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May 4, 2016
This is a useful history of the axiomatic-deductive method (ADM). I shall summarise it.

To Aristotle, ADM is the way of organising any science, starting from axioms that are primitive principles established by induction from experience. Deductive order should correspond to causal order. (9-10) Aristotle is rather equivocal on whether the principles also need to be obviously true. (12, 26, 59)

Galen placed more stress on the example of geometry specifically, and urged that it be followed in other fields. Thus he stressed that axioms should be obvious and absolutely certain. (10-11)

Proclus discussed ADM only in the context of geometry, but he too emphasised that axioms should be simple and obvious. (12)

Besides mathematics, other examples of ADM in Greek thought were Euclid's optics, Archimedes's statics, Aristarchus on sizes and distances of sun and moon, and Proclus's commentary on Aristotle's physics. (13-14)

Boethius applied ADM to a theological problem, deriving a proposition from 9 "common notions," and stating that he followed the method of mathematics and the other sciences. (14) Proclus also wrote a theological work with deductively organised propositions, though with less marked mathematical parallels and no explicit axioms. (14-15)

The text by Boethius was by far the most widely read in the early middle ages, and determined conceptions of ADM in this period. (16-19)

The situation improved when Adelard of Bath translated Euclid's Elements in the 12th century. He also included a basic introduction, explaining for example the division between, theorem and problems, and between postulates, common notions, and definitions. (20-21)

This refined Euclidean model of ADM was soon applied in theology by Nicholas of Amiens in 1189. The conception of ADM used is extrapolated from Euclid rather than based on philosophical authorities. (21-22)

Proclus's deductively organised theological work was also quite widely known at this time, and was used as a model by various theological writers. (24-25)

In the 13th century, Aristotle's scientific ADM rose to prominence and was invoked in theology, ethics, natural law, and Aristotelian physics. (26-28) Scholastic authors often employed a question-and-answer format of writing. They often incorporated arguments of ADM form within this structure, but rarely wrote entire treatises in ADM style. (29-33)

In the 16th century humanism replaced scholasticism. There was a renewed interest in classical sources themselves, and in subjects beyond theology, such as mathematics. Proclus's commentary on the Elements became available (1533 in Greek, 1560 in Latin). (35) It became popular to sing the praises of mathematics in orations and prefaces, focussing on its many practical uses yet noble character, and the exemplary certainty and model of its method. The famous inscription at Plato's Academy became high fashion and summed up the spirit of the age. (37-40)

It was generally accepted that mathematics was intended as one of the sciences in Aristotle's sense, but in some regards it was seen as exceptional. (41) Thomas Aquinas attributed the greater certainty to mathematics to the fact that it dealt with fewer aspects of the world than the other sciences, and concepts closer to sensory experience than theology. (41-42) Many authors also associated mathematics with purely syllogistic reasoning. (42-43)

It was also debated at length whether mathematics demonstrates, as an Aristotelian science should, "from causes," i.e., shows "why" the propositions are true, as opposed to merely proved that they must be as a matter of logic. (43-56) This was a very prominent debate in the years 1547-1620. By and large the view that mathematics was exceptional and not a typical Aristotelian science became more and more established as sophistication of mathematical understanding increased; especially Proclus's commentary strongly supported this view. However, the Aristotelian model remained undisputed as the norm of judgement for a long time and was only questioned by the most mathematically inclined authors in the early 17th century. (55-56)

But Proclus's commentary on the Elements also opened the door for attempts at understanding the mathematical method in its own right, independently of Aristotle's theory of ADM. (57) For instance, how is the Euclidean emphasis on constructions to be reconciled with the notion of geometry as abstract and theoretical? (58)

The 16th-century broadening of academic interests led to theories of ADM taking a turn for the practical and emphasing aspects that were useful when developing ADM-accounts for various new fields. (61-69) In this regard Galen proved an especially useful example and inspiration. (75) For example, Galen had employed the distinction between analysis and synthesis in his work on medicine, where analysis corresponds to inferring the illness and its causes from the symptoms, and synthesis to describing systematically first the parts of bodies, then causes of diseases, then their symptoms. (62-63)

But some still insisted on Aristotle's account, and defended it against charges of being overly subtle and not always applicable. (70) Lutheran reformers also found it useful to invoke Aristotle's authority in support of their cause, for instance by associating the protestant maxim of "sola scriptura" with the method of starting from undisputed primary principles. (71-72)

Mathematics was always considered a model of certainty, but in scholastic times only in a limited sense. Mathematics was seen as exemplifying certainty of demonstrations, but not certainty of objects. The objects of theology and metaphysics (such as God and the soul) were seen as more certain and obvious than the objects of mathematics. (76-77) This changed in the late 16th and early 17th century. More and more people now recognised mathematics as a model of certain knowledge without qualifications. One argument for this was the lack of differing opinions in mathematics as opposed to other fields (surely a much more prominent phenomena during this era of expanding intellectual interests than in scholastic times). (77) Related to this is a disillusionment at this time about the scholastic format of disputations, which was seen as geared toward verbally showing off in artificial disagreements rather than facilitating mutual agreement and convergence to truth. (78-81)

The above notwithstanding, there were not many full-fledged uses of ADM outside of mathematics in the 16th century. Nevertheless isolated examples exist in theology, music, natural philosophy, astronomy, ethics, and geography. (92-97) In part the humanistic tastes of the age led instead to a preference for more literary forms, such as dialog, poem, commentary, or overviews in tabular or question-and-answer form. In this context of literary forms, ADM stood out primarily for its brevity and clarity. (97-98)

More common were partial incorporations of ADM elements, rather than wholesale ADM treatises. The use of formal syllogisms was very widespread in theology. (84-91) Elsewhere, the terminology of ADM was employed (arguably inappropriately) more widely than the method itself. (99-100) In the natural sciences, using the "method of geometry" usually did not mean full ADM, but rather the use of geometrical figures and theorems to explain natural phenomena. (101-102)

The lack of widespread adoption of ADM was in some part due to the influential critique of it by Ramus. He first criticised Proclus's account of ADM, saying that Proclus overstates the parallels between Euclid's method and Aristotle's conception of ADM, and that the various distinctions between definitions/postulates/axioms and problems/theorems are meaningless and pointless and not adhered to even by Euclid himself, let alone others like Archimedes. (103-104)

Ramus also found many flaws with Euclid's Elements itself. He claims there are redundant definitions, and theorems that should be axioms. He disapproves of Euclid's reversal of the natural order that arithmetic is prior to geometry. The fact that Euclid gives all definitions and axioms at once at the beginning of each book Ramus compares to an architect building a city by first laying the all the foundations for each of the houses instead of completing one house at a time; thus a square should instead be defined right before the first theorem about squares etc. Euclid also does not strictly adhere to the natural order of first treating triangles, then squares, then higher rectilinear figures, then circles. Finally, Euclid often treats the particular before the general, contrary to an Aristotelian dictum. (104-105)

This critique paved the way for a more back-to-basics compendium style that trusted that simple, clear, and orderly presentation would convince the reader better than ADM subtleties. This "methodus Ramea" as an alternative to ADM had some followers (106-107) but also critics (108-109).
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