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by Corry Shores
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[Heyting’s Les fondements des mathématiques, 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. Also, my abilities with French are insufficient to translate reliably, so please again rely upon the quotations rather than my summarizations. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]
Summary of
Arend Heyting
Les fondements des mathématiques.
Intuitionnisme.
Théorie de la démonstration.
Première section:
Intuitionnisme
5.
L'intuitionnisme brouwérien
5.3
Continu. Suite de choix
5.3.1
“Calcul numérique”
Brief summary:
(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.
[Inequality and Continua]
Summary
[Inequality and Continua]
[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.]
[In rough form: Numerical calculation.– The use of choice sequences in mathematics is based on the possibility of establishing coordination between choice sequences. The sequence β can be coordinated with the sequence α by the fact that each choice of β can be defined by using a finite initial segment of α. The definitions of the operations of calculation constitute simple examples of such a coordination. Once these have been given, we then easily have the results of a pure calculation of elementary arithmetic and algebra; however, complications can appear as soon as inequalities are used. “a is different from b”, (a ≠ b), means, in BROUWER’s terminology, that a = b is impossible. 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. BROUWER has shown using examples [26] that there are real numbers for which we do not know if a = 0 or if a ≠ 0. We will write a ⩺ b when we know in the series of intervals for a an interval that is external and to the right of an interval of the sequence for.b. From a ⧣ b it follows either a ⩺ b, or b ⩺ a.]
]
Calcul numérique.– L’utilisation des suites de choix en mathématiques est fondée sur la possibilité d’établir des coordinations entre des suites de choix. La suite β peut être coordonnée à la suite α par le fait que chaque choix de β peut être défini au moyen d’un segment initial fini de α. Les définitions des opérations du calcul constituent des exemples simples de telles coordinations. Une fois celles-ci données, on a alors sans difficulté les résultats de pur calcul de l’arithmétique élémentaire et de l’algèbre ; pourtant des complications peuvent apparaître dès qu’on fait emploi d’inégalités. “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. a ≠ b résulte évidemment de a ⧣ b ; mais l’inverse ne peut pas être affirmé. De plus, on le voit facilement, la négation de a ≠ b, et aussi celle de a ⧣ b, est équivalente à a = b. BROUWER a montré par des exemples [26] qu’il y a des nombres réels pour lesquels on ne sait pas si a = 0 ou a ≠ 0. On écrira a ⩺ b quand on connaît dans la suite d’intervalles pour a un intervalle qui est extérieur et à droite d’un intervalle de la suite pour.b. De a ⧣ b il suit soit a ⩺ b, soit b ⩺ a.
(24-25)
[26] : Mathematik, Wissenschaft und Sprache. Mh, Math, Phys. 36 (1929), p.153–164 ;
(79) (boldface is mine)
Heyting, Arend. Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Paris / Louven: Gauthier-Villars / E. Nauwelaerts, 1955.
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