by Corry Shores
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[Mathematics, Calculus, Geometry, Entry Directory]
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[Heyting’s Intuitionism: An Introduction, 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 probably inelegantly articulated. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
Arend Heyting
Intuitionism: An Introduction
2.
Arithmetic
2.2
“Real Number Generators”
2.2.2
Inequality Relation Between Number-Generators
2.2.2.1,2,9,10
[Selections on inequality and negation]
Brief summary:
(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)
[The Contradiction of Equality as Inequality]
[The Contradiction of Inequality as Equality]
[Negation as Disproof and Negation as Lack of Proof]
[The Difference Between Constructing a Disproof for a Mathematical Assertion and Simply Not Having Constructed a Proof for It]
Summary
[The Contradiction of Equality as Inequality]
[“If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a ≠ b.”]
[ditto]
If a = b is contradictory (that means : if the supposition that a = b leads to a contradiction), we write a ≠ b.
(17)
[The Contradiction of Inequality as Equality]
[The first theorem says: “If a ≠ b is contradictory, then a = b.”]
[ditto]
Theorem 1. If a ≠ b is contradictory, then a = b [L. E. J. Brouwer 1925, p . 254].
(17)
BROUWER, L. E. J.
1925. Intuitionistische Zerlegung mathematischer Grundbegriffe. Jahresbericht deutsch. Math. Ver. 33, p. 251–256.
[2.2.2.3-8: skip]
[Negation as Disproof and Negation as Lack of Proof]
[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)]
[ditto]
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.
(18)
[The Difference Between Constructing a Disproof for a Mathematical Assertion and Simply Not Having Constructed a Proof for It]
[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)]
[ditto]
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.
(18-19)
Heyting, Arend. Intuitionism. An Introduction. Amsterdam: North-Holland, 1956.
.
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