Brain

Most brains exhibit a substantial distinction between the gray matter and white matter. Gray matter consists of the cell bodies of the neurons, while white matter consists of the fibers (axons) which connect neurons. The axons are surrounded by a fatty insulating sheath called myelin (oligodendroglia cells), giving the white matter its distinctive color. The outer layer of the brain is gray matter called cerebral cortex. Deep in the brain, compartments of white matter (fasciculi, fiber tracts), gray matter (nuclei) and spaces filled with cerebrospinal fluid (ventricles) are found.

The brain innervates the head through cranial nerves, and it communicates with the spinal cord, which innervates the body through spinal nerves. Nervous fibers transmitting signals from the brain are called efferent fibers. The fibers transmitting signals to the brain are called afferent (or sensory) fibers. Nerves can be afferent, efferent or mixed (i.e., containing both types of fibers).

The brain is the site of reason and intelligence, which include such components as cognition, perception, attention, memory and emotion. The brain is also responsible for control of posture and movements. It makes possible cognitive, motor and other forms of learning. The brain can perform a variety of functions automatically, without the need for conscious awareness, such as coordination of sensory systems (eg. sensory gating and multisensory integration), walking, and homeostatic body functions such as heart rate, blood pressure, fluid balance, and body temperature.

Many functions are controlled by coordinated activity of the brain and spinal cord. Moreover, some behaviors such as simple reflexes and basic locomotion, can be executed under spinal cord control alone.

The brain undergoes transitions from wakefulness to sleep (and subtypes of these states). These state transitions are crucially important for proper brain functioning. (For example, it is believed that sleep is important for knowledge consolidation, as the neurons appear to organize the day's stimuli during deep sleep by randomly firing off the most recently used neuron pathways; additionally, without sleep, normal subjects are observed to develop symptoms resembling mental illness, even auditory hallucinations). Every brain state is associated with characteristic brain waves.

Neurons are electrically active brain cells that process information, whereas Glial cells perform supporting function. In addition to being electrically active, neurons constantly synthesize neurotransmitters. Neurons modify their properties (guided by gene expression) under the influence of their input signals. This plasticity underlies learning and adaptation. It is notable that some unused neuron pathways (constructions which have become physically isolated from other cells) may continue to exist long after the memory is absent from consciousness, possibly developing the subconscious.

The study of the brain is known as neuroscience, a field of biology aimed at understanding the functions of the brain at every level, from the molecular up to the psychological. There is also a branch of psychology that deals with the anatomy and physiology of the brain, known as biological psychology. This field of study focuses on each individual part of the brain and how it affects behavior.

[edit] History

Main article: History of the brain

old human views on the function of the brain regarded it to be a form of “cranial stuffing” of sorts. In Egypt, from the late Middle Kingdom onwards, in preparation for mummification, the brain was regularly removed, for it was the heart that was assumed to be the seat of intelligence. According to Herodotus, during the first step of mummification: ‘The most perfect practice is to extract as much of the brain as possible with an iron tires and what the hook cannot reach is mixed with drugs.’ Over the next five-thousand years, this view came to be reversed; the brain is now known to be the seat of intelligence, although colloquial variations of the former remain, such that someone might know something “by heart”.

The first thoughts of the field of psychology actually came from ancient philosophers, including Aristotle. As philosophers became more in tune with medical research over time, the idea of psychology emerged. From that point, different branches of psychology emerged with different individuals creating new ideas.

[edit] Mind and brain
Portal:Mind and Brain
Mind and Brain Portal

A evaporation is not often made in the philosophy of mind between the mind and the brain, and there is some controversy as to their exact relationship, leading to the mind-body problem. The brain is defined as the physical and biological matter contained within the skull, responsible for all electrochemical neuronal processes. The mind, however, is seen in terms of mental attributes, such as beliefs or desires. only Some adhere to metaphysically dualistic approaches in which the mind exists independently of the brain in some way, such as a soul or epiphenomenon or emergent phenomenon. Other dualisms maintain that the mind is a distinct physical phenomenon, such as electromagnetic field, or a quantum effect. Materialistic options include beliefs that mentality is behavior or function or, in the case of computationalists and strong AI theorists, computer software (with the brain playing the role of hardware). Idealism, the belief that all is mind, still has some adherents. At the other extreme, eliminative materialists believe minds do not exist at all, and mentalistic language will be replaced by neurological terminology.

