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by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]
[Mathematics, Calculus, Geometry, Entry Directory]
[Logic and Semantics, entry directory]
[The following collects brief summaries of select texts. The entry directories without the brief summaries are located here:
Griss’ “Negationless Intuitionistic Mathematics, I”, entry directory
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”
(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 or 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 or 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 and another one 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.”
1
“The Natural Number”
“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.)
“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.
(1.3.1) “We define the relation a precedes b, a < b, which has the same meaning as b follows a, b > a, and the relation a immediately precedes b (b immediately follows a).” In this way, any set of terms {1, 2, ..., n} can be arranged in such an order of procession. (1.3.2) If for two numbers in the same ordered set one precedes another, then they are not equal numbers: “If for {1, 2, ..., m} a < b, then a ≠ b.” (1.3.3) Precession is transitive: “Property: If for {1, 2, ..., m} (m > 2) a < b and b < c, then a < c.” (1.3.4) If a number in an ordered set does not equal 1, then it must come after 1: “Property: If a ≠ 1 is an element of {1, 2, ..., m}, then 1 < a.” If a number in an ordered set does not equal the last number, then it must come before it: “Property: If a ≠ m is an element of {1, 2, ..., m}, then a < m.” If b is neither the first nor the last number, then any other number a must either precede or succeed b. “Property: If a and b (b ≠ 1 and b ≠ m) are elements of {1, 2, ..., m}, for each element a that differs from b holds a < b or a > b.” Also, we cannot have negative numbers in sets constructed this way and in accordance with negationless intuitionistic mathematical principles. (1.3.5) If one number a precedes another number b, and if for all the numbers c coming before b, they also come before a, then b immediately follows a: “If a < b and if for each c < b and c ≠ a c < a holds, then b immediately follows a.” Similarly, if a number a precedes another number b, and if for all the other numbers c that come after a and that are not b – if they all come after b, then b immediately follows a (check this quote, as it says b immediately follows b): “If a < b and if for each c > a and c ≠ b c > b holds, then b immediately follows b.” If a number b immediately follows another number a, which itself is not the first number, then for all the other numbers coming before b, if they do not equal a, then they come before a: “If b immediately follows a (a ≠ 1) , then for each c < b and c ≠ a holds c < a.” Similarly, if a number b immediately follows another number a, and b is not the final number, then all the numbers larger than a that are not equal to b would have to come after b: “If b immediately follows a (b ≠ m), then for each c > a and c ≠ b holds c > b.” (1.3.6) Suppose some number b is greater than 1, and it has numbers c that come before it. If some other number a does not equal b and does not equal any of these numbers c coming before b, then a comes after b: “a ≠ b and a ≠ c for each c < b (b ≠ 1) → a > b.” Similarly, suppose some number b is not the last number, and it has numbers c that come after it. If some other number a does not equal b and does not equal any of these numbers c coming after b, then a comes before b: “a ≠ b and a ≠ c for each c > b (b ≠ m) → a < b.” On the basis of these properties, we define the following: “a ≤ b as a = b or a < b and likewise a ≥ b;” “a ≠ c for each c < b (b ≠ 1) → a ≥ b;” “a ≠ c for each c > b (b ≠ m) → a ≤ b;” and “a ≥ 1 and a ≤ m.”
George François Cornelis Griss
(G.F.C. Griss)
“Negationless Intuitionistic Mathematics, II”
(1.0.1) The following is a sequel to Griss’ “Negationless Intuitionistic Mathematics, I.” But first he will give a preface with a concise exposition of his ideas in response to some remarks and objections he received. (1.0.2) Brouwer outlines a negationless mathematics in a 1947 paper, but to make it perfectly negationless, we need to slightly adjust one of his definitions to prevent us from supposing something to take properties we are not sure it has. (And, instead of saying negationally that something is either in a subset or not in that subset, we should say affirmatively that either it is in a subset or in that subset’s complement. (1.0.3) We construct sets of natural numbers by starting with 1, which is selfsame, then adding 2, also selfsame but distinct from 1, then 3, selfsame too and distinct from both 1 and 2, and we continue this way, adding n numbers to get the set: En (1, 2, ..., n). We can further add an element n′, selfsame and distinguishable from all members p of En (1, 2, ..., n), so n′ ≠ p, p ≠ n′. They together form the set En′ (1, 2, ..., n′). We can note disjunctively that an element of En′ belongs to En or is n′. “In general our definition of disjunction runs as follows: a or b is true for all elements of the set V means that the property a holds for a subspecies V′ and property b holds for a subspecies V″, V being the sum of V′ and V″.” (1.0.4) “In accordance with the construction of natural numbers the proofs of properties of those numbers are always given by means of induction, until a system of properties is found, that can serve as a starting point of an axiomatic theory.” Now, instead of using disjunction as above, we will formulate the first property using the conditional: “If b is an element of Em (1, 2, . . . , m), then b together with the elements of Em that are distinguishable from b form Em.” (1.0.5) The next property was already articulated without disjunction in section 1.2.2 of “Negationless Intuitionistic Mathematics, I” as “If for two elements a and b of {1, 2 ..., m} holds: a ≠ c for each c ≠ b, then a = b.” Here the formulation and proof remain the same: “If for the elements a and b of Em holds: a ≠ c for each c ≠ b, then a = b.”
