Some recent reading has lead me back to considering the philosophical problems of Mathematics. To arrive at a clear and proper understanding of mathematical principles, it is necessaryto build all mathematical constructs from abstraction of entities occuring in reality. The historical introduction of many (perhaps most) mathematical devices and techniques has been driven by their pragmatic utility in addressing problems in the existing framework of mathematics. Failure to tie these new constructs to a basis in reality (physical reality, or abstraction fromphysical reality), results in an incomplete understanding of the meaning of these constructs. This can lead to inappropriate application of the constructs to model phenomena, which in turn can undermine the validity of theory based upon the arbitrary application of the mathematics. Even if thescience used in developing and verifyinga theoryuses a valid method,if the mathematics used to describe the theory is based in floating abstraction, the theory remains vulnerable to misinterpretation and misapplication.
My inspiration for pursuing an investigation toward properly rooting fundamental mathematics derives from a small number of sources. I have previously read through the final two of a series of articles (published in TIA in the late 90’s through 2001) by Ron Pisaturo in which criticisms similar to those I’ve described above are presented, with a handful of specific examples. Meanwhile, I am currently reading The Road to Reality by Roger Penrose, which is a lengthy text seeking to present the major elements of modern physics to the general reader through emphasizing the underlying mathematics. At the moment, I’ve completed only about 120 pages, but it is already clear that Penrose’s philosophy of mathematics is otherworldly – he describes a Platonic realm which contains the ideals of mathematics, and presents a rather confusing, but not revolutionary, set of relationships between the human mind, the physical world, and this Platonic realm. This alone was enough to indicate that Penrose is personally comfortable with working with floating abstractions, but his repeated description of the “magic” of complexnumbers makes this much clearer.
This is not to say that the book is without merit – in fact, for the purpose of understanding how the philosophical errors of modern physics stem from the philosophical errors of the underlying mathematics, the book may be ideal. And beyond this purpose, I chose the book(without knowing its content) to attempt to come closer to at least being conversant in elements of modern physics. [For those reading this without knowing me personally, I shouldmention that I have undergraduate degrees in both physics and mathematics, so these topics are not – or at least should not – be alien territory for me]. As to whether Penrose accomplishes his goal of making the mathematics of modern physics approachable for the general public, I highly, highly doubt that a reader without at least an engineering degree is going to be able to make it through the first 100 pages without becoming intensely confused. And these 100 pages are only scratching the surface of the math he covers. A full critique of Penrose’s book will come later.