Elegi Menggapai "Logicism as the Epistemological Foundation of Mathematics"

By Marsigit
Yogyakarta State University

In his Principia Mathematica, Irvine A.D. elaborates that Logicism was first advocated in the late seventeenth century by Gottfried Leibniz and, later, his idea is defended in greater detail by Gottlob Frege.



Logicism 1 is the doctrine that Mathematics is reducible to Logic.

According to Irvine A.D., Logicism, as the modern analytic tradition, begins with the work of Frege and Russell for both of whom mathematics was a central concern.

He propounds that mathematicians such as Bernard Bolzano, Niels Abel, Louis Cauchy and Karl Weierstrass succeed in eliminating much of the vagueness and many of the contradictions present in the mathematical theories of their day; and by the late 1800s, William Hamilton had also introduced ordered couples of reals as the first step in supplying a logical basis for the complex numbers.

Further, Irvine A.D. (2003) sets forth that Karl Weierstrass 2, Richard Dedekind and Georg Cantor had also all developed methods for founding the irrationals in terms of the rationals; and using work by H.G. Grassmann and Richard Dedekind, Guiseppe Peano had then gone on to develop a theory of the rationals based on his now famous axioms for the natural numbers; as well as by Frege's day, it was generally recognized that a large portion of mathematics could be derived from a relatively small set of primitive notions.

For logicists 3, if mathematical statements are true at all, they are true necessarily; so the principles of logic are also usually thought to be necessary truths.

Frege 4 attempts to provide mathematics with a sound logical foundation.

On the other hand, Wilder R.L. persists that the effort to reduce mathematics to logic arose in the context of an increasing systematization and rigor of all pure mathematics, from which emerged the goal of setting up a comprehensive formal system which would represent all of known mathematics with the exception of geometry, insofar as it is a theory of physical space.

The goal of logicism 5 would then be a comprehensive formal system with a natural interpretation such that the primitives would be logical concepts and the axioms logical truths.

Eves H. and Newsom C.V. (1964) maintain that Russell, in his Principia of Mathematica starts with primitive ideas and primitive propositions to correspond the undefined terms and postulates of a formal abstract development.

Those primitive ideas and propositions 6 are not to be subjected to ίnterpretation but are restricted to intuitive cοncepts of logic; they are tο be regarded as, or at least are to be accepted as, plausible descriptions and hypotheses concerning the real world.

Eves H. and Newsom C.V. further specifies that the aim of Principia of Mathematica is to develop mathematical concepts and theorems from these prίmitive ideas and propositions, starting with a calculus οf propositions, proceeding up through the theory of classes and relations tο the establishment of the natural number system, and thence to all mathematics derivable from the natural number system.

Specifically, Eves H. and Newsom C.V. ascribe the following:
To avoid the contradictίons of set theory, Principia of Mathematica employs a theory of types that sets up a hierarchy of levels of elements. The primary elements constitute those of type 0; classes of elements of type 0 constitute those of type 1 ; classes of elements of type 1 constitute those of type 2; and so οn.

Ιn applying the theory of types, one follows the rule that all the elements of any class must be of the same type.

Adherence to this rule precludes impredίcative definitions and thus avoids the paradoxes of set theory.

As originally presented in Principia of Mathematica, hierarehies within hierarehies appeared, leading to the so-called ramίfied theory of types.

Ιn order to obtain the impredίcative definitions needed to establish analysis, an axiom of reducibility had to be introduced.

The nonprimitίve and arbitrary character of this axiom drew forth severe criticίsm, and much of the subsequent refinement of the logistic program lies in attempts to devise some method of avoiding the disliked axiom of reducibility. 7

On the other hand, Posy C. enumerates that as a logicist, Cantor is not concerned with what a number; instead, he wonders of the two sets of objects which have the same number.

Cantor 8 defines the notion of similarity of size i.e. equality of cardinal that two sets have the same cardinality if there exists a one to one mapping between them which exhausts them both.