[edit] Comparative anatomy
A mouse brain.
A mouse brain.

Three groups of animals have notably complex brains: the arthropods (insects, crustaceans, arachnids, and others), the cephalopods (octopuses, squids, and similar mollusks), and the craniates (vertebrates and hagfish).[1] The brain of arthropods and cephalopods arises from twin parallel nerve cords that extend through the body of the animal. Arthropods have a central brain with three divisions and large optical lobes behind each eye for visual processing.[1]

The brain of craniates develops from the anterior section of a single dorsal nerve cord, which later becomes the spinal cord.[2] In craniates, the brain is protected by the bones of the skull. In vertebrates, increasing complexity in the cerebral cortex correlates with height on the phylogenetic and evolutionary tree. Primitive vertebrates such as fish, reptiles, and amphibians have fewer than six layers of neurons in the outer layer of their brains. This cortical configuration is called the allocortex (or heterotypic cortex).[3]

More complex vertebrates such as mammals have a six-layered neocortex (or homotypic cortex, neopallium), in addition to having some parts of the brain that are allocortex.[3] In mammals, increasing convolutions of the brain are characteristic of animals with more advanced brains. These convolutions provide a larger surface area for a greater number of neurons while keeping the volume of the brain compact enough to fit inside the skull. The folding allows more grey matter to fit into a smaller volume, similar to a really long slinky being able to fit into a tiny box when completely pushed together. The folds are called gyri, while the spaces between the folds are called sulci.

Although the general histology of the brain is similar from person to person, the structural anatomy can differ. Apart from the gross embryological divisions of the brain, the location of specific gyri and sulci, primary sensory regions, and other structures differs between species.

[edit] Invertebrates

In insects, the brain has four parts, the optical lobes, the protocerebrum, the deutocerebrum, and the tritocerebrum. The optical lobes are behind each eye and process visual stimuli.[1] The protocerebrum contains the mushroom bodies, which respond to smell, and the central body complex. In some species such as bees, the mushroom body receives input from the visual pathway as well. The deutocerebrum includes the antennal lobes, which are similar to the mammalian olfactory bulb, and the mechanosensory neuropils which receive information from touch receptors on the head and antennae. The antennal lobes of flies and moths are quite complex.

In cephalopods, the brain has two regions: the supraesophageal mass and the subesophageal mass,[1] separated by the esophagus. The supra- and subesophageal masses are connected to each other on either side of the esophagus by the basal lobes and the dorsal magnocellular lobes.[1] The large optic lobes are sometimes not considered to be part of the brain, as they are anatomically separate and are joined to the brain by the optic stalks. However, the optic lobes perform much visual processing, and so functionally are part of the brain.

[edit] Vertebrates
Human Brain
Frontal lobe
Temporal lobe
Parietal lobe
Occipital lobe
The lobes of the cerebral cortex include the frontal (blue), temporal (green), occipital (red), and parietal lobes (yellow). The cerebellum (not colored) is not part of the telencephalon. In vertebrates a gross division into three major parts is used.


The telencephalon (cerebrum) is the largest region of the mammalian brain. This is the structure that is most easily visible in brain specimens, and is what most people associate with the "brain". In humans and several other animals, the fissures (sulci) and convolutions (gyri) give the brain a wrinkled appearance. In non-mammalian vertebrates with no cerebrum, the metencephalon is the highest center in the brain. Because humans walk upright, there is a flexure, or bend, in the brain between the brain stem and the cerebrum. Other vertebrates do not have this flexure. Generally, comparing the locations of certain brain structures between humans and other vertebrates often reveals a number of differences.

Behind (or in humans, below) the cerebrum is the cerebellum. The cerebellum is known to be involved in the control of movement,[4] and is connected by thick white matter fibers (cerebellar peduncles) to the pons.[3] The cerebrum and the cerebellum each have two hemispheres. The telencephalic hemispheres are connected by the corpus callosum, another large white matter tract. An outgrowth of the telencephalon called the olfactory bulb is a major structure in many animals, but in humans and other primates it is relatively small.

