Chapter 2: Q18 Thanks! R 2 THE BEGINNINGs oF MATHEMATICs IN GREECE examples of p

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Chapter 2: Q18 Thanks!

R 2 THE BEGINNINGs oF MATHEMATICs IN GREECE examples of peculiar truths for geometry are "the definitions of line and straight. By these he presumably meant that one postulates the existence ofstraight lines. Only for the most basic whenever ideas did Aristotle permit the postulation of the object defined. In general, however one defines an object, one must in fact prove its existence. "For example, arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, whereas the existence of these attributes is demonstrated by means of the axioms from Aristotle also listed certain basic principles of argument previous conclusions as premis principles that earlier thinkers had used intuitively. One such principle is that a given assertion cannot be both true and false. A second principle is that an assertion must be either true or false: there is no other possibility For Aristotle, logical argumentaccording to his methods is theonly certain way of attaining may be other ways of gaining knowledge, but demonstration via scientific knowledge. Th a series of syllogisms is the one way by which one can be sure of the results. Because one cannot prove everything, however, one must always be careful that the premises, or axioms are true and well known As Aristotle wrote syllogism there may indeed be without these conditions, but such syllogism, not being productive of scientific knowledge, will not be demonstration In other words, one can choose any axioms one wants and draw conclusions from them, but if one wants to attain knowledge, one must start with "true" axioms. The question then becomes, how can one be sure that one's axioms are true? Aristotle answered that these primary premises are learned by induction, by drawing conclusions from our own sense perception of numerous examples. This question of the "truth" of the basic axioms has been discussed by mathematicians and philosophers ever since Aristotle's time. On the other hand, Aristotle rules of attaining knowledge by beginning with axioms and demonstrations to gain new results has become the model for mathematicians to the present day Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematicians apparently never used them. They used other forms, as have most mathematicians down to the present. Why Aristotle therefore insisted on syllogisms is not clear. The basic forms of argument actually used in mathematical proof were analyzed in some detail in the third century BCE by the Stoics, of whom the most prominent was Chrysippus (280 206 BCE). This form of logic is based on propositions, statements that can be either true or false, rather than on the Aristotelian syllogisms. The basic rules of inference dealt with by Chrysippus, with their traditional names, are the following, where p, q, and r stand for propositions: (1) Modus ponens (2) Modus tollens If p, then q If p, then q Not

Explanation / Answer

18.

a) All Ferraris are cars. All Cars Have wheels. So, All Ferraris have wheels.

b) All dogs have four legs. Rover is a dog. Rover has four legs.

-a) Ostrich is a bird. Ostrich cannot fly. Some birds cannot fly.

-b) All Officers are Soliders. All soliders are not Lieutnents. All Lieutnents are not Officers.

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