Philosophy of Mathematics

Rating:
Summary: Needs an Editor
Review: James Robert Brown has exposed with excellence in his book a very good defense of Platonism in Philosophy of Mathematics. He shows the basic premises that are shared more or less by all Platonists, including Frege, Godel, and many others.

He shows why philosophers argue against Platonism, and which are the biases and confusions they make that apparently they show a rejection to it.

It discusses subjects such as numbers, sets, geometrical objects, graphs, and even fractals, and how Platonism can recognize all of them as abstract objects, and how pictures can help us psychologically to grasp these abstract objects.

However, with all of this I have only one problem. He proposes a kind of "mind's eye" by which we are able to "grasp" these abstract objects. Although he presents a very keen argument for refuting the argument that only through sensible experience we are able to know anything about the world, I still feel uneasy and not satisfied about his epistemological account. The hypothesis of this "mind's eye" is the reason why most philosophers find the Platonist proposal so objectionable. In order to account for our knowledge of abstract objects, we must posit a kind of myserious mystic faculty of the "mind's eye".

I think that Husserl's categorial intuition and categorial abstraction (which he proposes in his "Logical Investigations") or Katz proposal of intellectual intuition (in his "Realistic Rationalism") are much more acceptable proposals than Brown's "mind's eye".

Despite this difference, I highly recommend this book, along with Jerrold Katz's "Realistic Rationalism", as a great and serious exposition of the Platonist proposal in philosophy of mathematics that I have ever found. It also serves as a good introduction to philosophy of mathematics.

Rating:
Summary: Good Defense of Platonism in Phil. of Mathematics
Review: James Robert Brown has exposed with excellence in his book a very good defense of Platonism in Philosophy of Mathematics. He shows the basic premises that are shared more or less by all Platonists, including Frege, Godel, and many others.

He shows why philosophers argue against Platonism, and which are the biases and confusions they make that apparently they show a rejection to it.

It discusses subjects such as numbers, sets, geometrical objects, graphs, and even fractals, and how Platonism can recognize all of them as abstract objects, and how pictures can help us psychologically to grasp these abstract objects.

However, with all of this I have only one problem. He proposes a kind of "mind's eye" by which we are able to "grasp" these abstract objects. Although he presents a very keen argument for refuting the argument that only through sensible experience we are able to know anything about the world, I still feel uneasy and not satisfied about his epistemological account. The hypothesis of this "mind's eye" is the reason why most philosophers find the Platonist proposal so objectionable. In order to account for our knowledge of abstract objects, we must posit a kind of myserious mystic faculty of the "mind's eye".

I think that Husserl's categorial intuition and categorial abstraction (which he proposes in his "Logical Investigations") or Katz proposal of intellectual intuition (in his "Realistic Rationalism") are much more acceptable proposals than Brown's "mind's eye".

Despite this difference, I highly recommend this book, along with Jerrold Katz's "Realistic Rationalism", as a great and serious exposition of the Platonist proposal in philosophy of mathematics that I have ever found. It also serves as a good introduction to philosophy of mathematics.

Rating:
Summary: Needs an Editor
Review: The philosophy of mathematics contains many interesting issues. This makes it unavoidable that there should be some interesting material in JR Brown's _Philosophy of Mathematics_. On the other hand, while Brown's prose is certainly accessible, his treatment of the issues in the philosophy of mathematics does not aid but rather gets in the way of the reader's understanding and appreciation of them.

Brown is desperately in need of an editor, or at least a proofreader. There are countless grammatical and typographical errors which really ought not to be in a final edition; and while most of these do little more than make the text seem unpolished and amateurish, some obscure the subject matter: for example, because of Brown's extreme laziness in bracket-counting, his proof of Gödel's second incompleteness theorem (chapter 5) is quite incomprehensible unless the reader is adept enough at playing Sherlock Holmes to come up with the proof Brown *really* meant to give.

But Brown doesn't just need an editor to fix his mechanical errors: he needs someone to help him choose what to print. In chapter 8, on constructive mathematics, Brown fills up a full page with lengthy quotes from Brouwer, admitting that "This is pretty obscure stuff," (116) but declining to elucidate Brouwer's point except with more quotation, which he admits is "no better".

There is interesting material in this book, but the presentation is far from perfect, and often aggravating. Brown is to be credited with a sense for what is interesting, but he would put that sense to better use advising another writer than writing himself. If you can't find another introduction to the philosophy of mathematics, this book is worth reading; if you can, I strongly advise you to investigate your alternatives.