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From ancient times, the history of human beings has developed by a succession of steps and sometimes jumps, until reaching the relative sophistication of the modern brain and culture. Researchers are attempting to create systems that mimic human thinking, understand speech, or beat the best human chess player. Understanding the mechanisms of intelligence, and creating intelligent artifacts are the twin goals of Artificial Intelligence (AI). Great mathematical minds have played a key role in AI in recent years; to name only a few: Janos Neumann (also known as John von Neumann), Konrad Zuse, Norbert Wiener, Claude E. Shannon, Alan M. Turing, Grigore Moisil, Lofti A. Zadeh, Ronald R. Yager, Michio Sugeno, Solomon Marcus, or Lászlo A. Barabási. Introducing the study of AI is not merely useful because of its capability for solving difficult problems, but also because of its mathematical nature. It prepares us to understand the current world, enabling us to act on the challenges of the future.

The historical origin of Artificial Intelligence as a scientific field is usually believed to have been established at the Darmouth Conference (1956). In that year, John McCarthy coined the term, defined as “the science and engineering of making intelligent machines”. However, this definition does not cover the full breadth of the field. AI is multi-connected with many other fields, such as Neuroscience or Philosophy; and we can trace its origins further back, to arcane origins, perhaps to Plato, Raymond Lully (Raimundo Lulio, in Spanish), G. W. Leibniz, Blaise Pascal, Charles Babbage, Leonardo Torres Quevedo,

The history of the field is long and fruitful; just recall topics as diverse and important as the famous Turing test, the Strong

Frequently, AI requires Logic. However, Classical Logics show too many insufficiencies [

Among the things that AI needs to implement as representation are Categories, Objects, Properties, Relations, and so on. All of these are connected to Mathematics [

The problems in AI can be classified in two general types: Search Problems and Representation Problems. Then, we have Logics, Rules, Frames, Nets, as interconnected models and tools. As it is easy to see, all of them are very mathematical topics.

The origin of ideas about thinking machines [

The central purpose of AI would be to create an admissible model for human knowledge. Its subject is, therefore, the “pure form”. We try to emulate the reasoning of a human brain. Research directed to this goal can only happen in a succession of approximating steps, but attempts proceed always in this sense.

Initially, AI worked through idealizations of the real world. Its natural fields were, therefore, “formal worlds”. Search procedures operated in the Space of States, which contains the set of all states (or nodes, in the case of representation by graphs), that we can obtain when we apply all the available operators. Many early AI programs used the same basic algorithm. To achieve a certain goal (winning a game or proving a theorem), they proceeded step by step towards it (by each time making a move or a deduction) as if searching through a maze and backtracking whenever they reached a dead end. This paradigm was called “reasoning as search”.

The techniques for solving problems in AI are of two types:

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– Procedural: which itemizes the necessary paths to reach the solution to the problem (it is the

To pose problems is equivalent to constructing their solutions [

We may use a series of resources for approaching problems in AI, such as Logic, Rules, Associative Nets, Frames, and Scripts. The choice of methods must be based on the characteristics of the problem and our expectations about the type of solution. In many cases, we take two or more tools at a time, as is the case in Frame Systems with participation of Rules.

Inference in Rule Based Systems (RBS) consists of establishing the certainty of a certain statement from the available information in a Base of Facts (BF) or Knowledge Base. We have two methods to concatenate rules: going forward and going backwards.

Rules are a clear improvement over Classical Logics [

If there is more than one rule applicable at a certain point of the reasoning, which one should be executed first? The set of applicable Rules constitutes, at each step, the Conflict Set (which will be dynamic, obviously). The underlying decision problem is called Resolution of Conflicts, or Control of Reasoning.

Different strategies [

we must select R₂, because it is more specific than R₁.

We also have

The more recent studies of networks deal with Bayesian Networks. They were introduced in the context of systems for medical diagnosis, where classical statistical techniques, such as the Bayes Rule, had been employed previously.

In the modeling of such problems, the following hypotheses are assumed:

A

Inference in a BN consists of establishing on the net, for the known variables, their values, and for the unknown variables, their respective probabilities.

The objective of BNs in Medicine is to find the probability of success that we can give to certain diagnoses, knowing certain symptoms.

In AI, the problems can be classified according to their level [

The problems in AI can be finally classified in two principal types, Search and Representation Problems [

For representation, we need concepts such as Trees and Graphs; Structure of Facts, for instance: stacks, queues and lists; and the knowledge about the Complexity of Algorithms is also crucial.

