The Relationship of Mathematics and Philosophy

        Mathematics and philosophy has a history of engagement with each other since the days of Ancient Greece. Mathematics in addition to a source and inspiration for the philosophers, the method is also widely adopted to describe the philosophical thinking. We even know some mathematicians as well as philosophers alone, such as Descartes, Leibniz, Bolzano, Dedekind, Frege, Brouwer, Hilbert, G¨odel, and Weyl. In the last century in which logic is a study as well as the foundations of mathematics became an important study materials either by mathematicians as well as by philosophers. Mathematical logic has a role to philosophy to contemporary era in which many philosophers then studied logic. Mathematical logic has inspired the philosopher, then the philosopher is also trying to develop logical thinking such as "logic of capital", which was then further developed by mathematicians and beneficial to the development of a computer program and the analysis of language. One crucial point that a problem shared by mathematics and philosophy as the foundation of mathematics problems. Both mathematicians and philosophers together stakeholders to examine whether there is the foundation of mathematics? If there whether the foundation is singular or plural? If the sole is then whether the foundation is? If it is plural then how do we know that one or some of them more important or more important as a foundation? In the 20th century, Cantor continued by Sir Bertrand Russell, developing set theory and type theory, with a view to using it as the foundation of mathematics. However, the study of philosophy has found that there is a paradox here or inconsistencies later revived mathematicians motivation in finding the essence of a mathematical system.

        With the theory of incompleteness, Godel finally concluded that a complete system of mathematics if he shall surely will not be consistent, but if he is consistent then he must not be complete. The essence of truth together intensively studied both philosophy and mathematics. Truth value study intensively studied by the field of epistemology and philosophy of language. In mathematics, through formal logic, the truth value are also studied intensively. Kripke, S. and Feferman (Antonelli, A., Urquhart, A., and Zach, R. 2007) has revised theory of the value of truth, and in this work the math and philosophy have a problem with. On the other hand, in one study of philosophy, namely epistemology, formal epistemology also developed a formal approach as a research philosophy that uses inference as the primary method. Such inference is none other than formal logic that can be attributed to game theory, decision-making, basic computer and probability theory.

        The mathematicians and philosophers together is still involved in the debate on the role of intuition in mathematical understanding and the understanding of science in general. There are steps in the mathematical methods that can not be accepted by an intuitionist. An intuitionist can not accept the rules of logic that the phrase "a or b" is true to a true value and b is true. An intuitionist can not accept proof by refutation method of negation. An intuitionist also can not accept the numbers as infinite or infinite number Factual. According to an intuitionist, are potentially an infinite number. Hence the attempt to develop mathematical intuitionist only with numbers that are finite or infinite.

        Many philosophers have used mathematics to build a theory of knowledge and reasoning generated by using mathematical proofs can be considered to have produced an outstanding record. Mathematics has become a major source of inspiration for philosophers to develop epistemology and metaphysics. From the thoughts of the philosopher that sourced the mathematical thoughts or questions arise such as: Is it numbers or mathematical objects really exist? If they exist whether inside or outside of our minds? If they exist outside our mind how can we understand it? If they exist in our minds how we can differentiate them with our concepts are the others? How is the relationship between mathematical objects with logic? Question about the "no" it is a metaphysical question of mathematical objects whose position is almost the same as the question of the existence of other objects such as universality, the properties of objects, and values; according to some philosophers if these objects exist then whether he was associated with space and time? Whether he is an actual or potential? Is he abstract? Or concrete? If we accept that mathematical objects are abstract, the method or epistemology that can explain how these objects? Maybe we can use evidence to explain these objects, but the proof is always relying on axioms. In the end we will see the "Infinite Regress" because philosophically we still have to question the veracity and validity of an axiom.

        Hannes Leitgeb in (Antonelli, A., Urquhart, A., and Zach, R. 2007) on "Mathematical Methods in Philosophy" has been investigating the use of mathematics in philosophy. He concluded that the mathematical method has an important position in philosophy. At some level mathematics and philosophy have common problems. Hannes Leitgeb has investigated which aspects of mathematics and philosophy have the same degree as did the review namely similarity between objects, object properties, logic systems, the meaning of the sentence, the law of cause and effect, paradoxically, game theory and probability theory. Philosophers use the logic of causality for to know the implications of concepts or ideas, even to prove the truth of his expressions. Joseph N. Manago (2006) in his book "Mathematical Logic and the Philosophy of God and Man" demonstrates the philosophy of using mathematical methods to prove Lemma that there are some creatures that are "eternal". Living things are still said to be eternal life.