Mathematics and Logic in Ancient Greece Open University of Cyprus, Program of Studies in Hellenic Culture The question of the relation of mathematics and logic in ancient Greece has puzzled many historians, who viewed no connection between Euclidean geometrical demonstration and logical reasoning as conducted within Aristotle’s syllogistics and Stoic propositional logic.
The aim of this tutorial is to identify logical principles and modes of reasoning as applied in mathematics and in philosophical thinking. Logical thinking manifests itself in mathematical and philosophical reasoning over such fundamental questions, as the problem of the finite and the infinite and thereby of the finitary and infinitary methods of handling the infinite and the modes of reasoning about it, the problem of classes of finite objects and the status of their existence, and other relevant problems.
I. The finitary arithmetic of Euclid’s Elements (1) The “domain” of Euclid’s “Elements,” Book VI. The Euclidean number – arithmos – has the following formal structure: A , where E designates the unit and a is the number of times (multitude) that E is repeated to obtain the number A, denoted by a segment.Euclid constructs his arithmetic for the numbers-arithmoi, that is for the numbers designated as segments, while the arithmetic of multitudes is taken for granted. Thus, arithmetic is constructed as formal theory of numbers-arithmoi, while the concept of multitude or iteration number has a specific meta-theoretical character. (2) Equality. The concepts “equal,” “less,” “greater,” to which today are ascribed a purely quantitative meaning, in Euclid seems to be also associated with the geometric notion of relative position, but also applied to multitudes when Euclid compares two sets of numbers-arithmoi.
(3) Generality. Euclid sometimes uses quantificational words applied to numbers-arithmoi, although such expressions are very rare. The most common way by which Euclid expresses generality is to speak about arithmos without article. Thus, most enunciations in Euclid’s arithmetical Books state some property about numbers, where arithmos is used without article. However, when he proceeds to the ekthesis of a proposition, general statements about numbers are interpreted as statements about an arbitrary given (indicated) number. In virtue of the instantiation described above the process of proof takes places actually with an arbitrary given number. This “rule of specification” is considered inversible, although Euclid applies explicitly the inverse rule very rarely in the arithmetical Books. The degree of generality attained in this way is no higher than generality expressible by free variables ranging over numbers.
(4) Fundamental concepts. The basic undefined concept of Euclidean arithmetic is that of to measure (katametrein), which underlies most of the kinds of numbers defined by Euclid. The concept “a number B measures a number A” can be interpreted as follows: B measures A , that is A is obtained by n repetitions of B.
(5) ) Implicit assumptions concerning reasoning over infinite processes. In the proofs of Proposition 1 and 2, exposing the process of anthyphairesis, Euclid uses the following implicit assumptions:
i. The least number principle: the set of multiples , such that , has least element , such that , yet . ii. The infinite descent principle: the process of anthyphairesis will terminate in a finite number of steps, that is the chain is finite. iii. If X measures A and B, then X measures , that is if , , then . The first assumption is equivalent to the principle of mathematical induction if the following axiom is added: every number (except the unit) has a predecessor. The second assumption is equivalent to the principle of mathematical induction and is used in Proposition 31. However, the use of these principles has always finitary character in Euclid. (6) Introduction of entities of higher complexity. In Propositions 20-22, Euclid uses the “class” of all pairs that “have the same ratio.” Each such class is uniquely associated with one pair of numbers, namely the least pair of numbers that have the same ratio. Euclid gives an effective procedure for finding such a least pair.
(7) The finitary principle and the use of effective procedures. Euclidean arithmetic is constructed from below, beginning from the unit. Further, a number of arithmetical concepts are introduced in the Definitions of Book VII. From these, the concepts of part, multiple, parts, proportionality, and prime numbers are not defined effectively. However, they become effective in virtue of Propositions 1, 2, and 3 that provide an effective procedure for any numbers to find their common measure. In this way, the proofs of the Propositions 4-19 should be considered as effective either. The introduction of more complex objects is realised through the comparison of these objects and the establishment of an equality-type relation between them. Euclid always provides an effective procedure for finding the least pair of the objects found in equality-type relation.
