Pirates & Revolutionaries: principle of excluded middle

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Summary of

Daniel Smith

[Smith’s academia.edu page]

Essays on Deleuze

Ch.5

Pre- and Post-Kantianism

Logic and Existence: Deleuze on the Conditions of the Real

5.0

“[Introductory material]”

Brief summary (collecting those below):

(5.0.1) At the core of Deleuze’s interest in the rationalists, Leibniz especially, is the philosophical problem of using thought to distinguish the possible from the real. For, nothing would change among the predicates involved in the conception of something were it to exist instead of not exist (or not-exist instead of exist). (5.0.2) Smith will give a cinematic thematization of the material he presents in the text by characterizing its parts as if they were something like parts of a film. (5.0.3) The basic principles of logic, especially the three classic ones – Identity, Non-Contradiction, and Excluded Middle – tell us what is unthinkable and thus impossible. Under Deleuze’s formulations, the Principle of Identity is expressible as “A is A” or “A thing is what it is”; the Principle of Non-Contradiction is (unconventionally) formulated by Deleuze as “A is not non-A,” or “A thing is not what it is not”; and the Principle of Excluded Middle: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” They therefore offer some guidance regarding what is possible. Deleuze will conduct an investigation into the history of philosophy to see if these three laws especially had been reconceived to allow us to think beyond the possible to the real or existant itself. (5.0.4) Smith will follow how Deleuze tells a story in the history of philosophy that is about how philosophers of different times or sorts have reconceived the three classical principles of logic in order to think the real and existant itself and not merely the possible; the Principle of Identity: Pre-Kantian Rationalists, especially Leibniz; the Principle of Non-Contradiction: German Idealists, especially Hegel; and the Principle of Excluded Middle: “existentialists.” Lastly, Smith will explain Deleuze’s critique of these solutions and discuss the one Deleuze proposes instead.

5.0.1

[Deleuze’s Philosophical Question: How Can Thought Think the Real and Not Just the Possible?]

5.0.2

[Smith’s Cinematic Thematization]

5.0.3

[Deleuze’s Particular Formulations for the Three Classic Principles of Logic]

5.0.4

[Previewing the Text]

Bibliography

Summary

5.0.1

[Deleuze’s Philosophical Question: How Can Thought Think the Real and Not Just the Possible?]

[At the core of Deleuze’s interest in the rationalists, Leibniz especially, is the philosophical problem of using thought to distinguish the possible from the real. For, nothing would change among the predicates involved in the conception of something were it to exist instead of not exist (or not-exist instead of exist).]

[ditto] [Deleuze discusses this issue especially in his course of 1983.05.17, Part 1, Part 2, and Part 3.] As Smith explains so well, if we only have our thinking to rely upon, we cannot make a distinction, conceptually speaking, between the concept of a thing were it to exist and the concept of that same thing were it not to exist. [Deleuze says something similar to Smith’s 100 dollars and unicorn illustrations and Kant comment in Course 1983.05.17, Part 1: “Pourquoi est-ce que la pensée en tant que pensée ne dispose d’aucun moyen pour distinguer le possible et le réel ? C’est évident - ou : le possible et l’existant -, c’est évident si vous y réfléchissez. Considérez un concept quelconque, ou une représentation quelconque : représentation soit d’un bœuf, soit d’une chimère, soit d’un triangle. Cette représentation ou ce concept, c’est ce que la pensée pense. Rien n’est changé, que l’objet de la représentation existe ou n’existe pas. Tout est changé pour nous, rien n’est changé pour la pensée, c’est-à-dire : rien n’est changé dans la représentation. C’est ce que Kant disait déjà dans une page célèbre de la Critique de la raison pure. Vous vous faites la représentation de 100 francs - il disait, lui, pour des raisons de nationalité, 100 thalers. Vous vous représentez 100 francs : que ces 100 francs existent, bien plus, que vous les ayez ou que vous ne les ayez pas, c’est très important pour vous ; du point de vue de la représentation, rien n’est changé. Vous vous faites un concept de chimère, animal fabuleux. Qu’il y ait ou qu’il n’y ait pas des chimères, ça change beaucoup ; ça change rien du point de vue du concept, du point de vue de la représentation.” (00:07:17-00:09.32).] Let us work with Smith’s example of the unicorn. We have the idea of the unicorn. And we note that it does not exist. Suppose now that genetic scientists produce a unicorn (or suppose millions of years from now one evolves.) Would the existence of the unicorn change any of its conceptualizable properties? It would seem not. (It would still seem to be a horse with a horn. The fact that it does not presently exist does not change what defines it. It either could exist or it could not. Either way, a unicorn is still conceived as a horse with a horn pointing straight forward from its head.) Thus existence does not seem to be a predicate at least when it comes to predicates that we assign when conceiving concepts or ideas, especially in terms of their essential or defining features. This is because, as Smith explains, “The position of the real is outside the concept; the existing thing is external to the concept.” [Again, note Deleuze’s Course 1983.05.17, Part 1: “Ce qu’on a toujours traduit, en philosophie, en disant que l’existant, c’était extérieur à la représentation. L’existant, c’est la position de l’objet hors du concept. (00:09:35-00:09.44).]

