Mathematics is the science of measurement, of establishing quantitative relationships between the properties of entities. The entities being measured occupy the whole spectrum of abstractness, from first-level concepts, which are based on perceptual data obtained by direct observation, to high-level concepts, which are further up in the edifice of knowledge. Furthermore, being the science of measurement, mathematics provides the logical glue that cements and cross-connects the structural components of this edifice.
1. The effectiveness and the power of mathematics (and more generally of logic) in this regard arises from the most basic fact of nature: to be is to be something, i.e. to be is to be a thing with certain distinct properties, or: to exist means to have specific properties. Stated negatively: a thing cannot have and lack a property at the same time, or: in nature contradictions do not exist, a fact already identified by the father of logic-11 some twenty-four centuries ago.
Mathematics is based on this fact, and on the existence of a consciousness (a physicist, an engineer, a mathematician, a philosopher, etc.) capable of identifying it. Thus mathematics is neither intrinsic to nature (reality), apart from any relation to man's mind, nor is it based on a subjective creation of a man's consciousness detached from reality. Instead, mathematics furnishes us with the means by which our consciousness grasps reality in a quantitative manner. It allows our consciousness to grasp, in numerical terms, the microcosmic world of subatomic particles, the macro-cosmic world of the universe and everything in between. In fact, this is what mathematicians are supposed to do, to develop general methods for formulating and solving physical problems of a given type.
In brief, mathematics highlights the potency of the mind, its cognitive efficacy in grasping the nature of the world. This potency arises from the mind ability to form concepts, a process which is made most explicit by the science of mathematics.-12
2. Mathematics is an inductive discipline first and a deductive
discipline second. This is because, more generally, induction preceeds
deduction. Without the former, the latter is impossible. Thus, the
validity of the archetypical deductive reasoning process
``Socrates is a man. All men are mortal. Hence, Socrates is a mortal.''
depends on the major premise ``All men are mortal.'' It constitutes an identification of the nature of things. It is arrived at by a process of induction, which, in essence, consists of observing the facts of reality and of integrating them with what is already known into new knowledge - here, a relationship between ``man'' and ``mortal''. In mathematics, inductively formed conclusions, analogous to this one, are based on motivating examples and illustrated by applications.
Mathematics thrives on examples and applications. In fact, it owes its birth and growth to them. This is manifestly evidenced by the thinkers of Ancient Greece who ``measured the earth'', as well as by those of the Enlightenment, who ``calculated the motion of bodies''. It has been rightfully observed that both logical rigor and applications are crucial to mathematics. Without the first, one cannot be certain that one's statements are true. Without the second it does not matter one way or the other-13. These lecture notes cultivate both. As a consequence they can also be viewed as an attempt to make up for an error committed by mathematicians through most of history - the Platonic error of denigrating applications-14.
This Platonic error, which arises from placing mathematical ideas prior to their physical origin, has metastasized into the invalid notion ```pure'' mathematics'. It is a post-Enlightenment (Kantian) fig leaf-15 for the failure of theoretical mathematicians to justify the rigor and the abstractness of the concepts they have been developing. The roots of this failure are expressed in the inadvertent confession of the chairman of a major mathematics department: ``We are all Platonists in this department.'' Plato and his descendants declared that reality, the physical world, is an imperfect reflection of a purer and higher mystical dimension with a gulf separating the two. That being the case, they aver that ``pure'' mathematics - and more generally the `` a priori'' - deals only with this higher dimension, and not with the physical world, which they denigrate as gross and imperfect.
With the acceptance - explicit or implicit - of such a belief system, ``pure'' mathematics has served as a license to misrepresent theoretical mathematics as a set of floating abstractions cognitively disconnected from the real world. The modifier ``pure'' has served to intimidate the unwary engineer, physicist or mathematician into accepting that this disconnect is the price that mathematics must pay if it is to be rigorous and abstract.
Ridding a culture's mind from impediments to epistemic progress is a non-trivial task. However, a good first step is to banish detrimental terminology, such as ``pure'' mathematics, from discourses on mathematics and replace it with an appropriate term such as theoretical mathematics. Such a replacement is not only dictated by its nature, but it also tends to reinstate the intellectual responsibility among those who need to live up to their task of justifying rigor (i.e. precision) and abstractness.
3. Mathematics is both complex and beautiful. The complexity of mathematics is a reflection of the complexity of the relationships that exist in the universe. The beauty of mathematics is a reflection of the ability of the human mind to identify them in a unit-economical way-16 : the more economical the identification of a constellation of relationships, the more man's mind admires it. Beauty is not in the eyes of the beholder. Instead it is giving credit where credit is due - according to an objective standard. In mathematics that standard is the principle of unit economy. Its purpose is the condensation of knowledge, from the perceptual level all the way to the conceptual at the highest level.
4. Linearity is as fundamental to mathematics as it is to our mind in forming concepts. The transition from recognizing that to the act of grasping that is the explicit starting point of a conceptual consciousness grasping nature in mathematical terms with linearity at the center. Thus it is not an accident that linear mathematics plays its pervasive role in our comprehending the nature of nature around us. In fact, it would not be an exaggeration to say that ``Linearity is the exemplary method - simple and primitive - for our grasping of nature in conceptual terms''. The appreciation of this fact is found in that nowadays virtually every college and university offers a course in linear algebra, with which we assume the reader is familiar.
