As described in my previous post on My Project, I have tasked myself with collecting, organizing, and revising the central principles of math and physics. My intent is to to synthesize and analyze a comprehensive narrative on mathematical and physical methods. Ideally, by identifying the central principles of math and physics in a single and self contained narrative I will see how to extend these methods beyond the confines of math and physics. Practically, I hope to provide myself with a detailed foundation on which to grow my mathematical and physical intuitions so that I am prepared to solve future problems with greater clarity and certainty.
Since my goal is seemingly outlandish, as I have already described in the post on my project, I have begun with the utmost concern for making concrete and measurable progress. To this end I have succeeded in maintaining a consistent schedule of reading, writing, and reworking my way through some semisignificant mathematical texts.
Already I have completed a deep read of Lang and Murrow's Geometry: an advanced level high school text which moves from the concepts of distance and angles to transformations and isometries. A deep read is one in which I read, write, and react to the definitions, theorems, and exercises of a text. Since my goal is to extract the narrative which ties the elementary mathematical structures together, I keep a keen eye on the choice of logical and conceptual progressions chosen by the writer. For example, Lang and Murrow choose to emphasize the logical structure of geometric notions, providing a clear indication of what constitutes an assumption, a theorem, or a proof. The purpose of such an upper level high school text is to develop mathematical thinking by using the objects and relations of Euclidean geometry as a conceptual playground. Most elementary geometry texts take this perspective, and contain some shadow of the prototypical rigor of Euclid's Elements. Interestingly, it would a appear as if Lang and Murrow decided the SMSG axioms of geometry, or the so called "New Math" axioms for geometry, would form the skeletal structure of their logical development, though this is not strictly adhered to as it seems to be more of a remedy to the SMSG methods.
Such a decision is significantly dependent on the influence of the axiomatic method, though it is clearly far from a full fledged introduction to axiomatic structures. There are many reasons for this, well beyond being a pedagogical predisposition. The axiomatic method is, as a tool, often unappreciated by those with a large helping of "common sense." To a naive utilitarian it is only relevant up to that point at which it aids in organization while not completely eliminating one's intuitions. Often, it is not seen as a necessary mode of thought when navigating everyday mathematics. To present an axiomatic structure as a starting point is often not only a pedagogical error but is, I believe, a firm error in analysis.
To an extent, education takes a winding path through concepts via the methods of synthesis and analysis. The synthetic movements might be seen in the students application of old tools to new problems, where as the analytic methods are only rarely touched on below the undergraduate level. This is due in part to the lack of a significant mass of content to analyze. It seems that before students are able to exercise their analytic minds they must amass a sufficient amount of general knowledge about a topic, otherwise the analytic efforts are made in vein if at all.
In order to gather together enough concrete examples of these analytic and synthetic movements I have focused on the topics of math and physics. Lang and Murrow's geometry is just a simple starting point, a sort of soft summary of elementary geometric notions as they are presented at the high school level.
On the opposite end of the spectrum, I have completed a deep read of Clark's Axiomatic Geometry. Clark's treatment is short and dips into a whole method of education that is significantly different from that of Euclid. Rather than providing a student with a description of the general methods of a mathematical topic, the expectation is that a student must discover the basic methods on their own using only a well chosen list of definitions and theorems. Clark's presentation of axiomatic geometry is done via the Moore method, which takes a topic from the perspective of an alien who has recently arrived to earth and has been introduced to a subject without any of its historical or intuitive content. Surprisingly, a semisuccessful student of a Moore method class will discover that they are capable of making sense of mathematical customs and methods without being explicitly introduced to them by some omnipresent authority figure. That is, it is possible for them to extract intuition directly from the axiomatic structures given the appropriate statements of relevant theorems and definitions alone. This is startling to both the student and the teacher when it occurs as it amounts to lifting one's self by their own bootstraps. In that moment hides the essence of creativity and discovery, as it seems possible to get something from nothing.
For now it appears as if Geometry is one of the essential parts of any larger narrative of mathematics. It not only contains objects and relations which are principle to many mathematical theorems, but showcases the methods of mathematical thought using some of the more accessible parts of the mathematical world e.g. points and lines can be drawn on a board or a page, whereas the set of all cats is somewhat more difficult to depict to one not already familiar with that method of abstraction. In general, geometry is usually the first point during a students education where their intuition is used to develop definitions which are then used to destroy and rebuild new intuitions about geometric objects they once thought were familiar.
An example of this movement is found in a simple proposition on convex quadrilaterals. It is often startling to a beginning student of geometry that by connecting the midpoints of any convex quadrilateral one forms a parallelogram. The student is often familiar with the idea of a four sided figure, but surprised to find there is such well behaved structure to any such object. Often they respond by wondering how it is that ANY four sided figure contains this well behaved parallelogram, which leads to a need for proof. Specifically, it is easy for a student to imagine a four sided figure all of whose sides are not parallel with one another, and yet, even in this extreme case, there is hiding a well behaved parallelogram.
In addition to basic geometry, I have completed a deep read of half of Halmos' Naive Set Theory, a beautiful exposition of just what is naive set theory. I am also six chapters into Stewarts Calculus: Early Vectors and have learned much about the shortcomings of a book with disconnected and dissatisfying fragments of narratives. Thus far, Stewart's calculus represents a shadow of the lesser parts of the New Math era, where the content has been seemingly structured by a polycephalous madman at war with himself over just what constitutes calculus. I have found myself asking many times over whether Stewart's book wishes calculus to be a collection of concepts or a collection of methods, and have yet to see any benefit to what I hope is a conscious attempt at some sadistic compromise between the two. I wish not to imply that calculus is exclusively a collection of methods or a collection of concepts, but rather that the narrative chosen by Stewart is wanting regardless of what calculus is or might be. Furthermore, that such defects are not aesthetic, but impact the utility and relevance of such a text.
I do not wish to go any further into the faults of books from the past, there will always be such things in any critically examined text, but rather to move past them in search of the appropriate remedy. For now the remedy of my choosing is to see what is there and summarize what is there with the hope of discovering a relevant and meaningful solution.
I have begun a deep read of Knuth's Concrete Mathematics, Bourbaki's Elements of Mathematics, and Landau and Lifshitz General Physics, Mechanics and Molecular Physics. While I intend on finishing each of these, it will be some time before that task is completed. I have additional texts which I will be analyzing as I complete these, but in the mean time I have also taken an effort to work on short term projects of a more popular nature.
To most it must seem as if my project is a fantastical dream, one of a highly self indulgent nature, I have tried to address such doubts in my original post. The process of deep reading requires long and contiguous periods of focus without distractions, either internal or external. Worry of failure is often a form of internal distraction which tickles most minds. Some believe these concerns derive from the impossibility of defining success. To combat such nuisances I have been composing a wide and detailed collection of questions and problems asked throughout each of the texts I have studied thus far. It seems that to many the solutions to such problems, either in particular or in general, are more valuable than the narratives presented by each text. This can be seen in the predisposition of most universities in America to use graded homework and exams as a measure of a student's competence in a subject. As of yet I have not seen any simple way of testing a student's understanding of a narrative other than by requiring them to construct one of their own, and such things do not lend themselves so simply to the standard grading methods that are so popular at the moment.
It is my intent to wrap up these questions and problems with their relevant solutions in a form which is most easily digested by someone wishing to move past them as quickly as possible without condemning their examination grades. As such I am not writing a solutions manual, though many do desire such a thing and believe it to be synonymous with success. I hope to provide some tools for thought, but will not attempt any pretense at giving the reader knowledge or understanding.