Here is a philosophical problem that lies at the core of Deleuze’s interest in the rationalists, and particularly Leibniz.¹ By itself, thought has no means of distinguishing between the possible and the real. I can have a concept of 100 dollars in my mind, and while it may be important to me practically whether or not I actually have 100 dollars in my pocket, the existence of 100 dollars in reality changes nothing from the point of view of the concept: that is, from the viewpoint of pure thought. The position of the real is outside the concept; the existing thing is external to the concept. (This was Kant’s argument against the ontological argument: existence is not a predicate; from the viewpoint of the concept, an existing God is no more perfect than a non-existing God.) Even though I know that unicorns do not exist, I can still form a concept or a representation of a unicorn, or define the essence of a unicorn.

(72)

1. This paper was originally presented at the conference “Deleuze and Rationalism,” which took place on 16–17 March 2007 at the Centre for Research in Modern European Philosophy at Middlesex University, London.

(377)

5.0.2

[Smith’s Cinematic Thematization]

[Smith will give a cinematic thematization of the material he presents in the text by characterizing its parts as if they were something like parts of a film.]

In this paragraph, Smith explains different philosophical ways to approach this problem, and he describes and defends his cinematic thematization of the material. He firstly reformulates the philosophical problem Deleuze is working on here, namely: how can thought leave its domain of the possible and instead think the real itself (the existant)? He says “Pre-Kantians like Leibniz approached this problem in terms of the distinction between truths of essence (“A triangle has three sides”) and truths of existence (“Caesar crossed the Rubicon”), while post-Kantians like Maimon approached the problem in terms of the distinction between the conditions of possible experience and the conditions of real experience.” (I do not have textual substantiation for these claims, at the moment.) Smith next says that he will approach the problem from a “semi-cinematic perspective,” and he grounds this in a comment Deleuze makes regarding Godard making a film about philosophical texts. [Overall, what Smith is doing here is explaining why he will thematize the presentation of his text as if it were something like a film script. As we will see, it makes the text more playful and charming. It is not entirely obvious if it adds anything substantial to the philosophical material he presents, but it enlivens the text and makes it even more fun and exciting to read.]

For Deleuze, this is one of the fundamental problems of a theory of thought: How can thought leave this meager sphere of the possible in order to think the real: that is, to think existence itself, to think existing things. Pre-Kantians like Leibniz approached this problem in terms of the distinction between truths of essence (“A triangle has three sides”) and truths of existence (“Caesar crossed the Rubicon”), while post-Kantians like Maimon approached the problem in terms of the distinction between the conditions of possible experience and the conditions of real experience. I would like to approach this logical problem from a semi-cinematic perspective. “Theoretically,” Deleuze once mused, “Jean-Luc Godard would be capable of filming Kant’s Critique or Spinoza’s Ethics” (DI 141). In the 1990s, Godard did a multi-part film entitled Histoire(s) du cinéma; following Deleuze’s suggestion, I am imagining Godard undertaking a similar project entitled Histoire(s) de la philosophie. I have no idea, of course, what Godard might have done in such a film, but none the less I am presenting the first part of this essay as a possible scenario for a single sequence of that multi-part film, which has as its title Logic and Existence, which I am borrowing from a well-known book by Jean Hyppolite.²

