Pirates & Revolutionaries: Griss. G.F.C. Griss

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[Heyting, entry directory]

[Heyting’s ”G. F. C. Griss and His Negationless Intuitionistic Mathematics”, entry directory]

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

[Note, this post comes at the end of a series on negationless intuitionistic mathematics, and it in particular synthesizes all the other ones. See the collected brief summaries on this topic by Griss and Heyting.]

Summary of

Arend Heyting

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

“[Griss’ Negationless Mathematics and Real Numbers]”

Brief summary:

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

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.2, p.1132)

a ≠ c for each c ≠ b → a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3, p.1132)

] Rational numbers are defined as pairs of natural numbers. But real numbers are more complicated. They are defined as sequences of approximating intervals that converge upon a value. (They are Cauchy sequences of rational numbers that, as they go further down their sequence, form intervals between one another that eventually become arbitrarily small and convergent upon a particular value, which is the real number value expressed by that convergent series.

(Image from: Norman Wildberger, Math Foudations 111)

(Image source: wiki)

) (Following Edna Kramer, we could also call to mind another sort of narrowing, approximating intervals that are probably more intuitive for us, namely, the decimal expansion of a real number with non-terminating (and possibly non-repeating) decimals. Each additional decimal will have yet another coming after it. For instance, consider such a number that begins with 2 and will next have 2.6.

(Kramer, Nature and Growth of Modern Mathematics, section 2.x.1, p.34, boldface and underlining are mine)

That means previously at 2, it was really an interval spanning 2 and 3, because more precisely it will be at 2.6, which falls between 2 and 3. And after 2.6 is 2.63. So in fact, at 2.6 it is an interval between 2.6 and 2.7. Since yet another interval comes after the .63, that means it was an interval between 2.63 and 2.64. And so on. While this may not be a Cauchy sequence, it at least gives us the image of a series of narrowing, approximating intervals that ultimately converge upon a real number value, in a way that we are more familiar with.] Heyting calls such a series expressing a real number a “real number-generator”. When two such number-generators have terms (and approximating intervals) that all overlap, then they are the same. (Note: we are not yet at Griss’ definition of the equality of real numbers.) Next we will see Griss’ negationless conception of the inequality of two real numbers. The negational way that Griss rejects is to say that two real numbers are unequal if it is impossible that they are equal. For, this uses the negational notion of “impossibility” (and probably a reductio method of proof). Instead, the notion of inequality is understood positively as a distance or gap between them (between their approximating intervals). This apartness relation is symbolized with ‘⧣’. And it is defined in the following way: “two real numbers, defined by the number- generators a = {a_n} and b = {b_n} are apart from each other (a ⧣ b) if for some n, a_n and b_n are separated intervals” (Heyting 94). [Griss in one place words it: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another” (Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128). In other words, while two close real numbers may have many of their approximating intervals sharing common ‘space’, at some point along the sequence, the intervals will occupy space outside the other.

] So that defines the inequality of real numbers in a negationless, intuitionistic mathematics. But, the equality of two real numbers cannot then be defined negatively as the impossibility of their being apart. Instead, Griss defines the equality of two real numbers as their sharing distances to all the other real numbers. In Heyting’s wording: “if every real number c that is apart from a is also apart from b, then a = b” (94). [

]

Contents

[Griss’ Negationless Mathematics and Real Numbers]

Bibliography

Summary

[Griss’ Negationless Mathematics and Real Numbers]

[Griss, as a philosopher and mathematician, thought both theoretically about a negationless intuitionistic mathematics, and also constructed it formally. Griss constructed the natural numbers using a positive notion of difference (namely, being in a subset that is complementary to the other subset that contains all the rest of the numbers in the larger, whole set). Rational numbers are defined as pairs of natural numbers. But real numbers are more complicated. They are defined as sequences of approximating intervals that converge upon a value. (They are Cauchy series of rational numbers that, as they go further down their sequence, form intervals between one another that eventually become arbitrarily small and convergent upon a particular value, which is the real number value expressed by that convergent series.) Heyting calls such a series expressing a real number a “real number-generator”. When two such number-generators have terms (and approximating intervals) that all overlap, then they are the same. (We are not yet at Griss’ definition of the equality of real numbers.) Next we will see Griss conception of the inequality of two real numbers. The negational way that Griss rejects is to say that two real numbers are unequal if it is impossible that they are equal. For, this uses the negational notion of “impossibility” (and probably a reductio method of proof). Instead, the notion of inequality is understood positively as a distance or gap between them (between their approximating intervals). This apartness relation is symbolized with ‘⧣’. And it is defined in the following way: “two real numbers, defined by the number- generators a = {a_n} and b = {b_n} are apart from each other (a ⧣ b) if for some n, a_n and b_n are separated intervals” (Heyting 94). [Griss in one place words it: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another” (Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128).] So that defines the inequality of real numbers in a negationless, intuitionistic mathematics. But, the equality of two real numbers cannot then be defined negatively as the impossibility of their being apart. Instead, Griss defines the equality of two real numbers as their sharing distances to all the other real numbers. In Heyting’s wording: “if every real number c that is apart from a is also apart from b, then a = b” (94).]

