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Axiomatization

Axiomatization

In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. Every mathematical theory is based on a set of axioms. Usually these axioms are not mentioned when a mathematical equation is presented. Mathematicians know from their education on which axioms mathematical theories are based. Indeed, mathematical theories usually are based on very few axioms. Some of them are mentioned in the example below.

Example: The axiomatization of natural numbers

The mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system that was first written down by the mathematician Peano in 1901. He defined the axioms (see Peano axioms) for the set N of natural numbers as being:
- There is a natural number 0.
- Every natural number a has a successor, denoted by a + 1.
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a ≠ b, then a + 1 ≠ b + 1.
- If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers. Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but the truth is that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that is merely a limitation on the purposes that deductive logic serves. Category:Mathematics ko:공리주의 (수학)

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system; usually though the effort towards complete formalisation brings diminishing returns in certainty, and a lack of readability for humans. Therefore discussion of axiomatic systems is normally only semi-formal. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a proof within a formal system.

Properties

An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its negation from the system's axioms. In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent. Although independence is not a necessary requirement for a system, consistency is. An axiomatic system will be called complete if every statement is either derivable or its negation is derivable.

Models

A mathematical model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a concrete model
- proves the consistency of a system. Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem. Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called categorial, and the property of categoriality ensures the completeness of a system.
- A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems. The first axiomatic system was Euclidean geometry.

Axiomatic method

The axiomatic method is often discussed as if it were a unitary approach, or uniform procedure. With the example of Euclid to appeal to, it was indeed treated that way for many centuries: up until the beginning of the nineteenth century it was generally assumed, in European mathematics and philosophy (for example in Spinoza's work) that the heritage of Greek mathematics represented the highest standard of intellectual finish (development more geometrico, in the style of the geometers). This traditional approach, in which axioms were supposed to be self-evident and so indisputable, was swept away during the course of the nineteenth century, by the development of Non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's 'new' use of axiomatic method as a research tool. For example, group theory was first put on an axiomatic basis towards the end of that century. Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies. Therefore, there are at least three 'modes' of axiomatic method current in mathematics, and in the fields it influences. In caricature, possible attitudes are: #Accept my axioms and you must accept their consequences; #I reject one of your axioms and accept extra models; #My set of axioms defines a research programme. The first case is the classic deductive method. The second goes by the slogan be wise, generalise; it may go along with the assumption that concepts can or should be expressed at some intrinsic 'natural level of generality'. The third was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra. It is easy to see that the axiomatic method has limitations outside mathematics. For example, in political philosophy axioms that lead to unacceptable conclusions are likely to be rejected wholesale; so that no one really assents to version 1 above.

See also


- Axiomatization
- Model theory
- Gödel's incompleteness theorem Category:Mathematical logic Category:Systems

Axiom

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist. In mathematics, an axiom is not necessarily a self-evident truth but rather, a formal logical expression used in a deduction to yield further results. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.

Etymology

The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered self-evident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.

Mathematics

In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and non-logical axioms.

Logical axioms

These are certain formulas in a language that are universally valid, that is, formulas that are satisfied by every structure under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values. Usually one takes as logical axioms some minimal set of tautologies that is sufficient for proving all tautologies in the language.

Examples

In the propositional calculus it is common to take as logical axioms all formulas of the following forms, where \phi, \psi, and \chi can be any formulas of the language: #\phi \to (\psi \to \phi) #(\phi \to (\psi \to \chi)) \to ((\phi \to \psi) \to (\phi \to \chi)) #(\lnot \phi \to \lnot \psi) \to (\psi \to \phi) Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if A, B, and C are propositional variables, then A \to (B \to A) and (A \to \lnot B) \to (C \to (A \to \lnot B)) are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens. These axiom schemata are also used in the predicate calculus, but additional logical axioms are needed.
Example. Let \mathfrak\, be a first-order language. For each variable x\,, the formula
x = x
is universally valid.
This means that for any variable symbol x\,, the formula x = x\, can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by x = x\, (or, for all what matters, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol =\, has to be enforced, and mathematical logic does indeed do that. Another, more interesting example, is that which provides us with what is known as universal instantiation:
Example. Given a formula \phi\, in a first-order language \mathfrak\,, a variable x\, and a term t\, that is substitutable for x\, in \phi\,, the formula
\forall x. \phi \to \phi^x_t
is universally valid.
Informally speaking, this example allows us to state that if we know that a certain property P\, holds for every x\, and that if t\, stands for a particular object in our structure, then we should be able to claim P(t)\,. Again, we are claiming that the formula \forall x. \phi \to \phi^x_t is valid, that is, we must be able to give a "proof" of this fact, or more properly speaking, a metaproof. Actually, these examples are metatheorems of our theory of mathematical logic since we are dealing with the very concept of proof itself. Aside from this, we can also have existential generalization:
Axiom scheme. Given a formula \phi\, in a first-order language \mathfrak\,, a variable x\, and a term t\, that is substitutable for x\, in \phi\,, the formula
\phi^x_t \to \exists x. \phi
is universally valid.