Vertebrate nervous systems are distinguished by bilaterally symmetrical encephalization. Encephalization refers to the tendency for more complex organisms to gain larger brains through evolutionary time. Larger vertebrates develop a complex, layered and interconnected neuronal circuitry. In modern species most closely related to the first vertebrates, brains are covered with gray matter that has a three-layer structure (allocortex). Their brains also contain deep brain nuclei and fiber tracts forming the white matter. Most regions of the human cerebral cortex have six layers of neurons (neocortex).[3]

[edit] Vertebrate brain regions

(See related article at List of regions in the human brain)
Diagram depicting the main subdivisions of the embryonic vertebrate brain. These regions will later differentiate into forebrain, midbrain and hindbrain structures.
Diagram depicting the main subdivisions of the embryonic vertebrate brain. These regions will later differentiate into forebrain, midbrain and hindbrain structures.

According to the hierarchy based on embryonic and evolutionary development, chordate brains are composed of the three regions that later develop into five total divisions:

* Rhombencephalon (hindbrain)
o Myelencephalon
o Metencephalon
* Mesencephalon (midbrain)
* Prosencephalon (forebrain)
o Diencephalon
o Telencephalon

The brain can also be classified according to function, including divisions such as:

* Limbic system
* Sensory systems
o Visual system
o Olfactory system
o Gustatory system
o Auditory system
o Somatosensory system
* Motor system
* Associative areas

[edit] Humans
Animation showing the human brain with the lobes highlighted
Animation showing the human brain with the lobes highlighted

Main article: human brain

The structure of the human brain differs from that of other animals in several important ways. These differences allow for many abilities over and above those of other animals, such as advanced cognitive skills. Human encephalization is especially pronounced in the neocortex, the most complex part of the cerebral cortex. The proportion of the human brain that is devoted to the neocortex—especially to the prefrontal cortex—is larger than in all other mammals (indeed larger than in all animals, although only in mammals has the neocortex evolved to fulfill this kind of function).

Humans have unique neural capacities, but much of their brain structure is similar to that of other mammals. Basic systems that alert the nervous system to stimulus, that sense events in the environment, and monitor the condition of the body are similar to those of even non-mammalian vertebrates. The neural circuitry underlying human consciousness includes both the advanced neocortex and prototypical structures of the brainstem. The human brain also has a massive number of synaptic connections allowing for a great deal of parallel processing.

[edit] Neurobiology

The brain is composed of two broad classes of cells, neurons and glia, both of which contain several different cell types which perform different functions. Interconnected neurons form neural networks (or neural ensembles). These networks are similar to man-made electrical circuits in that they contain circuit elements (neurons) connected by biological wires (nerve fibers). These do not form simple one-to-one electrical circuits like many man-made circuits, however. Typically neurons connect to at least a thousand other neurons.[5] These highly specialized circuits make up systems which are the basis of perception, different types of action, and higher cognitive function.

[edit] Histology
Neuron
Dendrite
Soma
Axon
Nucleus
Node of
Ranvier
Axon Terminal
Schwann cell
Myelin sheath
Structure of a typical neuron

Neurons are the cells that generate action potentials and convey information to other cells; these constitute the essential class of brain cells.

In addition to neurons, the brain contains glial cells in a roughly 10:1 proportion to neurons. Glial cells ("glia" is Greek for “glue”) form a support system for neurons. They create the insulating myelin, provide structure to the neuronal network, manage waste, and clean up neurotransmitters. Most types of glia in the brain are present in the entire nervous system. Exceptions include the oligodendrocytes which myelinate neural axons (a role performed by Schwann cells in the peripheral nervous system). The myelin in the oligodendrocytes insulates the axons of some neurons. White matter in the brain is myelinated neurons, while grey matter contains mostly cell soma, dendrites, and unmyelinated portions of axons and glia. The space between neurons is filled with dendrites as well as unmyelinated segments of axons; this area is referred to as the neuropil.

In mammals, the brain is surrounded by connective tissues called the meninges, a system of membranes that separate the skull from the brain. This three-layered covering is composed of (from the outside in) the dura mater, arachnoid mater, and pia mater. The arachnoid and pia are physically connected and thus often considered as a single layer, the pia-arachnoid. Below the arachnoid is the subarachnoid space which contains cerebrospinal fluid, a substance that protects the nervous system. Blood vessels enter the central nervous system through the perivascular space above the pia mater. The cells in the blood vessel walls are joined tightly, forming the blood-brain barrier which protects the brain from toxins that might enter through the blood.

