Introduction

We start studying mathematics. A natural approach would be to first give a definition of our subject. However as mathematics is a broad and evolving subject this seems not a fruitful approach. Therefore we start with a short description of the nature of mathematics.

Science studies objects in the real world as opposed to mathematics which studies objects in our imagination. In the real world the laws of nature governs, but in the abstract world of mathematics there must also be some law and order. Otherwise the outcomes of this abstract world would never have been useful for applying in the real world. This law and order that governs mathematics is what we call deductive logic. The rules of logic guarantees that the results of mathematics make sense. A true mathematical statement, a theorem, can be proven to be true by deductive logic. However to deduct theorems the theory must start with some basic assumptions, the axioms. These axioms are theorems which are true by definition. Mathematics tries to minimize the number of axioms needed and to derive as many theorems as possible from this set of axioms. This approach of developing a theory is called the axiomatic method and such theories are called deductive or axiomatic theories. Mathematics is a deductive theory.

A deductive or axiomatic system $S$ consists of five constituents (L,P,A,R,T): a grammar for the propositions L, the set of all propositions P, the set of axioms A, the rules of inference R, and the set of derived theorems T. A proposition is a boolean function, associated with a truth value of either True of False. Axioms are by definition True. A theorem $t$ is a proposition whose truth value is True and which be derived via the rules of inference as an ordered list : $p_1,p_2,...,p_n=t$, where each $p_i$ is either an axiom or a theorem. The derivation is called a proof. Ultimately the truth of the theorems follows from the truth of the axioms.

But what does it mean that the axioms must be true ? To answer this question we must realize that each proposition of a language $L$ consists of semantic parts (the specific terms, or interpretation) and syntactical parts (the logical terms like: 'and' , 'or' , 'implies'). The truth of a proposition depends in principle on both parts. However to assign a truth value to the semantic part of a propostion a mapping with a certain subject matter must be made. This is called an interpretation. Then a proposition could be true under one interpretation and false under another. An interpretation for an axiomatic system which makes all of the axioms true is called a model of the axioms. A set of axioms is called consistent if it has at least one model otherwise it is called inconsistent. It is universally valid if true under all possible interpretations of its specific terms. The rules of interference guarantees that all theorems of an consistent axiomatic system $S$ are true for each model of $S$.

For first order (predicate logic) the interference rules are complete. Meaning that a logical contradiction $p \land \lnot p$ can be deduced from any inconsistent set of axioms. Thus, the axioms $A$ of a first-order system $S$ are inconsistent (have no model) if and only if a contradiction $p \land \lnot p$ is provable as a theorem of $S$. According to the logical rules of most systems (and first-order systems in particular), any proposition $p$ can be deduced from a formal contradiction.

It follows from the above that, for first-order systems, the consistency of a given set of propositions does not depend on the meaning of the specific terms in the propositions; it depends only on the form or structure of the propositions with regard to their logical parts. Moreover, formal deduction alone can be relied upon to detect inconsistency.

These results on logic, which have only been achieved in the twentieth century, have allowed for a much more general form of the axiomatic method, called the abstract axiomatic method. This method consists in leaving the specific terms of an axiomatic system uninterpreted (or "undefined") from the beginning. The only requirement is that the axioms be consistent, and thus true of some reality, but we no longer require that this reality be specified. In the formal axiomatic method, the role previously played by truth is now played by consistency, and consistency depends only on the syntactical form of the axioms with respect to their logical parts, not on their specific content (meaning) under a given interpretation. For example, it may turn out that the mathematical theory, say, of languages, of genetics and of machine computation is essentially the same. In such a case, each theorem $t$ of our system $S$ will have an interpretation as a true proposition in each of the respective models.

The developing of a mathematical theory is the process of precisely defining abstract objects and deducing relationships between them based on logic. Many mathematical theories directly evolved out of practical need. in fact quantity and space has been the major source for mathematical inspiration. The concept of quantity led to the development of the abstract object of numbers and a mechanism of counting. Along this line mathematics further developed in the 18th century to more and more abstract objects like functions and operations on functions like differentiation and integration. This mainly to support the development of classical mechanics. As of today the subject is so diverse and in depth that it is almost impossible to comprehend by a single individual.
Mathematics tries to solve problems and finds its starting points in concepts of the the real world but uses abstract objects and methods to describe and analyze the problem domain. We conclude with a quote from Bertrand Russell, the great mathematician and logician at the beginning of the 20th century. He argued that mathematics could be reduced to logic. It is now generally accepted that mathematics has its own existence separate form logic.

By the help of ten principles of deduction and ten other premises of a general logical nature (e.g. implication is a relation), all mathematics can be strictly and formally deduced; and all the entities that occur in mathematics can be defined in terms of those that occur in the above twenty premises. In this statement, Mathematics includes not only Arithmetic and Analysis, but also Geometry, Euclidean and non-Euclidean, rational Dynamics and an indefinite number of other studies still unborn or in their infancy. The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself. Bertrand Russell

With this short introduction of what mathematics is about we can start with our study.

The part of mathematics we study here, calculus, is very well applicable to the natural sciences. From the perspective of a natural scientist mathematics is a tool necessary for formulating models and solving problems. From the perspective of a mathematician calculus is the precise development of stating axioms, defining objects and deducing theorems. In calculus the world of objects is reduced to numbers and the relations to functions. A function is a specific relationship between numbers. In classical mechanics the function plays an important role because it defines the relation between a series of time measurements and a corresponding series of location measurements. Calculus makes mathematics not only more precise, but also extended the simple concept of counting to the multi-dimensional concept of space.

The next step in our mathematical journey is to define numbers and find out their properties. It will turn out that the real numbers require already a high level of abstraction. Irrational numbers fill in the holes, but are unreachable. But even before we start with numbers, we must first take a look at the most fundamental concepts of mathematics: logic and sets.