## 31 May 2019

### Heyting (5.1.1) Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Section 5.1.1, “Mathématique sans négation de Griss”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Heyting, entry directory]

[Heyting’s Les fondements des mathématiques, entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are possibly mistaken and probably inelegantly articulated. Also, my abilities with French are insufficient to translate reliably, so please again rely upon the quotations rather than my summarizations. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

Arend Heyting

Les fondements des mathématiques.

Intuitionnisme.

Théorie de la démonstration.

Première section:
Intuitionnisme

5.
L'intuitionnisme brouwérien

5.1
L'intuition mathématique

5.1.1
Mathématique sans négation de Griss

Brief summary:

(5.1.1.1) Griss devised a negationless intuitionistic mathematics. He thought there should be nothing like negation in it, because intuitive methods will not allow us to make a demonstration based on the falsity of an assumption, as we cannot clearly conceive a falsity in the first place. And we can only clearly conceive a property after constructing a mathematical entity that possesses that property, so we cannot introduce an empty species. For real numbers, Griss needs to avoid the negational (and exclusionary) notion of inequality, but he still needs to be able to say that one natural number is not identical to the other ones (and that two numbers are identical when they cannot be unequal). Instead of conceiving this in terms of not being equal (which cannot enter into intuitionistic thinking, because we can only conceive of positive properties), Griss (according to Heyting) recasts this inequality relation (≠) as a distance relation (⧣). On this basis, we can understand two (initially unidentified) numbers as being equal when they share the same distances to the same other numbers. (2 for instance is one away from 1 and one away from 3, and two away from 4, etc. If both a and b each likewise are one away from 1, one away from 3, etc., then they are equal. Here we are avoiding the non-intuitionistic notion of them not being unequal to one another by having them both positively sharing the same relational properties to all the other numbers in the set.) Heyting writes, in rough translation: In the theory of real numbers, the relation ≠, being negative, does not intervene. There is only the relation a = b and the distance relation ab (see “numerical calculation” below). The theorem “if ab is impossible, then a = b” is replaced by the following: “if a is distant from every number c which is distant from b, a = b”. (Heyting p.14). (5.1.1.2) But if we completely eliminate negation, then we cannot have a propositional logic in the normal sense. Nonetheless, both Griss and Destouches-Février attempt to construct such a propositional logic. But instead of being a logic of predicates, Griss here constructs a logic of classes that is unlike the intuitionistic logic of classes in that for Griss, two classes can intersect only if they have at least one element in common. (5.1.1.4) Van Dantzig outlined a formal system of affirmative mathematics. (5.1.1.5) Brouwer supports the role of negation by constructing theories that require it, and he articulated his ideas about the relationship of mathematics to experience, language, and wisdom.

Contents

5.1.1.1

[Griss’ Negationless Mathematics; His Elimination of Negational Structures, in Particular His Use of Distance (⧣) in Place of Inequality (≠)]

5.1.1.2

[Griss’ and Destouches-Février’s Negationless Intuitionistic Logic]

5.1.1.3

[Destouches-Février devises a logic of constructions, on the basis of which she defines a logic of problems and a propositional logic.]

5.1.1.4

[Van Dantzig’s Affirmative Mathematics]

5.1.1.5

[Brouwer and Negation]

Bibliography

Summary

5.1.1.1

[Griss’ Negationless Mathematics; His Elimination of Negational Structures, in Particular His Use of Distance (⧣) in Place of Inequality (≠)]

[Griss devised a negationless intuitionistic mathematics. He thought there should be nothing like negation in it, because intuitive methods will not allow us to make a demonstration based on the falsity of an assumption, as we cannot clearly conceive a falsity in the first place. And we can only clearly conceive a property after constructing a mathematical entity that possesses that property, so we cannot introduce an empty species. For real numbers, Griss needs to avoid the negational (and exclusionary) notion of inequality, but he still needs to be able to say that one natural number is not identical to the other ones (and that two numbers are identical when they cannot be unequal). Instead of conceiving this in terms of not being equal (which cannot enter into intuitionistic thinking, because we can only conceive of positive properties), Griss (according to Heyting) recasts this inequality relation (≠) as a distance relation (⧣). On this basis, we can understand two (initially unidentified) numbers as being equal when they share the same distances to the same other numbers. (2 for instance is one away from 1 and one away from 3, and two away from 4, etc. If both a and b each likewise are one away from 1, one away from 3, etc., then they are equal. Here we are avoiding the non-intuitionistic notion of them not being unequal to one another by having them both positively sharing the same relational properties to all the other numbers in the set.) Heyting writes, in rough translation: In the theory of real numbers, the relation ≠, being negative, does not intervene. There is only the relation a = b and the distance relation ab (see “numerical calculation” below). The theorem “if ab is impossible, then a = b” is replaced by the following: “if a is distant from every number c which is distant from b, a = b. (Heyting p.14).]

[In rough form: Griss’ Negationless Mathematics. Griss [1,2] expressed expressed doubts about the legitimacy of the use of negation in intuitionistic mathematics. According to him, one can never demonstrate by intuitive methods the falsity of an assumption, because if it is false one cannot conceive it clearly. Likewise, one can have a clear idea of a property only after the construction of a mathematical entity that possesses this property. Thus we cannot introduce an empty species. GRISS began the construction of a negationless intuitionistic mathematics. In the theory of real numbers, the relation ≠, being negative, does not intervene. There is only the relation a = b and the distance relation ab (see “numerical calculation” below). The theorem “if ab is impossible, then a = b” is replaced by the following: “if a is distant from every number c which is distant from b, a = b”. The demonstration of this theorem for real numbers creates difficulties; GRISS only succeeds by using BROUWER’s theorem on bounded expansions [déploiements bornés] (see “Bon ordre” below). Mme. DEQUOY studied projective geometry as understood without negation [1,2].] [With explanation: G.F.C. Griss formulated a negationless mathematics based on L.E.J. Brouwer’s intuitionism. Van Stigt explains in “Brouwer’s Intuitionist Programme” section 1.5.1.1 that “In the generation of the fundamental “mathematical entities,” such as the natural numbers and the Brouwer set or spread and its elements, there is no place nor immediate need for negation” (Van Stigt, section 1.5.1.1, p.14, boldface mine). But, negation does factor in when dividing a set into subsets, where to be in a set means it is impossible to be in the complementary set (Van Stigt, section 1.5.1.1, p.14; Griss “Negationless Intuitionistic Mathematics, II” section 1.0.2, p.457). Griss, we saw, was seeking a way to make such a designation without using logical negation or the law of excluded middle. Brouwer’s formulation uses the idea that an item is either in a particular subset or it is not. Such a move normally is founded in classical logic using excluded middle, and it need not be further proved. Griss’ formulation instead says that something is either in one subset or in its complement, and he needed to formulate a proof for this disjunction, which runs parallel to Brouwer’s formulation (see Griss’s “Negationless Intuitionistic Mathematics, I” sections 0 and 1.2 (and especially section 1.2.3 for the parallelism between his and Brouwer’s formulations); and see “Negationless Intuitionistic Mathematics, II” section 1.0). Heyting then writes, roughly, that according to Griss, one can never demonstrate by intuitive methods the falsity of an assumption, because if it is false one cannot conceive it clearly. As Griss writes in “Negationless Intuitionistic Mathematics, I,” section 0.2:

On philosophic grounds I think the use of the negation in intuitionistic mathematics has to be rejected. Proving that something is not right, i.e. proving the incorrectness of a supposition, is no intuitive method. For one cannot have a clear conception of a supposition that eventually proves to be a mistake. Only construction without the use of negation has some sense in intuitionistic mathematics.