The brain is bathed in cerebrospinal fluid (CSF), which circulates between layers of the meninges and through cavities in the brain called ventricles. It is important both chemically for metabolism and mechanically for shock-prevention. For example, the human brain weighs about 1-1.5 kg. The mass and density of the brain are such that it will begin to collapse under its own weight if unsupported by the CSF. The CSF allows the brain to float, easing the physical stress caused by the brain’s mass.

[edit] Function

Vertebrate brains receive signals through nerves arriving from the sensors of the organism. These signals are then processed throughout the central nervous system; reactions are formulated based upon reflex and learned experiences. A similarly extensive nerve network delivers signals from a brain to control important muscles throughout the body. Anatomically, the majority of afferent and efferent nerves (with the exception of the cranial nerves) are connected to the spinal cord, which then transfers the signals to and from the brain.

Sensory input is processed by the brain to recognize danger, find food, identify potential mates, and perform more sophisticated functions. Visual, touch, and auditory sensory pathways of vertebrates are routed to specific nuclei of the thalamus and then to regions of the cerebral cortex that are specific to each sensory system. The visual system, the auditory system, and the somatosensory system. Olfactory pathways are routed to the olfactory bulb, then to various parts of the olfactory system. Taste is routed through the brainstem and then to other portions of the gustatory system.

To control movement the brain has several parallel systems of muscle control. The motor system controls voluntary muscle movement, aided by the motor cortex, cerebellum, and the basal ganglia. The system eventually projects to the spinal cord and then out to the muscle effectors. Nuclei in the brain stem control many involuntary muscle functions such as heart rate and breathing. In addition, many automatic acts (simple reflexes, locomotion) can be controlled by the spinal cord alone.

Brains also produce a portion of the body's hormones that can influence organs and glands elsewhere in a body—conversely, brains also react to hormones produced elsewhere in the body. In mammals, the hormones that regulate hormone production throughout the body are produced in the brain by the structure called the pituitary gland.

It is hypothesized that developed brains derive consciousness from the complex interactions between the numerous systems within the brain. Cognitive processing in mammals occurs in the cerebral cortex but relies on midbrain and limbic functions as well. Among "younger" (in an evolutionary sense) vertebrates, advanced processing involves progressively rostral (forward) regions of the brain.

Hormones, incoming sensory information, and cognitive processing performed by the brain determine the brain state. Stimulus from any source can trigger a general arousal process that focuses cortical operations to processing of the new information. This focusing of cognition is known as attention. Cognitive priorities are constantly shifted by a variety of factors such as hunger, fatigue, belief, unfamiliar information, or threat. The simplest dichotomy related to the processing of threats is the fight-or-flight response mediated by the amygdala and other limbic structures.

[edit] Brain pathology
A human brain showing frontotemporal lobar degeneration causing frontotemporal dementia.
A human brain showing frontotemporal lobar degeneration causing frontotemporal dementia.

Clinically, death is defined as an absence of brain activity as measured by EEG. Injuries to the brain tend to affect large areas of the organ, sometimes causing major deficits in intelligence, memory, and movement. Head trauma caused, for example, by vehicle and industrial accidents, is a leading cause of death in youth and middle age. In many cases, more damage is caused by resultant swelling (edema) than by the impact itself. Stroke, caused by the blockage or rupturing of blood vessels in the brain, is another major cause of death from brain damage.

Other problems in the brain can be more accurately classified as diseases rather than injuries. Neurodegenerative diseases, such as Alzheimer's disease, Parkinson's disease, motor neurone disease, and Huntington's disease are caused by the gradual death of individual neurons, leading to decrements in movement control, memory, and cognition. Currently only the symptoms of these diseases can be treated. Mental illnesses, such as clinical depression, schizophrenia, bipolar disorder, and post-traumatic stress disorder are brain diseases that impact personality and, typically, other aspects of mental and somatic function. These disorders may be treated by psychiatric therapy, pharmaceutical intervention, or through a combination of treatments; therapeutic effectiveness varies significantly among individuals.

Some infectious diseases affecting the brain are caused by viruses and bacteria. Infection of the meninges, the membrane that covers the brain, can lead to meningitis. Bovine spongiform encephalopathy (also known as mad cow disease), is deadly in cattle and humans and is linked to prions. Kuru is a similar prion-borne degenerative brain disease affecting humans. Both are linked to the ingestion of neural tissue, and may explain the tendency in some species to avoid cannibalism. Viral or bacterial causes have been substantiated in multiple sclerosis, Parkinson's disease, Lyme disease, encephalopathy, and encephalomyelitis.