(72)

2. Jean Hyppolite, Logic and Existence [1952], trans. Leonard Lawlor and Amit Sen (Albany: State University of New York Press, 1997). This book completes the project Hyppolite began with Genesis and Structure of Hegel’s “Phenomenology of Spirit,” trans. Samuel Cherniak and John Heckman (Evanston, IL: Northwestern University Press, 1979), and examines the relation between the phenomenology and the logic. Deleuze wrote an important review of the book in 1954, “Jean Hyppolite’s Logic and Existence,” which is included as appendix to the English translation (191–5). (377)

[DI: Gilles Deleuze, Desert Islands and Other Texts, ed. Sylvère Lotinger, trans. Michael Taormina (New York: Semiotext(e), 2004).]

5.0.3

[Deleuze’s Particular Formulations for the Three Classic Principles of Logic]

[The basic principles of logic, especially the three classic ones – Identity, Non-Contradiction, and Excluded Middle – tell us what is unthinkable and thus impossible. Under Deleuze’s formulations, the Principle of Identity is expressible as “A is A” or “A thing is what it is”; the Principle of Non-Contradiction is (unconventionally) formulated by Deleuze as “A is not non-A,” or “A thing is not what it is not”; and the Principle of Excluded Middle: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” They therefore offer some guidance regarding what is possible. Deleuze will conduct an investigation into the history of philosophy to see if these three laws especially had been reconceived to allow us to think beyond the possible to the real or existant itself.]