(Griss, “Negationless Intuitionistic Mathematics, I,” section 0.2, p.1127)

Heyting next says, roughly: “Likewise, one can have a clear idea of a property only after the construction of a mathematical entity that possesses this property. Thus we cannot introduce an empty species.” Griss notes and emphasizes this in section 1.0.3 of “Negationless Intuitionistic Mathematics, II”

In 1947 Prof. L. E. J. BROUWER gave a formulation of the directives of intuitionistic mathematics 2). It is remarkable that negation does not occur in an explicit way, so one might be inclined to believe negationless mathematics to be a consequence of this formulation. The notion of species, however, is introduced in this way (translated from the Dutch text): “Finally in this construction of mathematics at any stage properties that can be supposed to hold for mathematical conceivabilities already obtained are allowed to be added as new mathematical conceivabilities under the name of species”. By this formulation it is possible that there are properties that can be supposed to hold for mathematical conceivabilities already obtained but that are not known to be true. With it negation and null-species are introduced simultaneously but at the cost of evidence. Whatever are the properties that can be supposed? What other criterion could there be than ‘to hold for mathematical conceivabilities already obtained’? In the definition of the notion of species the words “can be supposed” should be replaced by “are known”. One should restrict oneself in intuitionistic mathematics to mathematical conceivabilities and properties of those mathematical conceivabilities and one should not make suppositions of which one does not know whether it is possible to fulfil them. (The well-known turn in mathematics: “Suppose ABC to be rectangular” seems to be a supposition, but mostly means: “Consider a rectangular triangle ABC”).

(457)

2) L. E. J. BROUWER, Richtlijnen der intuïtionistische wiskunde. Proc. Kon. Ned. Akad. v. Wetensch., 50, (1947).

(Griss, “Negationless Intuitionistic Mathematics, II,” section 1.0.3, p.457, italics in the original)

(Also see Griss “Negationless Intuitionistic Mathematics, I,” section 0.10). The next idea is a bit harder to gather from the Griss texts we have examined. Roughly, Heyting says: In the theory of real numbers, the relation ≠, being negative, does not intervene. There is only the relation a = b and the distance relation ab (see “numerical calculation” below). The theorem “if ab is impossible, then a = b” is replaced by the following: “if a is distant from every number c which is distant from b, a = b”. (Heyting, p.14) But as we know from our sources, one of which Heyting cites in this paragraph, Griss does in fact use unequals (≠) for this formulation. Where in Griss is this notion of distance? I have not found it yet. My best current guess is the following. In section 1.3 of Griss’ “Negationless Intuitionistic Mathematics, I,” he discusses the ordering relation of the numbers in the set. Perhaps the notion of distance comes from the fact that if a and b in a set are identically the same number, then they share all the same numerical “distances” to the other numbers in the ordered sequence. But Heyting also notes that this presented difficulties for real numbers. Also, Dequoy worked on a negationless projective geometry.]

Mathématique sans négation de Griss.- ** GRISS [1,2] a exprimé des doutes sur la légitimité de l’emploi de la négation en mathématiques intuitionnistes. Selon lui on ne peut jamais démontrer par des méthodes intuitives la fausseté d’une supposition, car si celle-ci est fausse on ne saurait la concevoir clairement. Egalement on ne peut avoir une idée nette d’une propriété qu’après la construction d’un être mathématique qui possède cette propriété. On ne peut donc par introduire d’espèce vide. GRISS a commencé la construction des mathématiques intuitionnistes sans négation. Dans la théorie des nombres réels la relation ≠, étant négative, n’intervient pas. Il n’y a que la relation a = b et la relation de distance ab (voir ci-dessous “calcul numérique”). Le théorème “si ab est impossible, on a a = b” est remplacé par le suivant : “si a est distant de tout nombre c qui est distant de b, on a a = b”. La démonstration de | ce théorème pour les nombres réels fait des difficultés ; GRISS n’y parvient qu’en utilisant le théorème de BROUWER sur les déploiements bornés (voir ci-dessous, “Bon ordre”). Mlle DEQUOY a étudié la géométrie projective traitée sans négation [1,2].

(14-15)

GRISS, G.F.C.

 : Negatieloze intuitionistische wiskunde. Verslagen Akad. Wet. Amsterdam 53 (1944), p. 261–268. (Résumé dans Proc. Akad. Wet. Amsterdam 46, 47, 48, p. 128) ;

 Negationless mathematics. I Proc. Akad. Wet. Amsterdam 49 (1946), p. 1127–1133 = Indag. math. 8 (1946), p. 675-681, II ibidem 53 (1950), p. 456–463 = Indag. math. 12 (1950), p. 108–115 ;

(85)

DEQUOY, Mlle N.

 : La géométrie projective plane en mathématique intuitionniste sans négation. C.r. Acad. Sc. Paris 228 (1949) p. 1098–1100 ;

2 : Exposé d’un type de raisonnement en mathématique intuitionniste sans négation et résultats obtenus pour la géométrie projective plane. C.r. Acad. Sc. Paris 230 (1950), p. 357–359.

(84)

[contents]

5.1.1.2

[Griss’ and Destouches-Février’s Negationless Intuitionistic Logic]

[But if we completely eliminate negation, then we cannot have a propositional logic in the normal sense. Nonetheless, both Griss and Destouches-Février attempt to construct such a propositional logic. But instead of being a logic of predicates, Griss here constructs a logic of classes that is unlike the intuitionistic logic of classes in that for Griss, two classes can intersect only if they have at least one element in common.]

[In rough form: The suppression of negation has important consequences for logic. If only verified assumptions can be considered, such that any logical variable must represent a true proposition, a logic of propositions in the usual sense becomes manifestly impossible. However, attempts to construct a logic that allows for an interpretation that conforms to the requirements of GRISS have been made by Mrs. DESTOUCHES-FÉVRIER  and by GRISS [3,4]. Instead of the logic of predicates, Griss constructs a logic of classes that differs from the intuitionist logic of classes by the fact that the intersection of two classes can only be formed if they have at least one element in common.]

La suppression de la négation a des conséquences importantes pour la logique. Si l’on ne peut considérer que des suppositions vérifiées, de sorte que toute variable logique doit représenter une proposition vraie, une logique des propositions dans le sens usuel devient manifestement impossible. Cependant des tentatives pour construire une logique qui admet une interprétation conforme aux exigences de GRISS ont été faites par Mme. DESTOUCHES-FÉVRIER  et par GRISS [3,4]. Au lieu de la logique de prédicats GRISS construit une logique des classes qui diffère de la logique intuitionniste des classes par le fait que l’intersection de deux classes ne peut être formée que si elles ont au moins un élément en commun.

(15)

DESTOUCHES-FEVRIER, Mme P.

 : Logique de l'intuitionnisme sans négation et logique de l'intuitionnisme positif. C.r. Acad. Sc. Paris 226 (1948), p. 38–39 ;

(84)

GRISS, G.F.C.

 : Logique des mathématiques intuitionnistes. C.r. Acad. Sc. 227 (1948) p. 946–948 ;

 : Logic of negationless intuitionistic mathematics. Proc. Akad. Wet. Amsterdam 54 (1951), p. 41–49 = Indag, math. 13, p.41–49.

(85)

[contents]

5.1.1.3

[Destouches-Février’s Logic of Constructions, of Problems, and of Propositions]

[Destouches-Février devises a logic of constructions, on the basis of which she defines a logic of problems and a propositional logic.]

[In rough form: Mme. DESTOUCHES [3,5,6,7] defines the conception of a constructible mathematics in the strict sense, similar to GRISS. She outlines a logic of constructions, by means of which she defines a logic of problems and a logic of propositions. The notion of the realization of a construction is fundamental. In propositional logic, the conjunction p(a) ∧ q(a) can only be formed if the construction of an element a that satisfies p(x) and q(x) is feasible. DE BENGY-PUYVALLÉE  notes that this propositional logic is analogous to the logic to which we are led in wave mechanics.]