Non-logical axioms

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 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 formalized down to the bare language of logical formulas. This turned out to be impossible and proved to be quite a story (see below). Non-logical axioms are often simply referred to as axioms in mathematical discourse. This does not mean that it is claimed that they are true in some absolute sense. For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing (the more general) group theory, and we can even take its negation as an axiom for the study of non-commutative groups. Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.

Examples

This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms. Basic theories, such as arithmetic, real analysis (sometimes referred to as the theory of functions of one real variable), linear algebra, and complex analysis (a.k.a. complex variables), are often introduced non-axiomatically in mostly technical studies, but any rigorous course in these subjects always begins by presenting its axioms. Geometries such as Euclidean geometry, projective geometry, symplectic geometry. Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. Only under the umbrella of Euclidean geometry is this always true. The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory. The development of abstract algebra brought with itself group theory, rings and fields, Galois theory. This list could be expanded to include most fields of mathematics, including axiomatic set theory, measure theory, ergodic theory, probability, representation theory, and differential geometry.
Arithmetic
The Peano axioms are the most widely used axiomatization of arithmetic. They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem. We have a language \mathfrak_ = \\, where 0\, is a constant symbol and S\, is a unary function and the following axioms: # \forall x. \lnot (Sx = 0) # \forall x. \forall y. (Sx = Sy \to x = y) # ((\phi(0) \land \forall x.\,(\phi(x) \to \phi(Sx))) \to \forall x.\phi(x) for any \mathfrak_\, formula \phi\, with one free variable. The standard structure is \mathfrak = \langle\N, 0, S\rangle\, where \N\, is the set of natural numbers, S\, is the successor function and 0\, is naturally interpreted as the number 0.
Euclidean geometry
Probably the oldest, and most famous, list of axioms are the 4 + 1 Euclid's postulates of plane geometry. This set of axioms turns out to be incomplete, and many more postulates are necessary to rigorously characterize his geometry (Hilbert used 23). The axioms are referred to as "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.
Real analysis
The object of study is the real numbers. The real numbers are uniquely picked out (up to isomorphism) by the properties of a complete ordered field. However, expressing these properties as axioms requires use of second-order logic. The Löwenheim-Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in non-standard analysis.

Role in mathematical logic

Deductive systems and completeness

A deductive system consists of a set \Lambda\, of logical axioms, a set \Sigma\, of non-logical axioms, and a set \\, of rules of inference. A desirable property of a deductive system is that it be complete. A system is said to be complete if, for all formulas \phi,
if \Sigma \models \phi then \Sigma \vdash \phi
that is, for any statement that is a logical consequence of \Sigma there actually exists a deduction of the statement from \Sigma\,. This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly-used type of deductive system. Note that "completeness" has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms \Sigma\, of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement \phi\, such that neither \phi\, nor \lnot\phi\, can be proved from the given set of axioms. There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.

Further discussion

Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted, but that the grand parallels between axiomatic systems could be put to good use, as he algebraically solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born. In the modern view we may take as axioms any set of formulas we like, as long as they are not known to be inconsistent.

