A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms .
95-405: Mechanical or electronic devices (or, historically , people) that perform computations are known as computers . Computer science is an academic field that involves the study of computation. The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s , but agreement on a suitable definition proved elusive. A candidate definition
190-591: A Turing-complete machine in 1998 by Raúl Rojas . In two 1936 patent applications, Zuse also anticipated that machine instructions could be stored in the same storage used for data—the key insight of what became known as the von Neumann architecture , first implemented in 1948 in America in the electromechanical IBM SSEC and in Britain in the fully electronic Manchester Baby . Zuse suffered setbacks during World War II when some of his machines were destroyed in
285-446: A universal Turing machine . Putnam asks whether alien beings, artificially intelligent robots , and silicon -based life forms should be considered a priori incapable of experiencing pain merely because they do not have the same neurochemistry as humans. We can imagine that they might share our psychological states despite being made of different stuff. Our ability to conceive of that possibility means that multiple realizability
380-446: A common ancestor. Homoplasies are similar or identical characteristics shared by two or more species but not inherited from a common ancestor, having evolved independently. The feet of ducks and platypuses are an example of homoplasy, while the hands of humans and chimps are an example of homology. The fact that brain structures are homologous provides no evidence either for or against multiple realizability. The only way to empirically test
475-448: A computational system is a complex object which consists of three parts. First, a mathematical dynamical system D S {\displaystyle DS} with discrete time and discrete state space; second, a computational setup H = ( F , B F ) {\displaystyle H=\left(F,B_{F}\right)} , which is made up of a theoretical part F {\displaystyle F} , and
570-695: A correct circuit diagram for a 4 bit digital binary adder. Purely electronic circuit elements soon replaced their mechanical and electromechanical equivalents, at the same time that digital calculation replaced analog. Machines such as the Z3 , the Atanasoff–Berry Computer , the Colossus computers , and the ENIAC were built by hand, using circuits containing relays or valves (vacuum tubes), and often used punched cards or punched paper tape for input and as
665-414: A kind-predicate and therefore the entire statement cannot be a law of physics . The special sciences therefore cannot be reduced to physics. Functionalism, which attempts to identify mental kinds with functional kinds that are characterized exclusively in terms of causes and effects, abstracts from particle physics and hence seems to be a more suitable explanation of the relation between mind and body. As
760-560: A line, the slide rule was invented in the 1620s, shortly after Napier's work, to allow multiplication and division operations to be carried out significantly faster than was previously possible. Edmund Gunter built a calculating device with a single logarithmic scale at the University of Oxford . His device greatly simplified arithmetic calculations, including multiplication and division. William Oughtred greatly improved this in 1630 with his circular slide rule. He followed this up with
855-618: A method adapted from the Jacquard loom invented by Joseph Marie Jacquard in 1804, which controlled textile patterns with a sequence of punched cards. These cards became foundational in later computing systems as well. Babbage's machine would have featured multiple output devices, including a printer, a curve plotter, and even a bell, demonstrating his ambition for versatile computational applications beyond simple arithmetic. Ada Lovelace expanded on Babbage's vision by conceptualizing algorithms that could be executed by his machine. Her notes on
950-418: A minimally acceptable degree of generality in scientific theorizing. Any psychology that is sufficiently narrow to accommodate this level of multiple realizability required to account for neural plasticity will almost certainly not be general enough to capture the generalizations needed to explain human psychology. Some reductionists reply that this is not empirically plausible. Research and experimentation in
1045-618: A purely physical process occurring inside a closed physical system called a computer . Turing's 1937 proof, On Computable Numbers, with an Application to the Entscheidungsproblem , demonstrated that there is a formal equivalence between computable statements and particular physical systems, commonly called computers . Examples of such physical systems are: Turing machines , human mathematicians following strict rules, digital computers , mechanical computers , analog computers and others. An alternative account of computation
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#17327982411041140-512: A range of subsequent developments in computing hardware. Notably, in the 1890s, Herman Hollerith adapted the idea of punched cards for automated data processing, which was utilized in the U.S. Census and sped up data tabulation significantly, bridging industrial machinery with data processing. The Industrial Revolution's advancements in mechanical systems demonstrated the potential for machines to conduct complex calculations, influencing engineers like Leonardo Torres Quevedo and Vannevar Bush in
1235-414: A real part B F {\displaystyle B_{F}} ; third, an interpretation I D S , H {\displaystyle I_{DS,H}} , which links the dynamical system D S {\displaystyle DS} with the setup H {\displaystyle H} . History of computing hardware The history of computing hardware spans