Mme. DESTOUCHES [3,5,6,7] définit la conception des mathématiques du constructible au sens strict, voisine de celle de GRISS. Elle esquisse une logique des constructions, au moyen de laquelle elle définit une logique des problèmes et une logique des propositions. La notion de la réalisation d’une construction est fondamentale. Dans la logique des propositions la conjonction p(a) ∧ q(a) ne peut être formée que si la construction d’un élément a qui satisfait à p(x) et à q(x) est réalisable. DE BENGY-PUYVALLÉE  remarque que cette logique des propositions est analogue à la logique à laquelle on est conduit en mécanique ondulatoire.

(15)

DESTOUCHES-FEVRIER, Mme P.

 : Esquisse d'une mathématique intuitionniste positive. C.r. Acad. Sc. Paris 225 (1947) p.1241–1243;

 : Le calcul des constructions. C.r. Acad. Sc. Paris 227 (1948), p.1192–1193 ;

 : Connexions entre le calcul des constructions, des problèmes, des propositions. C.r. Acad. Sc. Paris 228 (1948), p.31–33 ;

 : Sur l’intuitionnisme et la conception strictement constructive. Proc. Akad. Wet. Amsterdam 51 (1951), p,80–86 = Indag. math. 13 p, 80–86.

(84)

BENGY-PUYVALLÉE, R.DE

 : Sur la relation de composabilité dans les logiques de complémentarité. C.r. Acad. Sc. Paris 228 (1949) p. 324–626.

(83)

[contents]

5.1.1.4

[Van Dantzig’s Affirmative Mathematics]

[Van Dantzig outlined a formal system of affirmative mathematics.]

[In rough form: VAN DANTZIG  has outlined a formal system of affirmative mathematics. He proposes to explicitly indicate in each proposition the construction that leads to it and shows by some examples how this can be done, but he does not indicate how one could execute his program in more complicated cases.]

VAN DANTZIG  a ébauché un système formel de mathématiques affirmatives. Il propose d’indiquer explicitement dans chaque énoncé la construction qui y conduit et montre par quelques exemples comment on peut faire cela, mais il n’indique pas comment on pourrait exécuter son programme en des cas plus compliqués.

(15)

DANTZIG, D. Van

 : On the principles of intuitionistic and affirmative mathematics. Proc. Akad. Wet. Amsterdam 50 (1947), I p. 918–929, II p. 1092–1103 = Indag. math. 4 (1947), p. 429–440 et p. 506–517 ;

(84)

[contents]

5.1.1.5

[Brouwer and Negation]

[Brouwer supports the role of negation by constructing theories that require it, and he articulated his ideas about the relationship of mathematics to experience, language, and wisdom.]

[In rough form: With regard to GRISS and VAN DANTZIG, BROUWER supports the essential role of negation; by broadening the notion of a series of choices, he constructs theories that cannot be formulated without negation (see the end of this chapter). In the conference  BROUWER made clear his fundamental ideas about the relationship of mathematics to experience, language, and wisdom.]

Vis-à-vis de GRISS et de VAN DANTZIG, BROUWER soutient le rôle essentiel de la négation ; en élargissant la notion de suite de choix il construit des théories qu’on ne peut pas formuler sans la négation (voir la fin de ce chapitre). Dans la conférence  BROUWER a exposé clairement ses idées fondamentales sur les relations des mathématiques avec l’expérience, avec le langage et avec la sagesse.**

(15)

BROUWER, L.E.J.

 Consciousness, Philosophy and Mathematics. Proc. 10th Congr. of Philosophy, Amsterdam 1948, p. 1235–1249.

(83)

[contents]

Heyting, Arend. Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Paris / Louven: Gauthier-Villars / E. Nauwelaerts, 1955.

.

## 29 May 2019

### Heyting (ED) Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration.

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Heyting, entry directory]

[The collected brief summaries can be found at that link.]

Entry Directory for

Arend Heyting

Les fondements des mathématiques.

Intuitionnisme.

Théorie de la démonstration.

Première section:
Intuitionnisme

5.
L'intuitionnisme brouwérien

5.1
L'intuition mathématique

5.3.1

Calcul numérique

Heyting, Arend. Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Paris / Louven: Gauthier-Villars / E. Nauwelaerts, 1955.

.

### Arend Heyting (ED), entry directory

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

Entry Directory for

Arend Heyting Topic:

Negationless Intuitionistic Mathematics

Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration.

Intuitionism: An Introduction

”G. F. C. Griss and His Negationless Intuitionistic Mathematics”

Image taken gratefully from:

.

### Griss (1.0) “Negationless Intuitionistic Mathematics, II” Section 1.0, “[Preface]”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Griss, entry directory]

[Griss, “Negationless Intuitionistic Mathematics, II”, entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are surely mistaken or inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

George François Cornelis Griss

(G.F.C. Griss)

“Negationless Intuitionistic Mathematics, II”

1.0

“[Preface]”

Brief summary:

(1.0.1) The following is a sequel to Griss’ “Negationless Intuitionistic Mathematics, I.” But first he will give a preface with a concise exposition of his ideas in response to some remarks and objections he received. (1.0.2) Brouwer outlines a negationless mathematics in a 1947 paper, but to make it perfectly negationless, we need to slightly adjust one of his definitions to prevent us from supposing something to take properties we are not sure it has. (And, instead of saying negationally that something is either in a subset or not in that subset, we should say affirmatively that either it is in a subset or in that subset’s complement. (1.0.3) We construct sets of natural numbers by starting with 1, which is selfsame, then adding 2, also selfsame but distinct from 1, then 3, selfsame too and distinct from both 1 and 2, and we continue this way, adding n numbers to get the set: En (1, 2, ..., n). We can further add an element n′, selfsame and distinguishable from all members p of En (1, 2, ..., n), so n′ ≠ p, p ≠ n′. They together form the set En′ (1, 2, ..., n′). We can note disjunctively that an element of En′ belongs to En or is n′. “In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.” (1.0.4) “In accordance with the construction of natural numbers the proofs of properties of those numbers are always given by means of induction, until a system of properties is found, that can serve as a starting point of an axiomatic theory.” Now, instead of using disjunction as above, we will formulate the first property using the conditional: “If b is an element of Em (1, 2, . . . , m), then b together with the elements of Em that are distinguishable from b form Em.” (1.0.5) The next property was already articulated without disjunction in section 1.2.2 of “Negationless Intuitionistic Mathematics, I” as “If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.” Here the formulation and proof remain the same: “If for the elements a and b of Em holds: a ≠ c for each cb, then a = b.

Contents

1.0.1

[Explanation of This Text]

1.0.2

[Brouwer’s Negationless Mathematics]

1.0.3

[(ad § 1.1.) Constructing Sets of Natural Numbers and Locating Them Disjunctively]

1.0.4

[(ad §1.2.) The First Property of Sets of Natural Numbers, Formulated Without Disjunction]

1.0.5

[The Second Property: Elements Sharing the Same Differences are The Same]

Bibliography

Summary

1.0.1

[Explanation of This Text]

[The following is a sequel to Griss’ “Negationless Intuitionistic Mathematics, I.” But first he will give a preface with a concise exposition of his ideas in response to some remarks and objections he received.]

[ditto]

In 1944 I gave a sketch of some parts of negationless intuitionistic mathematics in these Proceedings; afterwards I started on a more complete and systematic treatment 1) . This note is a sequal [sic] to it. As in the meantime, however, many remarks and objections reached me, I preface this note by a concise exposition of my point of view and some explanations to the second note.

(457)

1) G. F. C. Griss, Negatieloze intuïtionistische wiskunde. Versl. Ned. Akad. v. Wetensch., 53, (1944).

Negationless intuitionistic mathematics. Proc. Kon. Ned. Akad. v. Wetensch., 49, (1946).

(457)

[contents]

1.0.2

[Brouwer’s Negationless Mathematics]

[Brouwer outlines a negationless mathematics in a 1947 paper, but to make it perfectly negationless, we need to slightly adjust one of his definitions to prevent us from supposing something to take properties we are not sure it has. (And, instead of saying negationally that something is either in a subset or not in that subset, we should say affirmatively that either it is in a subset or in that subset’s complement.]