See also


- Axiomatic system
- Peano axioms
- Axiom of choice
- Axiom of countability
- Axiomatic set theory
- Parallel postulate
- Continuum hypothesis
- Axiomatization
- List of axioms

External links


- [http://us.metamath.org/mpegif/mmset.html#axioms Metamath axioms page] Category:AlgebraCategory:Logic ko:공리 ja:公理

Mathematician

A mathematician is a person whose area of study and research is mathematics. Today, most mathematicians are professors at a university or other research institution; however, a minority have a non-academic career and are often known as amateur mathematicians. While a number of misinformed people may believe mathematics is fully understood (as it is often presented this way in elementary textbooks), in fact, there is ongoing research into many areas of mathematics. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics). Unlike the other sciences, research in mathematics generally does not consist of performing experiments. Rather, mathematics is about problem-solving, where truths are deduced from other known truths. Computer experiments and other numerical evidence might be a part of this process, but in the end, mathematics research is about constructing proofs of theorems. In particular, calculation is not a big part of mathematics research, and mathematicians need not have any extraordinary ability in adding or multiplying numbers. See mental calculators to read about prodigies at performing such calculations.

Motivation

Mathematicians are typically interested in finding and describing patterns that may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. Problems have come from physics, economics, games, generalizations of earlier mathematics, and some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown the eventually applications are found. For example, number theory originally seemed to be without purpose, but after the invention of computers it gained countless applications to algorithms and cryptography.

Differences

Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question. Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal, a theorem that is proven true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematical theories are static. Once a statement is considered a theorem, it remains true forever.

Demographics

As is the case in many scientific disciplines, the field of mathematics has been disproportionately dominated by men. Among the minority of prominent female mathematicians are Emmy Noether (1882 - 1935), Sophie Germain (1776 - 1831), Sofia Kovalevskaya (1850 - 1891), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Mary Ellen Rudin, Eva Tardos, Émilie du Châtelet, Mary Cartwright and Marianna Csörnyei.

Quotes

...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. :-Saint Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such) A mathematician is a machine for turning coffee into theorems. :-Paul Erdős Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. (Mathematicians are [like] a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.) :-Johann Wolfgang von Goethe Some humans are mathematicians; others aren't. :-Jane Goodall (1971) In the Shadow of Man

Jokes

Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows: :An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep. :The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep. :Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs - and returns to his slumbers. Another joke goes thus: :Three men are flying in a hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing a field and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in a hot air balloon". One of the men in the air then says to his friends, "He was a mathematician - he thought before answering, his answer was totally right and totally useless" And another: :An astrologer, a chemist, and a mathematician are on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by. The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!" The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black." The mathematician then says, "Actually, we can only be sure there is at least one Scottish sheep of which at least one side is black" And finally: : An experiment is being made. A physicist (or an engineer) and a mathematician are asked to boil hot water, but the kettle is in the living room. The physicist goes to the living room, takes the kettle, returns to the kitchen and puts it on the stove and boils the water. The mathematician does the same. In the second stage, the kettle is in the kitchen and the two are again asked to boil hot water. The physicist simply puts the kettle on the stove and boils the water. However, the mathematician takes the kettle, puts it in the living room and declares: "We have already solved this problem!"

Links and references

References


- A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
  - Reprint edition, Cambridge University Press, 1992; ISBN 0521427061
  - First edition, 1940
- Dunham, William. The Mathematical Universe. John Wiley 1994.

See also


- mental calculator
- List of mathematicians
- List of amateur mathematicians
- Astronomers, Physicists, Philosophers, Scientists
- American Mathematical Society
- Mathematical Association of America

External links


- [http://www-history.mcs.st-and.ac.uk/history/index0.html The MacTutor History of Mathematics archive], a very complete list of detailed biographies.
- [http://genealogy.math.ndsu.nodak.edu/ The Mathematics Genealogy Project], which allows to follow the succession of thesis advisors for most mathematicians, living or dead. Category:Mathematical science occupations
- ja:数学者 ko:수학자 th:นักคณิตศาสตร์ __NOTOC__

Natural number

In mathematics, a natural number is either a positive integer (, , , , ...) or a non-negative integer (, , , , , ...). The former definition is generally used in number theory, while the latter is preferred in set theory. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), and they can be used for ordering ("this is the 3rd largest city in the country"). Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting, such as Ramsey theory, are studied in combinatorics.

