Platonism in Mathematics"The value of platonistically inspired mathematical conceptions is that they furnish models of abstract imagination. These stand out by their simplicity and logical strength. They form representations which extrapolate from certain regions of experience and intuition. Nonetheless, we know that we can arithmetize the theoretical systems of geometry and physics. For this reason, we shall direct our attention to platonism in arithmetic. But I am referring to arithmetic in a very broad sense, which includes analysis and set theory."
Theism & Mathematical Realism"Is mathematics a mere human invention? Or is it an exploration of an already existent realm? That is the basic question. Historically, most mathematicians have believed that mathematical truths such as '2+5=7' exist independently of human minds, being universally and eternally true. Mathematicians believe they are discovering properties of, say, prime numbers, rather than merely inventing them. This view of mathematics dates back to Pythagoras (500 BC) and Plato (400 BC). It is often called 'Platonism'. In order to avoid association with other features of Plato’s philosophy, I shall refer to the notion that mathematical truths are objective as mathematical 'realism'."
Mathematics and Metaphysics"In this paper, I argue that metaphysics, logic, and mathematics, as systematic investigations into the nature of order and knowledge, have much in common, and that mathematics as the way science quantifies data can be the vehicle science uses to investigate ultimate questions. Then referring to the work of George Lakoff and Rafael Núñez, I ask whether mathematics expresses something innate in the universe or something innate to the structure of the human brain. In raising this question, I argue that if the universe itself is mathematical, then dualism is affirmed and materialism falsified. However, if mathematics only expresses the cognitive structure of the human brain, as Lakoff and Núñez maintain, then it is compromised as a reliable guide for understanding the ultimate nature of the cosmos. In the later case, it follows that science will be unable to address metaphysical questions in any compelling way."