In the search process, we have two options: without information of the domain (Blind Search); and with information of the domain (Heuristic Search). In the first case, we can choose, according to the type of problem, between

There are other methods, derived from the previous method, such as

A different method, in this case not derived, is the

The searching process could be conducted in state spaces. Such a searching procedure has applicability on problems provided with some characteristics, when we can associate a state to each different situation of the domain. There is then a series of initial states: there are some operators, which allow us to take steps between the successive states and there is a final state. In such processes, the correspondence between State and Node of the graph, and between

Now, we introduce one new mathematical tool, the so-called

There are also strategies designed for the treatment of

All of these problems, their methods of solution or approximation tools, may be implemented in the classroom [

Tools that are in use currently in Mathematics and AI, as Graph Theory, may be used to introduce classical and very exciting questions, such as the Halting Problem, or the Traveling Salesman Problem (TSP), or an open question for the 21st century, P

The study of Mathematics can be supported by the introduction of games such as those mentioned above: Chess, Checkers, Stratego or Sudoku. Not only that: our students can be introduced to more subtle analyses; an example would be the Prisoner’s Dilemma. Students would have a large quantity of information readily available on the Web, so researching for the rules, tricks and hints to master games like Chess or Go can be an interesting activity in itself. Further motivation for the students may come in the reading of papers, or Web articles, on the history of the games; this would be illustrative and motivating for the students.

All of these techniques have been implemented [

I consider that this approach places Computer Science in the role Physics and its problems have played in the past, as a support of mathematical reasoning [

I propose showing such Methods through the parallel study of Mathematics and Computer Science foundations. Other Computer Science subfields could be carriers of this method too, but perhaps AI is the current better choice, given its characteristics, which practically coincide with many mathematical techniques and objectives.

The benefits of such an innovative educative method must consist in a more progressive adaptation of Mathematical Education to modern times [

The interest of the study of Artificial Intelligence is based on our increasing need for creating new manners of thinking and interpreting the mechanisms of the brain and of human reasoning. Inside this new and powerful science, stimulated by the constant advances of the Sciences of Computation and of the rapidity, efficiency and efficiency of the computational processes, the so-called Non-classic Logics are adequately placed. Among them, the one that finds more applications is Fuzzy Logic, and the reason for this is that it simulates the way in which we reason as human beings. Instead of the black-and-white simplified version of reality of Boolean binary classical logics, we are used to grades of truth, shades of grey: instead of the absolute truth or falsehood, yes or no, our way of thinking admits for gradation.

It is, therefore, crucial for the formation in Mathematics of the new generations of students that these innovative concepts become part of the curricula, as part of the subjects related to Mathematics, such as Physics, Chemistry, or Engineering.

We say that a language is formal when its syntax is precisely given. Mathematical Logic is the study of the formal languages. Usually, it is called Classical Logic, being dychotomic, or bi-valuated, only either True or False.

The modern study of Fuzzy Logic [

According to the old riddle, a Cretan asserts that all Cretans lie. So, is he lying? If he lies, then he tells the truth and does not lie. If he does not lie, then he tells the truth and so lies. Both cases lead to a contradiction because the statement is both True and False.

The set of all sets is a set, and so it is a member of itself. Yet the set of all apples is not a member of itself, because its members are apples and not sets. Perceiving the underlying contradiction, Russell then asked: “Is the set of all sets that are not members of themselves a member of itself?” An essential question, because: If it is, it isn’t; if it isn’t, it is.

Faced with such a conundrum, classical logic surrenders. But fuzzy logic says that the answer is half true and half false, a 50–50 divide. Fifty percent of the Cretan’s statements are true, and 50 percent are false. The Cretan lies 50 percent of the time and does not lie the other half. When membership is less than total, a bivalent system might simplify the problem by rounding it down to zero or up to 100 percent. Yet 50 percent does not round up or down.

In the 1920s, independently of Russell, the Polish mathematical logician Lukasiewicz worked out the principles of Multivalued Logic (MVL, by acronym), in which statements can take fractional truth values between the1’s and 0’s of Boolean logic (which is a binary logic).

The physicist and philosopher Max Black, in 1937, applied MVL to lists, or sets of objects, and in so doing drew the first fuzzy set curves. Following Russell’s lead, Black called the sets “vague” (equivalent term to “fuzzy”).

Three decades later, Lofti A. Zadeh published his very famous paper called “Fuzzy Sets”. Zadeh applied Lukasiewicz’s logic to every object in a set and worked out a complete algebra for fuzzy sets. Even so, fuzzy sets were not put to use until the middle of the 1970s, when Mamdani designed a fuzzy controller for a steam engine. Since then, the term “fuzzy logic” has come to mean any mathematical or computer system that reasons with fuzzy sets.