[Oftentimes philosophical ideas encounter problems when applied to concrete problems. Griss, however, did both the philosophical and applicational work, being both a philosopher and mathematician: “after a short philosophical introduction he begins the construction of mathematics” (93). His main aim is to “to find a substitute for reasonings which involve negation; simply banishing these he would leave but insignificant ruins.” [Recall from section 0.2 of “Negationless Intuitionistic Mathematics, I” that Griss writes:

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

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

] For this, he needs to see how difference operates for different kinds of numbers. Heyting begins with Griss’ non-negational intuitionistic formulation of the natural numbers. Let us quote Heyting first, then we will examine Griss’ relevant texts:

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

(Heyting 93)

First we need to be clear about what constitutes a negative mathematical formulation, conception, or proof. As we will see, it is one that involves a conception of negating a verb/predicate or whole proposition. In section 0 of Griss’ “Negationless Intuitionistic Mathematics, I” he fashions two sorts of mathematical proofs, one using negation and another that is negationless (section 0.5). Both will show that a certain line bisecting a triangle (constructed according to certain proportional conditions) will be parallel to one of the sides of the triangle. The first proof is negational, because it uses a reductio argument. It assumes that this bisecting line is not parallel:

If DE was not parallel with AB, ...

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.5, p.1128)

Then it finds a contradiction, from which we infer that the lines are parallel. But for the negationless proof, we first construct the line in question. We next construct another line that we know to in fact be parallel to the triangle side. Finally we show that this parallel line must necessarily be identical to the line in question. Since the second line is parallel, and since it is identical to the line in question, then the line in question is therefore parallel (section 0.5). Furthermore, the mathematical notion of parallel lines can be given either a negative or positive definition (section 0.6). The negative definition of parallel lines is:

parallel lines (in a plane) are lines which do not intersect

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

Here we see that the predicate, “intersect” is negated as “do not intersect.” But the positive definition would be:

parallel lines are such lines, that any point of one of them differs from any point of the other one. And this, again, presupposes a positive definition (i.e. a definition without negation) of difference relative to points.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128, boldface and underlining mine, italics in the original)

[But how we get a positive notion of difference is something we deal with in a short while.] Again we see that positive notion of difference in the following reformulation:

Which rational numbers satisfy x²– 2 = 0? The answer must be: No rational number satisfies. The question has been put in the wrong way. The fact is that x²– 2 differs positively from zero for every rational number.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

We will return to the next example later, because it uses notions we will work toward progressively. Before we move on, we should note one of Heyting’s explanations for negation, coming from his Intuitionism: An Introduction. He begins with the following negational yet intuitionistic propositions:

If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a ≠ b.

Theorem 1. If a ≠ b is contradictory, then a = b [L. E. J. Brouwer 1925, p . 254].

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BROUWER, L. E. J.

1925. Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht deutsch. Math. Ver. 33, p. 251–256.

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(Heyting, Intuitionism: An Introduction, sections 2.2.2.1 and 2.2.2.2, p.17 [Bib p.124])

Heyting then clarifies that there are two sorts of negation in this intuitionistic context. One is simply de facto negation, meaning that we do not yet have a proof for something. This is just lacking a proof. (On this “weaker” sort of negation, see Mancosu and van Stigt, From Brouwer to Hilbert, section 1.5.1.5.) The second kind is a de jure negation, which means you have a proof from which you can infer that some proposition is false. This is a disproof. (On this sort of stronger “Brouwer negation,” see see Mancosu and van Stigt, From Brouwer to Hilbert, sections 1.5.1.3 and 1.5.1.4.) But it is important to note that such a disproof is not formed by means of a reductio argument where the proposition is negated. So it is not like the reductio argument we noted above with the parallel dividing line of the triangle in Griss’ example. But in the non-negational case, after establishing that the line is parallel, you could then say, perhaps, that this constructed positive proof can serve to disprove that they are not non-parallel. As Mancosu and van Stigt explain in section 4.1.3 of From Brouwer to Hilbert:

“Intuitionist Splitting” is in fact such an exercise of creating new words, in this case words expressing the various relations between points and between points and species of points. In line with his own rules of correct logical practice, Brouwer starts from his concepts of mathematical truth and absurdity (i.e., proven impossibility), resulting immediately in the inapplicability of the Principle of the Excluded Middle and of what he calls “The Principle of Reciprocity of Complementary Species,” which asserts the equivalence of truth and double negation. He replaces the latter principle by a restricted form of complementarity: “Truth implies absurdity-of-absurdity, but absurdity-of-absurdity does not imply truth.”

(Mancosu and van Stigt, From Brouwer to Hilbert, section 4.1.3, p.276)

And from section 4.2.1:

In particular the excluded middle and the principle of double negation were singled out as especially problematic. By contrast, Brouwer remarked that the intuitionist accepts the following principles: A → ¬¬A

(Mancosu and van Stigt, From Brouwer to Hilbert, section 4.2.1, p.274)

Thus Heyting writes:

Strictly speaking, we must well distinguish the use of “not” in mathematics from that in explanations which are not mathematical, but are expressed in ordinary language. In mathematical assertions no ambiguity can arise: “not” has always the strict meaning. “The proposition p is not true”, or “the proposition p is false” means “If we suppose the truth of p, we are led to a contradiction”. But if we say that the number-generator ρ which I defined a few moments ago is not rational, this is not meant as a mathematical assertion, but as a statement about a matter of facts; I mean by it that as yet no proof for the rationality of ρ has been given. As it is not always easy to see whether a sentence is meant as a mathematical assertion or as a statement about the present state of our knowledge, it is necessary to be careful about the formulation of such sentences. Where there is some danger of ambiguity, we express the mathematical negation by such expressions as “it is impossible that”, “it is false that”, “it cannot be”, etc., while the factual negation is expressed by “we have no right to assert that”, “nobody knows that”, etc.

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There is a criterion by which we are able to recognize mathe- | matical assertions as such. Every mathematical assertion can be expressed in the form: “I have effected the construction A in my mind”. The mathematical negation of this assertion can be expressed as “I have effected in my mind a construction B, which deduces a contradiction from the supposition that the construction A were brought to an end”, which is again of the same form. On the contrary, the factual negation of the first assertion is: “I have not effected the construction A in my mind”; this statement has not the form of a mathematical assertion.

