Reuben
Hersh’s “What is Math, Really?” generally attempts to debunk the traditional
notion of how mathematics is ideally viewed or treated. These three mainstream
mathematical ideologies are Platonism, Formalism and Intuitionism. The book is
a good read for those who are already enthusiastic in Math but an unsteady
platform for those who are still finding their enthusiasm.
As mentioned in his book, Platonism asserts
that “mathematical entities exists outside space and time, outside thought and
matter, in an abstract realm independent of any consciousness, individual or
social”. It is something existing out there, maybe an answer to a problem still
waiting to be figured out. While Formalism, as what Hersh posits that
“mathematics is a meaningless game played by explicit arbitrary rules”. One may
understand the prescribed rules but it can be broken or be modified when
played. Intuitionism, on the other hand, is described by Hersh as “the
acceptance of the set of natural numbers as the fundamental datum of
mathematics from which all meaningful mathematics must be obtained through a
process of finite construction that does not make use of the law of the
excluded middle”. Perhaps, the knowledge on numbers is acquired naturally or
through intuition, with no consideration to rational processes.
All
of these traditional guiding principles were criticized by Hersh and that he
strongly argues that the better way to philosophize mathematics is through the Humanistic
approach. He debates that Platonism creates a disconnected aura of an “abstract
existence” and the “material reality”. Hence, “this independence of individuals,
makes mathematics seems immaterial and inhuman” as what he pointed out. He also
argue that the cold arbitrary rules mentioned in formalism is not valid as
rules are products of long “interactions and pressures of the society and the
physiological and biological environment of earth”. He then argues that natural
numbers “given by a supreme being” is not universally true because the
cognitive development of a person is based on his experiences from his or her
environment, “constructed in the person’s mind through coordinating concepts of
set inclusion and ordering”. His argument on intuitionism then continued on the
second part of his book, in which Hersh points out Piaget’s significant contribution
to cognitive psychology and its relationship to the philosophy of mathematics.
It
is agreeable that mathematics must be “understood as a human activity, a social
phenomenon, a part of human culture, historically evolved, and intelligible
only in a social context” or what he calls humanist perspective. It is indeed
unarguable that Mathematics was established because of the humans’ need to
self-actualization. But what Hersh failed to expound are the outcomes if one
adopts the concept of a humanist and how it differs from the outcomes of the
three ideologies he tried to debunk. This would left the readers a “so what?”
kind of disposition, wondering what unique and significant impact would this
contributes to the greater body of mathematical knowledge.
Although
Hersh uses simple mathematical terms or formulas that can be understood by an
average learner, it is still not appropriate for readers who are new to the Philosophy
of Math. He positions his book for readers who are already acquainted of the
ideas discussed in this book. Although there is a lengthy introduction
explaining the basic concepts of Platonism, Formalism and Intuitionism, still,
the language is a bit technical and intimidating creating an exclusive
environment for those who are highly acquainted of math. Beginners would have a
hard time understanding the merging of Philosophical and Mathematical idea as the
treatment of the writer is clearly intended for people with high interest in Mathematical
Philosophy.
From
a perspective of a student not from the field of mathematics, it is hard to
digest the ideas Hersh is trying to relay as it would take time to process the connection
of Philosophy and Math. And because comprehending his deep mathematical thoughts
would take a great amount of time and energy, it creates a demanding and
exhausting impression. It could have been a lot interesting if it was taken
into an open discussion inside the classroom rather than digesting it
individually (so the exhaustion could have been shared by everyone equally). It
looks like it is trying to homogenize the different complex functions of the left
and right hemispheres of the brain. This educational material could have been a
lot useful if the basics of these concepts are introduced, on a clear and
humble manner, fit for readers from different intellectual disciplines rather
than directly indulging the readers into the arguments of how math is viewed
and defined in different ideologies. Hence, applying these considerations would
make mathematics more “humane”.
No comments:
Post a Comment