[Griss next notes Brouwer’s 1947 “Richtlijnen der intuïtionistische wiskunde,” in which he gives a formulation of intuitionistic mathematics, but remarkably, “negation does not occur in an explicit way, so one might be inclined to believe negationless mathematics to be a consequence of this formulation.” He writes specifically:

The notion of species, however, is introduced in this way (translated from the Dutch text): “Finally in this construction of mathematics at any stage properties that can be supposed to hold for mathematical conceivabilities already obtained are allowed to be added as new mathematical conceivabilities under the name of species”. By this formulation it is possible that there are properties that can be supposed to hold for mathematical conceivabilities already obtained but that are not known to be true. With it negation and null-species are introduced simultaneously but at the cost of evidence.

(Griss, 457, see full quote below)

For context, consider the similar point that van Stigt makes in section 1.5.1.1 of “Brouwer’s Intuitionist Programme”

In the generation of the fundamental “mathematical entities,” such as the natural numbers and the Brouwer set or spread and its elements, there is no place nor immediate need for negation. The question of negation only arises at the level of species construction, at the point where the Subject is attempting to establish elementhood of a species S over a given domain of existing mathematical entities. Such attempt may lead to “successful fitting in”; that is, a particular mathematical entity is established as an element of S. The alternatives to “successful fitting in” are: (1) the constructed impossibility or “absurdity” of fitting in; and (2) the simple absence of the construction of elementhood or of its absurdity. Only negation in the first sense, of constructed impossibility, meets Brouwer’s strict requirements and can claim to be an act of mathematical construction.

(van Stigt, “Brouwer’s Intuitionist Programme,” section section 1.5.1.1, p.14)

I am not certain, but perhaps Brower defines subset membership in terms of it being impossible to fit into the complementary subset, and thus it would be a negational notion. And maybe Griss is saying something like we cannot think of something either fitting in to a set (or having a property) or not fitting into that set (not having a property) but rather as simply fitting into one set or its complement, thereby always knowing affirmatively where it is located. But I am not certain. The next point might be that that we should revise Brouwer’s formulation such that we should only suppose properties that are known (that we have evidence for). For, “One should restrict oneself in intuitionistic mathematics to mathematical conceivabilities and properties of those mathematical conceivabilities and one should not make suppositions of which one does not know whether it is possible to fulfil them.”]

In 1947 Prof. L. E. J. BROUWER gave a formulation of the directives of intuitionistic mathematics 2). It is remarkable that negation does not occur in an explicit way, so one might be inclined to believe negationless mathematics to be a consequence of this formulation. The notion of species, however, is introduced in this way (translated from the Dutch text): “Finally in this construction of mathematics at any stage properties that can be supposed to hold for mathematical conceivabilities already obtained are allowed to be added as new mathematical conceivabilities under the name of species”. By this formulation it is possible that there are properties that can be supposed to hold for mathematical conceivabilities already obtained but that are not known to be true. With it negation and null-species are introduced simultaneously but at the cost of evidence. Whatever are the properties that can be supposed? What other criterion could there be than ‘to hold for mathematical conceivabilities already obtained’? In the definition of the notion of species the words “can be supposed” should be replaced by “are known”. One should restrict oneself in intuitionistic mathematics to mathematical conceivabilities and properties of those mathematical conceivabilities and one should not make suppositions of which one does not know whether it is possible to fulfil them. (The well-known turn in mathematics: “Suppose ABC to be rectangular” seems to be a supposition, but mostly means: “Consider a rectangular triangle ABC”).

(457)

2) L. E. J. BROUWER, Richtlijnen der intuïtionistische wiskunde. Proc. Kon. Ned. Akad. v. Wetensch., 50, (1947).

(457)

[contents]

1.0.3

[(ad § 1.1.) Constructing Sets of Natural Numbers and Locating Them Disjunctively]

[We construct sets of natural numbers by starting with 1, which is selfsame, then adding 2, also selfsame but distinct from 1, then 3, selfsame too and distinct from both 1 and 2, and we continue this way, adding n numbers to get the set: En (1, 2, ..., n). We can further add an element n′, selfsame and distinguishable from all members p of En (1, 2, ..., n), so n′ ≠ p, p ≠ n′. They together form the set En′ (1, 2, ..., n′). We can note disjunctively that an element of En′ belongs to En or is n′. “In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.”]

[We are dealing now with some properties of the natural numbers, it seems, but I am not sure. Let us first recall how we constructed the natural numbers using negationless mathematics in section 1.1 of “Negationless Intuitionistic Mathematics, I.” The following is the brief summary from that section.

(1.1.1) We will construct the natural numbers using negationless intuitionistic mathematical principles (see section 0). We first simply imagine an object, call it “1”. It remains the same. Thus it is the same as 1. The symbolic formulation for this is: 1 = 1. (1.1.2) We next imagine another object that we call 2, which is also selfsame, meaning that, in symbolic formulation, 2 = 2; and, these two objects are distinguishable from one another, or in symbolic formulation, 1 ≠ 2, 2 ≠ 1. (1.1.3) Objects 1 and 2 (see sections 1.1.1 and 1.1.2) form a set. So 1 and 2 are members of the set {1, 2}. (For now, the set is simply these two.) If an object were to belong to this set, that object would be either 1 or 2. If that object is distinguishable from 1, then it is 2. If that object is distinguishable from 2, then it is 1. (1.1.4) We next imagine another object and set element. We call it 3. It remains selfsame, so in symbolic formulation, 3 = 3. Also, 3 is distinguishable from 1 and 2, so in symbolic formulation, 1 ≠ 3, 3 ≠ 1, 2 ≠ 3, 3 ≠ 2. (1.1.5) Objects 1, 2, and 3 (see sections 1.1.1, 1.1.2, and 1.1.4) form the set {1, 2, 3}. (The set is limited to these three.) Any object belonging to this set would  be either 1, 2, or 3.  So, “if it is distinguishable from 3, it is an element of {1, 2}.” (1.1.6) We can also imagine there being any additional number to the set that is selfsame and distinguishable from the rest of the members: “If, in this way, we have proceeded to {1, 2, …, n}, we can, again, imagine an element n′, remaining the same, n′ = n′, and distinguishable from each element p of {1, 2, ... , n}, in formula n′p, pn′.” (1.1.7) The set member n′ in addition to the set {1, 2, …, n} (see section 1.1.6) form the set {1, 2, …, n′}. Any number belonging to {1, 2, …, n′} either is a member of {1, 2, ... , n} or it is n′ itself. We can determine which in the following way. “If it is distinguishable from each element of {1, 2, ... , n}, it is n′; if it is distinguishable from n′, it is an element of {1, 2, ... , n}.” (1.1.8) We can obtain a finite set {1, 2, …, m} if we cease our additions with the mth element. Or we can obtain the countably infinite set {1, 2, …} by proceeding with the additions unlimitedly. (1.1.9) If we want large sets and we choose a new symbol for each one, then the symbolization can become difficult. (Either a large number of distinct simple symbols will need to be continuously invented, or redundancy methods, like simply combining strokes or even using numerative systems like decimal, will sooner or later create symbols that become unmanageably long.)

(brief summary of section 1.1 of “Negationless Intuitionistic Mathematics, I.”)