Smith next notes that we might turn to the principles of logic in order pursue this question of how can thought thing the real and existant? He then proceeds through the three classical principles of logic, identity, non-contradiction, and excluded middle. Smith will give verbal formulations for each of them. The Principle of Identity says that “A is A” or “A thing is what it is.” Smith notes that the next two can be seen as specifications of this principle [but in what way they are specifications is not explained yet]. The Principle of Non-Contradiction can be seen as saying that “A is not non-A,” or “A thing is not what it is not” [At this point, it is important that we take some care. To all appearances, Smith is closely following Deleuze’s course lecture of 1983.05.17 that we have and will continue to quote from. In this lecture, Deleuze verbally formulates the three classic principles of logic in a way that Smith also does here. But what I find problematic is Deleuze’s formulation of the Principle of Non-Contradiction. “A is not non-A” is not, as far as I know, the conventional way to formulate this principle, even going back to its early appearance in Aristotle. Normally we understand the Principle of Non-Contradiction as being verbally formulated as “It is not that A and not-A.” (See especially Graham Priest, Doubt Truth to be a Liar, 8-9). Deleuze’s and Smith’s formulation is much closer to the Principle of Double Negation (“A if and only if not not-A”) on account of the verb ‘to be’ normally functioning more like the biconditional operator. I have not discovered the source for Deleuze’s seeming confusion about how to formulate the Principle of Non-Contradiction. Given his later focus on Hegel when discussing non-contradiction, we might think that he is getting this formulation from Hegel. However, that is not how it seems to me at the moment. For instance, in Hegel’s Science of Logic, he gives a positive and negative formulation for the Principle of Identity: “the essential category of identity is enunciated in the proposition: everything is identical with itself, A = A. Or negatively: A cannot at the same time be A and not A” (Hegel, Science of Logic, 409). The negative formulation here is close to the normal formulation of the Principle of Non-Contradiction. Hegel later writes, “The other expression of the law of identity: A cannot at the same time be A and not-A, has a negative form; it is called the law of contradiction” (ibid., 416). Yet, Hegel does not next explain what the negative form of this expression is, which would presumably formulate precisely the Principle of Non-Contradiction. So he does not say here that it would be “A is not not-A,” as far as I can tell. I find this issue highly problematic for making more precise determinations of Deleuze’s logic. In other words, we cannot easily say, on the basis of what Deleuze says here, whether he rejects or accepts the Principle of Non-Contradiction. He will seemingly state that he accepts it, but what he would apparently be accepting rather is the Principle of Double Negation.] And the Principle of Excluded Middle is verbally formulable as: “‘either A or not-A,’ that is, between A or not-A, there is no middle term’.” These three laws tell us what is unthinkable, which thereby tells us what is impossible [presuming that anything unthinkable is impossible]. So, were something to break the Principle of Identity, then it would not be what it is. Were it to break the Principle of Non-Contradiction, then it would be what it is not. [This seems to me to be more a matter of breaking the Principle of Identity than of Non-Contradiction. But Deleuze is seemingly working with the Principle of Double Negation here instead of the Principle of Non-Contradiction. Hence, perhaps, this odd claim.] And finally, something breaks the Principle of Excluded Middle if it is “both what it is and what it is not.” [This seems to be what is more conventionally understood as breaking the Principle of Non-Contradiction. To break Excluded Middle, I think it would be something more like, “To be neither what it is nor what it is not.”] Deleuze’s question will be: “Is there any way in which these three classical principles can be used to exit the sphere of logic and penetrate existence itself?” [Again, see Course 1983.05.17, Part 1, at audio times (00:11:50-16:46): mais c’est très important de comprendre ça, c’est pour ça qu’il y a un problème de la pensée. Je dirais : le problème éternel de le pensée, ç’a été : moi, pensée, comment est-ce que je vais arriver à penser le réel et l’existant ? comment est-ce que je vais sortir de ma sphère des possibles ? comment penser autre chose que l’essence ? Je dirais presque, c’est à partir de là, bon... D’où... d’où, il me semble, la distinction de deux types de principes. La pensée par elle-même pense le possible. Au nom de quoi ? Au nom de certains principes qu’on appellera des principes logiques. Les principes logiques sont des principes qui fixent ce qui est possible et ce qui ne l’est pas ; qui déterminent ce qui est possible et ce qui n’est pas possible. Et ces principes logiques, je dirais : ce sont les principes des essences ou du possible, puisqu’ils discernent, ils distinguent le possible du non-possible ou de l’impossible, ces principes sont au nombre de trois dans la logique classique. → L’un, c’est le principe d’identité, A est A. Et puis deux petits principes qui semblent être comme des spécifications du grand principe d’identité, A est A, c’est-à-dire la chose est ce qu’elle est. → Second principe, dit de non-contradiction : A n’est pas non-A, la chose n’est pas ce qu’elle n’est pas. → Et puis troisième principe, dit du tiers-exclu : la chose est A ou non-A. Ou si vous préférez : entre A et non-A, il n’y a pas de tiers, d’où l’expression « principe du tiers exclu », A ou non-A. Ça m’intéresse déjà, parce que ces trois principes de pure logique, → l’un est un principe de position ou d’affirmation (A est A), → le second est un principe de négation (A n’est pas non-A), → le troisième est un principe d’alternative ou de disjonction (A ou non-A). Je sais donc ce qui est impossible, c’est-à-dire impensable. Ce qui est impossible ou impensable, c’est quelque chose qui ne serait pas ce qu’elle est (donc elle contredirait à l’identité), qui serait ce qu’elle n’est pas (elle contredirait à la non-contradiction), et qui serait à la fois ce qu’elle est et ce qu’elle n’est pas (elle contredirait au tiers exclu). Tout va bien. Sous ces trois principes, je pense les essences, le monde des essences ou le monde du possible, mais je retombe toujours là-dessus : comment penser quelque chose de réel ?]

Here’s the first shot: a radiant sphere hovering in the middle of nowhere. Nothing is written on it, but we know it is the sphere of logic. The film begins here | for an obvious reason: if thought, on its own, is only capable of thinking the possible, it does so on the basis of what can be called logical principles. Classical logic famously identified three such principles. These are the principle of identity (which says that “A is A,” or “A thing is what it is”), and then two smaller principles which seem to be specifications of the principle of identity: the principle of non-contradiction (which says that “A is not non-A,” or “A thing is not what it is not”) and the principle of the excluded middle (which says “either A or not-A,” that is, between A or not-A, there is no middle term). Taken together, these three principles determine what is impossible—that is to say, what is unthinkable without contradiction: something that would not be what it is (which would contradict the principle of identity); something that would be what it is not (which would contradict the principle of non-contradiction); and something that would be both what it is and what it is not (which would contradict the principle of the excluded middle). This sphere of logic would seem to enclose us within the domain of the possible, or what classical philosophy called the domain of essences. But this opening shot sets up the problem with a visual image: Is there any way in which these three classical principles can be used to exit the sphere of logic and penetrate existence itself?