Cantor 9 shows cardinality of Q = cardinality of N by showing one to one mapping; and he found that a denumerable set is one that can be put into a one to one correspondence with the set of natural numbers.

Cantor 10 conjectures that there are only two types of cardinal numbers: finite or infinite; thus all infinite sets would be of the same size; however, he proved this conjecture false because the set of R is larger than N; in fact there are more real numbers between zero and one than there are total natural numbers.

Furthermore, Posy C. indicates that the implication of Cantor’s investigation of infinity is that there is no longer taboo to learn it and infinity is accepted as a notion with rich content and central to mathematics as well as that a conceptual foundation for the calculus was provided that is all notions of mathematics was reduced to the ideas of natural numbers and the possibly infinite set.

By showing one to one mapping. Cantor proves that the set N x N = {(1,1), (2,1), (1,2), (1,3), (2,2), …} is denumerable. However, as Posy C. 11 claims there is also resistance of Cantor’s work.

Kronecker 12, for example criticizes that thought all Cantor did was nonsense because they just the artificial work of man; he wonders of mathematics has been reduced to natural numbers and sets and argues about the rigor behind natural numbers, what are natural numbers, why does the reduction stop there; and concluded that there is a general move towards creating a non-intuitive conceptual framework for natural numbers.

Still in the sphere of logicism, Zalta E.N. (2003) contends that Frege formulates two distinguished formal systems and used these systems in his attempt both to express certain basic concepts of mathematics precisely and to derive certain mathematical laws from the laws of logic; in his system, of 1879, he develops a second-order predicate calculus and used it both to define interesting mathematical concepts and to state and prove mathematically interesting propositions.

However, in his system of 1893/1903, Frege 13 addes (as an axiom) what he thought was a distinguished logical proposition (Basic Law V) and tried to derive the fundamental theorems of various mathematical (number) systems from this proposition.

According to Zalta E.N. (2003), unfortunately, not only did Basic Law V fail to be a logical proposition, but the resulting system proved to be inconsistent, for it was subject to Russell's Paradox.

Meanwhile, Folkerts M. (2004) designates that Logicist program was dealt an unexpected blow by Bertrand Russell in 1902, who points out unexpected complications with the naive concept of a set.

However, as it was stated by Irvine A.D that Russell’s famous of the logical or set-theoretical paradoxes arises within naive set theory by considering the set of all sets which are not members of themselves.

Such a set appears to be a member of it self if and only if it is not a member of itself.

Some sets, such as the set of teacups, are not members of themselves and other sets, such as the set of all non-teacups, are members of themselves.

Russell 14 lets us call the set of all sets which are not members of themselves S; if S is a member of itself, then by definition it must not be a member of itself; similarly, if S is not a member of itself, then by definition it must be a member of itself.

The paradox 15 itself stems from the idea that any coherent condition may be used to determine a set or class.

References:
Irvine, A.D., 2003, Principia Mathematica, Stanford Encyclopedia of Philosophy, Retrieved 2004
2Ibid.
3 Ibid.
4 Ibid.
5 Wilder, R. L. , 1952, “Introduction to the Foundation of Mathematics”, New York, p.206
6 Eves, H and Newsom, C.V., 1964, An Introduction to the Foundation & Fundamental Concepts of Mathematics, New York: Holt, Rinehart and Winston, p.287
7Posy, C. ,1992, “Philosophy of Mathematics”. Retreived 2004
8Ibid.
9 Ibid.
10 Ibid.
11In Posy, C. ,1992, “Philosophy of Mathematics”. Retreived 2004
12Ibid.
13Zalta, E.N., 2003, “Frege's Logic, Theorem, and Foundations for Arithmetic”, Stanford Encyclopedia of Philosophy, Retrieved 2004
14Irvine, A.D., 2003, “Principia Mathematica”, Stanford Encyclopedia of Philosophy, Retrieved 2004
15Ibid.