Appendix A: An Historical Overview We begin with an analogy between the history of calculus and the history of mathematical logic. 1 Calculus In Athens in the Golden Age of Classical Greece, Plato (c. 428-348 B.C.E.) made geometry a prerequisite for entrance to his philosophical academy, for a master of geometry was a master of correct and exact reasoning (see Thomas [1939, LID. Euclid (c. 300 B.C.E.) emphasized the importance of the axiomatic method, which proceeds by deduction from axioms. (From this point of view, logic is the study of deduction.) Euclid and Archimedes (287-212 B.C.E.) and their predecessors showed how to use synthetic geometry to calculate areas and volumes of many simple figures and solids. They also showed how to solve, using geometry, many simple mechanics, hydrostatics and geometrical optics problems. In the twenty centuries separating Euclid and Archimedes from Leibniz (1640­ 1710) and Newton (1640-1722), increasingly difficult problems of calculating areas and volumes and of mechanics and hydrostatics were solved one by one by special methods from Euclidean and Archimedian geometry. Each physical or mathematical advance made by the use of this geometric method required the extraordinary mathematical talent of a Galileo (1564-1642) or a Huygens (1629-1695). Things changed radically after Descartes's discovery, published as the appendix to his Discours de la Methode [1637,2.3], that geometric problems could be translated into equivalent algebraic problems. Geometric methods were replaced by algebraic computations. There were already strong hints ofsymbolic-algebraic methods of integration and differentiation in the work of Fermat (1601-1665) and in the works of Newton's teacher Barrow (1630-1677) and Leibniz's predecessor Cavalieri (1598-1647). The symbolic methods of differentiation and integration discovered by Ne\/ton and Leibniz made it possible for later generations to use the ordinary calculus to develop science and engineering without being mathematical geniuses. These methods are still the basis for understanding, modeling, simulating, designing 376 Appendix A: An Historical Overview and developing physical and engineering systems. Both Leibniz and Newton were aware of the breadth and importance of these discoveries for our understanding ofthe physical world. Calculus was well-named in that it reduced many problems of mathematics and physics to largely algebraic and symbolic calculation. 2 Logic Aristotle's (384-322 H.C.E.) theory of syllogistic also dates from the Golden Age of ancient Greece and the disputations of the Platonic Academy (see Plato's Euthydemus). It is found in the collection of his works, called by ancient editors the Oryanon. It consists ofthe Catagoriae, De Interpretatione, Analytica Priom, Analytica Posteriora, Topica and De Sophisticis Elenchis. We discuss only the elements of syllogistic from the Analytica Priom. This was the first successful calculus of reasoning with "all" and "some". In modern terminology, we translate "all" and "some" to the quantifiers "for all" and "there exists". To the modern eye, syllogistic looks quaint with its language of noun expressions, terms universal and particular. But it has solid motivation. For Aristotle, the world consisted of objects c which mayor may not possess a given property P. In our modern notation the letter P is called a predicate sym­ bol. A particular interpretation of P is given by specifying a nonempty domain C of objects and a set of these objects to be denoted by P. Then, with x a variable ranging over C, P(x) is a logical formula read "x has the property P" . Also, if c is a name for a particular object, then P(c) is a logical formula read "c possesses property P" . Now an object may simultaneously have many different properties. An object c may be simultaneously hard, round, red, lighter than water, in this room, on the floor, in the southeast corner. In the late seventeenth century, Leibniz thought that objects should be uniquely characterized by knowing all their properties. Leibniz called this idea the principle of identity of indiscernibles. Deducing that an object c has one property from the fact that c has some other properties is the kind ofquestion that Aristotle's syllogistic addressed. This use of logic is especially characteristic of the classificatory biology of Linnaeus (1707­ 1788) with its genus, species, and varieties. His system is a direct intellectual descendent of Aristotle's biology. Aristotle is often, for this reason, called the "father of biology". His conception of biology and his conception of syllogistic are intimately related. Syllogistic was taught in the standard college curriculum as part of the Trivium of Logic, Rhetoric and Grammar from the middle ages to 1900, and