FOUNDATIONS AND LOGIC
What are the foundations of mathematics?
The foundations of mathematics include how to formulate and analyze the language
(you have to “speak” the right mathematical language to make meaningful mathemat-
ical statements), axioms (a statement accepted as true without proof), and developing
logical methods in all mathematical studies. The most basic mathematical concepts in
the foundations include numbers, shapes, sets, functions, algorithms, axioms, defini-
tions, and proofs.
Why do so many philosophers studythe foundations of mathematics?
There are three underlying reasons why philosophers often study the foundations of
mathematics. First, these foundations have always been a part of scientific thought,
with the abstract nature of mathematical objects offering unusual and often unique
philosophical quandaries. Second, the subject offers a high level of technical sophisti-
cation, allowing philosophers to develop connections between models and patterns,
laying the groundwork for many other sciences. And finally, the foundations of mathe-
matics provides ways for philosophers (and mathematicians) to try out general philo-
sophical doctrines in a specific scientific context.
What is logic?
Although it closely resembles mathematics (and is sometimes used as a basis for it),
logic is a branch of knowledge or inquiry that is separate from mathematics and the sci-
ences, but it is still used by both fields in various ways. Simply put, logic is described as
the systematic study of well-founded inference, in which there is a definite distinction 103
FOUNDATIONS
OF MATHEMATICS