Take it or Leave it

*book review: What is Mathematics, Really? by Reuben Hersh

No one can exactly define math. I dare you. If you can let me know. Math's meaning seems like infinite.

      When I had a glance at the book (yes glance!), I don't want to read it, but I have to, I should!

       As the book, I mean Hersh imposed his ideas (blah, blah.. blah), I want to stop reading just because his' don't fit mine. But who am I to question his greatness? He's a mathematician, a philosopher, has a good shot in literature, an experienced man and I am just simple freshman studying Math 1 in UP. Do you feel me? It was like decoding computer codes, so hard to get.

      I don't know what he's tackling about. Philosophy here, there and almost the entire book. The ideas don't easily register into my head. I decided to stop, took a break, had a Kitkat. Back to work, reading, generating. My head was aching.I needed some help. Internet came for rescue. Feeling relieved. At least, I can already relate.

      Honestly, I have no idea regarding philosophies of math.Ignorant feeling. I love math! But philosophies, get out of my sight! Never had given it a shot. Where in this world did they get that?

      Many including myself are saying, " Why study philosophies of mat? What good willl it do to us? Will it change the country's economy? Will it feed the poor? Will it give employment? Will it help execute the corrupt (#salotngbayan)? Just a waste of time.

     As I get through reading it, I was encouraged by the statement, " Your studying history, philosophies of math and the like...simply because you do not want to just pass by. You'll make a difference someday.",quoted from our Math 1 instructor Cielo Fe Blasing. (Hi maam!).

     Philosophers are critical thinkers (I believe), if you can think as much as they can then you're not far from being one. It follows that you'll affect others, the country, and the world. Still thinking that studying it is useless?  

      Bull's eye! From that statement of our instructor, I was in great shame. That shame slowly fades out, replaced by determination. Shift mood. I said, " I'll finish this review! I will! Go fight! Reading this was really a challenge. Who knows I can make my own philosophical view. I guess?. Make a difference, maybe not now, but watch out!

      Who would have think that Mr. Math has a partner in life, Mrs. Philosophy. Am I just the one here don't know her? If you also then welcome to the club. I'm sure many still haven't met her.Have a ride. Let's know that Mrs. Philosophy.  I doubt if she's beautiful. I bet not. Kidding.

Reuben Hersh, the author of What is Mathematics, Really? get his way into making the book because of being displeased and discontented by the book of the authors Richard Courant and Herbert Robins, "What is Mathematics?" by

He started it with having the reader to think of the biggest number and who could count that number. Imagination now comes in,as a result philosophies about math were made. It tackles the whole idea of is mathematics invented or discovered. Later on, book’s point circled into an inquiry of what different thinkers perceive math and how will they react about it.

Hersh go into detail of the 3 classic mathematical philosophies: Platonism, formalism and constructivism which asserts that mathematics is secluded, undying, modern and unearthly.  He decline to agree in these philosophy and instead imposed the humanist philosophy that mathematics must be implied and viewed as a social emergence of human interaction and activity and not only mental and physical. Math is a societal sensation and occurrence, embraced in the human culture, historically progressed, and comprehensible in a communal context or the cultural aspect. According to him, mathematical matters are fashioned by humans, not random but originated from the commotion with the prevailing mathematical objects and from the natural phenomenon that humans use it in daily living and as tool in exploring the other disciplines say sciences.

In a deeper terms, Hersh perceive things as come from a social process of understanding real objects into a shared concepts, that real objects are identified as is because of how it does to the society. In the humanist perspective, mathematics is the analysis of mental objects with replicating properties, and intuition enters as we examine these internal, mental objects. This view is similar to the empiricist view. This view is far from constructivist, which views object in a mechanism known as reflective abstraction which utilizes of individual mental activity so with social interaction. Hersh’s humanist philosophy focuses mainly on social issues as conflicting to the strive of every individual to make a point based from experiences.

Platonism which considered to be the most universal philosophy concerning math, as what Hersh perceive it, it’s the idea that math is everywhere, exists even beyond space and time, thought and matter. In a simple say, math is already there before we knew it and have an idea of what it is. Hersh opposed to it for such reasons that it is far from a quantifiable reality or have connection with legally related mathematician and it defies the practicality of modern science, furthermore it demands on recognition of different existing realities: mathematical and physical. Objections to Platonism are mostly dealt with by having the ideal world as existing in the mind of God. Some mathematicians don’t take God accounted sought math to be separated from any godly involvement. For Hersh as he perceived it, the established philosophies are unsuccessful in reasonably dealing with the leading issues of mathematics, and this brought him to the humanist point of view.