So generally speaking, we constructed the first three natural numbers in the following way. We first assumed an object called ‘1’ that is understood to be self-same, so 1 = 1. We next assumed another self-same number, called ‘2’, so  2 = 2, but since it is distinguishable from 1, we have 1 ≠ 2 and 2 ≠ 1. They both belong to set {1, 2}. So if an object in this set is distinguishable from 1, it must be 2, and if it is distinguishable from 2, it must be 1. We assume a third distinct self-same set element, ‘3’, so likewise, 3 = 3 and 1 ≠ 3, 3 ≠ 1, 2 ≠ 3, 3 ≠ 2, now forming the set  {1, 2, 3}. If a member is distinguishable from 3, it must be in set {1, 2}. We can continue to imagine include additional self-same distinct objects/numbers, symbolized as n: {1, 2, …, n}. But no matter how many there are (no matter how large n), we can always add another self-same distinct object/number n′ that is equal to itself and unequal to all the other set members, thereby forming the set {1, 2, …, n′}. If a member of this set is distinguishable from all objects in the set {1, 2, ... , n}, then that member in question is n′. But if a member is instead distinguishable from n′, then this object in question is an element of the set {1, 2, ... , n}. Now in our current section, Griss returns us to this construction of natural numbers that we just reviewed above. One difference now seems to be that instead of saying “the set {1, 2, ... , n}” we say, “the set En (1, 2, ..., n),” and instead of  saying “the set {1, 2, …, n′)” we say, “the set En′ (1, 2, ..., n′).” So by adding n′ to the set En (1, 2, ..., n), we get the set En′ (1, 2, ..., n′). We thus call En′ the sum of En and n′. An element of En′ belongs to En or is itself n′. But then it gets tricky, and I may not summarize the next idea correctly, so see the quotation below. We notice that in the formulation, “An element of En′ belongs to En or is n′,” we have a logical disjunction. Griss then says, “It is evident the disjunction a or b in the usual meaning (the assertion a is true or the assertion b is true), does not occur in negationless mathematics, because there is no question of assertions that are not true.” We discussed the lack of the principle of excluded middle in section 1.2.3 of Griss’ “Negationless Intuitionistic Mathematics, I,” (where we also referenced the following places for more discussion on the issue of excluded middle in intuitionism: Priest, Introduction to Non-Classical Logic section 6.2.8, van Stigt’s “Brouwer’s Intuitionist Programme” sections 1.4.1.3 and 1.5.1.2, Mancosu & van Stigt’s “Intuitionistic Logic” sections 4.2.1, and Nolt’s Logics section 16.2, especially 16.2.7 and 16.2.29.) As I understand it, Griss is saying that although we have a disjunction where one disjunct may hold for a term and the other will not, we did not obtain that disjunction simply by using negation and appealing to the law of excluded middle. Rather, we had to construct a positive proof for it. A term is found either in one subset or its complement. We are not saying it is either found in one subset or not found in that same subset. Griss writes, “In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.” So we have a set V, which could be for instance {1, 2, 3}, and V′ would be the set {1, 2} and  V″ would be the set {3}. “a or b is true for all elements of the set V” means that both a and b hold for V, yet only a holds for one subset and b for another.]

ad §1.1.     After the introduction of the natural numbers 1, 2, 3 the | natural number n′ next to the natural number n was introduced by means of induction as follows:

“If, in this way, we have proceeded to En (1, 2, ..., n), we can again imagine an element n′, remaining the same, n′ = n′, and distinguishable from each element p of En (1, 2, ..., n), in formula n′ ≠ p, p ≠ n′. They form the set En′ (1, 2, ..., n′).”

En′ is called the sum of En and n′, in other words: An element of En′ belongs to En or is n′. In this way the disjunction is defined in a particular case. It is evident the disjunction a or b in the usual meaning (the assertion a is true or the assertion b is true), does not occur in negationless mathematics, because there is no question of assertions that are not true. In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.

(456-457)

[contents]

1.0.4

[(ad §1.2.) The First Property of Sets of Natural Numbers, Formulated Without Disjunction]

[“In accordance with the construction of natural numbers the proofs of properties of those numbers are always given by means of induction, until a system of properties is found, that can serve as a starting point of an axiomatic theory.” Now, instead of using disjunction as above, we will formulate the first property using the conditional: “If b is an element of Em (1, 2, . . . , m), then b together with the elements of Em that are distinguishable from b form Em.”]

[ditto]

ad §1.2.   In accordance with the construction of natural numbers the proofs of properties of those numbers are always given by means of induction, until a system of properties is found, that can serve as a starting point of an axiomatic theory. At the time I used the disjunction in the proofs of the two properties concerning the relations of identity and distinguishability. Now we will show, how it is possible to avoid the use of disjunction in accordance with the remark made ad §1.1. For that purpose I formulate the first property: If b is an element of Em (1, 2, . . . , m), then b together with the elements of Em that are distinguishable from b form Em.

Proof: The property holds for E2. Suppose the proof has proceeded to En. 1) Consider first an element b of En. The elements of En′ that differ from b are n′ and those elements of En that differ from b. The latter form together with b the set En and En together with n′ forms En′ . 2) Now consider b = n′. In this case the elements differing from b form the set En, so together with b the set En′ . So the property holds for the elements of En and for n′, so for all elements of En′ .

(457)

[contents]

1.0.5

[The Second Property: Elements Sharing the Same Differences are The Same]

[The next property was already articulated without disjunction in section 1.2.2 of “Negationless Intuitionistic Mathematics, I” as “If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.” Here the formulation and proof remain the same: “If for the elements a and b of Em holds: a ≠ c for each cb, then a = b.”]

[ditto]

The avoiding of the disjunction has little influence on the proof of the second property.

If for the elements a and b of Em holds: a ≠ c for each cb, then a = b.

Proof: The property holds for E2. Suppose the proof has proceeded to En. 1) If b = n′, then a is distinguishable from each element of En, so a = n′ and a = b. 2) b is element of En; choose c = n, then also a is an element of En, so a = b. The proof has been delivered for all elements of En and for n′, so for all elements of En′ .

[contents]

Griss, George François Cornelis. “Negationless Intuitionistic Mathematics, II.” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 53, no. 4 (1950): 456–463.

Journal PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00014669.pdf

Article PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00018796.pdf

Listing of Griss at this journal:

http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531

.

### Griss (1.3) “Negationless Intuitionistic Mathematics, I” Section 1.3, “The Order-Relation”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Griss, entry directory]

[Griss, “Negationless Intuitionistic Mathematics, I”, entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are surely mistaken or inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

George François Cornelis Griss

(G.F.C. Griss)

“Negationless Intuitionistic Mathematics, I”

1.3

“The Order-Relation”

Brief summary:

(1.3.1) “We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” In this way, any set of terms {1, 2, ..., n} can be arranged in such an order of procession. (1.3.2) If for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a b.” (1.3.3) Precession is transitive: “Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.” (1.3.4) If a number in an ordered set does not equal 1, then it must come after 1: “Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a.” If a number in an ordered set does not equal the last number, then it must come before it: “Property: If a m is an element of {1, 2, ..., m}, then a < m.” If b is neither the first nor the last number, then any other number a must either precede or succeed b. “Property: If a and b (b ≠ 1 and b m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.” Also, we cannot have negative numbers in sets constructed this way and in accordance with negationless intuitionistic mathematical principles. (1.3.5) If one number a precedes another number b, and if for all the numbers c coming before b, they also come before a, then b immediately follows a: “If a < b and if for each c < b and c a c < a holds, then b immediately follows a.” Similarly, if a number a precedes another number b, and if for all the other numbers c that come after a and that are not b – if they all come after b, then b immediately follows a (check this quote, as it says b immediately follows b): “If a < b and if for each c > a and c b c > b holds, then b immediately follows b.” If a number b immediately follows another number a, which itself is not the first number, then for all the other numbers coming before b, if they do not equal a, then they come before a: “If b immediately follows a (a ≠ 1) , then for each c < b and c a holds c < a.” Similarly, if a number b immediately follows another number a, and b is not the final number, then all the numbers larger than a that are not equal to b would have to come after b: “If b immediately follows a (b m), then for each c > a and c b holds c > b.” (1.3.6) Suppose some number b is greater than 1, and it has numbers c that come before it. If some other number a does not equal b and does not equal any of these numbers c coming before b, then a comes after b: “a b and a c for each c < b (b ≠ 1) → a > b.” Similarly, suppose some number b is not the last number, and it has numbers c that come after it. If some other number a does not equal b and does not equal any of these numbers c coming after b, then a comes before b: “a b and a c for each c > b (b m) → a < b.” On the basis of these properties, we define the following: “a b as a = b or a < b and likewise a b;” “a c for each c < b (b ≠ 1) → a b;” “a c for each c > b (b m) → a b;” and “a ≥ 1 and am.”