History of natural numbers and the status of zero

The natural numbers presumably had their origins in the words used to count things, beginning with the number one. The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. For example, the Babylonians developed a powerful place-value system based essentially on the numerals for 1 and 10. The ancient Egyptians had a system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to one million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. A much later advance in abstraction was the development of the idea of zero as a number with its own numeral. A zero digit had been used in place-value notation as early as 700 BC by the Babylonians, but it was never used as a final element.1 The Olmec and Maya civilization used zero as a separate number as early as 1st century BC, apparently developed independently, but this usage did not spread beyond Mesoamerica. The concept as used in modern times originated with the Indian mathematician Brahmagupta in 628 AD. Nevertheless, zero was used as a number by all medieval computists (calculators of Easter) beginning with Dionysius Exiguus in 525, but in general no Roman numeral was used to write it. Instead, the Latin word for "nothing," nullae, was employed. The first systematic study of numbers as abstractions (that is, as abstract entities) is usually credited to the Greek philosophers Pythagoras and Archimedes. However, independent studies also occurred at around the same time in India, China, and Mesoamerica. In the nineteenth century, a set-theoretical definition of natural numbers was developed. With this definition, it was more convenient to include zero (corresponding to the empty set) as a natural number. This convention is followed by set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers. The term whole number is used informally by some authors for an element of the set of integers, the set of non-negative integers, or the set of positive integers.

Notation

Mathematicians use N or \mathbb (an N in blackboard bold) to refer to the set of all natural numbers. This set is infinite but countable by definition. To be unambiguous about whether zero is included or not, sometimes an index "0" is added in the former case, and a superscript "
- " is added in the latter case: : N0 = ; N
- = . (Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R+ = [0,∞) and Z+ = , at least in European literature. The notation "
- ", however, is quite standard for nonzero or rather invertible elements.) Less frequently, W or \mathbb is used for the set of "whole numbers", which are sometimes identified with the natural numbers as defined here, sometimes with the integers. Set theorists often denote the set of all natural numbers by ω. When this notation is used, zero is explicitly included as a natural number.

Formal definitions

Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano postulates state conditions that any successful definition must satisfy. Certain constructions show that, given set theory, models of the Peano postulates must exist.

Peano axioms


- There is a natural number 0.
- Every natural number a has a natural number successor, denoted by S(a).
- There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.) It should be noted that the "0" in the above definition need not correspond to what we normally consider to be the number zero. "0" simply means some object that when combined with an appropriate successor function, satisfies the Peano axioms. There are many systems that satisfy these axioms, including the natural numbers (either starting from zero or one).

Constructions based on set theory

The standard construction

A standard construction in set theory is to define the natural numbers as follows: :We set 0 := :and define S(a) = a U for all a. :The set of natural numbers is then defined to be the intersection of all sets containing 0 which are closed under the successor function. :Assuming the axiom of infinity, this definition can be shown to satisfy the Peano axioms. :Each natural number is then equal to the set of natural numbers less than it, so that :
- 0 = :
- 1 = = :
- 2 = = = :
- 3 = = = :and so on. When you see a natural number used as a set, this is typically what is meant. Under this definition, there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) iff n is a subset of m. :Also, with this definition, different possible interpretations of notations like Rn (n-tuples vs. mappings of n into R) coincide.

Other constructions

Although the standard construction is useful, it is not the only possible construction. For example: :one could define 0 = :and S(a) = , :producing :: 0 = :: 1 = = :: 2 = = , etc. Or we could even define 0 = :and S(a) = a U :producing :: 0 = :: 1 = = :: 2 = , etc. For the rest of this article, we follow the standard construction described above.

Properties

One can recursively define an addition on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free monoid with one generator. This monoid satisfies the cancellation property and can therefore be embedded in a group. The smallest group containing the natural numbers is the integers. If we define S(0) := 1, then S(b) = S(b + 0) = b + S(0) = b + 1; i.e. the successor of b is simply b + 1. Analogously, given that addition has been defined, a multiplication × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (N, ×) into a commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. If we interpret the natural numbers as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that a + 1 = S(a) and a × 1 = a. For the remainder of the article, we write ab to indicate the product a × b, and we also assume the standard order of operations. Furthermore, one defines a total order on the natural numbers by writing a ≤ b if and only if there exists another natural number c with a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc. An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 we can find natural numbers q and r such that :a = bq + r and r < b The number q is called the quotient and r is called the remainder of division of a by b. The numbers q and r are uniquely determined by a and b. This, the Division algorithm, is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.

Generalizations

Two generalizations of natural numbers arise from the two uses: ordinal numbers are used to describe the position of an element in a ordered sequence and cardinal numbers are used to specify the size of a given set. For finite sequences or finite sets, both of these properties are embodied in the natural numbers. Other generalizations are discussed in the article on numbers.

