Wilkins’ and Hempel’s Notions of the Nature of Mathematics

By Marsigit
Yogyakarta State University

Wilkins, D.R., 2004, described some definitions of what is mathematics from different mathematicians.


The logician Whitehead 1 perceived that mathematics in its widest significance is the development of all types of formal, necessary, deductive reasoning; Boole 2 thought that it is not the essence of mathematics to be conversant with the ideas of number and quantity; Poincare 3 insisted that later generations will regard set theory as a disease from which one has recovered; Kant 4 argued that the science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain with the aid of experience; Von Neumann 5 believed that most of the best mathematical inspiration comes from experience; Gauss 6 stated that he has had the results for a long time, but he does not know yet know how to arrive at them; Riemann 7 claimed that if he only had the theorems, then he could find the proofs easily enough; Kaplansky 8 expressed that the most interesting moments are not where something is proved but where a new concept is involved; Weyl 9 stated that God exists since mathematics is consistent and the devil exists since we can't prove this consistency; Hilbert 10 concluded that mathematical science is an indivisible whole, an organization whose vitality depends on the connections between its parts, and the advancement in mathematics is made by simplification of methods, the disappearance of old procedures which have lost their usefulness and the unification of fields until then foreign; while Henkin 11 insisted that the number and rate of applications of mathematics is increasing and the equipment the students need to enable unforeseen applications, is not specialized mathematics but that core of the most general kind which will enable them to investigate new applications.

Hempel, C.G., 2001, advocated especially by John Stuart Mill that mathematics is itself an empirical science which differs from the other branches such as astronomy, physics, chemistry, etc., mainly in two respects: its subject matter is more general than that of any other field of scientific research, and its propositions have been tested and confirmed to a greater extent than those of even the most firmly established sections of astronomy or physics. Accordingly, the degree to which the laws of mathematics have been borne out by the past experiences of mankind is so overwhelming unjustifiably that we have come to think of mathematical theorems as qualitatively different from the well confirmed hypotheses or theories of other branches of science in which we consider them as certain, while other theories are thought of as at best as very probable or very highly confirmed. And of course this view is open to serious objections.

Hempel, C.G., 2001, elaborated that from a hypothesis which is empirical in character, it is possible to derive predictions to the effect that under certain specified conditions certain specified observable phenomena will occur; the actual occurrence of these phenomena constitutes confirming evidence, their non-occurrence disconfirming evidence for the hypothesis. He concluded that an empirical hypothesis is theoretically un-confirmable that is possible to indicate what kind of evidence would disconfirm the hypothesis; if this is actually an empirical generalization of past experiences, then it must be possible to state what kind of evidence would oblige us to concede the hypothesis was not generally true after all; and if any disconfirming evidence for the given proposition can be thought of. According to him, the mathematical propositions are true simply by virtue of definitions or of similar stipulations which determine the meaning of the key terms involved; their validation naturally requires no empirical evidence; they can be shown to be true by a mere analysis of the meaning attached to the terms which occur in them.

Hempel, C.G., 2001, argued so far that the validity of mathematics rests neither on its alleged self-evidential character nor on any empirical basis, but derives from the stipulations which determine the meaning of the mathematical concepts, and that the propositions of mathematics are therefore essentially "true by definition." He claimed that the rigorous development of a mathematical theory proceeds not simply from a set of definitions but rather from a set of non-definitional propositions which are not proved within the theory; these are the postulates or axioms of the theory and formulated in terms of certain basic or primitive concepts for which no definitions are provided within the theory; the postulates themselves represent "implicit definitions" of the primitive terms while the postulates do limit, in a specific sense, the meanings that can possibly be ascribed to the primitives, any self-consistent postulate system admits.

Hempel, C.G., 2001, further stated that once the primitive terms and the postulates have been laid down, the entire theory is completely determined. He summed up that every term of the mathematical theory is definable in terms of the primitives, and every proposition of the theory is logically deducible from the postulates; to be entirely precise, it is necessary also to specify the principles of logic which are to be used in the proof of the propositions. He claimed that these principles can be stated quite explicitly and fall into primitive sentences or postulates of logic. By combining the analyses of the aspects of the Peano system, Hempel 12 accepted the thesis of logicism that Mathematics is a branch of logic due to all the concepts of mathematics, i.e., of arithmetic, algebra, and analysis, can be defined in terms of four concepts of pure logic; and all the theorems of mathematics can be deduced from those definitions by means of the principles of logic.

References:
1 In Wilkins , D.R., 2004, Types of Mathematics, http://www.maths.tcd.ie/~dwilkins/
2Ibid.
3 Ibid.
4 Ibid.
5 Ibid.
6 Ibid.
7 Ibid.
8 Ibid.
9 Ibid.
10 Ibid.
11Ibid.
12Hempel, C.G., 2001, On the Nature of Mathematical Truth, http://www.ltn.lv/ ~podniek/gt.htm