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Constructibility as a criterion for existence
Published online by Cambridge University Press: 12 March 2014
Extract
Problems of construction have engaged the attention of mathematicians from the earliest times. The three famous unsolved problems of the Greeks were problems of construction. Apparently the attitude of the Greeks toward these problems was that, if a geometric entity exists, it must be constructible. For example they believed that given an arbitrary angle, there is another one equal to a third of the first. Hence there must be a way of constructing the second angle, that is, a way of trisecting the first angle. This attitude is very similar in formulation but vastly different in content from the attitude of the modern intuitionists, who say that a mathematical entity does not exist unless it is constructible. Had Brouwer been a Greek, he would probably have declared that, given an angle, it is non-sense to speak of another angle equal to a third of it until such a second angle has been constructed.
However, it seems scarcely necessary to remind the reader that, for the Greeks, constructibility meant a special kind of constructibility, namely, constructibility by ruler and compass. Other methods of constructing geometric entities are known, and by use of them, the problems of the Greeks can be solved. More than that, it is now known that ruler and compass constructions alone do not suffice to solve any one of the three famous problems of the Greeks. Therefore, we see that the Greeks erred in restricting themselves to ruler and compass constructions, even though they may have felt justified in making these restrictions, due to considerations of mathematical elegance, or to ignorance of other methods.
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- Copyright © Association for Symbolic Logic 1936
References
2 Because of the absence of an axiom of continuity, this does not follow from Euclid's axioms, but apparently no Greek thought of doubting its truth.
3 Heyting, A., Mathematische Grundlagenforschung, Institutionismus, Beweistheorie (Ergebnisse der Mathematik und ihrer Grenzgebiete), Berlin 1934. I shall refer to this tract as “Heyting.”Google Scholar
Heyting's, remark, “Aber auch diese Zergliederung ist wohl nicht endgültig,” near the top of p. 13Google Scholar, would indicate that he, however, does not hold that the constructions allowed by the intuitionists are the only allowable ones.
4 Heyting, p. 4, pp. 11–29.
5 Heyting, bottom of p. 12 and top of p. 13.
6 Often called the principle of the actual infinite.
7 Heyting, pp. 18–20.
8 Heyting, bottom of p. 18 and top of p. 19.
9 Heyting, pp. 20–29.
10 I have conversed with mathematicians who take this stand. Their attitude toward the contradictions obtainable by this principle is to ignore them, and trust that similar contradictions will not arise in their work.
11 The well-known contradictions in the theories of transfinite cardinals and transfinite ordinals are among these. Cf. Whitehead, and Russell, , Principia mathematica, 2nd. edn., vol. 1, pp. 60–65Google Scholar, or Fraenkel, A., Einleitung in die Mengenlehre, 2nd. edn., Berlin 1923, pp. 151–157CrossRefGoogle Scholar.
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14 Heyting, pp. 29–57.
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