Abstract
The following discusses the practice of mathematical argument or demonstration—at first based on what I shall speak of as “the locally obvious”, that is, presuppositions which the interlocutor—or, in case of writing, the imagined or “model” reader—will accept as obvious; next in its interaction with critique, investigation of the conditions for the validity of the seemingly obvious as well as the limits of this validity. This is done, in part through analysis of material produced within late medieval Italian abbacus culture, in part from a perspective offered by the Old Babylonian mathematical corpus—both sufficiently distant from what we are familiar with to make phenomena visible which in our daily life go as unnoticed as the air we breathe; that is, they allow Verfremdung. These tools are then applied to the development from argued procedure toward axiomatics in ancient Greece, from the mid-fifth to the mid-third century bce. Finally is discussed the further development of ancient demonstrative mathematics, when axiomatization, at first a practice, then a norm, in the end became an ideology. The whole is rounded off by a few polemical remarks about present-day beliefs concerning the character of mathematics.
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Notes
- 1.
When at home in Pisa, Dardi would obviously not be identified as coming from there. Where then did he write? The oldest manuscript (Vatican, Chigi M.VIII.170) is written in Venetian, which does not say much. However, this manuscript uses the characteristic Venetian spelling çenso and the corresponding abbreviation ç. So does the Arizona manuscript, whose orthography is also northern; the last two manuscripts, written in Tuscan, still use the abbreviation ç even though writing censo or cienso when not abbreviating (actually I have not seen the Florence manuscript, but Libri’s transcription of a short extract (1838, p. III, 349–359) uses “c,” probably standing for ç). There is thus no reasonable doubt that the original was written in Venetian or a related dialect.
The manuscripts are discussed in Hughes (1987) and in Franci (2001, p. 3–6). I thank Van Egmond for access to his personal transcription of the Arizona manuscript.
- 2.
I refer to the text edition in Franci (2001, p. 59); the Chigi manuscript (fol. 12v, original foliation; probably closer to Dardi’s own text) has \( \overset{\frown }{m} \) instead of meno and e (“and”) instead of più but is otherwise no different.
- 3.
“Abbacus” (abbaco, abbacho) has nothing to do with any variant of the reckoning board. It stands for practical arithmetic, but in the variant that was taught in the “abbacus school”, and it calculated with Hindu-Arabic numerals on paper. Abbacus schools, existing between Genova-Milan-Venice to the north and Umbria to the south from ca 1260 to c. 1600, were frequented by artisans’ and merchants’ sons (also sons of patrician-merchants like the Florentine Medici) for 2 years or less around the age of 11–12.
Abbacus algebra was not taught here, but flourished from ca 1310 onward in the environment of abbacus school teachers, serving to display their competence in the competition for pupils or for municipal employment.
- 4.
My translation, as other translations in the following unless otherwise stated.
- 5.
In contrast, the two alternative methods where division precedes multiplication can be explained meaningfully: q must cost p/P as much as Q; and Q/P is as much as can be bought for one monetary unit.
- 6.
Franci (2001, p. 44). The words are “dimostrare per numero” and “meno via meno fa più”—in the Chigi manuscript (fol. 5v) “demostrar per numero” and “men via men fa più”. Dardi distinguishes between mostrar, “to show”, and demostrar, “to prove”.
- 7.
Luca Pacioli, in (1494, p. I 113r) actually does no better—he adds yet another possibility to be ruled out, namely that (−2)⋅(−2) = −2 (Pacioli operates with negative, not just subtractive numbers), and is even more loquacious here than he usually is (to the point of being obscure).
- 8.
The “Old Babylonian period” is the period 2000–1600 bce (according to the “middle chronology”); the mathematical texts come from its second half.
- 9.
This notion of “broad lines” and its rather widespread occurrence is discussed in Høyrup (1995).
- 10.
A number of attempts have been made to save Plato by proving that Aristotle does not understand him—see, for instance, Tarán (1978) with references to others sharing his view, or the list in Cherniss (1944, p. X). Such attempts are misguided, what Aristotle does is to point out that the numbers Plato speaks about have nothing to do with what others mean by number—no more, indeed, than the “self-moving number” which the Pythagoreans identify with the soul (De anima 408b32f).
A different question, which however does not concern us here, is whether the traces we have of Plato’s views can be given a coherent and historically possible interpretation. Beyond the discussion and references in Tarán and Cherniss, see Mendell (2008, p. 128 n. 3).
- 11.
Thomas Heath (1921, p. I, 201) argues from Hippocrates’s text that he knew what was to become propositions III.20–22, 26–29 and 31 in Euclid’s Elements. This would not be amazing, they can be derived from the equality of the angles at the basis of an isosceles triangle by means of the same kind of counting as Hippocrates wields when applying the Pythagorean theorem. But it is equally possible—not least because Hippocrates makes use of these properties of figures without noticing that an argument might be needed—that he made use of what could “be seen” without having recourse to formulated propositions.
- 12.
Bekker (1831, p. 342b36–343a1). “Those around” was the standard way to refer to the circle of those who studied with a philosopher or similar teacher. Strangely, the Loeb as well as the Ross translation omits “those around”, even though the Loeb edition conserves it in its Greek text. The secondary literature on the other hand (including myself on earlier occasions) has spoken about Hippocrates’s teaching without questioning it.
- 13.
All that is needed is to measure the culmination of the sun at summer and winter solstice and to halve the difference.
- 14.
From Plato’s dialogues, too. But they are often (already, and perhaps mainly, because of the half-poetic genre) too ambiguous to be of much use in the present discussion.
- 15.
“Inspired”, not copying, already for the reason that Aristotelian syllogistic logic does not fit the way geometric proofs are argued. But also for other reasons, cf. McKirahan 1992, p. 135–143.
- 16.
- 17.
- 18.
Cf. also von Fritz 1937, p. 2265 f.
- 19.
One can argue from certain Platonic texts—but this would lead us astray—that this insight in “the good” is achieved via mystical experience. As a hint, observe the force of the images of light.
- 20.
Cf. Mueller 1981, p. 301.
- 21.
Genuinely critical stances had not disappeared—but they had become external, attacking the whole undertaking, not trying to save or to find the “possibility and limits” of mathematical knowledge. The best example is probably Sextus Empiricus (Bury 1933, p. IV, 244–321). This is harsh but informative and informed criticism—but not critique.
- 22.
These sweeping statements go beyond what can be documented in a few footnotes. But see Stedall (2010) for the development of algebra from Cardano to the early nineteenth century. Høyrup (2015, p. 29–33) covers an often overlooked aspect of the shaping and gradual reception of a Cartesian tool (the algebraic parenthesis). The painful advance in the foundation of infinitesimal calculus has been amply discussed; see, for example, Boyer (1949), Bottazzini (1986) and Spalt (2015)—not to speak of the innumerable publications dealing with particular aspects or figures.
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Høyrup, J. (2019). From the Practice of Explanation to the Ideology of Demonstration: An Informal Essay. In: Schubring, G. (eds) Interfaces between Mathematical Practices and Mathematical Education. International Studies in the History of Mathematics and its Teaching. Springer, Cham. https://doi.org/10.1007/978-3-030-01617-3_2
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