Therefore all propositions that involve existence of numbers appear, in Euclid’s arithmetic, associated with some effective procedure for finding the required number. This kind of arithmetic is constructed without assumptions of axiomatic character. It lacks the concept of absolute number or any elaborated concept of equality. (8) Reductio ad absurdum. Nowhere Euclid makes use of the assumption that all numbers form a fixed universe of discourse that is given beforehand. Hence, he never postulates or proves existence of numbers having a certain property, but always ‘constructs’ the required numbers by means of effective procedures. Existence of numbers is never deduced by strong indirect arguments. The use of reductio ad absurdum relies on a specific propositional form of the law of excluded middle and applies to decidable arithmetical predicates. Moreover, Euclid seems to avoid the law of excluded middle in the arithmetical proofs. All propositions of the form are proved by consideration of each part of the disjunction separately
(9) Underlying logic. The approach adopted by Euclid does not need any special predicate logic. Euclid’s arithmetic can be characterised as a finitary fragment of classical arithmetic; hence, it does not necessarily presuppose the full force of first-order predicate logic.
II. Arithmetic reasoning in the Neo-Pythagorean tradition
Pythagorean number theory in the form survived in the texts of later authors has the following distinctive features: (1) Arithmetical reasoning is conducted over a 3-dimensional “domain” that extends indefinitely in the direction of increase. (2) The monas, denoted by an alpha, is taken to be a designated object (yet, not a number), over which a (potentially infinite) iterative procedure of attaching an alpha is admitted. Numbers are defined as finite suites (finite instances of the natural series). Various kinds of numbers can be defined as suites constructed according to certain rules, following a finitary form of inductive definition. (3) Arithmetic is then developed by genetic constructions of various finite (plane or spatial) schematic patterns. Therefore, Pythagorean arithmetic represents a visual theory of counting over a distinctive combinatorial “domain.” (4) Arithmetical reasoning is conducted in the form of mental experiments over concrete objects of combinatorial character. Any assertion about numbers utters a law, which can be confirmed in each case by pure combinatorial means. (5) Arithmetic concerns affirmative sentences stating something ‘positive’ that can be confirmed by means of the construction of the corresponding configuration (deixis). No kind of ‘negative’ sentence is found. It is a ‘positive’ finitary fragment of classical arithmetic |
III. Self-reference in Plato and Aristotle: the Third Man Paradox
In Plato’s Parmenides 132a-133b, the widely known Third Man Paradox is stated, which has special interest for the history of logical reasoning, because of the self-reference involved. Many papers call attention to the violation of a metalogical principle – the type rules – because of the Third Man Paradox. This view is encouraged by the linguistic difficulties which Plato has faced in his attempt to formulate an ontology of abstract entities, i.e. that in Greek language abstract and concrete terms are formally indistinguishable: to leukon (literally ‘the white’) may signify both ‘the white thing’ and ‘whiteness’. The root of this misconception stems from the fact that in English literature the Platonic terms eidos and idea, are usually rendered by the same word: Form or Idea. However, we find a clear distinction in Plato’s texts, that corresponds to a fundamental logical distinction between class-as-many, distributed to its elements by predication, and class-as-one, standing as an individual in an extended sense and capable of being an element of a further class-as-many (Russel), or between distributive and collective class (Leśniewski), etc. The Third Man paradox is obtained, speaking in modern terms, as follows: out of all the things of an initial domain of particulars to which the property ‘… is large’ (idea) applies is formed an eidos (‘the large’). Further, this eidos is added to the initial domain of particulars and the scope of the universal quantifier ‘all’ is extended over it, taken for individual. The construction results in an impredicative generation of a (potentially) infinite sequence of new eide (infinite regress).
Plato puts the following solution into Parmenides’ mouth. The eidos is defined as a paradigm, which expresses the form of instances of the eidos, considered as a singular thing ‘found’ in nature. Further, participation in an eidos is identified with instantiation of the eidos. Further, the eidos is compared with a fixed instance of it and the following question is posed: can we conclude that an eidos is similar to an instance of it on the basis that the latter is an instantiation of the eidos?
Plato defines similarity in such a way that leads to a negative answer to the above question: entities are similar to each other if and only if they participate in one and the same eidos. In this way, what is today called domain of the class (the domain of ‘participants’ of the eidos) is taken into consideration. This domain consists of homogeneous things (“similar” to each other). Therefore, neither a thing is “similar” (homogeneous) to an eidos, nor an eidos to another thing that participates in it; otherwise, if an eidos is “made similar” to a thing, we obtain the Third Man Paradox. In this way, Parmenides makes a clear demarcation between two kinds of homogeneous entities: the level of particulars and the level of eidon, and the confusion between them is ad hoc removed.
The Peripatetic commentaries of the Third Man Paradox focus primarily on the statement of the argument and the premises on which it is grounded, rather than on its solution by means of the predicate of similarity. The first scholar of antiquity who explicitly ascribes a solution to Plato is Proclus.
Bibliography
Useful Links
Back to the 6th Universal Logic School ! | ||||