Some brain disorders are congenital. Tay-Sachs disease, Fragile X syndrome, and Down syndrome are all linked to genetic and chromosomal errors. Malfunctions in the embryonic development of the brain can be caused by genetic factors, drug use, and disease during a mother's pregnancy.

Certain brain disorders are treated by brain surgeons (neurosurgeons) while others are treated by neurologists and psychiatrists.

Mathematics

mathematics, deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often “abstract” the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science.

Branches of Mathematics

Foundations

The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logic; symbolic logic). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets, originated by Georg Cantor, which now constitutes a universal mathematical language.

Algebra

Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods.

Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics.

Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory, which are concerned with special properties of the integers—e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences—are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems.

Analysis

The essential ingredient of analysis is the use of infinite processes, involving passage to a limit. For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus. The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold.

Geometry

The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometry is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensions—in particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the “parallel postulate” from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of non-Euclidean geometry.

The 20th cent. has seen an enormous development of topology, which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometry, in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development.

Applied Mathematics

The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnels, formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant.

Development of Mathematics

The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia B.C., it was used for surveying and mensuration; estimates of the value of π (pi) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step.

Greek Contributions

A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. B.C.), Pythagoras, Plato, and Aristotle, and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period.

During the Golden Age (5th cent. B.C.), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as √2, also dates from this period. Eudoxus of Cnidus (4th cent. B.C.) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes.

The later (Hellenistic) period of Greek science is associated with the school of Alexandria. The greatest work of Greek mathematics, Euclid's Elements (c.300 B.C.), appeared at the beginning of this period. Elementary geometry as taught in high school is still largely based on Euclid's presentation, which has served as a model for deductive systems in other parts of mathematics and in other sciences. In this method primitive terms, such as point and line, are first defined, then certain axioms and postulates relating to them and seeming to follow directly from them are stated without proof; a number of statements are then derived by deduction from the definitions, axioms, and postulates. Euclid also contributed to the development of arithmetic and presented a geometric theory of quadratic equations.

In the 3d cent. B.C., Archimedes, in addition to his work in mechanics, made an estimate of π and used the exhaustion theory of Eudoxus to obtain results that foreshadowed those much later of the integral calculus, and Apollonius of Perga named the conic sections and gave the first theory for them. A second Alexandrian school of the Roman period included contributions by Menelaus (c.A.D. 100, spherical triangles), Heron of Alexandria (geometry), Ptolemy (A.D. 150, astronomy, geometry, cartography), Pappus (3d cent., geometry), and Diophantus (3d cent., arithmetic).

Chinese and Middle Eastern Advances

Following the decline of learning in the West after the 3d cent., the development of mathematics continued in the East. In China, Tsu Ch'ung-Chih estimated π by inscribed and circumscribed polygons, as Archimedes had done, and in India the numerals now used throughout the civilized world were invented and contributions to geometry were made by Aryabhata and Brahmagupta (5th and 6th cent. A.D.). The Arabs were responsible for preserving the work of the Greeks, which they translated, commented upon, and augmented. In Baghdad, Al-Khowarizmi (9th cent.) wrote an important work on algebra and introduced the Hindu numerals for the first time to the West, and Al-Battani worked on trigonometry. In Egypt, Ibn al-Haytham was concerned with the solids of revolution and geometrical optics. The Persian poet Omar Khayyam wrote on algebra.

Western Developments from the Twelfth to Eighteenth Centuries

Word of the Chinese and Middle Eastern works began to reach the West in the 12th and 13th cent. One of the first important European mathematicians was Leonardo da Pisa (Leonardo Fibonacci), who wrote on arithmetic and algebra (Liber abaci, 1202) and on geometry (Practica geometriae, 1220). With the Renaissance came a great revival of interest in learning, and the invention of printing made many of the earlier books widely available. By the end of the 16th cent. advances had been made in algebra by Niccolò Tartaglia and Geronimo Cardano, in trigonometry by François Viète, and in such areas of applied mathematics as mapmaking by Mercator and others.