(Heyting, Intuitionism: An Introduction, sections 2.2.2.9 and 2.2.2.10, pp.18-19)

So to be clear, Griss thinks that all proofs must be positively constructive and all properties must be positively conceived and stated. This matter of positive conceptions we turn to now, and it brings us to the part of our current Heyting text we left of at a while ago, namely, the part reading,

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

(Heyting 93)

Griss writes in “Negationless Intuitionistic Mathematics, I”:

To construct negationless mathematics one must begin with the elements and a positive definition of difference must be given instead of a negative one (ex. 1 and 3).

But even from a general intuitionistic point of view a positive construction of the theory of natural numbers must be given: one cannot define 2 is not equal to 1 (i.e. it is impossible that 2 and 1 are equal), for from this one could never conclude that 2 and 1 differ positively.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 0.10, p.1130, boldface mine, italics in the original)

So we are first going to define the natural numbers in a positive, constructive way. But this will require a non-negational notion of difference. We will see how Griss does this also with the real numbers, and Deleuze uses Heyting’s formulation of it in Difference and Repetition. Griss begins with two primitive notions, being identical and being distinguishable (see section 3.2 of Griss’ “Logic of Negationless Intuitionistic Mathematics”). We need a positive definition of difference to define the natural numbers, because each one needs to be different from the others. It will be based on distinguishability, which is not defined, as we just noted. However, it will be given a conceptual formation and precise mathematical formulation using notions of sets or “species.” We begin by imagining a selfsame object.

Imagine an object, e.g. 1. It remains the same, 1 is the same as 1, in formula 1 = 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.1, p.1131)

Next we imagine another object that is distinguishable from the first one. We call it 2.

Imagine another object, remaining the same, and distinguishable ⁴) from 1. e.g. 2; 2 = 2; 1 and 2 are distinguishable (from one another), in formula 1 ≠ 2, 2 ≠ 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.2, p.1131)

They are distinguishable from one another, and they form a set. So if we can distinguish one of them from 1, then it is 2, and vice versa.

They form the set {1, 2}; 1 and 2 belong to the set. If conversely an object belongs to this set, it is 1 or 2. If it is distinguishable from 1, it is 2; if it is distinguishable from 2, it is 1.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.3, p.1131)

We repeat this for 3, giving us the set {1, 2, 3} (see section 1.1.5). What is important here is that we can now regard this set as being made of complementary sets. We say that if an item is distinguishable from each element in the set {1, 2}, then it is 3, etc.

They form the set {1, 2, 3}. If an element belongs to {1, 2, 3}, it belongs to 1, 2 or it is 3. If it is distinguishable from each element of {1, 2}, it is 3; if it is distinguishable from 3, it is an element of {1, 2}.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.5, p.1131)

We can keep adding members, going up to some number n: {1, 2, ... , n}, and we can always imagine an additional n :

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′.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.1.6, p.1131)

They form the set {1, 2, …, n′}. If an element belongs to {1, 2, …, n′}, it belongs to {1, 2, ... , n} or it is n′. 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}.

(1131)

At this point we need to emphasize very strongly that one element of this intuitionistic rejection of negation is a disjunctive sort of exclusion. Let us move to Griss’ “Negationless Intuitionistic Mathematics, II”, where he addresses this. But we need to modify our notation a little, although it will yield the same structure above, namely, we will call the set of natural numbers {1, 2, …, n} as E_n (1, 2, ..., n), and {1, 2, …, n′} as E_n′ (1, 2, ..., n′). I will begin with the full quote, and then we will analyze the key parts, as we need this for filling out our notion of negation:

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″.

(Griss, “Negationless Intuitionistic Mathematics, II”, section 1.0.3, pp.456-457)

What we have seen in these Griss texts is that what makes a number be unique is if it can be distinguished from the set of all the remaining numbers, within the larger set it is a part of. And for the larger whole set to be divided in this way, we need a disjunction that says a number is either in the one or in the other. Again:

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.

Now let us narrow in on the key passage (again):

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.

The usual meaning of ‘a or b’ is: either a is true or b is true. That much is fine. But he claims that this involves a conception of untrue assertions. The problem is that he never mention untrue assertions or falsity in that formulation. What might it be? It would seem to be a sort of disjunctive syllogism where the untruth of a allows us to know the truth of b. He does not say this. But as we will see with his own definition, he will convert this disjunction into a conjunction of mutually affirmative conjuncts, even though it articulates a distinction between the conjoined parts. He writes (again):

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 to be clear, disjunction here is not understood in the classical, negational sense as meaning that the falsity of one disjunct entails the truth of the other. [In intuitionistic negationless mathematics, we cannot conceive a falsity and we cannot blindly assert it. So perhaps this clarifies two reasons for the intuitionistic prohibition of the principle of excluded middle, at least in this negationless mathematics context. The first is that we can have a proposition and its negation both being false (in the weak sense), if it has not yet been proven. And so the falsity of one cannot be seen as exclusive to the falsity of the other. The second is that we cannot conceive a (strong) falsity in the first place. We can only infer it by means of constructive, positive proofs. Also note this quotation from Griss:

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”).

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

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

There Griss explains why there can be no null-set or property. We cannot conceive of a property that no thing can have.] We thereby can understand disjunction as a conjunction of terms that are in different sets or that thus have different properties. However, the exclusive element here is built into the notion of complementarity, but it is not initially conceived as such. First we say that a is in one subset and thus has some property, then we say b is in another subset and thus has some other property, and finally, the fact that one plus the other makes the larger set entails the members of one not being in the other. We did not begin with that exclusionary notion or definition of complementarity, although we arrived upon it. We will next look at how equality can be defined in a negationless way. The negational definition for equality says,

If it is impossible, that a is not the same as b, then a is the same as b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3, p.1132)

[Note that there is a similar one for inequality:

“a est différent de b”, (a ≠ b), signifie, dans la terminologie de BROUWER, que a = b est impossible.