(72-73)

5.0.4

[Previewing the Text]

[Smith will follow how Deleuze tells a story in the history of philosophy that is about how philosophers of different times or sorts have reconceived the three classical principles of logic in order to think the real and existant itself and not merely the possible; the Principle of Identity: Pre-Kantian Rationalists, especially Leibniz; the Principle of Non-Contradiction: German Idealists, especially Hegel; and the Principle of Excluded Middle: “existentialists.” Lastly, Smith will explain Deleuze’s critique of these solutions and discuss the one Deleuze proposes instead.]

Smith next outlines how the text will proceed. First he looks at how pre-Kantian rationalists, especially Leibniz, reconceive the Principle of Identity and extend it to the whole of existence. Secondly, he examines how German Idealists, especially Hegel, do this with the Principle of Non-Contradiction. And thirdly he looks at how “existentialist” sorts of philosopher do this with the Principle of Excluded Middle. He lastly will explain why Deleuze thinks they all fail and how he offers his own response to the problem.

The response to this question will take us through three scenes, which correspond to three broad sequences in the history of philosophy, three attempts to resolve this problem using one of these logical principles. Scene one focuses on the pre-Kantians, the rationalists; its star is Leibniz, since it was he who attempted to extend the principle of identity to the whole of existence. Scene two focuses on the post-Kantians, primarily the German Idealists; its story culminates in Hegel, since it was he who attempted to extend the principle of non-contradiction to the whole of existence. Scene three, finally, looks at that loosely related group of thinkers that often tend to be called, precisely, “existentialists,” since it is they who attempted to extend the principle of the excluded middle to existence. The screenplay reaches its climax with Deleuze: at the end, it briefly examines the reasons why Deleuze is at once fascinated with all three of these philosophical attempts to “think existence,” but none the less thinks they fail, and why he ultimately charts out his own response to the problem. The ending, alas, is somewhat truncated, since the production went over budget, which meant that entire scenes wound up being consigned to the editing room floor.

(73)

Bibliography:

Smith, Daniel. “Logic and Existence: Deleuze on the Conditions of the Real.” In Essays on Deleuze, 72–85. Edinburgh: Edinburgh University, 2012.

(or simply:)

Smith, Daniel. Essays on Deleuze. Edinburgh: Edinburgh University, 2012.

https://www.academia.edu/20805798/Essays_on_Deleuze

Note that an earlier version of this chapter text (which is nearly but not precisely identical) is found in:

Smith, Daniel. “Logic and Existence: Deleuze on the ‘Conditions of the Real.’” Chiasmi International 13 (2011): 361–77.

Smith’s Academia.edu page

Other sources, if otherwise noted:

Deleuze, Gilles. Course 1983.05.17, Part 1. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=236 ; https://www.webdeleuze.com/textes/204.

Deleuze, Gilles. Course 1983.05.17, Part 2. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=250 ; https://www.webdeleuze.com/textes/204.

Deleuze, Gilles. Course 1983.05.17, Part 3. Online recording at Bibliothèque nationale de France/Gallica; recording and transcript at La voix de Gilles Deleuze en ligne, Université Paris 8 (No transcriber is named); transcript at Web Deleuze (Transcription by François Zourabichvili. The Voix transcript is identical to Web Deleuze, which is perhaps the original source, and thus the Voix one perhaps is also transcribed by Zourabichvili). Paris, 1983. https://gallica.bnf.fr/ark:/12148/bpt6k128342x ; http://www2.univ-paris8.fr/deleuze/article.php3?id_article=251 ; https://www.webdeleuze.com/textes/204.

Hegel, G. W. F. Science of Logic. Translated by A. V. Miller. Oxford/ New York: Routledge, 2002.

Priest, Graham. Doubt Truth to Be a Liar. Oxford: Oxford University, 2006.

29 May 2019

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

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[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: E_n (1, 2, ..., n). We can further add an element n′, selfsame and distinguishable from all members p of E_n (1, 2, ..., n), so n′ ≠ p, p ≠ n′. They together form the set E_n′ (1, 2, ..., n′). We can note disjunctively that an element of E_n′ belongs to E_n 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 E_m (1, 2, . . . , m), then b together with the elements of E_m that are distinguishable from b form E_m.” (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 E_m holds: a ≠ c for each c ≠ b, then a = b.”

