topics: |  mathematics | history | philosophy | biography

One of my most valuable books on the history of science.  Their definition
of mathematics is well respected: The study of "the study of mental objects
with reproducible properties". (e.g. quoted in Edelman's Bright air,
Brilliant mind [Edelman 92], p. 162)

Review: AMS, Kenneth C. Millett

... [we are] introduced to the context in which mathematics exists and the
incredible magnitude of words devoted to communicating mathematics (hundreds
of thousands of theorems each year). How much mathematics can there be?
Raises questions re: underlying logical and philosophical issues, the role of
mathematical methods and their origins, the substance of contemporary
mathematical advances, the meaning of rigor and proof in mathematics, the
role of computational mathematics, and issues of teaching and learning. How
real is the conflict between “pure” mathematics, as represented by
G. H. Hardy’s statements, and “applied” mathematics?  they may ask.

    This is a book about the human experience of mathematics, connecting with
each person’s own experience doing mathematics.

    However, as a collection of essays about mathematics from these different
perspectives it is not entirely consistent. Asking for a definition or
description of “mathematics” is comparable to asking physicists for a
definition of “particle” or seeking the meaning of “love” from your
neighbors. ... What we understand seems to depend on our individual
experience and the experiences of others with whom we interact.  Often what
we understand is altered by how we say it and by how it is heard. Is not
mathematics much the same? The authors state, “Most writers on the subject
seem to agree that the typical working mathematician is a Platonist on
weekdays and a formalist on Sundays.” The substance of the mathematics
appears to change with experience and depends on the person recounting the
story. But it has an objective reality that is independent of the person.

    In a chapter entitled “Mathematical Reality” there is a discussion of the
human aspect of mathematical proof, the impact of computers (the fourcolor
problem, the distribution of primes, the Riecommmann hypothesis), and the
robustness of mathematical programs encompassing thousands of lines of code.
... It can help bring [readers] from a vision of mathematics as arithmetic
and memorization to an understanding of mathematics as an intellectually
challenging and creative experience— one in which there are surprises at
every turn, one in which today’s understanding is never sufficient but more
like a foundation upon which to build.  The more one learns, the more one
knows how little is known. An appreciation for the accomplishments of the
past is important if one is to understand the potential for the future.