(Heyting, Les fondements des mathématiques, section 5.3.1.1, p.24)

] Griss reformulates the above negational definition for equality into the following negationless kind:

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

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.2, p.1132)

a ≠ c for each c ≠ b → a = b.

(Griss, “Negationless Intuitionistic Mathematics, I”, section 1.2.3, p.1132)

So we begin with two unidentified numbers, and we want to know if they are equal. If they are both distinguishable (unequal) from precisely every other same number in the set, then they are equal to one another. In other words, two numbers are equal if they share the same differences or distinguishability relations to the other members. This means that they stand outside the set of all the other numbers but that one. If we think of a simplistic case where we have three numbers, with 1 and 3 being included, a and b would be equal if they are each different from those other terms.

So this completes the section on natural numbers. We see now that in negationless intuitionistic mathematics, natural numbers can be constructed member-by-member in a positive way on the basis of a distinguishability from the all other natural numbers already in the set. And, this is not a matter of not being in the other set, but rather of being in the additional set, which, when combined with the first, completes the whole set. Furthermore, properties like equality can be defined without the notion of an impossibility of it being otherwise but rather as an affirmation of all their shared differences or distinguishabilities to the other numbers. Let us return to the text at hand, picking up where we left off:

For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept.

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But matters are more difficult for rational numbers. Yet, we will need this conception if we want to fully grasp Deleuze’s interest in negationless intuitionistic mathematics.

For real numbers the case is different. A real number is defined by a convergent, contracting sequence of rational intervals; for the sake of brevity I shall call such a sequence a number-generator. Two number-generators a = {a_n} and b = {b_n} coincide, if a_n and b_n overlap for every n. Coinciding number-generators define the same real number;

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We will need to unpack this. The first concept is rational numbers, which are ones that can be expressed as an integer over an integer. (See Wildberger, Math Foundations 13, 01.30.) [Perhaps this is what Heyting means when he says that they “are defined as pairs of natural numbers”, but I am not sure.] The next concept is real number, which will get a special definition here. [For a discussion of the conventional definitions, see Wildberger’s Math Foundations 115, 02.50. Recall that real numbers can be expressed in decimal form, whether it be terminating or not, and repeating or not; that real numbers include the rational and the irrational; and that real numbers are ones that can be understood, as wikipedia says, as a “value of a continuous quantity that can represent a distance along a line.”] Heyting defines a real number as “a convergent, contracting sequence of rational intervals”, and he construes them as “real number-generators” (or just “number-generators” in this context). So what is a number-generator? In Heyting’s Intuitionism: An Introduction, section 2.2.1.4, he says that they are Cauchy sequences of rational numbers.

Definition 1. A Cauchy sequence of rational numbers is a real number-generator.

(Heyting, Intuitionism: An Introduction, section 2.2.1.4, p.16)

What is a Cauchy sequence? Here is Heyting’s definition:

A sequence {a_n} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k), such that |a_n_+p – a_n| < 1/k for every natural number p.

(Heyting, Intuitionism: An Introduction, section 2.2.1.2, p.16)

This is quite complex. Generally speaking, a Cauchy sequence is one with a series of rational numbers that progressively tend toward an ultimate value, with the gap between successive numbers narrowing upon that ultimate value. Wildberger, in Math Foundations 111.6, shows this gradual, interchanging convergence of the values with this diagram:

(Image from: Norman Wildberger, Math Foudations 111)

The green line is the value that the series of rationals are tending toward. The idea was that no matter how small an interval you choose, you will be able to find a place in the sequence after which the gaps between successive values (the space above and below the green line) will be less than that arbitrarily small interval. This implies that it is always moving toward some specific value (the green line) that it converges upon. So let us look again at the more formal definition again:

‘A sequence {a_n} of rational numbers is called a Cauchy sequence, if for every natural number k we can find a natural number n = n(k), such that |a_n_+p – a_n| < 1/k for every natural number p.’

(Heyting, Intuitionism: An Introduction, section 2.2.1.2, p.16)

Here, the 1/k is the arbitrarily small interval. The larger the k value, the smaller the interval. The definition here says that no matter how large the k value (and thus no matter how small the interval), there will be some point along the sequence, some n^thterm, after which no matter what further point you select (no matter what p), the difference between successive terms will be smaller than that arbitrarily small interval. We see that narrowing of values also in this diagram from wikipedia of a Cauchy sequence:

(Image source: wiki)

We might also think of this narrowing of intervals in a related (but probably not equivalent) way as a progressive determination of intervals in a decimal expansion, which Edna Kramer does in Nature and Growth of Modern Mathematics, section 2.x.1. There we said that a real number can be considered as a series of approximating intervals, getting smaller and smaller, and converging upon a particular point on the number line (and thus to an exact value), even if the decimals are non-terminating and non-repeating. Each new decimal, when taken along with the decimal value of one higher, creates an interval, with each one being nested within the prior one and all shrinking down to a particular point. She writes:

2.6314 ... . The decimal gives us a sequence of rational approximations to the real number, namely, 2, 2.6, 2.63, 2.631, 2.6314, ... . In other words, the first approximation in the sequence places the real number in the interval (2, 3), and then 2.6 gives the approximating interval (2.6, 2.7), etc. Thus we have the sequence of nested intervals, (2, 3), {2.6, 2.7), (2.63, 2.64), ... , illustrated in Figure 2.9.

