“A worked exercise Polya’s
heuristic” and the “inquiry into mathematical existence” is the opening tactic
of the book. Now the problem is the counting the different parts of a
4-dimensional cube and indicate on what kind of calculation could be done.
Then, Hersh immediately changed it to 3-cube and count its edges, vertices and
faces. And count the same style in 2D and 1D object. The three sets of objects
show a pattern that can be deduced for a 4D object. This followed by a list of
questions about the 4-D object. If it
exists, where is it? If it doesn’t exist, how could we obtain such detailed
information about it? What about a 3-cube? Does it exist in ordinary space,
given that we can’t produce a perfect 3-cube as a physical object? Including
his own humanism, he imparts the possible answers to the questions and to help
explain the philosophy of mathematics. After these examples, he then switches
to the topic about his rejection to the three mainstream philosophies namely Platonism,
Formalism and Intuitionism and believes that his humanism is superior.
Platonism or the idea of the
existence of mathematical entities outside a space and time, thought, matter,
outside in an abstract realm of independent consciousness of individual or
social. It is really impossible for a student or a researcher to find answer to
the odd-numbered problems or to prove a theorem. When taking in an activity, a
Platonist view is required. Platonism, as Hersh point out, does not mean that
this is an adequate philosophy of mathematics. Indeed, he rejected this
philosophy for some reasons: it violates the experiential modern science; it
insists on acceptance between the two realities-physical and mathematics as a
strange parallel of existence; and it doesn’t make contact to the
flesh-and-blood of a mathematician.
Formalism says that mathematics
is a meaningless game played with precise but irrational rules. So Hersh
objected this philosophy, he argues that the rules are not irrational instead it
is determined by the society and interactions between social groups and the
physical entity of the earth.
Intuitionism accepts that the
set of natural numbers is a fundamental datum of mathematics which all
meaningful mathematics must be met by a process called finite construction that
doesn’t make use of law of the excluded middle. So Hersh adopts the
anthropological point of view in objecting this philosophy, that the intuition
of natural numbers are not that simply universal. Then Hersh make use of a
research and use Piaget (who proposed that children construct natural numbers
based on their experience), according to him natural numbers are not given by
God, but it is a construction of the coordination of concepts of ordering through
individual’s mind.
Hersh then offers the
humanist point of view for mathematics. He said that mathematics is beyond its
social-historic-cultural meaning. Then he added the third standard kinds of
existence- social. He also considered a pair of examples, the concept of “two”
and the existence of the 4-cube.To understand the concept of “two” is to see
that the word is used as an adjective and as noun. For adjective, it represents
counting. Hersh argues that the set of counting numbers is actually finite
because no one can count (1010)10. Also “two” is a noun.
According to the Hersh Philosophy the existence of an object in which this noun
referred “comes from a social process of disconnecting “from real objects, to
exist as shared concepts in the minds of the people who know elementary arithmetic.”
But this view is opposed by constructivism view or Piagetian, that a “two”
object is constructed using reflective abstraction. In both cases one can focus
on both mental activity and social interaction.
4-cube can be used to
illustrate the differences of the philosophies mentioned. For Platonist a 4-cube exists as a
“transcendental, immaterial, inhuman abstraction,” for the formalist and
intuitionist, there is no 4 cube but it is only a representation of nothing,
and for humanist it exist “at the social-cultural-historic level, in the shared
consciousness of people or a kind of shared thought or idea.”
The combination of mental
construction and formal expression is what the individual utilizes to have a
meaning in an abstract concept of an n-cube. Shared conceptualization is needed
in this concept because mental activity generally takes place in a social
context.
In chapters 3 and 4 Hersh
suggested that mathematics has “front and “back”. Front is consists of polished
results/concepts that we are now studying at school and “back” is what we do to
obtain those results. According to him, Platonism and formalism focused only on
the front and Humanism focuses on the back. Hersh also finds that mathematics
is not always right because mathematicians do mistakes. Mathematics, according
to Humanism, is not unique because every mathematician has different approach
to study the same phenomena. For example, Euclid proofs are incomplete and
there are alternatives to his geometry. He claimed that formalism doesn’t
describe mathematical results, yet he stated that before a mathematician writes
down the formal proof he or she already knew the results.
Intuition is an important
issue that a mathematician must consider and here Hersh provides some
interesting insights. For the Platonist, it is a mechanism in accessing ideal
world; connecting human into mathematical world. Then Hersh criticized
Platonism for not answering these questions: How intuition can be acquired? Why
does this vary in every individual? Does it really perceive an ideal
reality?
For Formalism, Intution
is the source of theorems which formal proofs are devised. Humanist, intuition
consists of objects of mathematical mental representation which are acquired
through repeating experience. These mathematical representations now are
checked by teachers or mathematical colleagues. Thus, according to Hersh, it is
a “mutually congruent mental representation” which is also called a set shared
concept.
In the Part two of the
book, Hersh give the capsule account, the introduction is written from the 50
individuals’ point of view from Aristotle to Wittgenstein which can be helpful
if an individual wants to be a mathematician. Hersh is less interested with
arithmetic/number theory/ algebra or the writers of mathematical philosophy yet
he pays attention to Piaget’s writings on cognitive psychology.
Mathematics comes from
many thought and logical foundations. For others, mathematics is formalism; and
for still others, mathematics is intuitionism. For Hersh, these philosophies
failed to deal with the questions of philosophy of mathematics and this brought
him into the Humanist point of view. For me, this book is valuable because of
its sketch of philosophical ideas. But for me, he should utilize effectively
his Humanist philosophy in order to attack the philosophical problems of
mathematics. He did not also tell about
the mechanism of between the interactions of human in that mathematical world
in Platonism. But I now wondered on what new philosophy can be deduced on
Hersh’s Humanist approach on Mathematics.
The totally of the book is not that understandable indeed but picking up the author's point of view made it a little bit easy for me to comprehend.
ReplyDeleteI agree, Hersh should've discussed his Humanist philosophy further! He still left some unanswered questions. Great read though!
ReplyDeleteI want to read the review,,, but I cant... hehehe Jorge the font is too small, konting laki namn jan.
ReplyDelete