Abstract
The paper shows that, contrary to what has been held since the sixteenth-century mathematician Christoph Clavius, there is no application of consequentia mirabilis (CM) in Greek mathematical works. This is shown by means of a detailed discussion of the logical structure of the proofs where CM is allegedly employed. The point is further enlarged to a critical assessment of the unsound methodology applied by many interpreters in seeking for specific logical rules at work in ancient mathematical texts.
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Notes
See Clavius (1611, 365).
Of course, this paper is Unguru (1975–1976).
One uses p → (q ∧ r)⟷(p → q) ∧ (p → r) and then crosses out ¬p → ¬p.
These terms are, for instance, “straight line”, “triangle”, schematic letters, etc.
See the beginning of the next section for a definition of this notion.
At In Euc., 203.1–207.25 Friedlein. Proclus’ scheme takes an idealized geometrical “theorem” as its model. This is a mathematical proposition that is neither a problem nor concerned with number theory, in which only one result is proved, and whose construction is not interspersed with deductive steps. Note that just a strict minority of propositions in the Elements in the text edited by Heiberg (who follows the manuscript Vat. gr. 190, see below) do have a “conclusion”: these are 192 out of 465; theorems give the ratio 176/370, problems 16/95.
See Bobzien (1997).
Such suppositions are not necessarily included in the setting-out or in the initializing clause of the reductio (see below).
But not in the setting-out, as in principle this specific part does not add any information with respect to the enunciation.
This is a non-simple statement first singled out as a logical entity by the Stoic Crinis, as asserted in Diogenes Laertius VII.71.
An elliptical antecedent may assume several forms. It may contain a negation, like in “in fact, if not” (61 occurrences in the Euclidean corpus, not only in reductiones), or a modal operator of possibility, like in “in fact, if possible” (62 occurrences), mainly when negative enunciations are reduced to the impossible, or in “in fact not, but if possible”. Strictly speaking, the initializing clause announced by the latter expression is not a conditional, but a severely abridged conjunction: the first conjunct, the only vestige of which is “in fact not” (“in fact, let so-and-so not be the case”), is a supposition with the verb in the imperative; the second conjunct is the conditional we read in the text. For our purposes, this complication is immaterial. I shall occasionally restore between angular brackets the words understood in the elliptical antecedent.
Thus, the statement to be proved is provided three times: in the “determination” immediately preceding the proof by reductio, in negated and elliptical form in the antecedent of the initializing conditional, in fully fledged, negated, and possibly equivalent form in the consequent of the initializing conditional. We shall check this in El. III.2, to be read presently.
In the Elements, the former clause is more frequent than the latter: 119 occurrences versus 29; in Archimedes we only find the former.
The former in IX.10, 13, 34, X.9, 42, X.13 and 39 (supplementary material); the latter only in IX.10. In IX.13, the framing argument is identified by “absurd”, the 5 nested reductiones are identified by “impossible” or (the third one) by the formula just seen.
The 9 reductiones without a canonical closing formula are El. VIII.6, X.44, XII.18 (1 argument out of 2), and, among the supplementary material, III.11 (1 out of 2), XI.23 (1 out of 4), lemma after XIII.2 (1 out of 2), lemma after XIII.18 (3 out of 5). The argument in El. VIII.6 is in fact a direct proof, as we shall see in Sect. 2.3. The closing formulas of El. X.44 and XII.18 have certainly dropped by some accident of transmission.
I will translate this negative particle “it is not the case that”. All translations are mine.
Greek text at EOO I, 168.17–170.17. The reader is urged to reconstruct the diagram.
The case in which the straight line joined from Α to B coincides with the circumference is similar but not identical to the case proved in full in the theorem: the construction should be rewritten, since point Z is no longer necessary; the proof would end with “therefore DB is greater than DE; but they are also equal, which is really impossible”.
Greek text at EOO II, 362.17–364.2 and 366.1–12.
This can be applied because A, B, C, D are continuously proportional, an assumption that I have not included in my transcription.