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)

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)

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: E_n (1, 2, ..., n). We can further add an element n′, selfsame and distinguishable from all members p of E_n (1, 2, ..., n), so n′ ≠ p, p ≠ n′. They together form the set E_n′ (1, 2, ..., n′). We can note disjunctively that an element of E_n′ belongs to E_n 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, p ≠ n′.” (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 E_n (1, 2, ..., n),” and instead of saying “the set {1, 2, …, n′)” we say, “the set E_n′ (1, 2, ..., n′).” So by adding n′ to the set E_n (1, 2, ..., n), we get the set E_n′ (1, 2, ..., n′). We thus call E_n′ the sum of E_n and n′. An element of E_n′ belongs to E_n 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 E_n′ belongs to E_n 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 E_n (1, 2, ..., n), we can again imagine an element n′, remaining the same, n′ = n′, and distinguishable from each element p of E_n (1, 2, ..., n), in formula n′ ≠ p, p ≠ n′. They form the set E_n′ (1, 2, ..., n′).”

E_n′ is called the sum of E_n and n′, in other words: An element of E_n′ belongs to E_n 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)

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 E_m (1, 2, . . . , m), then b together with the elements of E_m that are distinguishable from b form E_m.”]

[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 E_m (1, 2, . . . , m), then b together with the elements of E_m that are distinguishable from b form E_m.

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

(457)

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 E_m holds: a ≠ c for each c ≠ b, 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 E_m holds: a ≠ c for each c ≠ b, then a = b.

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

Bibliography:

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

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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 = b → b = 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.

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. [1] both a and b equal n′ (they both are 3). Thus the proposition holds, because the two elements are the same. [2] 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. [3] 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 [4], 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)

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. [1] 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.) [2] 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)

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 = b → b = 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.

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 = b → b = 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 = b → b = 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)

Bibliography:

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

1 Jan 2019

van Stigt (1.4.1) “Brouwer’s Intuitionist Programme” part 1.4.1, “Logic”, summary

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[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations. Bracketed comments are my own. Proofreading is incomplete, so please forgive my mistakes.]

Summary of

Walter P. van Stigt

“Brouwer’s Intuitionist Programme”

From Brouwer to Hilbert:

The Debate on the Foundations of Mathematics in the 1920’s

Part I.

L.E.J. Brouwer

Ch1.:

“Brouwer’s Intuitionist Programme”

1.4

“Mathematics, Language, and Logic”

1.4.1

“Logic”

Brief summary:

(1.4.1.1) Brouwer has a notion of “theoretical logic.” It is an application of mathematics in which one uses a “mathematical viewing” of a given mathematical record to see some regularity in that symbolic representation. (1.4.1.2) What we consider classical laws or principles of logic are regularities that we discern secondarily after genuine (intuitive) mathematical constructions are created. (1.4.1.3) But the operation of mathematics in the verbal or symbolic domain in fact is operating outside mathematical reality. And logic itself cannot generate new mathematical truths. In fact, we cannot even apply all the principles of logic to mathematics. Brouwer’s famous example is the invalidity of the Principle of the Excluded Middle (PEM) in mathematical applications. He identifies the Principle of the Excluded Middle with the principle of the solvability of every mathematical problem. Brower takes a strict interpretation of affirmation and negation: “True mathematical statements, affirmative and negative, express the completion of a constructive proof; in particular the negative statement expresses what Brouwer calls ‘absurdity,’ the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence” (9-10). But for infinite systems there is no absolute guarantee that a complete constructed affirmative proof or completed absurd construction can be formulated for them. (Thus, it can be that for such an infinite system, neither an affirmative nor a negative construction can be given for it. Now, the Principle of the Excluded Middle says that for any formula, it is either affirmative or negative, or understood another way, it is either true or false. But infinite systems are neither affirmable nor negatable.) Hence, “the Logical Principle of the Excluded Middle is not a reliable principle” (10).

