OUR MATH-DICTIONARY | etwinning-2015-2016

Mission:

MATH -DICTIONARY

Greek-English-Italian

"We collaborate once more about the translation using Maths tems in three languages: Greek, English and Italian.

That was an excellent activity espesially for those students who want to become Foreign Language Teachers. It was intresting for them to find out that Maths and similar subjects can be used in future as a special translation field.

Please press the buttons below to use the dictionaries we formed together. "

by Greek-Italian Translation Team

We desided to present some Math-Terms extensively....

Algorit
Axiom
Incompleteness Theorem
Principia Mathematica
nfinity
Logic
Proof
Madness
Russell's paradox
Set Theory

“Words belong to each other,” Virginia Woolf said in the only surviving recording of her voice, a magnificent meditation on the beauty of language. But what happens when words are about Maths?

For our team words about Math were the motive for this mission. Let me tell you how did it happen.

I want το be a lawyer and I don't like math as a school subject. I join eTwinning "Math Investigation" because I like new challenges and I love communicate with new friends. I proposed this mission to the rest in an online meeting and as the majority found it an excellent idea during the middle-evaluation , we included it as a mission in the eTwinning project! Both teachers, Ms Atmatzidou and Ms Aimetta helped our team to translate mathematical terms in three languages: English, Greek and Italian.

Finally I realized that this is what I like the most about eTwinning projects:

You can find a way to include your own dreams in a school activity!

by the Team....

Axiom

Since the time of Euclid, who was working in the wake of Aristotle’s philosophy of logic, mathematicians agree that a workable theory must rest on some (few) agreed-upon first principles that don’t require proof. This is a logical necessity if one wants to avoid, on the one hand, infinite regression (endlessly having to base something on something else) and, on the other, circuitous thinking (constructing proofs for statements which, however indirectly, assume the original statement to be true in the first place). Up to the 19th century, axioms were generally considered to be self-evident truths about the world. After Hilbert, however, and under the influence of the mathematico-philosophical school of formalism, which developed from his ideas, axioms came to be seen as existing independently of any outside reality, the only requirements of an axiomatic system being: for the individual axioms their grammatical correctness (in other words, their being well-formed according to the rules of the logical language in which they are expressed), and independence (their not being derivable from the other axioms of the particular theory); and, for the whole set of axioms, its internal consistency (not containing axioms which contradict one another).

Infinity

(Bertrand Russell, Autobiography, p. 149)

At the end of the Lent Term, Alys and I went back to Fernhurst, where I set to work to write out the logical destruction of mathematics which afterwards became Principia Mathematica. I thought the work was nearly finished, but in the month of May I had an intellectual set-back almost as severe as the emotional set-back which I had had in February [Evelyn Whitehead’s seemingly near-death experience]. Cantor had a proof that there is no greatest number, and it seemed to me that the number of all the things in the world out to be the greatest possible. Accordingly, I examined his proof with some minuteness, and endeavoured to apply it to the class of all the things there are. This led me to consider those classes which are not members of themselves, and to ask whether the class of such classes is or is not a member of itself. I found that either answer implies its contradictory. At first I supposed that I should be able to overcome the contradiction quite easily, and that probably there was some trivial error in the reasoning. Gradually, however, it became clear that this was not the case.

Madness

(Bertrand Russell, Greek Exercises)

March 9, 1888

I read an article in the Nineteenth Century today about genius and madness. I was much interested by it. Some few of the characteristics mentioned as denoting genius while showing a tendency to madness I believe I can discern in myself. Such are, sexual passion which I have lately had great difficulty in resisting, and a tinge of melancholy which I have often had lately and which makes me anxious to go to this tutor’s as there I shall probably be too much occupied to indulge such thoughts. Also he mentions a desire to commit suicide, which though hitherto very slight, has lately been present more or less with me in particular when up a tree. I should say it is quite possible I may develop more or less peculiarity if I am kept at home much longer. The melancholy in me is I think chiefly caused by the reserve which prompted the writing of this, and which is necessary owing to my opinions.