The adjective nested describes the fact that each interval lies within the preceding one. We observe also that the lengths of successive intervals are. 1, 0.1, 0.01, 0.001, 0.0001, ... . Since we are considering a nonterminating decimal, the nest of intervals will ultimately contain an interval of length 0.000 000 001 and then there will be still smaller intervals, so that interval length shrinks toward zero. As the innermost intervals get smaller and smaller, one can imagine their bounding walls approaching collision or, at any rate, getting close enough to “trap” a point of the number line. It is postulated, that is, assumed, that there is a unique point contained in all intervals of the nest. If there is such a point, we see that it must be unique, for if there were another distinct point, it would be separated from the first by some distance, 0.000 01, say. But ultimately some interval of the nest will be smaller than that number, and the first point must be contained in that very small interval. Then the second point would be too far away to be inside the interval and hence would not be contained in every interval of the nest. Since every nonterminating decimal will give rise to a sequence of nested intervals like the one described, there will always be a unique point of the number line corresponding to every real number.

(Kramer, Nature and Growth of Modern Mathematics, section 2.x.1, p.34, boldface and underlining are mine)

So a real number-generator is a real number as defined as being a Cauchy sequence of rationals, in other words, as a series of approximating intervals narrowing down and converging upon a singular value, even if the decimal expansion is non-terminating and non-repeating. Let us return to our current text and pick up on the next notion:

Two number-generators a = {a_n} and b = {b_n} coincide, if a_n and b_n overlap for every n. Coinciding number-generators define the same real number

(Heyting 93)

So here we see that two number-generators (two real numbers) coincide if the series “overlaps” for every term in the series. This brings us to some complexities, but we will simplify them eventually. In Heyting’s Intuitionism: An Introduction, section 2.2.1.5, he defines the identity of number-generators in a similar way:

Two number-generators a ≡ {a_n} and b ≡ {b_n} are identical, if a_n = b_n for every n. We express this relation by a ≡ b. The following notion of coincidence is more important.

(Heyting, Intuitionism: An Introduction, section 2.2.1.5, p.16)

Here the difference is that instead of every term “overlapping,” they are identical. But his definition of coinciding number generators in this other text is also similar, but it is technical and not entirely within my grasp:

The number-generators a ≡ {a_n} and b ≡ {b_n} coincide, if for every k we can find n = n(k) such that |a_n_+p – b_n_+p| < 1/k for every p. This relation is denoted by a = b.

(Heyting, Intuitionism: An Introduction, section 2.2.1.6, p.16)

In other words, perhaps, although the terms of the two sequences may not be identically the same, if they are coincident, then they still converge upon the same value, and this is because, after a certain point, their corresponding n^th terms will always fall within a gap smaller than any arbitrarily given one. For, it is saying |a_n_+p – b_n_+p| < 1/k for every p. In other words, his notion of the n terms “overlapping” may be made more mathematically precise, even though I am not exactly sure about its meaning. Or maybe what he is calling coincide here and “overlap” are equivalent to being identical and equaling, in the technical definitions. At any rate, we can say that one way or another, two number-generators coincide, and thus express the same value, when their series of terms are at least arbitrarily close if not equal. We will now look at his technical definition for the apartness relation.

‘For real number-generators a and b, a lies apart from b, a ⧣ b, means that n and k can be found such that |a_n_+p – b_n_+p| > 1/k for every p.’

(Heyting, Intuitionism: An Introduction, section 2.2.3.1, p.19)

It seems to mean that two real number-generators are apart if after some nth term in their series, the succeeding corresponding terms will always be separated by some gap and thus each number-generator is converging upon a different value. And we symbolize the apartness relation between a and b as: a ⧣ b. Let us look at some other definitions, moving to the simplest. This is from Heyting’s Les fondements des mathématiques, section 5.3.1.1:

a est différent de b”, (a ≠ b), signifie, dans la terminologie de BROUWER, que a = b est impossible. Pour le continu, on a en outre la relation “a est positivement différent de b” ou “a est | écarté de b” (a ⧣ b). Celle-ci est remplie quand, dans les suites d’intervalles qui définissent a et b, on connaît deux intervalles extérieurs l’un à l’autre.

(Heyting, Les fondements des mathématiques, section 5.3.1.1, p.24-25)

Here we define the apartness in terms of being “positively different” or being “apart from (écarté de).” a and b are apart when the series of intervals that define a and b, two external intervals can be found from one to the other. The series of intervals here seems to be the narrowing approximations we mentioned earlier. If there is an external interval or gap between a’s and b’s internal intervals, then they lie apart.

Now we will give a positive definition of their equality. So again, rather than saying (or proving) that their inequality is impossible, we will consider a positive formulation.

Dans la théorie des nombres réels la relation ≠, étant négative, n’intervient pas. Il n’y a que la relation a = b et la relation de distance a ⧣ b (voir ci-dessous “calcul numérique”). Le théorème “si a ≠ b est impossible, on a a = b” est remplacé par le suivant : “si a est distant de tout nombre c qui est distant de b, on a a = b”.

(Heyting, Les fondements des mathématiques, section 5.1.1.1, p.14)

So if number a is distant to all numbers c, which themselves are distant to b, then a equals b.

For Griss’ formulations, we return to the other example in “Negationless Intuitionistic Mathematics, I” that in the above we set aside temporarily. Here he gives a formulation for different real numbers (in boldface):

Has the equation ax + by = 0 a solution for x and y, different from zero. i.e. a solution with at least x or y different from zero? The letters represent real numbers.

In intuitionistic mathematics they make a distinction between positively and negatively different with regard to real numbers. Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another; they differ negatively, if it is impossible that they are equal; you can only divide by a real number if it differs positively from zero. In negationless mathematics the idea negatively different is, of course, omitted. Therefore we mean henceforth by different positively different.

[...]

The result is:

ax + by = 0 has a solution different from zero, if at least one of the coefficients a and b differs from zero or if both are zero.