5. Twentieth century mathematics is characterized by an inflationary version of Moore's Law. Moore's Law expresses the observation that the number of transistors that fit onto a microchip doubles every two years. This achievement has been a boon to everybody. It put a computer into nearly every household.
The mathematical version of Moore's Law expresses the observation that, up to the Age of Enlightenment, all of Man's mathematical achievements fit into a four-volume book; the achievements up to, say, 1900 fit into a fourteen- volume tome, while the mathematical works generated during the twentieth century take up a whole floor of a typical university library.
Such abundance has its delightful aspects, but it is also characterized by repetitions and non-essentials. This cannot go on for too long. Such an increase ultimately chokes itself off.
One day, confronted with an undifferentiated amorphous juxtaposition of mathematical works, a prospective scientist/engineer/physicist/mathematician might start wondering: ``I know that mathematics is very important, but am I learning the right kind of mathematics?''
Such a person is looking for orientation as to what is essential, i.e. what is fundamental, and what is not. It has been said that the value of a book-17 , like that of a definition-18, can be gauged by the extent to which it spells out the essential, but omits the nonessential, which is, however, left implied. With that in mind, this text develops from a single perspective six mathematical jewels (in the form of six chapters) which lie at the root twentieth century science.
Another motivation for making the material of this text accessible to a wider audience is that it solves a rather pervasive problem. Most books which the author has tried to use as class texts either lacked the mathematics essential for grasping the nature of waves, signals, and fields, or they misrepresented it as a sequence of disjoint methods. The student runs the danger of being left with the idea that the mathematics consists of a set of ad hoc recipes with an overall structure akin to the proverbial village of squat one-room bungalows instead of a few towering skyscrapers well-connected by solid passage ways.
6. We extend and then apply several well-known ideas from finite dimensional linear algebra to infinite dimensions. This allows us to grasp readily not only the overall landscape but it also motivates the key calculations whose purpose is to connect and cross-link the various levels of abstraction in the constructed edifice. Even though the structure of these ideas have been developed in linear algebra, the motivation for doing so and then using them comes from engineering and physics. In particular, the goal is to have at one's disposal the constellation of mathematical tools for a graduate course in electromagnetics and vibrations for engineers or electrodynamics and quantum mechanics for physicists. The benefits to an applied mathematician is the acquisition of nontrivial mathematics from a cross-disciplinary perspective.
All key ideas of linear mathematics in infinite dimensions are already present with waves, signals, and fields whose domains are one-dimensional. The transition to higher dimensional domains is very smooth once these ideas have been digested. This transition does, however, have a few pleasant surprises. They come in the form of special functions, whose existence and properties are a simple consequence of the symmetry properties of the Euclidean plane (or Euclidean three-dimensional space). These properties consist of the invariance under translations and rotations of distance measurements and of the shapes of propagating waves.
7. What is the status of the concept ``infinite'' appearing in the title of this text? Quite generally the concept ``infinite'' is invalid metaphysically but valid mathematically.
In the sense of metaphysics (i.e. pertaining to reality, to the nature of things, to existence) infinity falls into the category of invalid concepts, namely attempts to integrate errors, contradictions, or false propositions into something meaningful. Infinity as a metaphysical idea is an invalid concept because metaphysically it is only concretes that exist, and concretes are finite, i.e. have definable properties. An attempt to impart metaphysical significance to infinity is an attempt to rewrite the nature of reality.
However, in mathematics infinity is a well defined concept. It has a definite purpose in mathematical calculations. It is a mathematical method which is made precise by means of the familiar - process of going to the limit. This text develops only concepts which by their nature are valid. Included is the concept ``infinity'', which, properly speaking, refers to a mathematical method.
8. The best way to learn something is to teach it. In order to facilitate this motto of John A. Wheeler, the material of this book has been divided into fifty lecture sessions. This means that there is one or two key ideas between ``Lecture '' and ``Lecture '', where . Often the distance between and extends over more pages than can be digested in a forty-eight minute session. However, the essentials of the th Lecture are developed in a small enough time frame. Thus the first four or five pages following the heading ``Lecture '' set the direction of the development, which is completed before the start of lecture ``Lecture ''.
Such a division can be of help in planning the schedule necessary to learn everything.
9. It is not necessary to digest the chapters in sequential order. A desirable alternative is to start with Sturm-Liouville theory (chapter 3) before proceeding systematically with the other chapters. Moreover, there is obviously nothing wrong with diving in and exploring each chapter according to one's background and predilections. The opening remarks of each one point out how linear algebra relates it to the others.
10. Acknowledgments: The author would like to thank Danai Torrungrueng for valuable comments and Wayne King from the Speech and Hearing Science Department for many fruitful conversations on wavelet theory and multiresolution analysis.
Ulrich H. Gerlach
Columbus, Ohio, March 24, 2009
``The mathematician is the exemplar of conceptual integration. He does professionally and in numerical terms what the rest of us do implicitly and have done ever since childhood, to the extent that we exercise our distinctive human capacity''.