For this reason, the omitted portion of the text is noted [… (3)…].
See EOO II, 367 n. 1.
With a slip in step (4), as we have seen.
Translation: “The first proof of this proposition is wonderful. For in it, Euclid shows, by means of a direct proof, that E measures A itself from the fact that E is said not to measure A itself, what apparently cannot happen. […] Cardanus carries out something similar for magnitudes in prop. 201 of Book 5 on proportions, and boasts that he himself first of all found this demonstrative method: I guess he would not have said this, if he had carefully worked out the gist of this proof—and if he indeed did so, he certainly would have recalled it, since Euclid also used this way of proving well before him, as is evident from theorem 12. Theodosius used the same kind of proof in prop. 12 of Book 1 of the Spherics, as we noted there.”
Translation: “E measures A itself and A measures H itself the same number of times, which is against the supposition, for it was assumed that E does not measure A itself. Thus, it is false that E does not measure A itself. For from the fact that E is said not to measure A itself it will always be proved that E measures A itself, which is absurd. Therefore, E measures A itself.”
Greek text at EOO II, 410.15–21, 410.23–412.5, 412.11–16, and 412.21–414.1.
Here and in the similar step below, one must argue by conversion: if E does measure P, then O measures P according to the same number by the definition of numerical proportion in VII.def.21.
As a matter of fact, O cannot be either of D or E and otherwise P would be the other, so that ZH would straightforwardly be measured by a number (P itself) that is the same with some of A, B, C, D, E, FK, L, M. Thus, including or not including D and E in the partition does not make any difference.
Greek text at EOO II, 270.8–11.
Greek text at EOO II, 372.17–18 and 366.26–368.1, respectively.
A lemma between for example propositions XII.4 and 5 is here noted XII.4/5.
Cf. EOO IV, 212.13–16, with ibid., 407.22–23.
Greek text at EOO II, 288.9–290.6. The bracketed clause is a later addition, as we shall see in a moment.
See EOO II, 288.14–15 in app.
The version in Vat. gr. 190 already has two (identical) “determinations”. These are kept in Theon’s rewriting.
De Young (1981, 123–125 (transl.)).
See the survey in Vitrac (2009).
See, for instance, Apollonius, Con. I.5 and II.48, where it is shown that a particular conic section is a circle.
Cf. Castagnoli (2000, esp. 317–327).
The claim is at Aristotle, APr. 2.4, 57b9–17.
See Łukasiewicz (1957, 49–50).
See Ebrey (2015) on the fact that Aristotle did not employ conditional clauses in his syllogistic.
See Bobzien (1999, 84–86 and 106–108), on these three types of conditional.
Sextus Empiricus, Pyrrh. Hyp. II.111.
Bobzien (1996, 183–184)—two of these arguments correspond to instances of the rules known as modus ponens and modus tollens; Chrysippus set the number of basic undemonstrated to five, by adding further rules regulating the use of conjunction and disjunction.
Cf. Sextus Empiricus, Adv. Math. VIII.281–284 and Pyrrh. Hyp. II.131 (sign), Adv. Math. VIII.292–294, 463–469 (proof), IX.205 (cause).
At Ammonius, in APr. I, xi.13–26 Wallies.
The “compound statements” are the conditional, the conjunction, and the disjunction. The preceding argument is an example of inference “in virtue of three compound ‹statements›”.
At Galen Plac. Hipp. Plat. II.3.18–19.
And they actually were, as Galen attests; a reduction can be found in Bobzien (1996, 169–170).
Sextus Empiricus, Pyrrh. Hyp. II.189.
Take, for instance, Bellissima and Pagli (1996, 7–8).
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Acerbi, F. There is no consequentia mirabilis in Greek mathematics. Arch. Hist. Exact Sci. 73, 217–242 (2019). https://doi.org/10.1007/s00407-019-00223-1
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DOI: https://doi.org/10.1007/s00407-019-00223-1