Book Review: "What is Mathematics, Really" by Reuben Hersh

   Reuben Hersh’s “What is Mathematics, Really?” is a book which views (philosophically) the exploration of the true nature of mathematics. Basically, Hersh wants to deviate from the already constructed “status quo” of mathematics which is a reference of “indubitable truth”.  Hersh explains that we need to see mathematics as more of a science which also advances because of mistakes and corrections. He stresses that mathematics could not exist only in the external form (objects which exist outside space and time) but needs to coexist materially and become part of human culture. As he explains his understanding on the true nature or philosophy of mathematics, he deducts the other philosophies-Platonism, Formalism and Intuitionism, to debunk them as philosophies of mathematics.

Debunking the Three Philosophies

“Mathematics is part of human culture and history, which are rooted in our biological nature and our physical and biological surroundings. Our mathematical ideas in general match our world for the same reason that our lungs match earth's atmosphere.” –Reuben Hersh 

As we go along the book, we see an in-depth explanation of the three philosophies and why, according to Hersh, they are unfit to be the definitions of the true nature of maths. Hersh tries to explain the nature of these philosophies and tries to compare them to his humanist philosophy of mathematics.

Source: http://m-phi.blogspot.com/2011/05/peanuts-and-platonism.html

In the first philosophy Hersh tries to debunk is the fundamental foundation of why 2 plus 2 would always be 4. This could be considered as the Platonism way of approaching math. Hersh quotes that “Platonism says mathematical objects are real and independent of our knowledge. Space-filling curves, uncountably infinite sets, infinite-dimensional manifolds—all the members of the mathematical zoo—are definite objects, with definite properties, known or unknown. These objects exist outside physical space and time. They were never created. They never change.” Because of Hersh’s viewpoint of mathematics as a social concept or human culture, mathematics should “make-contact” with the mathematics in our physical world. Mathematics existing without material reality would violate his stand on advancing mathematics through corrections and mistakes thus debunking the idea of Platonism as a philosophy of mathematics. He also stresses that mathematics is formed through concepts. These are matched to what we see physically in our world. Once we see reference, we immediately picture out a shape or a variable which would closely become the absolute truth. This then make the existence of mathematics as part of human culture.

“Mathematics is a meaningless game”. Above Platonism, Hersh strongly disagrees to the philosophy of Formalism as a way to define the true nature of mathematics. As the quote implies, mathematics is a game- formed by rules and people to play by the rules. However, these rules can be arbitrary. For formalism, everything is meaningless unless interpreted. As long as one plays the game, the rules would simply follow. Hersh however, disagrees to the fact that these rules are all based on whims but are “historically determined by the workings of society that evolve under pressure of the inner workings and interactions of social groups, and the physiological and biological environment of earth.” Also, because of the existence of these rules, it would make all moves (as long as it abides the rule) the absolute truth. For Hersh, this does not apply to real life.

            Lastly, Intuitionism is a philosophy of mathematics which deals with ‘mathematics a creation of the mind’ (Iemhoff, 2008). In Intuitionism, there is a fact that a set of natural numbers exist and that it objects the law of the excluded middle. This would imply that a statement once proven true can be in time proven false and vice versa. The argument of Intuitionism over Humanism has not been very clear but as quoted by Hersh, “The mathematical knowledge of one generation is rooted in that of its parent's generation.” This presumes that the existence of mathematics cannot be only from the conception of the mind but are based in the physical acts of collecting, matching, ordering and counting.

Why Humanism above all else?

            Hersh has debunked the three philosophies of mathematics to pave way for his stand on the humanistic approach of understanding mathematics. To Hersh, “There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning.” Apart from the physical and mental existence of math, there should exist its social entity. Mathematics cannot continue to be disconnected as physical or mental entities. It should then be considered as part of the human culture that no one can deem it and its rules arbitrary and of no basis.

             Hersh has also defined that the Humanistic approach allows us to see that mathematics is not perfect. In the book he has shown a lot of proofs which makes certain equations questionable because of its complexity. This would allow the advancement of math by allowing the acceptance of corrections and mistakes. That mathematics is like all other modern sciences.

Conclusion

            Hersh’s book is a serious work in explaining the true nature of math. Although given proofs, it is somewhat difficult to understand because there seems to have no strong foundations between his arguments. Although I must say that his in-depth search between the short-comings of the other three philosophies were great, the grounds for the humanistic approach in dealing with the philosophy of mathematics is obscure and weak. From the book, I would have gone to say that the philosophy of mathematics might as well lean to intuitionism rather than the humanistic approach. To Hersh, he only considered dealing mathematics through a sense of culture or comparison to other groups, but he has not dealt mathematics in an individual perspective, where (based on my experience) is more done so.

References:

Iemhoff, R. 2008. Intuitionism in the philosophy of mathematics. 10 December 2013. http://plato.stanford.edu/entries/intuitionism/