1330-624: A result of these arguments and others that build upon them, the dominant theory in philosophy of mind since the 1960s has been a version of non-reductive physicalism based on multiple realizability. In 1988, Hilary Putnam used multiple realizability to argue against functionalism. Noting that functionalism is essentially a watered-down reductionist or identity theory in which mental kinds are ultimately identified with functional kinds, Putnam argues that mental kinds are probably multiply realizable over functional kinds. The same mental state or property can be implemented or realized by different states of
1425-435: A rule. "Medium-independence" requires that the property can be instantiated by multiple realizers and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable . In short, medium-independence allows for the use of physical variables with properties other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that which occurs in
1520-694: A scale, rotation of a shaft, or a specific voltage level. Numbers could also be represented in the form of digits, automatically manipulated by a mechanism. Although this approach generally required more complex mechanisms, it greatly increased the precision of results. The development of transistor technology, followed by the invention of integrated circuit chips, led to revolutionary breakthroughs. Transistor-based computers and, later, integrated circuit-based computers enabled digital systems to gradually replace analog systems, increasing both efficiency and processing power. Metal-oxide-semiconductor (MOS) large-scale integration (LSI) then enabled semiconductor memory and
1615-405: A variety of ways. Just as the logical states of a Turing machine can be realized by different structural states in different mechanisms, so, by analogy, the mental states of a human being can be realized by different physical states in different individuals. Pain, for example, is correlated with different physical states of the nervous system in different organisms, but the organisms all experience
1710-416: A vast number of administrative uses. The Astronomical Computing Bureau, Columbia University , performed astronomical calculations representing the state of the art in computing . By the 20th century, earlier mechanical calculators, cash registers, accounting machines, and so on were redesigned to use electric motors, with gear position as the representation for the state of a variable. The word "computer"
1805-499: Is Boolean algebra , developed by the British mathematician George Boole in his work The Laws of Thought , published in 1854. His Boolean algebra was further refined in the 1860s by William Jevons and Charles Sanders Peirce , and was first presented systematically by Ernst Schröder and A. N. Whitehead . In 1879 Gottlob Frege develops the formal approach to logic and proposes the first logic language for logical equations. In
1900-461: Is temperature . The temperature of a gas is identical to mean molecular kinetic energy. Temperature in a solid is identical to mean maximal molecular kinetic energy because the molecules of a solid are more restricted in their movements. Temperature in a plasma is a mystery because the molecules of a plasma are torn apart. Therefore, temperature is multiply realized in a diversity of microphysical states. Jaegwon Kim has argued that disjunction —
1995-418: Is found throughout the works of Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account." Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system and the computation such that the "microphysical states [of the system] mirror the state transitions between
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#17327982411042090-444: Is highly improbable that this would happen, if not for constraints on the type of physical system that can realize mental phenomena. This, however, would not completely refute the possibility of realizability of mental states in radically different physical systems such as non- carbon based life forms or machines. Observations of multiple realizability — and of its relation to functionalism — predate their use in philosophy beginning in
2185-667: Is one example. The abacus was early used for arithmetic tasks. What we now call the Roman abacus was used in Babylonia as early as c. 2700 –2300 BC. Since then, many other forms of reckoning boards or tables have been invented. In a medieval European counting house , a checkered cloth would be placed on a table, and markers moved around on it according to certain rules, as an aid to calculating sums of money. Several analog computers were constructed in ancient and medieval times to perform astronomical calculations. These included
2280-728: Is possible. Putnam cites examples from the animal kingdom as evidence for the multiple realizability of mental states. Evolutionary biology — including evolutionary neuroscience — and comparative neuroanatomy and neurophysiology have demonstrated that mammals, reptiles, birds, amphibians, and mollusks have different brain structures. These animals can only share the same mental states and properties if these mental states and properties can be realized by different physical states in different species. Putnam concludes that type-identity and other reductive theories make an extremely "ambitious" and "highly implausible" conjecture that can be disproven with just one example of multiple realizability. On
2375-448: Is provably capable of computing anything that is computable by executing a program stored on tape, allowing the machine to be programmable. Von Neumann acknowledged that the central concept of the modern computer was due to this paper. Turing machines are to this day a central object of study in theory of computation . Except for the limitations imposed by their finite memory stores, modern computers are said to be Turing-complete , which