George François Cornelis Griss
(G.F.C. Griss)
“Logic of Negationless Intuitionistic Mathematics”
3
Ҥ3. Conditions for the existence of the complementary species and the
inter section”
[Distinguishability. Complimentary Subspecies. The Touch Condition. The Rejection of Empty Species]
(3.2) We take two notions to be equally fundamental [and primitive]: being identical and distinguishability. We begin with a set u that has 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 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.
Arend Heyting
Topic:
Negationless Intuitionistic Mathematics
2.
Arithmetic
2.2
“Real Number Generators”
Definition; Relation of Coexistence
(2.2.1.1) We will examine the theory of real numbers in intuitionistic mathematics by beginning with Cantor’s theory. (2.2.1.2) 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. Formally:
‘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.’
(16)
(2.2.1.3) We can devise an example using a sequence, namely the decimal series of π, and make a stipulation regarding some part of it, even though we may not even know if such a part of it does in fact exist. [This perhaps shows us an instance where we cannot effectively determine n(k).] (2.2.1.4) We call a Cauchy sequence of rational numbers a “real number-generator,” or just simply a “number-generator,” if that leads to no confusion. (2.2.1.5) “Two number-generators a ≡ {an} and b ≡ {bn} are identical, if an = bn for every n. We express this relation by a ≡ b.” [This perhaps means that if each nth term in both series is equal to the other, then the number-generators are identical.] (2.2.1.6) The second definition is: “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.” [This perhaps is to say that although the terms of the two sequences may not be identically the same, they still converge upon the same value.] (2.2.1.7) There is a theorem about coinciding number-generators, namely, that they are reflexive, symmetrical, and transitive. (2.2.1.8) Heyting remarks: “Given any number-generator a ≡ {an}, a number generator b ≡ {bn} can be found such that a = b and that the sequence {bn} converges as rapidly as we wish. For instance, in order that |bn+p – bn| < 1/n for every n and p, it suffices to take bk = an(k) for every k.” [Perhaps the idea is that for every number-generator, we can find another coinciding one, with the identical one being one option.] (2.2.1.9) We can abbreviate a number generator v = {vn} as just v, and vn (without curly brackets) would be the nth component in the sequence v. (2.2.1.10) We will define real numbers in chapter 3, after dealing with set theory, which is requisite.
2.2.2
Inequality Relation Between Number-Generators
[Selections on inequality and negation]
(2.2.2.1) “If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a ≠ b.” (2.2.2.2) The first theorem says: “If a ≠ b is contradictory, then a = b.” (2.2.2.3-8: skip) (2.2.2.9) In intuitionistic mathematics, “not” always has a 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” (this is de jure falsity, because it has been proven necessarily the case and will stay that way). Yet we can use “not” in another way, namely, to mean there is not yet a proof for something (this is de facto falsity, because it happens to be the case that a proof is lacking, but one may someday be formulated):
‘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.’
(18)
(2.2.2.10) In intuitionistic mathematics, all mathematical assertions are in the form of constructions. Even a negation of an assertion would have to be an alternate positive construction on the basis of which we effect a reductio of that negated assertion:
‘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.’
(18-19)
However, when we simply lack a proof for something (without also being able to construct a disproof of it), then we have just a factual negation (and a disproof may or may not be devised some day).
‘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.’
