The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics.
The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people's lives.
The logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts.
Platonism is the philosophy of Plato or the name of other philosophical systems considered closely derived from it. In a narrower sense the term might indicate the doctrine of Platonic realism. The central concept of Platonism is the distinction between that reality which is perceptible, but not intelligible, and that which is intelligible, but imperceptible; to this distinction the Theory of Forms is essential. The forms are typically described in dialogues such as the Phaedo, Symposium and Republic as transcendent, perfect archetypes, of which objects in the everyday world are imperfect copies. In the Republic the highest form is identified as the Form of the Good, the source of all other forms, which could be known by reason. In the Sophist, a later work, the forms being, sameness and difference are listed among the primordial "Great Kinds". In the 3rd century BC, Arcesilaus adopted skepticism, which became a central tenet of the school until 90 BC when Antiochus added Stoic elements, rejected skepticism, and began a period known as Middle Platonism. In the 3rd century AD, Plotinus added mystical elements, establishing Neoplatonism, in which the summit of existence was the One or the Good, the source of all things; in virtue and meditation the soul had the power to elevate itself to attain union with the One. Platonism had a profound effect on Western thought, and many Platonic notions were adopted by the Christian church which understood Platonic forms as God's thoughts, whilst Neoplatonism became a major influence on Christian mysticism.
Nominalism is a metaphysical view in philosophy according to which general or abstract terms and predicates exist, while universals or abstract objects, which are sometimes thought to correspond to these terms, do not exist.
Thus, there are at least two main versions of nominalism.
One version denies the existence of universals—things that can be instantiated or exemplified by many particular things (e.g. strength, humanity).
The other version specifically denies the existence of abstract objects—objects that do not exist in space and time.
Fictionalism is a methodological theory in philosophy that suggests that statements of a certain sort should not be taken to be literally true, but merely as a useful fiction.
Two important strands of fictionalism are modal fictionalism developed by Gideon Rosen, which states that possible worlds, regardless of whether they exist or not, may be a part of a useful discourse, and mathematical fictionalism advocated by Hartry Field, which states that talk of numbers and other mathematical objects is nothing more than a convenience for doing science. Also in meta-ethics, there is an equivalent position called moral fictionalism.
Fictionalism consists in at least the following three theses:
Claims made within the domain of discourse are taken to be truth-apt; that is, true or false.
The domain of discourse is to be interpreted at face value--not reduced to meaning something else.
The aim of discourse in any given domain is not truth, but some other virtue(s) (e.g., simplicity, explanatory scope).
Next session: 26 November 2012 (12 weeks long) Workload: 5-6 hours/week
About the Course Reasoning is important. This course will teach you how to do it well. You will learn some simple but vital rules to follow in thinking about any topic at all and some common and tempting mistakes to avoid in reasoning. We will discuss how to identify, analyze, and evaluate arguments by other people (including politicians, used car salesmen, and teachers) and how to construct arguments of your own in order to help you decide what to believe or what to do. These skills will be useful in dealing with whatever matters most to you.
Next session: September 2012 (10 weeks long) Workload: 4-8 hours/week About the Course We live in a complex world with diverse people, firms, and governments whose behaviors aggregate to produce novel, unexpected phenomena. We see political uprisings, market crashes, and a never ending array of social trends. How do we make sense of it? Models. Evidence shows that people who think with models consistently outperform those who don't. And, moreover people who think with lots of models outperform people who use only one. Why do models make us better thinkers? Models help us to better organize information - to make sense of that fire hose or hairball of data (choose your metaphor) available on the Internet. Models improve our abilities to make accurate forecasts. They help us make better decisions and adopt more effective strategies. They even can improve our ability to design institutions and procedures. In this class, I present a starter kit of models: I start with models of tipping points. I move on to cover models explain the wisdom of crowds, models that show why some countries are rich and some are poor, and models that help unpack the strategic decisions of firm and politicians. The models covered in this class provide a foundation for future social science classes, whether they be in economics, political science, business, or sociology. Mastering this material will give you a huge leg up in advanced courses. They also help you in life.
All of us have some sort of "philosophy of life," even though we may not have verbalized it. Here you can get ideas for your own philosophy of life. You can see what others think of your own philosophical ideas, and you can help others to become clearer in their own thinking.
When there is difference of opinion, we have an opportunity for "friendly debate," a very growth-promoting experience.