(Griss’ “Negationless Intuitionistic Mathematics, I”, section 0.6, p.1128)

So again: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another.” Here, perhaps, we are saying that two real numbers are different if they have approximating intervals, perhaps something like parts of that triangular sort of shape of narrowing intervals in the Cauchy sequence, where one lies completely outside the other. So while two close real numbers may have many approximating intervals that overlap, at some point down the chain, there will be ones that do not overlap, that is to say, they lie completely apart from one another. Thus we can return to our current Heyting text:

Now the notion of different real numbers occurs in intuitionistic mathematics in two ways. In the first place it can be defined as meaning simply the negation of equality: two real numbers are unequal if it is impossible that they are equal; in the second place it can be defined in a positive way: two real numbers, defined by the number- generators a = {a_n} and b = {b_n} are apart from each other (a ⧣ b) if for some n, a_n and b_n are separated intervals. Of course the second definition must be so understood, that we can actually find the number n. For Griss the first definition is useless, so he defines the relation of difference between real numbers as being that of apartness.

(94)

So here we see that two real numbers (two real number-generators) are apart from each other if there is some approximating interval that is separate from the corresponding one in the other series. Heyting then discusses a problem that I do not quite get. But I think it may be the following. Above, we defined inequality non-negatively, as being apartness. But, Heyting notes, there is then the danger of defining equality negationally as the impossibility of being apart. Instead, Heyting explains, Griss offered a positive formulation like we saw above, namely, two real numbers are equal if they are both apart from all the other real numbers:

One of the main properties of the apartness relation is: if it is impossible that a ⧣ b, then a = b. This contains again the negation and hence must be replaced by a positive property. Griss found out that the following can take its place: if every real number c that is apart from a is also apart from b, then a = b. Let us call this property E.

(93-94)

Here is the full quote.]

The touchstone of a philosophical conception on the foundation of science is the actual development of the science in question on the basis of that conception; even the philosophical ideas which are involved gain in clearness and determination by their application to concrete problems. Too often philosophers content themselves with general ideas and leave the elaboration to specialists in the science; but in most cases the real difficulties occur in the application. Griss had the advantage to be at the same time a philosopher and a mathematician; after a short philosophical introduction he begins the construction of mathematics. Of course the main problem is to find a substitute for reasonings which involve negation; simply banishing these he would leave but insignificant ruins. In the first place the notion of difference must be examined, for each sort of mathematical entities separately. For natural numbers the notion of difference is only apparently negative; in the concept of natural number that of different natural numbers is enclosed, and after two natural numbers have been defined, we are always able to decide either that they are equal or that they are different. Hence difference for natural numbers is a positive concept. For rational numbers, which are defined as pairs of natural numbers, there is no more difficulty. For real numbers the case is different. A real number is defined by a convergent, contracting sequence of rational intervals; for the sake of brevity I shall call such a sequence a number-generator. Two number-generators a = {a_n} and b = {b_n} coincide, if a_n and b_n overlap for every n. Coinciding number-generators define the same real number; thus a real number may be defined as the class (in Brouwerian terminology the species) of number-generators which coincide with a given number-generator. All this is the same as in classical mathematics and has nothing to do with intuitionism or negation. Only Brouwer gave a larger interpretation of the word “sequence”; for reasons which I cannot explain here he admits that the members of a sequence are not determined | beforehand by some fixed law, but that they become determined one after the other, no matter how, for instance by free choices. Now the notion of different real numbers occurs in intuitionistic mathematics in two ways. In the first place it can be defined as meaning simply the negation of equality: two real numbers are unequal if it is impossible that they are equal; in the second place it can be defined in a positive way: two real numbers, defined by the number- generators a = {a_n} and b = {b_n} are apart from each other (a ⧣ b) if for some n, a_n and b_n are separated intervals. Of course the second definition must be so understood, that we can actually find the number n. For Griss the first definition is useless, so he defines the relation of difference between real numbers as being that of apartness. But here a new difficulty arises. One of the main properties of the apartness relation is: if it is impossible that a ⧣ b, then a = b. This contains again the negation and hence must be replaced by a positive property. Griss found out that the following can take its place: if every real number c that is apart from a is also apart from b, then a = b. Let us call this property E.

(93-94)

[contents]

Bibliography:

Heyting, Arend. “G. F. C. Griss and His Negationless Intuitionistic Mathematics.” Synthese 9, no. 2 (1953-1955): 91–96.

6 Jun 2019

Griss (3.2) “Logic of Negationless Intuitionistic Mathematics”, Section 3.2, “[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]”, summary

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

[Griss, “Logic of Negationless Intuitionistic Mathematics”, entry directory]

[Griss, entry directory]

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

Summary of

George François Cornelis Griss

(G.F.C. Griss)

“Logic of Negationless Intuitionistic Mathematics”

“§3. Conditions for the existence of the complementary species and the
inter section”

3.2

[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]

Brief summary:

(3.2) We take two notions to be equally fundamental [and primitive]: being identical and distinguishability. We begin with a set u that has at least two distinguishable elements. “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” [So a is a proper subspecies if it is a set that contains just members of u but not all of them.] Then, the complementary species or compliment as those other u elements that are the remainder: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and it results from the rejection of there being any empty species.

Contents

[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]

Bibliography

Summary

[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]

[We take two notions to be equally fundamental [and primitive]: being identical and distinguishability. We begin with a set u that has at least two distinguishable elements. “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” [So a is a proper subspecies if it is a set that contains just members of u but not all of them.] Then, the complementary species or compliment as those other u elements that are the remainder: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and it results from the rejection of empty species.]

