An excerpt from the preface:
Looks promising...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.
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 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 in grasping the nature of the world.
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. 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.
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 fig leaf 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 the physical universe is an imperfect reflection of a purer and higher reality 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 reality, 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.
