti-LEM; then also, at some future time, anti-LEM constructivists can perhaps be persuaded, by force of overwhelming evidence, to abandon anti-LEM and switch to numerical constructivism of the companion variety.

The nature of such evidence will be discussed below. Richman writes that, even in the countable case, the centerpiece of the subject, the classification theorem for countable torsion abelian groups, cannot even be stated without ordinal numbers Richman, To Richman, the non-constructivity of the arguments appears to be compounded by the non-constructive formulation of the very statements of the results of torsion abelian group theory.

Richman apparently perceived a lack of numerical meaning of even the post-LEM kind, as discussed in Subsection V. What is the nature of the evidence in favor of a numerical constructivism of a companion variety? A mathematician working in the tradition of Archimedes, Leibniz, and Cauchy owes at least a residual allegiance to the idea that the most important mathematical problems are those coming from physics, engineering, and science more generally. Is constructive mathematics part of classical mathematics?

Tait argues that, unlike intuitionism, constructive mathematics is part of classical mathematics. In this sense, intuitionism is not a rival, but an offspring, of classical mathematics. To quote J. Avigad, [ t] he syntactic, axiomatic standpoint has enabled us to fashion formal representations of various foundational stances, and we now have informative descriptions of the types of reasoning that are justified on finitist, predicative, constructive, intuitionistic, structuralist, and classical grounds Avigad, This idea, as applied to intuitionism, was expressed by J.

Gray in the following terms: Intuitionistic logics were developed incorporating the logical strictures of Brouwer; constructivist mathematics still enjoys a certain vogue. But these are somehow contained within the larger framework [ emphasis added— authors] of modern mathematics. Gray, , p. In a similar vein, no less an authority than M. Heidegger wrote almost simultaneously with Kolmogorov quoted in Subsection II. Lest one should doubt whether Heidegger meant for his comments to apply to mathematics, he continues:.

Lest one should doubt whether he had Brouwer in mind, Heidegger continues: In the controversy between the formalists and the intuitionists,. What Heidegger appears to be saying is that if we take the supposed objects to be sets or integers, the issue becomes whether a primary way of. Yet, we will show that Heyting unequivocally sides with the view of intuitionism as a companion, rather than. The first part of the book is written in the form of a dialog among representatives of some of the main schools of thought in the foundations of mathematics.

The protagonist named. Class makes the following point: Intuitionism should be studied as part of mathematics. In mathematics, we study consequences of given hypotheses. The hypotheses assumed by intuitionists may in fact be interesting, but they have no right to a monopoly Heyting, , p. In other words, emptying our logical toolkit of the law of excluded middle is one possible foundational framework among others.

In this connection, S. One can legitimately pose the question whether Int is merely being polite in his response to Class. He does this, obviously, not as a way of disparaging intuitionism, but as a way of underscoring its intrinsic value as an intellectual pursuit, as distinct from a scientific pursuit. Maddy claims that Heyting [. Int replies as follows: As to the mutilation of mathematics of which you accuse me, it must be taken as an inevitable consequence of our standpoint. It can also be seen as an excision of noxious ornaments, [.

His comments show a clear recognition of the potency of the mutilation challenge, as well as a necessity to compensate for the damages. Heyting was able to, so to speak, rise above differences of Class and Int. What would Heyting have thought of a rigorous justification of infinitesimals in the framework of classical logic? Already in , Heyting allowed his protagonist Form to cite a reference by A. Robinson in a comment dealing with the use of metamathematics for the deduction of mathematical results Heyting, , p. Shapiro , in a retort to N. Here Heyting semantics refers to the constructive interpretation of the quantifiers discussed in Subsection II.

Shapiro writes as follows: Let x be any predicate that applies to natural numbers. It is a routine theorem of classical arithmetic that. Shapiro, , p. Shapiro continues: Under Heyting semantics, this proof amounts to a thesis that there is a computable function that decides whether holds. So under Heyting semantics [. Namely, the constructive existence of y necessarily entails being able to find a computational procedure capable of identifying such a y as a function of x , and hence a decision procedure for the predicate.

Does this prove that classical mathematics and intuitionism cannot get along, as Shapiro puts it? In fact, the philosophical exchange between Tennant and Shapiro has an uncanny parallel— a quarter century earlier, in the exchange between Bishop and Timothy G. McCarthy, writing for Math Reviews It is not plausible to suppose that A is classically true false if and only if [ the proposition] LPO! A [ respectively,] LPO! Such seems to be the position adopted by the mature Heyting, as well, as we analyze in the next section. Robinson Heijting, are fascinating and worth reproducing in some detail: 1.