[We take the notion of being identical as fundamental. (So we assume that it is a matter of sameness or perhaps as having the traditional properties of reflexivity, symmetry, and transitivity. Perhaps we are just saying it is a primitive notion that is expressed using the = sign.) We also take the notion of distinguishability as equally fundamental. (It seems we would assume that there are things that we can distinguish from one another, meaning that they have some kind of uniqueness in relation to other things, or a separation of some sort from them.) We begin with a set u, and we will assume that it has at least two distinguishable elements. There are no empty subsets (subspecies). We next define proper subspecies: a is a proper subspecies of u if it is a subset of u where some element(s) of u are distinguishable from those of a (and thus lie outside it): “a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a.” So if we have a proper subspecies a of u, that means there is a set of u members outside of a but that in addition to a complete the set u. We define the complementary species or compliment as those other u elements: “If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint.” (Disjoint here might be a non-negational way of dealing with disjunction like we saw in section 1.0.3 of Griss’ “Negationless Intuitionistic Mathematics, II”. For the two subspecies to be disjunct, that means an item is in either one or the other. What is excluded from this conception is a disjunctive synthesis whereby we would say that we know an that an item is in one subspecies on account of it not being in the other.) In order for two sets to intersect, a ∩ b, they need to share at least one common element, which is called the touch condition, a χ b, and this results from the rejection of there being empty species.]

In negationless intuitionistic mathematics the notion of distinguisha- | bility is equally fundamental as the notion of identity. In the following we shall suppose that u contains at least two distinguishable elements. Then we can define: a proper subspecies a of u is a subspecies so that at least one element of u is distinguishable from all elements of a. If a is a proper subspecies of u, the complementary species (complement) ¬a is the species of all elements that are distinguishable from the elements of a. Each element of a is distinguishable from each element of ¬a, a and ¬a are disjoint. a ≠ u, in words: “a is a proper subspecies of u” is the condition that is necessary to form the complement ¬a. There is also a condition for the existence of an intersection a ∩ b, a so-called touch condition, a χ b ⁴) expressing that a common element of a and b can be indicated. The appearance of these two conditions, a ≠ u and a χ b, is essential in negationless mathematics. It results from the rejection of empty species.

(44-45)

⁴) “Condition de composabilité” in the papers quoted sub ¹).

(45)

¹) PAULETTE DESTOUCHES-FÉVRIER; Logique de l'intuitionisme sans négation et logique de l'intuitionisme positif, C. R. de l'Ac. des Sc. Paris, 226 (1948); RENAUD DE BENGY-PUYVALLÉE, Sur les règles de composabilité dans la logique de la mathématique intuitioniste sans négation. C. R. de l'Ac. des Sc. Paris, 226 (1948).

(41)

[contents]

Bibliography:

Griss, G.F.C. “Logic of Negationless Intuitionistic Mathematics.” Indagationes Mathematicae (Proceedings) 54 (1951): 41–49.

Griss (ED) “Logic of Negationless Intuitionistic Mathematics”, entry directory

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

[Griss, entry directory]

[You can find the collected brief summaries at that link.]

Entry Directory for

George François Cornelis Griss

(G.F.C. Griss)

“Logic of Negationless Intuitionistic Mathematics”

“§3. Conditions for the existence of the complementary species and the
inter section”

[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]

Griss, G.F.C. “Logic of Negationless Intuitionistic Mathematics.” Indagationes Mathematicae (Proceedings) 54 (1951): 41–49.

(CBS) Negationless Intuitionistic Mathematics, collected brief summaries

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

Griss’ “Negationless Intuitionistic Mathematics, I”, entry directory

[Griss, entry directory]

[The following collects brief summaries of select texts. The entry directories without the brief summaries are located here:

Griss’ “Negationless Intuitionistic Mathematics, II”, entry directory

Heyting’s Intuitionism: An Introduction, entry directory

Heyting’s Les fondements des mathématiques, entry directory

]

[One of these posts itself contains a synthesis of all the other ones:

Arend Heyting. “G. F. C. Griss and His Negationless Intuitionistic Mathematics”, section 4, “[Griss’ Negationless Mathematics and Real Numbers]”

]

Collected Brief Summaries for the

Topic:

Negationless Intuitionistic Mathematics

George François Cornelis Griss

(G.F.C. Griss)

“Negationless Intuitionistic Mathematics, I”

“Introduction”

(0.1) Griss will discuss negationless intuitionistic mathematics. (0.2) In intuitionistic mathematics, we have philosophical reasons for needing to reject negation. For, “Proving that something is not right, i.e. proving the incorrectness of a supposition, is no intuitive method. For one cannot have a clear conception of a supposition that eventually proves to be a mistake. Only construction without the use of negation has some sense in intuitionistic mathematics” (1127). (0.3) From this point forward, we no longer consider our philosophical justifications for negationless intuitionistic mathematics and instead we are concerned with the purely mathematical problem of formulating it. (0.4) We begin with some examples. (0.5) Griss gives an example to show two ways to construct proofs. {1} The first one uses negation: it is a reductio argument, so it negates the conclusion. The premise is that we have a triangle ABC whose CA and CB sides are bisected by a fourth line DE such that it brings about the following proportional relation:

