by Corry Shores
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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: 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 c ≠ b, then a = b.”
[Explanation of This Text]
[Brouwer’s Negationless Mathematics]
[(ad § 1.1.) Constructing Sets of Natural Numbers and Locating Them Disjunctively]
[(ad §1.2.) The First Property of Sets of Natural Numbers, Formulated Without Disjunction]
[The Second Property: Elements Sharing the Same Differences are The Same]
Summary
[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)
[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)
[(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, 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 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)
[(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)
[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 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 Em holds: a ≠ c for each c ≠ b, 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′ .
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
.
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