Contents

1.3.1

[The Ordering Relation Defined]

1.3.2

[Precession Entails Inequality]

1.3.3

[The Transitivity of Precession]

1.3.4

[Some Ordering Properties]

1.3.5

[Some More Ordering Properties]

1.3.6

[Yet More Ordering Properties]

Bibliography

Summary

1.3.1

[The Ordering Relation Defined]

[“We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” In this way, any set of terms {1, 2, ..., n} can be arranged in such an order of procession.]

[We will define the ordering relation. We have items – numbers – in a set, and we will order them according to which ones precede or follow which other ones; and we will use the less than or greater than symbols. We also more specifically designate when one item immediately proceeds or follows another: “We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” So in {1, 2}, 1 immediately precedes 2. In {1, 2, 3}, 2 immediately precedes 3, and also 1 precedes 3. (There is more to this, see the quotation below.) In this way, we have a series of ordered terms, such that for {1, 2, ..., n} and {1, 2, .... n′}, we have p < n′ for each p of {1, 2, ..., n}. ]

We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).

For {1, 2} we have 1 < 2. If a and b are elements of {1, 2} and if a < b, then a = 1 and b = 2. 1 immediately precedes 2. |

For {1, 2, 3} we have 2 < 3 and 1 < 3. If a and b are elements of {1, 2, 3} and if a < b, then b = 3 and a belongs to {1, 2} or a and b belong to {1, 2}. 2 immediately precedes 3.

If, in this way, we have proceeded to {1, 2, ..., n}, for {1, 2, .... n′} we have p < n′ for each p of {1, 2, ..., n}. If a and b are elements of {1, 2, ..., n′} and if a < b, then b = n′ and a belongs to {1, 2, ..., n} or a and b belong to {1, 2, ..., n}. n immediately precedes n′.

(1132-1133)

[contents]

1.3.2

[Precession Entails Inequality]

[If for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a b.”]

[The next idea is that if for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a b.” (See the proof in the quotation below.)]

If for {1, 2, ..., m} a < b, then a b.

Proof: For {1, 2} the proposition holds. Let the proof have proceeded to {1, 2, ..., n}. If a and b are elements of {1, 2, …, n′} and if a < b, then b = n′ and a belongs to {1, 2, ..., n}, so that a b or a and b belong to {1, 2, ..., n}, so that a b.

(1133)

[contents]

1.3.3

[The Transitivity of Precession]

[Precession is transitive: “Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.”]

[ditto]

Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.

Proof: {1, 2, 3}: as a < b b belongs to {2, 3} and as b < c b belongs to {1, 2}. So b = 2, a = 1 and c = 3, so that a < c. Let the proof has proceeded to {1, 2, ..., n}. For the elements of {1, 2, ..., n′} is a < b and b < c. c = n′ or c belongs to {1, 2, ..., n}; in both cases b belongs to {1, 2, ..., n}, so a too and a < c.

(1133)

[contents]

1.3.4

[Some Ordering Properties]

[If a number in an ordered set does not equal 1, then it must come after 1: “Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a.” If a number in an ordered set does not equal the last number, then it must come before it: “Property: If a m is an element of {1, 2, ..., m}, then a < m.” If b is neither the first nor the last number, then any other number a must either precede or succeed b. “Property: If a and b (b ≠ 1 and b m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.” Also, we cannot have negative numbers in sets constructed this way and in accordance with negationless intuitionistic mathematical principles.]

[ditto]

Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a. Property: If a m is an element of {1, 2, ..., m}, then a < m. Property: If a and b (b ≠ 1 and b m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.

These properties, just as the following ones, can easily be proved by induction. The condition b ≠ 1 (also b m) is necessary in the last property, for if b = 1, there is no element a which would satisfy a < b. This cannot be allowed in negationless mathematics (Cf. Introduction). We return to this subject in the theory of sets.

(1133)

[contents]

1.3.5

[Some More Ordering Properties]

[If one number a precedes another number b, and if for all the numbers c coming before b, they also come before a, then b immediately follows a: “If a < b and if for each c < b and c a c < a holds, then b immediately follows a.” Similarly, if a number a precedes another number b, and if for all the other numbers c that come after a and that are not b – if they all come after b, then b immediately follows a (check this quote, as it says b immediately follows b): “If a < b and if for each c > a and c b c > b holds, then b immediately follows b.” If a number b immediately follows another number a, which itself is not the first number, then for all the other numbers coming before b, if they do not equal a, then they come before a: “If b immediately follows a (a ≠ 1) , then for each c < b and c a holds c < a.” Similarly, if a number b immediately follows another number a, and b is not the final number, then all the numbers larger than a that are not equal to b would have to come after b: “If b immediately follows a (b m), then for each c > a and c b holds c > b.”]

[ditto]

If a < b and if for each c < b and c a c < a holds, then b immediately follows a.

If a < b and if for each c > a and c b c > b holds, then b immediately follows b.

If b immediately follows a (a ≠ 1) , then for each c < b and c a holds c < a.

If b immediately follows a (b m), then for each c > a and c b holds c > b.

(1133)

[contents]

1.3.6

[Yet More Ordering Properties]

[Suppose some number b is greater than 1, and it has numbers c that come before it. If some other number a does not equal b and does not equal any of these numbers c coming before b, then a comes after b: “a b and a c for each c < b (b ≠ 1) → a > b.” Similarly, suppose some number b is not the last number, and it has numbers c that come after it. If some other number a does not equal b and does not equal any of these numbers c coming after b, then a comes before b: “a b and a c for each c > b (b m) → a < b.” On the basis of these properties, we define the following: “a b as a = b or a < b and likewise a b;” “a c for each c < b (b ≠ 1) → a b;” “a c for each c > b (b m) → a b;” and “a ≥ 1 and am.”]

[ditto]

Finally:

a b and a c for each c < b (b ≠ 1) → a > b.

a b and a c for each c > b (b m) → a < b.

From the preceding properties follows, if we define

a b as a = b or a < b and likewise a b.

a c for each c < b (b ≠ 1) → a b.

a c for each c > b (b m) → a b.

a ≥ 1 and am.

[contents]

Griss, G.F.C. (1946). “Negationless Intuitionistic Mathematics, I,’’ Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 49, 1127–1133.

Journal PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00014659.pdf

Article PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00018278.pdf

Listing of Griss at this journal:

http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531

.

### Griss (1.2) “Negationless Intuitionistic Mathematics, I” Section 1.2, “Properties of the Relations ‘The Same’ and ‘Different’”, summary

[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]

[Mathematics, Calculus, Geometry, Entry Directory]

[Logic and Semantics, entry directory]

[Griss, entry directory]

[Griss, “Negationless Intuitionistic Mathematics, I”, entry directory]

[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are surely mistaken or inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

George François Cornelis Griss

(G.F.C. Griss)

“Negationless Intuitionistic Mathematics, I”

1.2

“Properties of the Relations ‘The Same’ and ‘Different’”

Brief summary:

(1.2.1) The first property of sameness and difference for our intuitionally and non-negationally constructed sets of natural numbers is that: Two elements of the set {1, 2, ..., m} are the same or distinguishable. (1.2.2) The second property of sameness and difference is that if two numbers (which may either be the same or different numbers, but we do not determine that initially) share all the same differences to all the other numbers, then they are the same number (or if they are unequal to all the other same numbers, then they are equal to one another): “If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.” (1.2.3) The complementary set of the element a of the set {1, 2, ..., m} is denoted by A. And “The complement of A is a and the sum of a and A is {1, 2, ..., m}”. The “main proposition of arithmetic” would be formulated here as: “If there is a one to one reciprocal correspondence between {1, 2, ..., m} and {1, 2, ..., p}, then m = p.” “For the elements of the set {1, 2, ..., m} the following propositions hold now:

I   a = a

II   a = bb = a

III  a = b and b = c a = c

IV   a b b a

V   a = b and b c a c

VI   a = b or a b

VII   a c for each c b a = b.