1.4.1.1

[Theoretical Logic]

1.4.1.2

[The Secondariness of Principles or Laws of Logic to Mathematical Constructions]

1.4.1.3

[The Unreliability of the Principle of the Excluded Middle in Infinite Mathematical Systems]

Summary

1.4.1.1

[Theoretical Logic]

[Brouwer has a notion of “theoretical logic.” It is an application of mathematics in which one uses a “mathematical viewing” of a given mathematical record to see some regularity in that symbolic representation.]

[ditto]

The Foundations (B1907) defines “theoretical logic” as an application of mathematics, the result of the “mathematical viewing” of a given mathematical record, seeing a certain regularity in the symbolic representation: “People who want to view everything mathematically have done this also with the language of mathematics ... the resulting science is theoretical logic ... an empirical science and an application of mathematics ... to be classed under ethnography rather than psychology” (p. 129).

(9)

1.4.1.2

[The Secondariness of Principles or Laws of Logic to Mathematical Constructions]

[What we consider classical laws or principles of logic are regularities that we discern secondarily after genuine (intuitive) mathematical constructions are created.]

[ditto]

The classical laws or principles of logic are part of this observed regularity; they are derived from the post factum record of mathematical constructions. To interpret an instance of “law like behavior” in a genuine mathematical account as an application of logic or logical principles is “like considering the human body to be an application of the science of anatomy” (p. 130).

(9)

1.4.1.3

[The Unreliability of the Principle of the Excluded Middle in Infinite Mathematical Systems]

[But the operation of mathematics in the verbal or symbolic domain in fact is operating outside mathematical reality. And logic itself cannot generate new mathematical truths. In fact, we cannot even apply all the principles of logic to mathematics. Brouwer’s famous example is the invalidity of the Principle of the Excluded Middle (PEM) in mathematical applications. He identifies the Principle of the Excluded Middle with the principle of the solvability of every mathematical problem. Brower takes a strict interpretation of affirmation and negation: “True mathematical statements, affirmative and negative, express the completion of a constructive proof; in particular the negative statement expresses what Brouwer calls ‘absurdity,’ the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence” (9-10). But for infinite systems there is no absolute guarantee that a complete constructed affirmative proof or completed absurd construction can be formulated for them. (Thus, it can be that for such an infinite system, neither an affirmative nor a negative construction can be given for it. Now, the Principle of the Excluded Middle says that for any formula, it is either affirmative or negative, or understood another way, it is either true or false. But infinite systems are neither affirmable nor negatable.) Hence, “the Logical Principle of the Excluded Middle is not a reliable principle” (10).]

[ditto]

The cunning application of such principles in the verbal or symbolic domain produces nothing but “verbal edifices” outside the mathematical reality:

Linguistic edifices, sequences of sentences which follow one another according to the laws of logic .... Even if it appears that these edifices can never show up the linguistic figure of a contradiction, they are only mathematics as linguistic constructions and have nothing to do with mathematics, which is outside this edifice. (B1907, p. 132)

Brouwer reiterates Descartes’ observation that logic cannot generate new mathematical truths. In his “Unreliability of the Principles of Logic” (B1908C) he goes one step further and questions the validity of the principles of logic when applied to mathematics. To prove his general point he singles out the Principle of the Excluded Middle (PEM) – identified with the principle of the solvability of every mathematical problem – as flawed and an obvious misstatement of fact. The argument here is based on the lack of guarantee of a solution for an infinite system and on his strict interpretation of affirmation and negation. True mathematical statements, affirmative and negative, express the completion of a constructive proof; in partic- | ular the negative statement expresses what Brouwer calls “absurdity,” the constructed incompatibility of two mathematical constructions represented, respectively, by the subject and predicate of the sentence. In B1908C Brouwer simply states that for an infinite system there is no guarantee that such a construction can be completed and “that for infinite systems the Principle of the Excluded Middle is not a reliable principle” (p. 157). His major campaign against the use of the PEM starts after the publication of “The Foundations of Set Theory Independent of the Logical Principle of the Excluded Middle” (B1918B and B1919A), when in a number of papers he challenges the proofs of certain classical theorems, analyzes the logic of negation, and tries to prove the invalidity of the PEM by means of his counterexamples of essentially unsolvable mathematical problems (see further Section 4.1).

(9-10)