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What, When, Where, How, Who?


Introduction, Important Definitions and Related Concepts:

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be self-evident. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths. In mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else they logically follow from (otherwise they would be classified as theorems). Logical axioms are usually statements that are taken to be universally true (e.g., (A ∧ B) → A), while non-logical axioms (e.g, a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in that sense, "axiom," "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain. Outside logic and mathematics, the term "axiom" is used loosely for any established principle of some field. The word "axiom" comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (axios), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof. The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologies excluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles of valid inference and demonstration. As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to specialized analysis of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory. [1] Traditionally, logic was considered a branch of philosophy, a part of the classical trivium of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied in the context of foundations of mathematics, where it was often called symbolic logic. In 1903 Alfred North Whitehead and Bertrand Russell attempted to establish logic formally as the cornerstone of mathematics with the publication of Principia Mathematica.[2] However, except for the elementary part, the system of Principia is no longer much used, having been largely superseded by set theory. As the study of formal logic expanded, research no longer focused solely on foundational issues, and the study of several resulting areas of mathematics came to be called mathematical logic. The development of formal logic and its implementation in computing machinery is fundamental to computer science. Form is central to logic. It complicates exposition that 'formal' in "formal logic" is commonly used in an ambiguous manner. Symbolic language is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions.

  • Informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic. The dialogues of Plato [3] are a good example of informal logic. Formal logic is the study of inference with purely formal content, where that content is made explicit. (An inference possesses a purely formal content if it can be expressed as a particular application of a wholly abstract rule, that is, a rule that is not about any particular thing or property. The first rules of formal logic that have come down to us were written by Aristotle. [4] In many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal language captures all of the nuance of natural language. Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference.[2][5] Symbolic logic is often divided into two branches, propositional logic and predicate logic. Mathematical logic is an extension of symbolic logic into other areas, in particular to the study of model theory, proof theory, set theory, and recursion theory.

    In epistemology (theory of knowledge), a self-evident proposition is one that is known to be true by understanding its meaning without proof. Some epistemologists deny that any proposition can be self-evident. For most others, the belief that oneself is conscious is offered as an example of self-evidence. However, one's belief that someone else is conscious is not epistemically self-evident. The following metaphysical propositions are often said to be self-evident:

    • A finite whole is greater than any of its parts It is impossible for the something to be and not be at the same time in the same manner.

    Certain forms of argument from self-evidence are considered fallacious or abusive in debate. For example, if a proposition is claimed to be self-evident, it is an argumentative fallacy to assert that disagreement with the proposition indicates misunderstanding of it. It is sometimes said that a self-evident proposition is one whose denial is self-contradictory. It is also sometimes said that an analytic proposition is one whose denial is self-contradictory. But these two uses of the term self-contradictory mean entirely different things. A self-evident proposition cannot be denied without knowing that one contradicts oneself (provided one actually understands the proposition). An analytic proposition cannot be denied without a contradiction, but one may fail to know that there is a contradiction because it may be a contradiction that can be found only by a long and abstruse line of logical or mathematical reasoning. Most analytic propositions are very far from self-evident. Similarly, a self-evident proposition need not be analytic: my knowledge that I am conscious is self-evident but not analytic. An analytic proposition, however long a chain of reasoning it takes to establish it, ultimately contains a tautology, and is thus only a verbal truth: a truth established through the verbal equivalence of a single meaning. For those who admit the existence of abstract concepts, the class of non-analytic self-evident truths can be regarded as truths of the understanding--truths revealing connections between the meanings of ideas. Claims of self-evidence also exist outside of epistemology. Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".[2] Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere.[3][4] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions.[5] Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.[6] Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later.[7]

    The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art. The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical".[8] In English, however, the noun mathematics takes singular verb forms.

    according to or agreeing with the principles of logic: a logical inference.
    2. reasoning in accordance with the principles of logic, as a person or the mind: logical thinking.
    3. reasonable; to be expected: War was the logical consequence of such threats.
    4. of or pertaining to logic.

