Rationalism In Math Research Paper

1.0 INTRODUCTION

Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, and complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analyzing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced.

Logicism created by mathematicians Richard Dedekind and GottlobFrege(1893) in Die Grundlagen der Arithmetik (1884) and other works.However, they are few of Mathematicians claim about logicism such asG.Leibiniz, B.Russell (1919), A.N.Whitehead and R.Carnap(1931). The purposeof logicism was to show classical mathematics is part of logic.

John L.Bell (n.d) state that the logicism of Bertrand Russell was in certain respects evenmore radical than that of Frege, and closer to the views of Leibniz. In The Principles of Mathematics (1903), Russell was asserts that mathematics and logic are identical. He claim that Pure mathematics is the class of all propositions of the form “pimplies q” where p and q are propositions ... and neither p nor qcontains any constants except logical constants(Russel, 1903).Paul Ernest (1991) stated thatthere are two claims from Russell about logicism,

( i )All the concepts of mathematic can ultimately be reduced to logical concepts,provided that these are taken to include the concepts of set theory or some system of similar power, such as Russell’s Theory of Types.

( ii )All mathematical truths can be proved from the axioms and rules of inference of logic

(Brouwer 1907, pg 179)

This means the Brouwer state that primordial intuition is linked to the priori intuition of time. Priori intuition in this sense are a thinking activity of human mind, independent language, that it cannot be adequately described by any prescribed rules, and it has objective reality, that it is the same in all thinking beings (Curry, 1963). When stated that all humans have primordial intuition, this means humans already understand what the meaning of number 1 is and secondly we can do the mental process of forming the number 1 can be repeated. When we do the repetition of forming the number 1, we will get the second number, which is number 2, repeat again and we will get number 3 and so on. This way we as human beings can construct any finite numbers after the other for any natural numbers. This mental construction is only possible if we have awareness of time within us. This immediate awareness is where the name intuitionism comes from (Snapper. E,