Incompleteness Theorem

In 1931, the 25 year-old Kurt Gödel proved two theorems that are sometimes referred to as “the” Incompleteness Theorem ― though occasionally this form is used to denote the first of these. The completeness of a logical system is the property that every well-formed (i.e. grammatically correct by the rules of the system) proposition in it can be proved or disproved from the system’s axioms. Gödel’s earlier Completeness Theorem shows that there is a simple such axiomatic system for first-order logic. However, the holy grail of Hilbert’s Program was to create a complete and consistent axiomatic system that can support arithmetic, i.e. the mathematics of whole numbers. Gödel shocked the mathematical world by proving, in his famous paper “On Undecidable Propositions in the Principia Mathematica and Related Systems”, that any consistent axiomatic system for arithmetic, in the form developed in the Principia, must of necessity be incomplete. More precisely, the first of the two Incompleteness Theorems establishes that in a logical axiomatic system rich enough to describe properties of the whole numbers and ordinary arithmetic operations, there will always be propositions that are grammatically correct by the rules of the system, and moreover true, but cannot be proven within the system. The second Incompleteness Theorem states that if such a system were to prove its own consistency it would be inconsistent.

Logic

The term covers a broad spectrum of disciplines ― not unexpectedly, as it derives from one of the semantically richest Greek words, logos, some of whose meanings are word, speech, thought, reason, ratio, rationality, and/or concept ― but can perhaps be best described as the study of methodical thinking, deduction and demonstration. The books of Aristotle’s Organon present an extensive study of the deductive patterns called syllogisms, which for over two millennia were considered practically synonymous with logical thinking. Until the middle of the 19th century, logic was considered a branch of philosophy. But with the advent of Boole and his algebra of propositions and, more importantly, Frege and his “concept script” which led to a predicate calculus, it increasingly came within the province of mathematics. The new logic revealed both the underlying mathematical nature of the subject and its potential role in the creation of solid foundations of mathematics. The basic claim of the school in the philosophy of mathematics known as logicism ― the school founded by Frege, of which Bertrand Russell was one of the primary exponents ― was that all of mathematics can be reduced to logic or, in other words, that mathematics is essentially a branch of logic. After the years of the foundational quest, however, and especially after Gödel’s results, logic became a well-developed, diversified field in the interface between philosophy and mathematics.

Principia Mathematica

The extremely influential, but highly controversial, essentially unfinished work in which Alfred North Whitehead and Bertrand Russell attempted to rescue Frege’s grand project to create foundations of mathematics built on logic, in the wake of the crisis brought on by Russell’s Paradox. The title Principia Mathematica (i.e. “Principles of Mathematics”) in itself provoked controversy, as it is the exact same as that of Newton’s greatest work; many in the British mathematical community thought this choice to be in bad taste, if not actually blasphemous. The three volumes of the Principia, published in 1910, 1912 and 1913, were based on a developed version of Russell’s theory of types, the so-called “ramified”, which imposed a hierarchical structure on the objects of set theory. This could not be made to yield the required results, however, without the addition of what Russell called an axiom of reducibility, which eventually became one of the main reasons for negative criticism of the whole work. Logicians found this axiom extremely counter-intuitive, a far-fetched and basically artificial method to sweep the very problem it was trying to solve under the rug. Despite the fact that the Principia fell short of its authors’ immense ambition, it had a huge influence on the shaping of modern logic, its greatest effect possibly being the inspiration and context it provided Kurt Gödel for his groundbreaking discovery, the Incompleteness Theorem.

Proof

The process of arriving at the logical verification of a mathematical or logical statement, starting from a set of agreed-upon first principles (these could be either axioms or already proven statements, deriving from these axioms), and proceeding by totally unambiguous and unabridged logical steps or rules of inference. The demonstrations of geometric propositions in Euclid’s Elements were considered for over two millennia to set the standard of excellence to which mathematical proof should aspire. Yet, towards the end of the 19th century his method came under logical and philosophical scrutiny and was found to lack, principally, in two directions: a) in its sense of the logical “obviousness” of the axioms, and b) in its logical gaps, where intuition ― which, in Euclid’s case was mostly visual-geometric ― took over from strict application of a formal system of rules. In a sense, Frege’s and Russell and Whitehead’s logicist project was developed as a reaction to the imperfections found in Euclid’s proofs, as well as all those developed in his wake. The logicists, as well as the formalists working on the foundations of mathematics, aimed at a fully developed theory and practice of rigorous proof, by which arithmetic (as the basis of all mathematics) would begin from a small number of consistent axioms, and eventually lead, via proof, to the full range of truth.