2470-491: Is termed computable , while the statement or calculation itself is referred to as a computation . Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements , and all statements written in modern computer programming languages. Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes
2565-567: Is to say, they have algorithm execution capability equivalent to a universal Turing machine . The era of modern computing began with a flurry of development before and during World War II. Most digital computers built in this period were built with electromechanical – electric switches drove mechanical relays to perform the calculation. These mechanical components had a low operating speed due to their mechanical nature and were eventually superseded by much faster all-electric components, originally using vacuum tubes and later transistors . The Z2
2660-465: Is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used." However, Leibniz did not incorporate a fully successful carry mechanism. Leibniz also described the binary numeral system , a central ingredient of all modern computers. However, up to the 1940s, many subsequent designs (including Charles Babbage 's machines of 1822 and even ENIAC of 1945) were based on
2755-479: The Analytical Engine in 1833. This concept, far more advanced than his Difference Engine, included an arithmetic logic unit , control flow through conditional branching and loops, and integrated memory. Babbage's plans made his Analytical Engine the first general-purpose design that could be described as Turing-complete in modern terms. The Analytical Engine was programmed using punched cards ,
2850-456: The Norden ( United States Army Air Forces ). The art of mechanical analog computing reached its zenith with the differential analyzer , built by H. L. Hazen and Vannevar Bush at MIT starting in 1927, which built on the mechanical integrators of James Thomson and the torque amplifiers invented by H. W. Nieman. A dozen of these devices were built before their obsolescence became obvious;
2945-575: The a priori argument . Some philosophers accept the thesis that mental states are multiply realizable but deny that multiple realizability gives rise to functionalism or other forms of non-reductive physicalism. Early objections to multiple realizability were limited to the narrow, "across structures-type" version. Starting with David Kellogg Lewis , many reductionists argued that it is very common in scientific practice to reduce one theory to another via local, structure-specific reductions. A frequently cited example of this sort of intertheoretic reduction
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3040-664: The astrolabe and Antikythera mechanism from the Hellenistic world (c. 150–100 BC). In Roman Egypt , Hero of Alexandria (c. 10–70 AD) made mechanical devices including automata and a programmable cart . The steam-powered automatic flute described by the Book of Ingenious Devices (850) by the Persian-Baghdadi Banū Mūsā brothers may have been the first programmable device. Other early mechanical devices used to perform one or another type of calculations include
3135-536: The brain or in a quantum computer . A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system. In the theory of computation , a diversity of mathematical models of computation has been developed. Typical mathematical models of computers are the following: Giunti calls the models studied by computation theory computational systems, and he argues that all of them are mathematical dynamical systems with discrete time and discrete state space. He maintains that
3230-444: The microprocessor , leading to another key breakthrough, the miniaturized personal computer (PC), in the 1970s. The cost of computers gradually became so low that personal computers by the 1990s, and then mobile computers ( smartphones and tablets ) in the 2000s, became ubiquitous. Devices have been used to aid computation for thousands of years, mostly using one-to-one correspondence with fingers . The earliest counting device
3325-515: The planisphere and other mechanical computing devices invented by Al-Biruni (c. AD 1000); the equatorium and universal latitude-independent astrolabe by Al-Zarqali (c. AD 1015); the astronomical analog computers of other medieval Muslim astronomers and engineers; and the astronomical clock tower of Su Song (1094) during the Song dynasty . The castle clock , a hydropowered mechanical astronomical clock invented by Ismail al-Jazari in 1206,
3420-556: The telephone exchange network into an electronic data processing system, using thousands of vacuum tubes . In the US, in 1940 Arthur Dickinson (IBM) invented the first digital electronic computer. This calculating device was fully electronic – control, calculations and output (the first electronic display). John Vincent Atanasoff and Clifford E. Berry of Iowa State University developed the Atanasoff–Berry Computer (ABC) in 1942,
3515-417: The " cryptologic bomb " ( Polish : "bomba kryptologiczna" ). In 1941, Zuse followed his earlier machine up with the Z3 , the world's first working electromechanical programmable , fully automatic digital computer. The Z3 was built with 2000 relays , implementing a 22- bit word length that operated at a clock frequency of about 5–10 Hz . Program code and data were stored on punched film . It
3610-413: The "inventor of the mechanical calculator" and the range of issues to be considered is discussed elsewhere. Gottfried Wilhelm von Leibniz invented the stepped reckoner and his famous stepped drum mechanism around 1672. He attempted to create a machine that could be used not only for addition and subtraction but would use a moveable carriage to enable multiplication and division. Leibniz once said "It
