Since when did Mathematics meet Philosophy, really

BOOK REVIEW ON “What is Mathematics, really”

Since when did philosophy joined Mathematics?  If you study Math then you also studied Philosophy? What is Mathematics, really.  In his preface, Hersh recalls how he felt cheated by the explanation of the book “What is Mathematics” by Courant and Robbins which drives his curiosity and boost up his inspiration on writing the said book.  I don’t know but why certain people do seek more information out of the preexisting knowledge? Is it for recognition? Money? Power?  The power of human mind is just not understandable, if we base on curiosity a person will ask so many questions on how the world works and why the existence of certain objects exist. I feel I’m not just inclined to Mathematics ultimately philosophy that’s why I ask so much irrelevant question after reading a mind boggling preface. Something tells me to stop reading but since this is a school project I’ll force myself.

So much more of the introduction, a challenge was prepared just this early at chapter 1 introducing at “the inquiry of mathematical existence”. The problem is to count the various parts of a four dimensional cube and reflect on what kind of sense the calculations would make. Sounds tricky right? Hersh gave mathematical clues on how to solve this problem. He considered a 3 dimensional cube, which was easier, to have its vertices, edges and faces. He does the same for the 2 cube and 1 cube. He lets us see what pattern we could deduce on this and make a formula out of it to solve our 4 dimensional cube. Congratulations, we have assumingly solved the problem but does this prove its existence? If it exists, where is it? If it didn’t exist how could we obtain detailed information about it? How about the other cubes we solved earlier, does it really assure such “perfection”. With this, we used possible answers to these questions or oftentimes ignored the question that helped explain various philosophies including his own perception of “mathematical philosophy” which he called humanism.

As the chapter progresses, Hersh points out why the three common philosophies, Platonism, formalism and intuitionism, is not suitable for Mathematics and his humanism is superior. Along the way he considers a number of generally accepted properties of mathematics and tries to disprove them.

An experimental science is what Platonism viewed about Mathematics, studying objects that really exist, though they clearly don't exist in a physical or material sense. It think it is impossible for anyone who has “actually” done Mathematics because there is a feeling of trying to find something that has an existence independent of what anyone might be thinking or doing. Hersh stressed that this philosophy is not adequate on Mathematics. Indeed, he rejects it for some reasons; it does not relate to material reality and it violates modern science.  The formalists argued that mathematics was really simply the formal manipulation of symbols. The intuitionists had the most radical point of view; essentially, they saw all mathematics as a human creation and therefore as essentially finite. In everyday life, we speak as Platonists, treating the objects of our study as real things that exist independently of human thought. If challenged on this, however, we retreat to some sort of formalism, arguing that in fact we are just pushing symbols around without making any metaphysical claims. Most of all, however, we want to do mathematics rather than argue about what it actually is. Just leave it to the philosophers.

As an alternative to the common philosophies of mathematics which he rejects. He offers humanistic type of philosophy that says “there is no need to look for a hidden meaning or definition of mathematics beyond its social-historical-cultural meaning”. This means a person can answer a big question by looking and treating it like a mathematical situation in everyday life. Thereby, mental and physical aspects are not enough to be a standard on the existence of philosophy of mathematics. It will only be complete if social aspect is added according to a humanist view.

To relate this, the example of the 4 cube can be used to apply the differences of the philosophies. For the Platonists, the 4D cube exists as a “transcendental in material on inhuman abstraction” an our ideas about it are representations of this ideal; for the intuitionist as well as the formalist, there is no real 4D cube, but only a representation “without a represented”; and for the humanist, the 4D cube exists “at the social-cultural-historic level” in the shared consciousness of the people. This may seem hard to beginners who are not introduced to the logic of philosophy but just deal with it.

The book concludes with an invocation of the story of the blind men and the elephant "as a metaphor for the philosophy of mathematics, with its Wise Men groping at the wondrous beast, Mathematics." All accept, it seems the humanist, who can see the whole elephant and laugh at the fallibility of all the others. Hersh doesn't seem to realize how arrogant this attitude is. On the other hand, his analysis of other philosophies is very useful, but they don’t rely on the humanist point of view and could easily have been made without it. The book missed on the realization that philosophy of mathematics is indeed philosophy, and not science. Anyhow, the book is still amazing no matter how mind blowing it is.