Proposition “VI replaces the negative proposition: Two natural numbers are the same or not,” which holds in non-intuitionistic mathematics but not in intuitionistic mathematics, on account of the principle of excluded middle or excluded third not holding. Proposition VII is functionally correspondent with its negational counterpart, which is: “If it is impossible, that a is not the same as b, then a is the same as b.” And our positive theory replaces the following other negational propositions regarding sameness and difference:

different ⇄ not the same.

the same ⇄ not different.

the same and different exclude one another.

two natural numbers are either the same or different.

Contents

1.2.1

[The Being Either Same or Different for Natural Numbers]

1.2.2

[Equality as Shared Difference]

1.2.3

[Remaining Properties]

Bibliography

Summary

1.2.1

[The Being Either Same or Different for Natural Numbers]

[The first property of sameness and difference for our intuitionally and non-negationally constructed sets of natural numbers is that: Two elements of the set {1, 2, ..., m} are the same or distinguishable.]

[In the previous section 1.1, we constructed sets of natural numbers {1, 2, 3, ..., n} inductively by beginning with a selfsame object, 1, which is the same as itself, so 1 = 1. Then we added another number, 2, which is also selfsame, so 2 = 2, but it is distinguishable from 1, so 1 ≠ 2, 2 ≠ 1. When we added 3, which is also selfsame and also distinct from the other two, so 3 = 3 and 1 ≠ 3, 3 ≠ 1, 2 ≠ 3, 3 ≠ 2. Now, we think of some object whose identity we do not assume beforehand but which we will deduce on the basis of its samenesses and differences. We consider a set of any number of members, {1, 2, …, n} and more broadly a set that is one member larger, {1, 2, …, n}. And we say that if some given number whose identity we do not begin by assuming is distinguishable from each element of {1, 2, …, n}, then it is n′, but if it is distinguishable from n′, then it is a member of {1, 2, …, n}. Also, if we cease our additions with an mth element, we get a finite set {1, 2, …, m}. Now Griss will outline some of the properties of these same and different relations. (They seem closely tied to equality and inequality of quantity, but I am not sure how to distinguish sameness and equality, and difference and inequality. The other idea at work here is distinguishability. But that still does not help me, because how would we distinguish sameness, equality, and indistinguishability from one another, and how would we distinguish difference, inequality, and distinguishability from one another?) The first property of sameness and difference is that any two members of a finite set of natural numbers is either the same or distinguishable. (I am not sure how any two can be the same, if we have fashioned the set on the basis of adding distinct terms consecutively. Perhaps the idea is that we can have both 2 and 2, or maybe, we have two variables, a and b, whose identity is not initially assumed but whose sameness is deduced.) (The proof for this I may not get right, but it seems to work in the following way.) We will show that regardless of what might be in the set, in any case any two elements a and b will be either the same or different. We begin by saying that the proposition holds for {1, 2}, (but no explanation is given for that. Perhaps the idea is that with only two members, there can only be two possibilities for a and b, either they are both 1 or both 2, or one is 1 and the other is 2.) The proof proceeds to the larger set {1, 2, ..., n} and it seems also to a one-larger set {1, 2, ..., n′}. So with this larger set, we denote two elements as a and b. Since they are both in the larger set {1, 2, ..., n′}, that means they can be either in the subset {1, 2, ..., n} or be the only remaining item n′. For our convenience of illustration, we will have the set {1, 2, 3} with 2 exemplifying n, and 3 being n′. There are four possible situations that may hold.  both a and b equal n′ (they both are 3). Thus the proposition holds, because the two elements are the same.  a is n′ (a is 3), and b belongs to the set of other numbers (b is either 1 or 2). In this case, the proposition holds, because the two elements are different.  a belongs to the set of other numbers (a is either 1 or 2), and b is n′. Again, the proposition holds, because here a and b are different. And , a and b both belong to the set of other numbers (they are either 1 or 2), in which case either a is the same as b (they are both either 1 or both 2), or they are different (a is one of the two options, and b is the other option.) Here we again see that the proposition holds, because in this case a and b are either the same or different. As these are the only possibilities, it is necessarily the case that two elements of such a set are either the same or distinguishable.]

Proposition: Two elements of the set {1, 2, ..., m} are the same or distinguishable.

Proof: For {1, 2} the proposition holds. Let the proof have proceeded to the set {1, 2, ..., n}. Denote two elements of {1, 2, ..., n′} by a and b. a is an element of {1, 2, ..., n} or it is n′, likewise b. There are 4 possibilities: 1) a = n′ and b = n′, so a = b; 2) a = n′ and b belongs to {1, 2, ..., n}, so a b; 3) a belongs to {1, 2, ..., n} and b = n′, so a b; 4) a and b belong to {1, 2, ..., n}, then a = b or a b.

(1131)

[contents]

1.2.2

[Equality as Shared Difference]

[The second property of sameness and difference is that if two numbers (which may either be the same or different numbers, but we do not determine that initially) share all the same differences to all the other numbers, then they are the same number (or if they are unequal to all the other same numbers, then they are equal to one another): “If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.”]

[The next proposition regarding the properties of sameness and difference is something like the following. If two unidentified elements of a set of natural numbers share all the same differences to the other numbers (if they are different to all the other same numbers), then they are the same number. So suppose our set has three numbers, {1, 2, 3}. And we have two unidentified numbers in that set, a and b, which are either the same number or are two different numbers. We will show the conditions under which they are the same, non-negationally. For this, we need to consider a third number c, and ask about c’s relation to a and b. For the sake of illustration, let us just assume that both a and b are the number 2, so that we can see how this proposition works. So b is 2. We now want to see all the other terms that b does not equal, that is to say, all the c’s that it is distinct from. In this case, they are 1 and 3. The proposition says, “If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.” So we have determined each c b, with those c’s being 1 and 3. Now, for each such inequality, namely, 1 ≠ b and 3 ≠ b, where 1 and 3 are the c’s, we also have that a as well does not equal those same c’s: 1 a and 3 ≠ a. So since a and b are different from the same set of all the other members, they must be identical to each other. For, there is only one option left that both can be, which is 2. Now let us look at the actual proof. We begin by asserting that the proof holds for the set {1, 2}, but I again am not sure the reasoning why. Perhaps it is because c can never be a third number. So if both a and b are different than c, then they must be the number other than c, and with there only being one option, both a and b would have to be that same one. But I am not sure. We then proceed to a larger set, {1, 2, ..., n} and it seems more so to {1, 2, ..., n′}. We can say that b either belongs to {1, 2, ..., n} or b = n′. So we can now say that there are then two possibilities with regard to the difference relations between a and b with respect to c.  One possibility is that b = n′. (From the text below, which you should consult, I do not grasp all the reasoning, but I am guessing it is the following. We are assuming that  a c for each c b. And we are saying here that b = n′, which means that b cannot be in the set {1, 2, ..., n}. From this we are supposed to conclude that a is distinguishable from each element of {1, 2, ..., n}. But I cannot see how that can be inferred from b = n′. This is why I wonder if we need to include the other assumptions. So perhaps {1, 2, ..., n} gives us all the c’s that do not equal b. And we are also assuming that a does not equal any of these c’s. That means a cannot be in that set, leaving the only other option, it being n′, which b is also, and thus a = b.)  The second possibility is that b does in fact belong in the set {1, 2, ..., n}. We next want to know about the numbers c that b does does not equal. We know at least one, namely n′, because it lies outside the set that b is a member of. So we can say that one such c would have to be n′. Now, one of our assumptions was that a c for each c b. So we can say that a is also not n′ and that a is a member of {1, 2, ..., n} just like b is. But from this, we are to conclude that a = b, which is not yet obvious to me. Could not a = 1 and b = 2? Both are in that set. So to come to that conclusion, I wonder if you repeat the exercise. After establishing that both a and b are not n′, then perhaps we must then establish all the other numbers in the set {1, 2, ..., n} that b is not, going one by one, each time noting that a is not equal to that number, until finally we arrive upon the one same number that they both must be. I am not sure actually. So please consult the quotation below.)]