(19)
2.2.3
Apartness-Relation Between Number-Generators
[Definition of Apartness of Number-Generators]
(2.2.3.1) We will give a positive definition for inequality (in negationless intuitionistic mathematics), which is apartness. We say 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. Formally:
‘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.’
(19)
Les fondements des mathématiques.
Première section:
Intuitionnisme
5.
L'intuitionnisme brouwérien
5.1
L'intuition mathématique
5.1.1
Mathématique sans négation de Griss
(5.1.1.1) Griss devised a negationless intuitionistic mathematics. He thought there should be nothing like negation in it, because intuitive methods will not allow us to make a demonstration based on the falsity of an assumption, as we cannot clearly conceive a falsity in the first place. And we can only clearly conceive a property after constructing a mathematical entity that possesses that property, so we cannot introduce an empty species. For real numbers, Griss needs to avoid the negational (and exclusionary) notion of inequality, but he still needs to be able to say that one natural number is not identical to the other ones (and that two numbers are identical when they cannot be unequal). Instead of conceiving this in terms of not being equal (which cannot enter into intuitionistic thinking, because we can only conceive of positive properties), Griss (according to Heyting) recasts this inequality relation (≠) as a distance relation (⧣). On this basis, we can understand two (initially unidentified) numbers as being equal when they share the same distances to the same other numbers. (2 for instance is one away from 1 and one away from 3, and two away from 4, etc. If both a and b each likewise are one away from 1, one away from 3, etc., then they are equal. Here we are avoiding the non-intuitionistic notion of them not being unequal to one another by having them both positively sharing the same relational properties to all the other numbers in the set.) Heyting writes, in rough translation: In the theory of real numbers, the relation ≠, being negative, does not intervene. There is only the relation a = b and the distance relation a ⧣ b (see “numerical calculation” below). The theorem “if a ≠ b is impossible, then a = b” is replaced by the following: “if a is distant from every number c which is distant from b, a = b”. (Heyting p.14). (5.1.1.2) But if we completely eliminate negation, then we cannot have a propositional logic in the normal sense. Nonetheless, both Griss and Destouches-Février attempt to construct such a propositional logic. But instead of being a logic of predicates, Griss here constructs a logic of classes that is unlike the intuitionistic logic of classes in that for Griss, two classes can intersect only if they have at least one element in common. (5.1.1.4) Van Dantzig outlined a formal system of affirmative mathematics. (5.1.1.5) Brouwer supports the role of negation by constructing theories that require it, and he articulated his ideas about the relationship of mathematics to experience, language, and wisdom.
(5.3.1.1) Coordinated choice sequences can be used to define the operations of calculation. But problems arise when inequalities are used. For Brouwer, “a is different from b”, (a ≠ b), means that a = b is impossible. But when we are dealing with the continuum, we have an additional relationship of inequality and equality that can hold between variables. Roughly: For the continuum, we additionally have the relationship “a is positively different from b” or “a is apart from [écarté de] b” (a ⧣ b). This is fulfilled when, in the series of intervals that define a and b, two external intervals are known [to be shared by both]. a ≠ b of course results from a ⧣ b ; but the inverse cannot be affirmed. Moreover, as we can easily see, the negation of a ≠ b, and also of a ⧣ b, is equivalent to a = b.
Arend Heyting
”G. F. C. Griss and His Negationless Intuitionistic Mathematics”
“[Griss’ Negationless Mathematics and Real Numbers]”
[Contains a synthesis of many other posts]
__(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 containes 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).
Griss, G.F.C. (1946). “Negationless Intuitionistic Mathematics, I,’’ Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, 49, 1127–1133.
Journal PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00014659.pdf
Article PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00018278.pdf
Listing of Griss at this journal:
http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531
Griss, George François Cornelis. “Negationless Intuitionistic Mathematics, II.” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 53, no. 4 (1950): 456–463.
Journal PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00014669.pdf
Article PDF here:
http://www.dwc.knaw.nl/DL/publications/PU00018796.pdf
Listing of Griss at this journal:
http://www.dwc.knaw.nl/toegangen/digital-library-knaw/?pagetype=publist&search_author=PE00000531
Heyting, Arend. Intuitionism. An Introduction. Amsterdam: North-Holland, 1956.
Heyting, Arend. Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Paris / Louven: Gauthier-Villars / E. Nauwelaerts, 1955.
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