For everyone who attended class, thank you very much. I would like to apologize for the disorganization; the room I reserved became unavailable at the last moment, and I find it very difficutl to organize and present material without a blackboard. I am making arrangements for a different room.

The class began with a discussion of the definition of mathematics. Although many quality definitions were proposed, for the remainder of the class, I will be considering mathemtics as a formalized language. It is a set of symbols and rules about how to use them. This defintion, while consistent with a formalist approach to mathemtics, does not exclude other foundational schools.

What is all this talk of "foundations"? The study of the foundations of mathematics, a field which runs thru most mathemical and philosophical fields, concerns establishing notions of truth, provability, and verifiablity. What sort of objects are mathemtical? What sort of methods is it acceptable to use in mathemtics? What does it mean to say a statement is true? What does it take to show that something is true (or false)? Among modern mathemticians and philosophers, there are four broad answers these questions. These are the four traditional schools of the foundation of mathematics:

Formalism is the most widely accepted school of thought among non-practicing mathematicians. It says that mathematics is an "empty game", that is, that mathematics is a set of symbols and rules for their use. These symbols have no referants in the real world; they are deviod of meaning. This means that all mathemtics is content-free; when you make a mathemtical statement you assert nothing about the world.

Logicism is very similar to formalism. It states that all mathemtics is really just logic. This view is popular among logicists (wonder why?) and philosophers.

Intuitionism was very popular for the first part of the century, but has fallen out of favor as of late. I am in many ways an intuitionist. Intuitionism claims that mathematics is a natural activity of the human intellect, and that all languages, formal and otherwise, are poor attempts to comunicate these most basic of thought-patterns.

Platonism has also fallen out favor. However, a number of leadingm mathematicians are vehemant Platonists. Goedel (the best logician of the past century), for example, was a strong proponent of Platonsim. Platonism in philosophy of mathematics, as in any other field of philosophy, holds that there is an ideal world of forms, and that words in language (formal or otheriwse) are direct pointers to this worlds. So, in the world of forms, there is an ideal "redness", and every red thing "partakes" of this form.

At theis point the conversation turned to a discussion of infinity. I laid the groundwork for serious discussion of the topic with some socratic conversation about cardinality and the nature of "size" and "infinity". however, by this point, the class was devoloving. I will repeat this material at the next class.