The Philosophy of Mathematics


This material on the philosophy of mathematics is mainly oriented toward foundational issues.
Not just a bag of facts, mathematics is a highly structured edifice resting on deep foundations.
Some very brief notes on what a selection of philosophers have contributed to the philosophy of mathematics.
Mention of some of the interesting problems which arise in the philosophy of mathematics.
We give special prominence to this philosophical position on the relationship between mathematics and logic.


Mathematics is a logical science, cleanly structured, and well-founded. Here we look at those foundations.
What is a "foundation" for mathematics?
We discuss some of the ways the word "foundation" is used in relation to mathematics.
History of Foundations
An assortment of historical cameos, starting in ancient Greece, running through the turbulent present into an exiting future.
Logical Foundation Systems
The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics.
Alternative Foundation Systems
A bewildering variety of alternative foundation systems are superficially surveyed.


An introduction to some of the interesting philosophical problems which concern mathematics.
Mathematicians use special languages for talking about strange things, out of this world. What does it all mean?
What are these strange things that mathematicians talk about? Do they really exist? How can we tell? Does it matter?
Mathematics has often been presented as a pardigm of precision and certainty, but some writers have suggested that this is an illusion. How can we know the truth of mathematical propositions?
How can knowledge of abstract mathematics be applied in the real world?
Mathematics is a highly structured logical science, but dig deep enough and you can find some sand. Making the best foundations for mathematics involves philosophy.
What are the implications for mathematics of the information revolution? What can mathematics contribute?


Some very brief notes on what a selection of philosophers have contributed to the philosophy of mathematics.


Logicism is a philosophical theory about the status of mathematical truths, to wit, that they are logically necessary or analytic.
What it is
Just the claim that the theorems of mathematics are logically necessary or analytic.
Some Original Readings
The primary sources of the logicist doctrine are Frege and Russell.
What it isn't
Not a claim about formalisability, either in principle or in any specific logical system. Not a claim about how mathematical truths are discovered or applied. Not concerned any aspect of mathematics other than the theorems that mathematicians seek to demonstrate.
Arguments Against Logicism
It is generally understood that set theory is required to do modern mathematics. The existence of sets is, however, not logically necessary. The truths of mathematics are therefore contingent.
Arguments For Logicism
The truths of mathematics are the same in all possible worlds, since they do not depend upon the existence of sets, just upon the consistency of the supposition that the required sets exist. Since true in every possible world, mathematics must be logically necessary.

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