Russell's Paradox

Discovered in 1901, as Russell was working on his first book on the foundations of mathematics, the Principles of Mathematics (published in 1903), the Paradox, in the form originally expressed, shows an essential flaw in Cantor’s set theory, developed from Bolzano’s simple concept of a “collection of elements with a common property”. By the generality of this definition, which Frege extended to the realm of logic, one can speak of a “sets of sets” and thus, eventually of the “set of all sets”. Of the elements of this all-encompassing set one defines the property of “self-inclusiveness”, i.e. of a set containing itself as an element. Thus, for example, the set of all sets is a set (and thus contained in itself), as is the set of all entries in a list (it can appear as an entry in a list), but the set of all numbers is not a number and thus not contained in itself. By virtue of this property, we can define the “set of all sets which don’t contain themselves”, and ask, with the young Russell, the question: “Does this set contain itself or not?” See what happens: if it does contain itself, it follows that it is one of the sets which don’t contain themselves (as this is the property that characterizes elements of this set) and thus cannot contain itself. But if it doesn’t contain itself, then it does not have the property of not containing itself, and thus does contain itself. This situation, in which assuming something implies its negation, and vice versa, is called a paradox. When a paradox, such as Russell’s, arises in a theory, it is a sign that one of its basic premises, definitions or axioms is faulty.

Set theory

The study of collections of objects united by a common property ― in some cases this property can be nothing more than the fact that they are defined to be members of the same set, as for example in the arbitrarily defined set whose elements are the numbers 2, 3, 8, 134, 579. Sets were first studied by the Czech mathematician Bernard Bolzano (1781-1848), who also introduced the term Menge (‘set’) and defined the notion of a set’s cardinality, i.e. of its “size” in a way not directly involving measurement. The advanced mathematical discipline of set theory was arguably born on December 7, 1873, when Georg Cantor wrote to his teacher, Richard Dedekind describing his proof of the non-denumerability of the real numbers (the set of the whole numbers, decimals, zero and the negative numbers), as opposed to the denumerability of the rationals (all fractions), which Cantor also proved ― denumerability is defined as a one-to-one correspondence with the natural numbers 1, 2, 3… etc. The concept of a set is almost too primitive to merit a mathematical definition, and is practically impossible to define informally without the use of some synonym (here we used the word “collection”). It is precisely this “naturalness” of the concept in Bolzano’s and Cantor’s work that led to Russell’s Paradox. To overcome it, and to rule out the flawed concept of “the set of all sets” it allowed for, one has to come up with bottom-up constructions and axioms for sets, as in the Principia Mathematica and, later, the system called “ZFC”, from the names of its two creators, Ernst Zermelo and Abraham Fraenkel, and the Axiom of Choice, a necessary additional axiom that allows the theory to deal with infinite sets. Set theory is considered by some the most basic branch of mathematics, as all others can be defined in terms of it. This was the gist of an over-ambitious project undertaken, from the 1930s onwards, by the group of brilliant French mathematicians writing under the pen name of “Nicolas Bourbaki”.

From the final evaluation of the project:

"I want to be a Foreign Language Teacher. I loved the activity about the Math Dictionary in three languages. I realized that even thought I "Hate" Maths and Science as school subjects I probably used them in the future. Why not? I could be a new dictionary famous writer on Math terms! This thought surprised me and it was the ...start point to join possitivelly Math and Science classes from now on! Thanks eTwinning!"

D.M Schimatari

Algorithm

A methodical, step-by-step procedure described in terms of totally unambiguous instructions, which starts at a specified initial condition and eventually terminates with the desired outcome. Though there is no reason why a well-written cooking recipe, or the instructions for finding a certain geographical location or address cannot be called algorithms, the term originated in mathematics, where it is still mostly used. The word “algorithm” comes from a European transcription of the name of the 9th century astronomer and mathematician Al Khwarizmi of Baghdad, who catalogued and championed these methods, having invented many of them. His compendium of algorithms, the Hisab al-jabr w’al-muqabala, is generally considered to be the first algebraic treatise, the very words al-jabr in it also providing the root for our word “algebra”. An example of a simple mathematical algorithm is the method we learn in elementary school for adding two integers: “write the two numbers one under the other with their rightmost digits justified to the right; add their last digits; if the sum is less than 10, write that number right under the other two; if it is greater than 10, write the second digit of the sum right under the other two, and add the first digit to the sum of the digits immediately to the left ...” and so on. Algorithms gained prominence in the West in the 15th century with the introduction of the decimal system, which, in stark contrast with the Roman numerical system, was amenable to fast calculations, such as the one described above. Today, algorithms are usually coded in advanced notations called programming languages. They are often transmitted over the Internet, and constitute the software that is the workhorse, platform, and backbone of computers and the Internet.