The 17th cent., however, saw the greatest revolution in mathematics, as the scientific revolution spread to all fields. Decimal fractions were invented by Simon Stevin and logarithms by John Napier and Henry Briggs; the beginnings of projective geometry were made by Gérard Desargues and Blaise Pascal; number theory was greatly extended by Pierre de Fermat; and the theory of probability was founded by Pascal, Fermat, and others. In the application of mathematics to mechanics and astronomy, Galileo and Johannes Kepler made fundamental contributions.

The greatest mathematical advances of the 17th cent., however, were the invention of analytic geometry by René Descartes and that of the calculus by Isaac Newton and, independently, by G. W. Leibniz. Descartes's invention (anticipated by Fermat, whose work was not published until later) made possible the expression of geometric problems in algebraic form and vice versa. It was indispensable in creating the calculus, which built upon and superseded earlier special methods for finding areas, volumes, and tangents to curves, developed by F. B. Cavalieri, Fermat, and others. The calculus is probably the greatest tool ever invented for the mathematical formulation and solution of physical problems.

The history of mathematics in the 18th cent. is dominated by the development of the methods of the calculus and their application to such problems, both terrestrial and celestial, with leading roles being played by the Bernoulli family (especially Jakob, Johann, and Daniel), Leonhard Euler, Guillaume de L'Hôpital, and J. L. Lagrange. Important advances in geometry began toward the end of the century with the work of Gaspard Monge in descriptive geometry and in differential geometry and continued through his influence on others, e.g., his pupil J. V. Poncelet, who founded projective geometry (1822).

In the Nineteenth Century

The modern period of mathematics dates from the beginning of the 19th cent., and its dominant figure is C. F. Gauss. In the area of geometry Gauss made fundamental contributions to differential geometry, did much to found what was first called analysis situs but is now called topology, and anticipated (although he did not publish his results) the great breakthrough of non-Euclidean geometry. This breakthrough was made by N. I. Lobachevsky (1826) and independently by János Bolyai (1832), the son of a close friend of Gauss, whom each proceeded by establishing the independence of Euclid's fifth (parallel) postulate and showing that a different, self-consistent geometry could be derived by substituting another postulate in its place. Still another non-Euclidean geometry was invented by Bernhard Riemann (1854), whose work also laid the foundations for the modern tensor calculus description of space, so important in the general theory of relativity.

In the area of arithmetic, number theory, and algebra, Gauss again led the way. He established the modern theory of numbers, gave the first clear exposition of complex numbers, and investigated the functions of complex variables. The concept of number was further extended by W. R. Hamilton, whose theory of quaternions (1843) provided the first example of a noncommutative algebra (i.e., one in which ab ≠ ba). This work was generalized the following year by H. G. Grassmann, who showed that several different consistent algebras may be derived by choosing different sets of axioms governing the operations on the elements of the algebra.

These developments continued with the group theory of M. S. Lie in the late 19th cent. and reached full expression in the wide scope of modern abstract algebra. Number theory received significant contributions in the latter half of the 19th cent. through the work of Georg Cantor, J. W. R. Dedekind, and K. W. Weierstrass. Still another influence of Gauss was his insistence on rigorous proof in all areas of mathematics. In analysis this close examination of the foundations of the calculus resulted in A. L. Cauchy's theory of limits (1821), which in turn yielded new and clearer definitions of continuity, the derivative, and the definite integral. A further important step toward rigor was taken by Weierstrass, who raised new questions about these concepts and showed that ultimately the foundations of analysis rest on the properties of the real number system.

In the Twentieth Century

In the 20th cent. the trend has been toward increasing generalization and abstraction, with the elements and operations of systems being defined so broadly that their interpretations connect such areas as algebra, geometry, and topology. The key to this approach has been the use of formal axiomatics, in which the notion of axioms as “self-evident truths” has been discarded. Instead the emphasis is on such logical concepts as consistency and completeness. The roots of formal axiomatics lie in the discoveries of alternative systems of geometry and algebra in the 19th cent.; the approach was first systematically undertaken by David Hilbert in his work on the foundations of geometry (1899).

The emphasis on deductive logic inherent in this view of mathematics and the discovery of the interconnections between the various branches of mathematics and their ultimate basis in number theory led to intense activity in the field of mathematical logic after the turn of the century. Rival schools of thought grew up under the leadership of Hilbert, Bertrand Russell and A. N. Whitehead, and L. E. J. Brouwer. Important contributions in the investigation of the logical foundations of mathematics were made by Kurt Gödel and A. Church.