Footnote

¹ "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place." [http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html] Category:Elementary mathematics Category:Integers Category:Number theory Category:Numbers Category:Set theory ko:자연수 ja:自然数 th:จำนวนธรรมชาติ

1901

1901 (MCMI) was a common year starting on Tuesday (see link for calendar).

Events

January-March


- January 1 - World celebrates what is regarded as the start of the new century. (Zero-ists' argument that new century should be celebrated in 1900 rejected worldwide).
- January 1 - The British colonies of New South Wales, Queensland, South Australia, Tasmania, Victoria and Western Australia federate as the Commonwealth of Australia. Edmund Barton becomes first Prime Minister.
- January 1 - Nigeria becomes a British protectorate
- January 7 - Alferd Packer is released from prison after serving 18 years for cannibalism
- January 10 - The first great Texas gusher, oil discovered at Spindletop in Beaumont, Texas Beaumont, Texas
- January 22 - Death of Queen Victoria. Her eldest son, Prince Albert Edward, Prince of Wales becomes King, reigning as King Edward VII. His son, Prince George, Duke of York becomes Duke of Cornwall.
- February 20 - The legislature of Hawaii Territory convenes for the first time.
- February 25 - J.P. Morgan incorporates the United States Steel Corporation.
- March 2 - The U.S. Congress passes the Platt amendment, limiting the autonomy of Cuba as a condition for the withdrawal of American troops.
- March 6 - In Bremen an assassin attempts to kill Wilhelm II of Germany.
- March 17 - A showing of 71 Vincent van Gogh paintings in Paris, 11 years after his death, creates a sensation.

April-June


- April 25 - New York State becomes the first to require automobile license plates.
- May 5 - Official end of the Caste War of Yucatàn, although mayan skirmishers will continue sporadic fighting for the next decade.
- May 9 - Australia opens its first parliament in Melbourne.
- May 27 - In New Jersey, the Edison Storage Battery Company is founded.
- June 2 - Katsura Taro becomes Prime Minister of Japan
- June 12 - Cuba becomes US protectorate

July-September


- July 4 - The 1,282 foot (390 meters) covered bridge crossing the St.John River at Hartland, New Brunswick, Canada opens. It is the longest covered bridge in the world.
- July 24 - O. Henry is released from prison in Columbus, Ohio after serving three years for embezzlement from the First National Bank in Austin, Texas.
- August 21 - The Cadillac Motor Company formed in Detroit, Michigan, USA
- September 2 - Vice President Theodore Roosevelt utters the famous phrase, "Speak softly and carry a big stick" at the Minnesota State Fair.
- September 5 - The National Association of Professional Baseball Leagues (later renamed Minor League Baseball), is formed in Chicago, Illinois.
- September 6 - American anarchist Leon Czolgosz shoots and fatally wounds US President William McKinley at the Pan-American Exposition in Buffalo, New York. McKinley dies there eight days later.
- September 7 - The Boxer Rebellion in China officially ends with the signing of the Peking Protocol.
- September 9 - Hendrik Frensch Verwoerd, was prime minister of South Africa from 1958 - 1966 (d. September 6 1966)
- September 14 - With the death of William McKinley, Theodore Roosevelt succeeds him as President of the United States.

October-December

President of the United States
- October 2 - Royal Navy's first submarine launched at Barrow
- October 24 – Michigan schoolteacher Annie Taylor goes down Niagara Falls in a barrel and survives
- October 29 - In Amherst, Massachusetts nurse Jane Toppan is arrested for murdering the Davis family of Boston with an overdose of morphine.
- October 29 - Capital punishment: Leon Czolgosz, the assassin of US President William McKinley, is executed by electrocution.
- November 9 - Prince George, Duke of Cornwall becomes Prince of Wales and Earl of Chester.
- November 15 - Miller Reese Hutchinson patents Acousticon, a heavy hearing-aid prototype
- November 27 - U.S. Army War College is established.
- December 3 - US President Theodore Roosevelt delivers a 20,000-word speech to the House of Representatives asking Congress curb the power of trusts "within reasonable limits".
- December 10 – Marie Curie receives doctorate. The first Nobel Prize ceremony is held in Stockholm.
- December 12 - Guglielmo Marconi receives the first trans-Atlantic radio signal in Newfoundland, Canada; it is Morse code for the letter "S."

