Poles of Mathematics

“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.