Plato: Parmenides 149a7-c3. A Proof by Complete Induction?

A MATHEMATICAL REVIEW: Hintikka, Jaakko Aristotelian induction. Rev. Internat. Philos. 34 (1980), no. 3-4, 422--439. MR0630683 (82m:00016) The author considers the question of how Aristotle explains knowledge of axioms, i.e., basic premises of axiomatic science. "Induction" is the conventional translation of Aristotle's term "epagogé" indicating a process through which knowledge of the truth of axioms is drawn from experience. Thus, it has no special relation to mathematical induction, the number-theoretic principle. The fact that Aristotle traced all knowledge of axioms to experience is already clear and well known to scholars despite contrary information persistently appearing in popularizations. It is true, however, as pointed out in this paper, that Aristotle gave apparently conflicting accounts of how induction functions and also that some of Aristotle's remarks about axioms may seem to conflict with his view that knowledge of them is based on experience. This paper proposes to solve these problems through a plausible premise, perhaps original with the author, to the effect that Aristotle's induction involves a combination of empirical examination of examples with conceptual analysis of exemplified notions. Reviewed by J. Corcoran NOTE ADDED 121715: Given the various ways ‘induction’ is used today, it is misleading to use "induction" as the translation of Aristotle's term "epagoge" indicating a process through which knowledge of the truth of axioms is drawn from experience. “Experience-based intuition” might work better. On this subject, one should note that there are many sentences, propositions, rules, processes, and so on that are all called ‘mathematical induction’—an expression totally unsuited for any of them. It the first place, the adjective ‘mathematical’ is far too general, ‘arithmetic’, ‘numerical’, or ‘number-theoretic’ would be better. In the second place, the noun ‘induction’ in ‘mathematical induction’, if justified at all, cannot be justified by reference to the most common uses of the word in normal English, in science, in philosophy, or in methodology of science. In current senses, mathematical induction does not produce knowledge of mathematical axioms. In fact, in some current senses, mathematical induction is a mathematical axiom, knowledge of which is produced by induction in the sense associated with Aristotle. I am glad to point out that Hintikka’s use of the word ‘axiom’ for “fundamental truth” is in full accord with its use by Tarski, Church, and other giants of mathematical logic. Moreover, it reveals the pathetic inadequacy of ill-advised and unjustified attempts to use the word for “arbitrary assumption”.

2008

THE CONSTRUCTION OF MATHEMATICAL KNOWLEDGE: CURIOSITIES IN A LACON AND PHILOSOPHICAL PERSPECTIVE (Atena Editora), 2022

When he stopped being a nomad and started to live a principle of society, man began to feel the need to count. The first idea gave rise to one-to-one correspondence (basis for working with concepts involving function). The fingers of the hands, stones (from the Latin, calculus) become instruments used in this correspondence, whose decimal traces are preserved and considered to this day. However, groups of stones have ephemeral characteristics that make it impossible to preserve numerical information in the long term. The first innovative idea was to register values ​​through marks on bones and sticks. Thus begins a language that was developed within principles, whose objective was to facilitate the work of the man who calculated (even without him knowing it). “The tendency of language to develop from the concrete to the abstract can be seen in many of the measures of length in use today. A horse's height is measured in “spans” and the words “foot” and “ell” (elbow, elbow) also derive from body parts” (BOYER, 1996, p. 04). All this contributed to the development of mathematics, transforming it into something much bigger than just counting and measuring. The origin of the first indications of a structure, which would later be called mathematics, is lost in times without records. Sometimes it is due to practical necessity (in Herodotus' view - the “rope-drawers” ​​of ancient Egypt, for example), sometimes to priestly and ritual leisure (in Aristotle's view). Therefore, our chronological observation will have as a principle to highlight important points according to personal motivations, arranged on a firmer ground in the history of mathematics, recorded in documents that have been preserved.

29th Novembertagung on the History of Mathematics, 28th -30th November 2018

We are interested in some aspects of the very early mathematical conception and use of the infinite in Ancient Greece. Following a suggestion of Fabio Acerbi (2000), we examine the extant corpus of and about Zeno of Elea in the context of the practice of reasoning ad infinitum (eis apeiron). Zeno introduces in philosophy what could be called “indefinite iterative reasoning”, which is based on the a priori recognition that a certain operation, when performed, will ineluctably replicate the conditions for it to be performed again (in the way a continuous magnitude is always cut in new cutable continuous magnitudes, or a number is always followed by a followable number). Just as classical reductio ad absurdum, iterative reasoning secures its conclusion thanks to a meta-logical step back, that in the present case acknowledges the unachievability of the operation. Complete and incomplete inductions, as well as reductio ad infinitum or infinite descent are all conceptually different arguments, to be found both in early maths and early philosophy, all based on such iterative reasoning. The Aristotelian analysis of the infinite in Physics, III, reflects the vivid difference between this conception of infinity as logical unachievability and the more substantial conceptions of infinite objects or multiplicity to be found in the Presocratic and Atomistic traditions. Following again the lead of Fabio Acerbi, we want to reflect on the way these conceptual differences allow or forbid certain of the aforementioned concepts or logical moves to be introduced and maintained in early mathematical practices.