The Fuzzy Logic is a successful generalization of the Mathematical or Classical Logic [

With the introduction of concepts and methods of Fuzzy Logic, the ideas of sets, relations and so on, are modified in the sense of covering adequately the indetermination or imprecision of the real world. We define the “world” as a complete and coherent description of how things are or how they could have been. In the problems related with this “real world”, which is only one of the “possible worlds”, the Monotonic Logic seldom works. Such type of Logic is classical in formal worlds, such as Mathematics. However, it is necessary to provide our investigations with a mathematical construct that can express all the “grey tones”, not the classical representation of real world as either black or white, either all or nothing, but as in the common and natural reasoning, through progressive gradation.

Let A and B be two fuzzy sets, not necessarily on the same universe of discourse. The implication between them is the relation R: A→B such that A → B ≡ A * B, where * is an outer matrix product using the logical operator AND. To each Fuzzy Predicate, we can associate a Fuzzy Set, defined by such property, that is, composed of the elements of the universe of discourse such that totally or partially verify such conditions. So, we can prove that the class of fuzzy sets, with the operations union (∪), intersection (∩), and

Fuzzy Rules are linguistic IF-THEN constructions that have the general form “IF A THEN B”, where A, B are propositions containing linguistic variables. A is called the premise, or antecedent, and B is the consequent (or action) of the fuzzy rule. In effect, the use of linguistic variables and fuzzy IF-THEN rules exploits the tolerance for imprecision and uncertainty [

In this respect, Fuzzy Logic imitates the ability of the human mind to summarize data and focus on decision-relevant information. For these reasons, it is a very interesting way to advance, as an interesting access point to many new fields of Mathematics.

As for the games and mathematical riddles, there are many very good texts that contain them. We all have in mind some of the main ones, in particular on this topic and in general on mathematical divulgation, are the works of George Pólya, [

Another very important aspect, which is in the habit of instigating many disagreeable “surprises”, resides in the skill to adapt the material to the audience. This is an art and it is often learned only with time [

The rejection and the difficulties that this incorrect route has caused to many teachers tends to the infinite. The audience must be in the center of our planning, their knowledge, skills and possibilities have to be born in mind, instead of trying to organize an imitation of the way the old subject was presented to us at university. It is a good exercise to take a written topic to a higher, or more difficult level, or to choose an article from a scientific magazine and to try to turn it into something that the pupils could understand. Even more, it could motivate them, predispose them in favor, and not against, the study of Mathematics in the future.

We may denominate Information and Communication Technologies (ICTs) the set of technologies which allow us to acquire, produce, store, treat, communicate, and register information, by voice, images and data, all of them translated to optical, acoustical, or electromagnetic signals.

New technologies permit the development of new didactic materials through different supports: Internet, digital discs,

Amongst the immense range of Internet applications, the World Wide Web will be the space with more possibilities. Web pages allow us to publish any digital element (text, photo, video,

Furthermore, the didactic material available on the Web can be accessed globally, independently of place and time. Not only the quantity of available material on the Web is increasing rapidly, also its access is more free, thanks to the so-called Free Software, really free for any user.

The use of ICTs as part of the learning methods is advantageous with respect to the very old resources of more classic educational schools. Such advantages are:

Such set of ideas is very tempting indeed, according to the aforementioned advantages. Nevertheless the use of new ICTs inside the classrooms has potential disadvantages, which must be considered to plan for its reasonable use:

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Through problem solving we attempt to evaluate the degree of acquisition of certain mathematical competencies with ICTs and without ICTs. Such educational competencies are: resolution of Problems; use of Resources and Tools; to communicate; to interpret; to represent; to think, and Reasoning and Argumentation.

In the Mathematical area, we may distinguish between two categories: Attitudes towards Mathematics, and Mathematical Attitudes.

With “Attitudes towards Mathematics” we refer to the valuation and high regard to such scientific discipline and language, and the increasing interest to learn this useful tool, where it dominates the affective component over the cognitive component; so stating in terms of interest, satisfaction, curiosity,

We must avoid the very extended, but erroneous because of its artificiality, division of Mathematics between Pure Mathematics (also called Fundamentals), and Applied Mathematics. Such territories are, in any case, evolving with time, and that is a totally subjective boundary. Because we can only observe two types of Mathematics: the good and the bad Mathematics, independent of when they are to be applied—just now, or in the more distant future—and also by Fuzzy Theory interpretation, with all the intermediate degrees of “goodness”.

We have briefly covered some of the aspects of how Fuzzy Logic mimics the ability of the human mind to summarize data and focus on decision-relevant information. For these reasons, the introduction to the study of Fuzzy Logic is an interesting access point for a number of new fields for Mathematical Education.

Studying Artificial Intelligence is interesting not only because of its potential to tackle many open problems, both inside the field and in application to others scientific areas, and even the study of the humanities, but also because it is a new and strongly creative branch of Mathematics, and it prepares us to understand the current world, enabling us to act on the challenges of the future.