    Of, relating to, in accordance with, or of the nature of logic. Based on earlier or otherwise known statements, events, or conditions; reasonable: Rain was a logical expectation, given the time of year. Reasoning or capable of reasoning in a clear and consistent manner.

    capable of or reflecting the capability for correct and valid reasoning; "a logical mind" [ant: illogical
    2.  based on known statements or events or conditions; "rain was a logical expectation, given the time of year" [syn: legitimate
    3.  marked by an orderly, logical, and aesthetically consistent relation of parts; "a coherent argument" [syn: coherent] [ant: incoherent
    4.  capable of thinking and expressing yourself in a clear and consistent manner; "a lucid thinker"; "she was more coherent than she had been just after the accident" [syn: coherent] (thinking or acting) according to the rules of logic
    Example: It is logical to assume that you will get a higher salary if you are promoted; She is always logical in her thinking. Of or pertaining to logic; used in logic; as, logical subtilties. --Bacon. According to the rules of logic; as, a logical argument or inference; the reasoning is logical. --Prior. Skilled in logic; versed in the art of thinking and reasoning; as, he is a logical thinker. --Addison. (From the technical term "logical device", wherein a physical device is referred to by an arbitrary "logical" name) Having the role of. If a person (say, Les Earnest at SAIL) who had long held a certain post left and were replaced, the replacement would for a while be known as the "logical" Les Earnest.

    Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.

    Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story. This is the role of non-logical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not. Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.


    Examples Arithmetic, Euclidean geometry, linear algebra, real analysis, topology, group theory, set theory, projective geometry, symplectic geometry, von Neumann algebras, ergodic theory, probability, etc. All these theories are based on their respective set of non-logical axioms.

    Arithmetic In all this formalism, the Peano axioms constitute the most widely used axiomatization of arithmetic; these are a set of non-logical axioms strong enough to prove several relevant facts of number theory and they allowed G�del to establish his second incompleteness theorem The language is <math>\mathfrak{L}_{NT} = \{0, S\}\,<math> where <math>0\,<math> is a constant symbol and <math>S\,<math> is a unary function. The postulates are: <math>\forall x \lnot (Sx = 0) <math> <math>\forall x \forall y (Sx = Sy \to x = y) <math> <math>((\phi(0) \land \forall x\,(\phi(x) \to \phi(Sx))) \to \forall x\phi(x)<math> for any <math>\mathfrak{L}_{NT}\,<math> formula <math>\phi\,<math> with one free variable.

    There is a standard structure is <math>\mathfrak{N} = <\N, 0, S>\,<math> where <math>\N\,<math> is the set of natural numbers, <math>S\,<math> is the successor function and <math>0\,<math> is naturally interpreted as the number 0.


    Probably the most famous very early set of axioms is the 4 + 1 postulates of Euclid. This turns out to be incomplete, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23). "4 + 1" because for nearly two millennia the fifth (parallel) postulate (through a point outside a line there is exactly one parallel) was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly or more than a straight line respectively and are known as elliptic, Euclidean and hyperbolic geometries. In mathematical logic, a theorem is a type of abstract object, one token of which is a formula of a formal language which can be derived from the rules of the formal system that is applied to the formal language; another token of which is a statement in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions. In all settings, an essential property of theorems is that they are derivable using a fixed set of inference rules and axioms without any additional assumptions. This is not a matter of the semantics of the language: the expression that results from a derivation is a syntactic consequence of all the expressions that precede it. In mathematics, the derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical. Although they can be written in a completely symbolic form, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organised and clearly worded informal arguments intended to demonstrate that a formal symbolic proof can be constructed. Such arguments are typically easier to check than purely symbolic ones — indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgements vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's last theorem is a particularly well-known example of such a theorem. Logically most theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true, within the accepted standards of the field. A proof is a logical argument, not an empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture. Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of informal logic. Purely formal proofs are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.

    Regardless of one's attitude to formalism, the result that is proved to be true is a theorem; in a completely formal proof it would be the final line, and the complete proof shows how it follows from the axioms alone by the application of the rules of inference. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma if it is used as a stepping stone in the proof of a theorem. The axioms are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.






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