What to do in the absence of instinctive knowledge?

            Reuben Hersh, the author of ‘What is Mathematics, really?’ wrote it as he was determined to convey what he thought was not conveyed by the authors of ‘What is Mathematics?’, Richard Courant and Herbert Robbins. He insisted that Mathematics came first before its philosophy. He believed that the philosophy of Mathematics is as the philosophy of other sciences wherein it is subject to change and growth.

            The conversation between the author and Laura, a 12-year-old child, relays that a man could not be a hundred percent sure of what he can perceive through his senses, but can be a hundred percent sure towards one added with one is two. He also rejected the three main mathematical philosophy which are Platonism, Formalism, and Constructivism. He said that mathematical philosophy should be humanistic. His aim was to prove that the philosophy he proposed was better than the ones he rejected.

            Part one started by introducing an exercise in Polya’s heuristic wherein the question was if a 4-cube exists. He started by making a pattern from the count of the vertices, edges, and faces of a 3-cube, 2-cube, and a 1-cube. The reason why Hersh debunk the Platonism is because it stated that Mathematics is situated outside the space and realm we don’t even know. Even at our present understanding of Mathematics, we usually deal with the problems given to us and find solutions for it. It is unusual, we have to admit, that we look for mathematical problems on our own and seek for its answer. Another mathematical philosophy he rejected was Formalism. He strongly reacted to this because it stated that Mathematics is just a game and that its rules are random, unsystematic. I affirm to Hersh’s stand and contradict to Formalism such that Mathematics was taught to us in a very systematic way. There are various steps we need to overcome in solving problems. Though each of us uniquely differs in our approach, still it is systematic. As for intuitionism or constructivism, Hersh rejected this as it only employs the context of natural numbers as basis of fundamental data. It constricts what he said that Mathematical philosophy is subjected to change and growth.

            It is true, may it be Philosophy or any other fields that it is a challenge whether it can be used as a tool for further studies or just a useless pursuit. Hersh acknowledges that there is a front and back parts of Math. The front is the one exposed to the world for view because it contains the answers, and the back part which has the steps on how you came up with the answers. Here comes another reason why Hersh rejected the three mathematical philosophies: all three focuses on the front part of math, whereas humanism insists on examining the back part where we came up with the answers. With this, he elaborated more that mathematicians uses or may use different approaches as to how they would solve a problem.

            Intuition is the instinctive knowledge of a person. This means that when a person is introduced with a problem, he or she already has an idea on what the answer would be and how to come up with that answer. Hersh contradicts with this idea. He explains that instinctive knowledge is not available always. An example of this is the western children wherein they don’t acquire this until a certain age. He also criticizes the formalist that they could not prove the theorems they come up with.

            Hersh acknowledges the existence of infinity but he does not consider it as a number. He also underestimates the brain; that it cannot have a capacity to hold infinity. It may be true that no computers invented can get hold of a very large number but it is also wrong to underestimate the capacity of a brain. No one studying the brain had ever been able to say in figure the capacity of one’s brain. Hersh considers his mathematical philosophy superior among other philosophies. He mentioned that the book would be an aid to mentors and students in understanding more about mathematics, but I can see that his main point is to attract readers to accepting his proposed philosophy.

            As a student, the way I studied Mathematics was secluded in the four corners of the classroom not realizing that it is and was more than that. I think I needed to change my conception about Math because I have to accept the fact that I could never avoid it. It would always be of existence in all fields of science. That is, one characteristic of Math; it is interdisciplinary. It coexists and gives strong proofs to other fields of study.

            To answer the question stated in the title, I do not think that there is a state wherein a person does not have an instinctive knowledge. One thing I know, our Creator would not give us problems we could not answer. Biologically speaking, our mind is way complex that any inventions would not, I believe, surpass. Though the author has said that Mathematics would be ever-growing, I know that there is this side of math that will always stay constant and true. Overall, the book was able to convey what it aimed to and had met its objectives. I appreciated the part where the author uses words easily understood. Kudos for the author.