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 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”
4
“[Griss’ Negationless Mathematics and Real Numbers]”
Brief summary:
__(4)__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). And we can also determine when natural numbers are equal. [Their equality can be established, not in the negational way of saying that it is impossible that they are unequal, but rather in the negationless way of saying that they share differences to exactly the same other numbers.
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 = {an} and b = {bn} are apart from each other (a ⧣ b) if for some n, an and bn 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). [
]
[Griss’ Negationless Mathematics and Real Numbers]
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 = {an} and b = {bn} are apart from each other (a ⧣ b) if for some n, an and bn 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}.
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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 En (1, 2, ..., n), and {1, 2, …, n′} as En′ (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 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″.
(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:
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.
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 = {an} and b = {bn} coincide, if an and bn 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 {an} 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 |an+p – an| < 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 {an} 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 |an+p – an| < 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 nth term, 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 = {an} and b = {bn} coincide, if an and bn 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 ≡ {an} and b ≡ {bn} are identical, if an = bn 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 ≡ {an} and b ≡ {bn} coincide, if for every k we can find n = n(k) such that |an+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 nth terms will always fall within a gap smaller than any arbitrarily given one. For, it is saying |an+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 |an+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 = {an} and b = {bn} are apart from each other (a ⧣ b) if for some n, an and bn 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 = {an} and b = {bn} coincide, if an and bn 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 = {an} and b = {bn} are apart from each other (a ⧣ b) if for some n, an and bn 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)
Heyting, Arend. “G. F. C. Griss and His Negationless Intuitionistic Mathematics.” Synthese 9, no. 2 (1953-1955): 91–96.
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