CA : CB = CD : CE

The conclusion is that the bisecting line DE is parallel to line AB. We then negate this conclusion and see what follows logically. We then see how the premises plus the negated conclusion yields a contradiction, which proves that the experimentally negated conclusion is false and thus that the originally proposed conclusion is true. {2} The second proof does not use negation. Here we start with the line which fulfills the proportions. Next we construct another line that we know is parallel to the undivided side. Finally we show that this parallel line must necessarily be identical to the line which fulfills the proportions. Thus lines that fulfill the proportions are parallel to the undivided line. (0.6) In the negationless proof, we will need to define our concepts without negation. Parallel lines cannot be defined as “which do not intersect” (for here the “do not intersect” is a negation of “do intersect”). Rather, they must be defined without a negation, as for instance, “parallel lines are such lines, that any point of one of them differs from any point of the other one.” This formulation then requires a positive definition of “difference relative to points.” One reason that negation is used has to do with the triangle figure requiring an additional, different DE line being drawn even though it is identical to that line. (Maybe Griss is saying that this notion of something being both different and identical is counter-intuitive, so people might prefer the reducio proof instead.) Another reason people use negation has to do with how we formulate our mathematical questions. We might take x²– 2 and form the question, which rational numbers satisfy x²– 2 = 0, with the answer being the negative “no rational numbers satisfies it.” But we need not think of x²– 2 in such a formulation that leads us to a negative conceptualization. We can instead say that “x²– 2 differs positively from zero for every rational number.” (In other words, its value is not seen as not being a rational number when it is equated with zero but rather that given any rational number for x, its value will be another value that is always different from zero.) (0.7) In the second illustration, we wonder if the equation ax + by = 0 has a solution for x and y where x and y are different from zero and the letters represent real numbers? And we will compare the negationless and negative way of answering this question. To do this, we first note the following distinction between real numbers understood as different either positively or negatively: “Two real numbers differ positively, if there can be indicated two approximating intervals which lie outside one another; they differ negatively, if it is impossible that they are equal; you can only divide by a real number if it differs positively from zero.” {1} We begin with the negationless way. We do this first by assuming in one case x has a non-zero value and seeing how that gives a non-zero result for y, secondly we likewise assume that y has a non-zero value and see how that gives a non-zero result for x, and lastly we give both a and b the value zero and see how that yields a non-zero value for both x and y. We conclude from this non-negative approach to the question that “ax + by = 0 has a solution different from zero, if at least one of the coefficients a and b differs from zero or if both are zero.” {2} We next look at how we can use negation to formulate this positive result in a negative way, namely as: “It is impossible that no solution different from zero exists.” We learn this by assuming the only possible solution is zero (“there were no solution different from zero”), which logically yields the contradictory claim that there is such a non-zero solution. Thus it is impossible that there is no solution different from zero. Now, what we learn by comparing the two results is that “The negative formulation is shorter, but distorted, and the details of the positive result are lost. In non-intuitionistic mathematics ax + by = 0 has always a resolution different from zero. In this formulation the positive result has vanished entirely.” (0.8) The third illustration is: “If ax + b ≠ 0 for each value of x, then a = 0.” In our exploration of the proof for it, we make use either of a positive definition for equality or a negative one. {1} The positive definition of equality: “If a differs from c for each value c that differs from b, then a = b.” (In other words, two values are equal if they are both different from all other values.) {2} The negative definition of equality: If a does not differ from b, then a = b. (In the first case, the two equal things share all the same differences to other things. In the second case, they simply are not different to each other). From this Griss concludes that “The positive proposition has to be proved for the different sorts of numbers, to begin with the natural numbers. But therefore again it proves to be necessary to construct the whole of negationless intuitionistic mathematics from the beginning.” (0.9) We can compare a positive formulation, “Two triangles are congruent, if they have equal one side, the angle opposite that side and the sum of the two other sides, while of one of the adjacent angles is known that they are either equal or different” with a negative formulation, “If two triangles have equal one side, the angle opposite that side and the sum of the two other sides, it is impossible, that they are not congruent.” (0.10) Griss now summarizes the results of these illustrations. {1} From example 1 (see section 0.5) we learn that “In some cases it is simpler to avoid the use of the negation.” {2} From example 2 (see section 0.7) we learn that “Positive properties can sometimes be formulated more briefly in a negative way, but details get lost.” {3} From example 4 (see section 0.9) we learn that “The parts of intuitionistic mathematics which in a positive construction are disposed of are less important, for probably examples cannot be constructed for which a negative property could be applied and a corresponding positive property could not.” {4} From examples 1 and 3 (see section 0.5 and section 0.8), we learn that “To construct negationless mathematics one must begin with the elements and a positive definition of difference must be given instead of a negative one.” Moreover, “But even from a general intuitionistic point of view a positive construction of the theory of natural numbers must be given: one cannot define 2 is not equal to 1 (i.e. it is impossible that 2 and 1 are equal), for from this one could never conclude that 2 and 1 differ positively. Conversely one could define in a positive way negation by means of difference, e.g. not equal means different, etc., but, for the present, this seems unfit.” (Perhaps then we might note the following. We need numbers in our negationless mathematics. But to get those numbers, we need more than just inequality (the impossibility of being equal) to tell us that each number is different from the others. We rather need a positive construction of the numbers that does not involve the impossibility of equaling. In section 1 to follow, we learn that there is a notion of distinguishability that grounds inequality.) (0.11) Griss lastly has us “consider the property: If a and b are elements of the set of natural numbers, and if a ≠ b, then a b for each element a of the set. If we apply this property to b = 1, we get: For each element a ≠ 1 of the set of natural numbers we have a < 1 or a > 1. a < 1, however, has not any sense in negationless mathematics. If we say: a b for each a of a set, we mean 1) that for each a at least one of these conditions is fulfilled, 2) that conversively at least one element fulfils the condition a b.” We then note that “Negationless intuitionistic logic will differ much from the usual intuitionistic logic by the absence of the negation and the altered meaning of the disjunction” and also that “‘Affirmative’ mathematics is something quite different from the negationless intuitionistic mathematics.”

“The Natural Number”

1.1

“Construction of the Natural Numbers”

(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.)

1.2

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

(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.

1.3

“The Order-Relation”

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

George François Cornelis Griss

(G.F.C. Griss)

“Negationless Intuitionistic Mathematics, II”

1.0

“[Preface]”

(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.”

George François Cornelis Griss

(G.F.C. Griss)

“Logic of Negationless Intuitionistic Mathematics”

“§3. Conditions for the existence of the complementary species and the
inter section”