This paper is based on a series of three lectures that I gave during the LC 2000, in the context of the “tutorials” which have now become a tradition at the European meetings of the ASL. I have kept fairly close to the actual format and style of the talks.

It is always difficult to identify precisely the audience such a tutorial should address. A fair number of broad and ambitious surveys have already been published on the subject of the applications of model theory to algebraic geometry (see section 4.4). I did not, during this tutorial, choose to address the specialists of the subject. The audience I had in mind consisted of both young “inexperienced” researchers in model theory and more “mature” logicians from other parts of logic. Rather than attempting one more broad survey, I tried to present some of the main concerns of “geometrical model theory” by looking at concrete examples and this is what I will try to do also in the present paper.

We will discuss three algebraic examples, algebraically closed fields, differentially closed fields and difference fields (fields with automorphisms). The geometric application we will take up as illustration is Hrushovski's approach to the Manin-Mumford conjecture. This is based on a fine study of the model theory of difference fields and is quite emblematic of the method. Perhaps the key technical notion is that of “localmodularity”(or “one-basedness”), which arises in a purely model theoretic setting. We will see that the Diophantine conjectures of the Manin-Mumford type can be rephrased in terms of this notion. Furthermore, as one thinks through the rephrasing process, one realizes the need for the introduction of auxiliary algebraic theories such as the theory of difference fields.

I would like to thank the anonymous referee, despite my temporary shock at the initial suggestion that the paper be totally rewritten and turned into a survey of a completely different type. Fortunately, he/she also provided a long list of detailed comments and less draconian suggestions, in case I did not choose to follow this first drastic piece of advice. I have found these comments very helpful and have followed most of these suggestions.

The psychologist's problem. Since its inception in the 19th century, psychological science has made steady progress investigating perceptual and motoric abilities—how the visual system encodes color, for example, or how we shift our gaze to peripheral events. Much less is understood about abilities thatmake us distinctly human. Some information is available about themechanisms of natural language. But there is hardly any insight into how people create scientific theories about the world. This is an embarassing gap for Psychology since scientific achievement is themost distinctive and remarkable feature of our species.

What's blocking progress is that the most natural account of this ability seems to face an insuperable difficulty. According to the account in question, most everyone has an innate disposition to reason in rough conformity with normatively correct principles of deductive and inductive logic—just as most everyone is endowed with perceptual mechanisms that give us a roughly accurate picture of the environment. How else could our ancestors have met the challenges of survival? It is the twin pillars of natural reasoning—deductive and inductive—that allow people to draw out the consequences of rival scientific theories and assign sensible credibilities to each in the light of data.

These vague remarks are just an attempt to prepare for sharper theories. But we stumble even at this initial step, because one of the twin pillars seems to be absent. The problem is not so much with deductive logic. It can be challenging to communicate the informal concept of logical necessity. But once this is achieved, most people distinguish validity from invalidity on an intuitive basis across a broad class of arguments.

§1. Introduction. Logical calculi were invented to model mathematical thinking and to formalize mathematical arguments. The calculi of Boole [8] and of Frege [15] can be considered as the first mathematical models of logical inference. Their work paved the way for the discipline of metamathematics, where mathematical reasoning itself is the object of mathematical investigation. The early calculi, the so-called Hilbert type- and Gentzen-type calculi [25], [17] developed in the 20th century served the main purpose to analyze and to reconstruct mathematical proofs and to investigate provability. A practical use of these calculi, i.e., using them for solving actual problems (e.g., for proving theorems in “real” mathematics), was not intended and even did not make sense.

But the idea of a logical calculus as a problem solver is in fact much older than the origin of propositional and predicate logic in the 19th century. Indeed this idea can be traced back toG.W. Leibniz with his brave vision of a calculusratiocinator [29], a calculus which would allow solution of arbitrary problems by purely mechanical computation, once they have been represented in a special formalism. Today we know that, even for restricted languages, this dream of a complete mechanization is not realizable—not even in principle (we just refer to the famous results of Gödel [20] and Turing [39]). That does not imply that we have to reject the idea altogether. Still it makes sense to search for a lean version of the calculus ratiocinator. Concerning the logical language, the ideal candidate is first-order logic; it is axiomatizable (and thus semidecidable), well-understood and sufficiently expressive to represent relevant mathematical structures. By Church's result [10] we know that there is no decision procedure for the validity problem of first-order logic; thus there is no procedure which is 1. capable of verifying the validity of all valid formulas and 2. terminating on all formulas. So, even in first-order logic, we have to be content with the verification of problems. The only thing we can hope for is a calculus which offers a basis for efficient proof search. It is not surprising that the invention of the computer lead to a revival of Leibniz's dream.

We organized the chapters in this book to be read in sequence. However, each chapter begins with a clear statement of what we assume you know before reading the chapter, so you can jump around a little bit, depending on your background and experience. You can use the book as a reference and jump in anywhere once you have the fundamentals.

Chapter 1 presents reasons why you should learn Java and describes the many similarities between COBOL and Java.

Chapter 2 explains what OO is, and what it is not. OO terms and concepts are described using several everyday examples.

Chapter 3 describes the overall structure and format of a Java program. Several small programs are developed to show you how to create objects and call methods.

Chapter 4 shows you how to define Java data items and use them in a program. Java data definition is somewhat different than COBOL and these differences are clearly explained and demonstrated in the program examples.

Chapter 5 introduces Java computation and, again, several small programs are written to illustrate the ideas and concepts present. You will see that some Java computation is nearly identical to COBOL.

Chapter 6 illustrates how to use the Java decision-making statements.