Modern Platonism Mathematical Platonism is the form of realism that suggests that mathematical entities are abstract, have no spatiotemporal or causal properties, and are eternal and unchanging. This is often claimed to be the view most people have of numbers. Both Plato's cave and Platonism have meaningful, not just superficial connections, because Plato's ideas were preceded and probably influenced by the hugely popular Pythagoreans of ancient Greece, who believed that the world was, quite literally, generated by numbers. A major question considered in mathematical Platonism is:
Introduction The dispute between rationalism and empiricism takes place within epistemology, the branch of philosophy devoted to studying the nature, sources and limits of knowledge.
The defining questions of epistemology include the following.
What is the nature of propositional knowledge, knowledge that a particular proposition about the world is true? To Indispensability thesis a proposition, we must believe it and it must be true, but something more is required, something that distinguishes knowledge from a lucky guess.
A good deal of philosophical work has been invested in trying to determine the nature of warrant. How can we gain knowledge? We can form true beliefs just by making lucky guesses. How to gain warranted beliefs is less clear.
Moreover, to know the world, we must think about it, and it is unclear how we gain the concepts we use in Indispensability thesis or what assurance, if any, we have that the ways in which we divide up the world using our concepts correspond to divisions that actually exist.
What are the limits of our knowledge? Some aspects of the world may be within the limits of our thought but beyond the limits of our knowledge; faced with competing descriptions of them, we cannot know which description is true.
Some aspects of the world may even be beyond the limits of our thought, so that we cannot form intelligible descriptions of them, let alone know that a particular description is true.
The disagreement between rationalists and empiricists primarily concerns the second question, regarding the sources of our concepts and knowledge. In some instances, their disagreement on this topic leads them to give conflicting responses to the other questions as well. They may disagree over the nature of warrant or about the limits of our thought and knowledge.
Our focus here will be on the competing rationalist and empiricist responses to the second question. Some propositions in a particular subject area, S, are knowable by us by intuition alone; still others are knowable by being deduced from intuited propositions.
Intuition is a form of rational insight. Deduction is a process in which we derive conclusions from intuited premises through valid arguments, ones in which the conclusion must be true if the premises are true.
We intuit, for example, that the number three is prime and that it is greater than two. We then deduce from this knowledge that there is a prime number greater than two.
Intuition and deduction thus provide us with knowledge a priori, which is to say knowledge gained independently of sense experience. Some rationalists take mathematics to be knowable by intuition and deduction.
Some place ethical truths in this category. Some include metaphysical claims, such as that God exists, we have free will, and our mind and body are distinct substances. The more propositions rationalists include within the range of intuition and deduction, and the more controversial the truth of those propositions or the claims to know them, the more radical their rationalism.
Rationalists also vary the strength of their view by adjusting their understanding of warrant.
mathematical entities such as sets (the indispensability thesis).1 Moreover, scientists are ontologically committed to all those entities that are indispens- able to the best current scientiﬁc theories. In general, an indispensability argument is an argument that purports to establish the truth of some claim based on the indispensability of the claim in question for certain purposes (to be specified by the particular argument). The indispensability argument in the philosophy of mathematics transfers evidence from natural science to mathematics. Thus, this argument is an inter-theoretic indispensability argument. One might apply inter-theoretic indispensability arguments in other areas.
Some take warranted beliefs to be beyond even the slightest doubt and claim that intuition and deduction provide beliefs of this high epistemic status. Others interpret warrant more conservatively, say as belief beyond a reasonable doubt, and claim that intuition and deduction provide beliefs of that caliber.
Still another dimension of rationalism depends on how its proponents understand the connection between intuition, on the one hand, and truth, on the other.
Some take intuition to be infallible, claiming that whatever we intuit must be true. Others allow for the possibility of false intuited propositions.
The second thesis associated with rationalism is the Innate Knowledge thesis. The Innate Knowledge Thesis: We have knowledge of some truths in a particular subject area, S, as part of our rational nature.
The difference between them rests in the accompanying understanding of how this a priori knowledge is gained. The Innate Knowledge thesis offers our rational nature. Our innate knowledge is not learned through either sense experience or intuition and deduction. It is just part of our nature.
Experiences may trigger a process by which we bring this knowledge to consciousness, but the experiences do not provide us with the knowledge itself.
It has in some way been with us all along. According to some rationalists, we gained the knowledge in an earlier existence. According to others, God provided us with it at creation.MATHEMATICS, INDISPENSABILITY AND SCIENTIFIC PROGRESS 87 to the reasonableness of the Indispensability Thesis.
My aim in this paper is to show that the Indispensability Thesis is . Indispensability argument for realism. This argument, associated with Willard Quine and Hilary Putnam, is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets.
The form of the argument is as follows. mathematical entities such as sets (the indispensability thesis).1 Moreover, scientists are ontologically committed to all those entities that are indispens- able to the best current scientiﬁc theories.
Quine's views on indispensability arguments in mathematics are scrutinised. A weak indispensability argument is distinguished from a strong indispensability thesis. The weak argument is the combination of the criterion of ontological commitment, holism and a mild naturalism.
It is used to refute. The second thesis of abstractness follows close behind, because clearly we cannot prove the existence of mathematical objects in the same . 1 Justin Clarke-Doane Columbia University Debunking and Indispensability Harman: “In explaining the observations that support a physical theory, scientists [must] appeal to mathematical principles (The Indispensability Thesis).On the other hand, one never seems to need to.
Indispensability argument for realism. This argument, associated with Willard Quine and Hilary Putnam, is considered by Stephen Yablo to be one of the most challenging arguments in favor of the acceptance of the existence of abstract mathematical entities, such as numbers and sets. The form of the argument is as follows. MATHEMATICS, INDISPENSABILITY AND SCIENTIFIC PROGRESS 87 to the reasonableness of the Indispensability Thesis. My aim in this paper is to show that the Indispensability Thesis is . Let the Indispensability Thesis be the claim that the contents of our typical mathematical beliefs figure into the best explanation of every observation (in Harman’s .