The two subjects become more and more intertwined. For a discussion of the transfer principle of non-standard analysis, see Appendix A. Heyting does not stop there, and makes the following additional points: 1. This is the case in differential geometry and in many parts of applied mathematics such as hydrodynamics and electricity theory, where physicists use infinitesimals without a twinge of conscience. You showed how these theories can be made precise by your method. This mysterious object became lucid in the light of non-archimedean analysis [ emphasis added— authors]. On the subject of whether there is meaning after model theory, Heyting has this to say: 1.

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Once you had shown by the paradigm of the calculus how it can be used, many other applications were found by yourself as well as by many other mathematicians. In a sense your work can be considered as a return from this abstraction to concrete applications. The general non-constructive theory of non-standard models links its applications together into a harmoni[ ous] whole. He recognizes that, while the model may be non-constructive, its applications may indeed be concrete. Such a thesis is consonant with what we called numerical constructivism in Subsection V.

Such an awareness is therefore not at odds with a recognition of the coherence of post-LEM numerical meaning. It is at odds, however, with an anti-LEM radical constructivist stance, as enunciated by E. Constructivism, physics, and the real world. In this section we deal with challenges to anti-LEM constructivism stemming from natural science. Bishop Bishop, , p. Here is a question that has bothered me ever since the first time I read Bishop.

In what sense can your [ constructive] real line be used as a model for either space or time in physics? Richman, , p. It needs to be understood what Richman meant by this cryptic remark. A classical mathematician believes that the circle can be decomposed as the disjoint union of the lower halfcircle with one endpoint included, and the other excluded and its antipodal image upper halfcircle.

## Troelstra, A. S. (Anne Sjerp)

Meanwhile, the constructivist believes that it is impossible to decompose the circle as the disjoint union of a pair of antipodal sets. Colyvan , or G. In fact, Bishop himself was challenged on the relationship to physics by G. Mackey, in the following terms: Consider the foundational question in physics: what is the real mathematics that the physicists are doing?

Rather, it involves the relation of the results to the real world [ emphasis added— authors]. However, what Bishop is emphasizing in his essay is a notion of meaning. The kind of unabashedly LEM-dominated mathematics that a physicist practices is, as Bishop seems to admit matter-of-factly, successful in relating to the real world, and therefore apparently meaningful.

In his Constructivist manifesto, Bishop wrote that Weyl, a great mathematician who in practice suppressed his constructivist convictions, expressed the opinion that idealistic [ i. Bishop does not elaborate as to why he feels H. The challenge to constructivism stemming from the Quine-Putnam indispesability thesis has been extensively pursued by G.

Hellman expresses an appreciation of the foundational significance of constructivism in the following terms: A turning point [. Hellman a, p. However, one can recognize the coherence of the intuitionist critique of the foundations of mathematics, while rejecting the intellectual underpinnings of its insurrectional narrative, in its mutually contradictory realist and anti-realist versions, as discussed in Subsection VI. In the context of general relativity theory, Hellman argues that, very likely, the Hawking— Penrose singularity theorems for general relativistic spacetimes are essentially non-constructive Hellman, , see also Billinge, , see Subsection VIII.

The philosopher of mathematics M. The battle imagery is typical of the anti-realist type of insurrectional narrative, already discussed in Subsection VI. Meanwhile, the philosopher of mathematics G. Figure VIII. More generally, variational problems tend to be resistant to efforts at constructivizing, a point apparently acknowledged by Beeson when he writes that Calculus of variations is a vast and important field which lies right on the frontier between constructive and non-constructive mathematics Beeson, , p.

The problem is that the extreme value theorem is not available in the absence of the law of excluded middle. The extreme value theorem is at the foundation of the calculus of variations. As concrete examples, one could mention general existence results for geodesics, minimal surfaces, constant curvature mean surfaces, and therefore soap films and soap bubbles.

Question VIII. Classically, one has a description of the resulting object in terms of a minimizing possibly non-unique closed geodesic. Providing a constructive description is not immediate, as it depends on general results of the calculus of variations, as already described above. What Novikov makes excruciatingly clear is the disastrous effect of such a divorce on mathematics itself. A broad range of subjects illustrating the symbiotic relationship between the two fields can be found in Grattan-Guinness Grattan-Guinness emphasizes the role of analogies drawn from other theories,.

What is worse, the gap is growing wider, according to Novikov. Traditional mathematical foundations, whether classical or intuitionist, are proving to be inadequate for the job of accounting for the progress in physics. Accordingly, Novikov is critical of set-theoretic foundations, going as far as criticizing Kolmogorov himself, for systematic efforts to introduce a set-theoretic approach in secondary education. The Hawking— Penrose theorem in relativity theory has been the subject of something of a controversy in its own right, see G.

Hellman , H. Billinge , and E.