A Course in the Philosophy and Foundations of Mathematics

An examination of mathematical methods and the search for mathematical meaning.

During your studies of mathematics, physics and engineering, you may find yourself distracted or troubled by meta questions about mathematics — questions that fall outside the syllabi of most of the coursework that you’ll take.

For those for whom this itch is persistent, what follows is an outline and reading list for a Course in the Philosophy and Foundations of Mathematics. Among the topics included are the relation of mathematics to science, the examination of mathematical method, and the search for mathematical meaning.

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Good mathematical technique and the case for mathematical insight

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the solution of the general question itself. … Good mathematical technique has built within it the mathematical insight of the best of previous generations.

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Catalysts in the Development of Mathematics

The development of mathematics has had many encouraging forces: societal, technological, cultural. These have served to accelerate mathematics and have been accelerated in turn, in many cases the pair becoming locked into a mutually beneficial resonance that has dramatically energized both.

In this article, I look at some of the significant catalysts, from the rise of the leisured class in ancient times to the impact of computing in modern times.
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The Development of Mathematics

… in a nutshell

The development of mathematics is intimately interwoven with the progress of civilization, influencing the course of history through its application to science and technology.

But mathematics has changed. Even the mathematics of the 1800s can seem quite strange now, so greatly has mathematics evolved in the past 100 years and so thoroughly has it been reworked in the post-modern approach.

Despite its arcane appearance from the outside looking in, the present, abstract and highly specialized state of mathematics is the natural evolution of the subject, and there is much ahead that is exciting.

Here, then, is the story of mathematics, in a nutshell…
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Characteristics of Modern Mathematics

What are the characteristics of mathematics, especially contemporary mathematics?

I’ll consider five groups of characteristics:

  1. Applicability and Effectiveness,
  2. Abstraction and Generality,
  3. Simplicity,
  4. Logical Derivation, Axiomatic Arrangement,
  5. Precision, Correctness, Evolution through Dialectic…

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What is Mathematics?

Can a definition be given that captures the meaning of Mathematics across the millennia of its recorded history? What unites the practice of mathematics throughout its history and into the present time?

In this article, I will try for a short answer by proceeding iteratively — convergence will be reached in two iterations….
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