Proposition: If for two elements a and b of {1, 2 ..., m} holds: a c for each c b, then a = b.

Proof: For {1, 2} the proposition holds. Let the proof have proceeded to {1, 2, ..., n}. There are two possibilities: In {1, 2, ..., n′} b = n′ or b belongs to {1, 2, ..., n}. 1) If b = n′, then a is distinguishable from each element of {1, 2, ..., n}, so a = n′ and a = b. 2) b belongs to {1, 2, ..., n}; take c = n′, then a also belongs to {1, 2, …, n}, so a = b.

(1132, boldface is mine)

[contents]

1.2.3

[Remaining Properties]

[The complementary set of the element a of the set {1, 2, ..., m} is denoted by A. And “The complement of A is a and the sum of a and A is {1, 2, ..., m}”. The “main proposition of arithmetic” would be formulated here as: “If there is a one to one reciprocal correspondence between {1, 2, ..., m} and {1, 2, ..., p}, then m = p.” “For the elements of the set {1, 2, ..., m} the following propositions hold now:

I a = a

II a = bb = a

III a = b and b = c a = c

IV a b b a

V a = b and b c a c

VI a = b or a b

VII a c for each c b a = b.

Proposition “VI replaces the negative proposition: Two natural numbers are the same or not,” which holds in non-intuitionistic mathematics but not in intuitionistic mathematics, on account of the principle of excluded middle or excluded third not holding. Proposition VII is functionally correspondent with its negational counterpart, which is: “If it is impossible, that a is not the same as b, then a is the same as b.” And our positive theory replaces the following other negational propositions regarding sameness and difference:

different ⇄ not the same.

the same ⇄ not different.

the same and different exclude one another.

two natural numbers are either the same or different.]

[The complementary set of an element plus that element is their sum, making the full set. “If we denote the complementary set of the element a of the set {1, 2, ..., m} by A, the complement of A is a and the sum of a and A is {1, 2, ..., m}.” The “main proposition of arithmetic” says that “If there is a one to one reciprocal correspondence between {1, 2, ..., m} and {1, 2, ..., p}, then m = p.” Griss then provides seven propositions that hold in this negationless mathematics. Let us go through them.

I   a = a

This perhaps follows from the idea that the members of our set are all, by stipulation of their construction, self-same. It seems to be like a reflexive property.

II   a = bb = a

This is a sort of symmetrical property. It would seem to be a contradiction otherwise.

III   a = b and b = c a = c

Here is transitivity, which also makes sense especially in this mathematical context.

IV  a b b a

This also would seem to be a contradiction if otherwise.

V  a = b and b c a c

This seems to following the notion of equality.

VI   a = b or a b

For this one, Griss writes that it “replaces the negative proposition: Two natural numbers are the same or not, which in non-intuitionistic mathematics holds in virtue of the principium tertii exclusi, but which in intuitionistic mathematics must be proved.” Throughout all this, it was never very clear to me how unequals is not a negative conception. It seems to be because not-equals is defined not as a lack or negation of being equal, but rather as belonging to a different set of numbers (or perhaps, has a distinguishing additional property or sense, like being larger than, as it might be the n′ number or in the set lower than it). In the wording for the negative proposition, two numbers are either the same or they are not the same. This is somehow different from them being either equal or unequal. It is still hard to articulate the distinction, but being unequal is somehow a positive property. Perhaps it can be understood as each unequal number having some additional distinguishing trait. At any rate, the second point is that the negational formulation holds in a classical setting on account of the principle of excluded third, but in intuitionism that does not work. As we know, the law of excluded middle does not hold in intuitionist logic and mathematics (see Priest, Introduction to Non-Classical Logic section 6.2.8, van Stigt’s “Brouwer’s Intuitionist Programme” sections 1.4.1.3 and 1.5.1.2, Mancosu & van Stigt’s “Intuitionistic Logic” sections 4.2.1, and Nolt’s Logics section 16.2, especially 16.2.7 and 16.2.29.) This means, it would seem, that we cannot simply say, on the basis of logic itself, that either it is the case that two numbers are the same or it is not the case that they are the same. Instead we must offer a positive proof that the two numbers are either equal or not-equal, which we did above in section 1.2.1 on the basis of possible set memberships.

VII  a c for each c b a = b.

This one we proved above in section 1.2.2. About it Griss says, “VII runs with negation: If it is impossible, that a is not the same as b, then a is the same as b.” I do not know what the “runs with” negation means. As far as I know, it is supposed to be not negational. So I suspect the “runs with” means is not negational but corresponds functionally with the negational formulation, which would say that if it is impossible for two things to not be the same then they must be the same. Proposition VII says something more like if a and b share all the same differences to everything else, then they are the same. Finally Griss next gives four negative propositions regarding sameness and difference that he has replaced in the positive theory. The first is:

different ⇄ not the same.

I am supposing that the ⇄ means biconditional. So perhaps he is saying that formerly, different meant not being the same. Now it means belonging to different parts of a set.

the same ⇄ not different.

Previously perhaps, sameness was understood negationally as not being different. Now it means sharing the same differences.

the same and different exclude one another.

Previously it may have been said that two things are the same, then they cannot be different. Now perhaps, but I am not sure, we are saying that things can be the same or they can be different, with the fact that in all cases they will not be both being a result of their construction and not an assumption about them. I am guessing.

two natural numbers are either the same or different.

This one is very tricky for me, because I do not know how to distinguish it from the proposition from section 1.2.1 above:

Two elements of the set {1, 2, ..., m} are the same or distinguishable.

And also, I do not see the negation in the formulation. Perhaps in that formulation, “different” is understood as “being not the same” but in the non-negational formulation, “distinguishable” is understood as belonging to different parts of a set. I am guessing wildly, so please see the quotation below.)]

We can also formulate these propositions in the following way, though we anticipate the general theory of sets we hope to treat of in a following paragraph.

If we denote the complementary set of the element a of the set {1, 2, ..., m} by A, the complement of A is a and the sum of a and A is {1, 2, ..., m}.

In this connection I mention the so-called main proposition of arithmetic.

If there is a one to one reciprocal correspondence between {1, 2, ..., m} and {1, 2, ..., p}, then m = p.

I do not repeat proofs which have been already given without using the negation.

For the elements of the set {1, 2, ..., m} the following propositions hold now:

I a = a

II a = bb = a

III a = b and b = c a = c

IV a b b a

V a = b and b c a c

VI a = b or a b

VII a c for each c b a = b.

VI replaces the negative proposition: Two natural numbers are the same or not, which in non-intuitionistic mathematics holds in virtue of the principium tertii exclusi, but which in intuitionistic mathematics must be proved.

VII runs with negation: If it is impossible, that a is not the same as b, then a is the same as b.

I briefly enumerate the negative propositions concerning the relations “the same” and “different” which have been replaced in a positive theory.

different ⇄ not the same.

the same ⇄ not different.

the same and different exclude one another.

two natural numbers are either the same or different.

(1132)

[contents]

Griss, G.F.C. (1946). “Negationless Intuitionistic Mathematics, I,’’ Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 49, 1127–1133.

Journal PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00014659.pdf

Article PDF here:

http://www.dwc.knaw.nl/DL/publications/PU00018278.pdf

Listing of Griss at this journal:

http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531

.