On the other hand, Formalism as defined by Hersh is then irrelevant pastime by clear but illogical rules. He extremely contest against it. For him math is not arbitrary but rather emerge historically from the forces and act of the people that progress by the continuing effort of the communal group and so with the biological and physiological nature of the Earth. Furthermore, Hersh defended that math is very far from the philosophy of formalism. It is only an illusion for which the idea of meticulously obeying rules without a confrontation from any sorts of judgment. The last one is the constructivism or known also as the intuitionism. It is the tool for getting into the proposed ideal world. Relating societal awareness with the reality surrounding math. Considered to be the result in the mind of from the initial phases into larger ones and later, operating mental images, problems, and make sense of discovering things in our own.  It considers all set of natural numbers as the primary figures of math and by way of a finite construction process it can obtain all significant mathematics. In protesting to this philosophy, Hersh makes use of the anthropological point of view, which have the idea that the instinct of natural numbers is not universal. In other terms, Hersh established that the conception of natural numbers come from experiences and modes of thinking of every individual.

Explaining further Hersh’s idea using the 4- cube. For the Platonist, the 4-cube happens to be mystical, unconfined, irrelevant and far from humanist view, while for the intuitionist and formalist, there’s no such thing as 4-cube but only a mere representation. For the humanist view, 4-cube exists as a kind of communal idea from the consciousness of people at the social cultural-historic level. But as I see it, it’s more of perceptual formations of every individual instead of their societal collaborations.

To test the humanist philosophy, it must be able to aid as means for studying relevant inquiries. One of these is what he perceived to be the unseen part of mathematics. If we are going to relate it in the real world, for instance in a restaurant, we only see the menu and the product food they made, but we weren’t able to view the whole process of how the food was made. To make sense, Hersh wanted to tell that to completely appreciate math, you have to go into detail of its "back portion" and not only consider the front part, getting into the background of its philosophies and anything that will uncover the mystery behind it. For Hersh, the 3 philosophies only covers the front while humanism demands to get into the back of it.

From the idea of the back of math, Hersh imposed that math is not perfect that even the greatest among all mathematicians made mistakes. And some of the proofs are lengthy and very complex to the extent that it is not already clear that anyone can have a doubt of it. Also math is not unique because different mathematicians deal with mathematical situations in different manner and approaches. Say for instance Humanism contends to Euclid’s proofs that he considers to be incomplete and there are alternatives to Euclidean geometry. As he concludes, Math doesn’t comprise truths that are pure, well defined and conclusive.

I can honestly say I don’t fully understand the book and which I think, cannot easily be absorbed by anyone who don’t have a deep perspective in math. I was like a newbie working in a factory that needs to have ample time to study precisely what that field is and how it relates to the world. What disturbs me most is that, he is imposing anything about his social-historic humanist approach but he was not able to utilize effectively to argue the great problems of the philosophy of math. Hersh’s analyses and criticisms of philosophies he rejected to take into account are very useful but they don’t depend one Humanist view and could easily have been made without it. I was left pondering of other new results on the philosophy of math that will arise from Hersh’s humanist view.

One thing that instil to me was Hersh’s statement in the preface that he’s upto a discussion of teaching but I failed to see it. The preface also claims that the book can assist education reform and again it fails to establish it. Depressing fact about Hersh is that he is somewhat a mathematician expressing an opinion about education and thinking that the fact that he has this view means that it has been established for all.

Math is no simple argument that has minute effect to the world. Math is big deal. In the context of the book, he links to the different philosophies. And it seems that several segments seem to be recurring and unnecessary. Correspondingly, Hersh tried to put all the mathematicians in a contemporary American political context: for who are the lefties and on the right side in which he entailed to label the lefties for whom he agrees with on the subject of math.

Hersh has some very good points in his humanist philosophy, but he seems to pay most time tackling the competing views of Platonism and formalism. And as I see it, Hersh's historical review of the philosophy of math was not fully satisfied. Also, his shallow survey of other philosophers’ views is somewhat bias into his own preference.

The marginal thing about Hersh’s proposition in the end part of the book was that associating other theorists view with their diplomatic stand just to impose that Platonist’s view is bounded by political illiberalism and is autocratic while humanist is of liberal, unrestricted legislation.

Another thing also, Hersh’s failed to realize that philosophy of math undeniably and actually philosophy and is different from any sciences and disciplines, which then he don’t have the authority and he’s view cannot disregard nor discount the encompassing philosophical issues that associated to it.

After all its philosophy, it depends on how and what you believe in.The book is very good at many points but also strikes critical points concerning the philosophy of math.

What's your judgment? Will you take it or just leave it?Just make sure you're doing the right thing.