Unknown dates


- In the United Kingdom, Factory Act forbids child labor under 12
- Two typhoid outbreaks in USA
- Winston Churchill enters the House of Commons
- In Germany, Eugen Hollander makes the first known facelift to a Polish noblewoman
- Scotland Yard creates a fingerprint archive
- Cleveland Indians founded
- Europium discovered by Eugène-Antole Demarçay
- First prototype Harley-Davidson created
- Okapi discovered (previously known only to local natives)
- Independent Maya of Eastern Yucatán surrender to Mexico
- American Standard Version Bible first published.
- Intercollegiate Prohibition Association established in Chicago, Illinois.
- Mordecai Ham, American evangelist enters ministry.

Births

January-March


- January 3 - Ngo Dinh Diem, 1st President of South Vietnam (d. 1963)
- January 4 - CLR James, Trinidad-born writer and journalist (d. 1989)
- January 14 - Bebe Daniels, American actress (d. 1971)
- January 16 - Frank Zamboni, American inventor (d. 1988)
- January 26 - Stuart Symington, American politician (d. 1988)
- January 29 - E. P. Taylor, Canadian business tycoon (d. 1989)
- January 30 - Rudolf Caracciola, German race car driver (d. 1959)
- February 1 - Clark Gable, American actor (d. 1960)
- February 2 - Jascha Heifetz, Lithuanian violinist (d. 1987)
- February 10 - Stella Adler, American actress (d. 1992)
- February 25 - Zeppo Marx, American comedian (d. 1979)
- February 27 - Horatio Luro, Argentine horse trainer (d. 1991)
- February 28 - Linus Pauling, American chemist, recipient of the Nobel Prize in Chemistry and Peace (d. 1994)
- March 4 - Charles Goren, American bridge player (d. 1991)
- March 17 - Alfred Newman, American film composer (d. 1970)
- March 21 - Karl Arnold, German politician (d. 1958)
- March 22 - Greta Kempton, American artist (d. 1991)
- March 24 - Ub Iwerks, American cartoonist (d. 1971)
- March 27 - Carl Barks, American cartoonist (d. 2000)
- March 27 - Erich Ollenhauer, German politician (d. 1963)
- March 27 - Eisaku Sato, Prime Minister of Japan, recipient of the Nobel Peace Prize (d. 1975)
- March 27 - Kenneth Slessor, Australian poet (d. 1971)

April-June


- April 1 - Whittaker Chambers, American spy (d. 1961)
- April 29 - Emperor Hirohito of Japan (d. 1989)
- April 30 - Simon Kuznets, Ukrainian-born economist, Nobel Prize laureate (d. 1985)
- May 5 - Blind Willie McTell, American singer (d. 1959)
- May 7 - Gary Cooper, American actor (d. 1961)
- May 17 - Werner Egk, German composer (d. 1983)
- May 18 - Vincent du Vigneaud, American chemist, Nobel Prize laureate (d. 1978)
- May 20 - Max Euwe, Dutch chess player (d. 1981)
- May 21 - Horace Heidt, American bandleader (d. 1986)
- May 21 - Sam Jaffe, American film producer (d. 2000)
- June 3 - Chang Hsüeh-liang, Chinese military leader (d. 2001)
- June 17 - F. F. E. Yeo-Thomas, English World War II hero (d. 1964)
- June 18 - Grand Duchess Anastasia of Russia (d. 1918)
- June 24 - Harry Partch, American composer (d. 1974)
- June 29 - Nelson Eddy, American singer and actor (d. 1967)

July-September


- July 9 - Dame Barbara Cartland English novelist (d. 2000)
- July 17 - Bruno Jasieński, Polish poet (d. 1938)
- July 20 - Heinie Manush, baseball player (d. 1971)
- July 31 - Jean Dubuffet, French painter (d. 1985)
- August 4 - Louis Armstrong, American jazz musician (d. 1971)
- August 8 - Ernest Lawrence, American physicist, Nobel Prize laureate (d. 1958)
- August 10 - Franco Dino Rasetti Italian scientist (d.2001)
- August 18 - Jean Guitton, French writer and philosopher (d. 1999)
- August 20 - Salvatore Quasimodo, Italian writer, Nobel Prize laureate (d. 1968)
- September 9 - James Blades, English percussionist (d. 1999)
-