still persists in many Catholic colleges undiminished and unchanged as the main training in logical reasoning, even though outdated by modern mathe­ matical logic. Syllogistic's chief function was as a check that the quantifiers "for all x" and "there exists an x" were being used correctly in arguments. The aim was to eliminate incorrect arguments which use principles that look like logically true 2 Logic 377 principles but are not. We follow Aristotle's follower, Chryssipus (d. 207 B.C.E.), in writing syllogistic entirely as rules of inference. An example of a valid mode of syllogism in this style is the rule of inference called "mode Barbara". From "All Pare Q" and "All Q are R" infer "All Pare R". In contemporary logical notation we write P(x) for "x is a P", Q(x) for "x is a Q", R(x) for "x is an R", (V'x) for "for all x", (3x) for "there exists an x" and" -+" for "implies". Translated to a modern rule of inference, mode Barbara becomes From (V'x)(P(x) -+ Q(x)) and (V'x)(Q(x) -+ R(x)) infer (V'x)(P(x) -+ R(x)). In this notation P, Q, R are called unary predicate (or relation) symbols. An invalid Aristotelian mode is From "Some P are Q" and "Some Q are R" infer "Some Pare R". In modern notation, using /\ for "and" this "rule" would be translated as follows: From (3x)(P(x) /\ Q(x)) and (3x)(Q(x) /\ R(x)) infer (3x)(P(x) /\ R(x)). (Give a counterexample to this mode!) Syllogistic treated four forms of propositions, called categorical propositions, whose medieval names were A, E, I and 0: A) "Every P is a Q" E) "NoPisaQ" I) "Some P is a Q" 0) "Some P is not a Q" . The valid modes were given mnemonic names based on the sequence of proposi­ tions. Thus the valid mode listed above which has the sequence AAA as the two hypotheses and the conclusion was called "Barbara". 378 Appendix A: An Historical Overview Consider the sequence EAE which had the mnemonic name "Celarent": From "No Q is R" and "Every P is Q" infer "No Pis R". The modern notation for "not" is -', so this mode could be translated to modern notation as From -,(3x)(Q(x) 1\ R(x» and (V'x)(P(x) -> Q(x» infer -,(3x)(P(x) 1\ R(x». Exercise 1. Reconstruct the syllogisms from the vowels of the medieval mnemonics, which represent the two premises and the conclusion of the syllogism and then translate them into modern notation. 1. Darii 2. Ferio 3. Cesare 4. Camestres 5. Festino 6. Baroco 7. Darapti 8. Disamis 9. Datisi 10. Felapton 11. Bocardo 12. Ferison 13. Bramantip 14.Camenes 15. Dimaris 16. Fesapo 17. Fresison. (Warning: Aristotle assumes that predicates are nonvacuous. This means that, with modern conventions, the validity of 7, 10, 13 and 16 require that such assumptions be made explicitly.) Aristotle gave systematic derivations of some of these modes from others. His work was the first axiomatic system for deriving some logical truths from oth­ ers. In addition, he constructed counterexamples for false modes and rules for negations (contradictories). Aristotle also gave the first systematic discussion of modal logics (discussed in IV) based on the connectives (>p, "it is possible that p", and Dp, "it is necessary that p". These connectives, unlike those mentioned above, are not "truth functional". That is, the truth or falsity of p does not determine the truth or falsity of Dp or Op : p may be possible and false or p may be possible and true. (For a more detailed analysis of the work of the ancient Greek logicians, see Lukasiewicz [1957, 2.2J, Bochenski [1951, 2.2] and Mates [1961, 2.2J.) Even though syllogistic was useful in clarifying philosophical discussions, it had no substantial influence on mathematicians. Mathematicians had reasoned very tightly even before the time of Aristotle. Indeed, their work was traditionally the model of exact reasoning. However, their reasoning was not fully described by syllogistic. What was missing? With the benefit of hindsight, a short answer is: the rest of propositional logic and the notion of a relation with many arguments. Developing the work of Philo of Megara (c. 300 B.C.E.), the Stoic Chryssipus of Soli introduced implication (now written -», conjunction (now written 1\) and exclusive disjunction ("P or Q but not both"). Nowadays, instead of the last 3 Leibniz's Dream 379 mentioned connective we use inclusive disjunction, "P or Q or both" , written P V Q. Chryssipus understood the characteristic property of propositional logic: the truth or falsity of compound propositions built from these connectives is determined by knowing the truth or falsity of the parts. As for relations, Euclidean geometry is based on the relation of incidence R(x, y), meaning that x is incident with y, where x, yare points, lines, or planes. Thus a point may be incident with (lie on) a line; a line may be incident with (lie on) a plane.