3705-517: The 1930s and working independently, American electronic engineer Claude Shannon and Soviet logician Victor Shestakov both showed a one-to-one correspondence between the concepts of Boolean logic and certain electrical circuits, now called logic gates , which are now ubiquitous in digital computers. They showed that electronic relays and switches can realize the expressions of Boolean algebra . This thesis essentially founded practical digital circuit design. In addition Shannon's paper gives
3800-686: The 1930s that could add, subtract, multiply and divide. In 1948, the Curta was introduced by Austrian inventor Curt Herzstark . It was a small, hand-cranked mechanical calculator and as such, a descendant of Gottfried Leibniz 's Stepped Reckoner and Thomas ' Arithmometer . The world's first all-electronic desktop calculator was the British Bell Punch ANITA , released in 1961. It used vacuum tubes , cold-cathode tubes and Dekatrons in its circuits, with 12 cold-cathode "Nixie" tubes for its display. The ANITA sold well since it
3895-653: The 1950s and 1960s, and later in some specialized applications. The principle of the modern computer was first described by computer scientist Alan Turing , who set out the idea in his seminal 1936 paper, On Computable Numbers . Turing reformulated Kurt Gödel 's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with the formal and simple hypothetical devices that became known as Turing machines . He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm . He went on to prove that there
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3990-835: The 1960s, Hilary Putnam used multiple realizability as an argument against type-identity theory . Specifically, Putnam noted that the multiple realizability of pain entails that, contrary to type-identity theory , pain is not identical to C-fibre firing. More generally, multiple realizability shows that psychological attributes are not the same as physical attributes. Psychological attributes, rather, are disjunctions of physical attributes. Fodor, Putnam, and others noted that, along with being an effective argument against type-identity theories, multiple realizability implies that any low-level explanation of higher-level mental phenomena would be insufficiently abstract and general. Jerry Fodor (1975) deployed multiple realizability more generally as an argument against any reductionist account of
4085-425: The 1960s. Alan Turing remarked on multiple realizability in 1950, for example, writing: "The fact that Babbage's Analytical Engine was to be entirely mechanical will help us rid ourselves of a superstition. Importance is often attached to the fact that modern digital computers are electrical, and the nervous system is also electrical. Since Babbage's machine was not electrical, and since all digital computers are in
4180-421: The 1970s. In 1804, French weaver Joseph Marie Jacquard developed a loom in which the pattern being woven was controlled by a paper tape constructed from punched cards . The paper tape could be changed without changing the mechanical design of the loom. This was a landmark achievement in programmability. His machine was an improvement over similar weaving looms. Punched cards were preceded by punch bands, as in
4275-484: The Analytical Engine, written in the 1840s, are now recognized as the earliest examples of computer programming. Lovelace saw potential in computers to go beyond numerical calculations, predicting that they might one day generate complex musical compositions or perform tasks like language processing. Though Babbage's designs were never fully realized due to technical and financial challenges, they influenced
4370-509: The Z3, but was not quite Turing-complete. The term digital was first suggested by George Robert Stibitz and refers to where a signal, such as a voltage, is not used to directly represent a value (as it would be in an analog computer ), but to encode it. In November 1937, Stibitz, then working at Bell Labs (1930–1941), completed a relay-based calculator he later dubbed the " Model K " (for " k itchen table", on which he had assembled it), which became
4465-400: The behavior or characteristics of another species (or organism of the same species; or in the generalized form, even the same organism). Sungsu Kim has recently responded to this objection using the distinction between homology of brain structures and homoplasy . Homologies are characteristics of physiology, morphology, behavior, or psychology shared by two or more species and inherited from
4560-501: The calculation of first, second, third and quarter degrees can be avoided. Guidobaldo is the first to document the use of gears for mechanical calculation. Wilhelm Schickard , a German polymath , designed a calculating machine in 1623 which combined a mechanized form of Napier's rods with the world's first mechanical adding machine built into the base. Because it made use of a single-tooth gear there were circumstances in which its carry mechanism would jam. A fire destroyed at least one of
4655-401: The computational states." Philosophers such as Jerry Fodor have suggested various accounts of computation with the restriction that semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of
4750-535: The concept in these terms: "Two systems are functionally isomorphic if there is a correspondence between the states of one and the states of the other that preserves functional relations." Two computers, for example, are functionally isomorphic if the sequential relations among states in one are exactly mirrored by those in the other. A computer made of electrical components and a computer made of cogs and wheels can be functionally isomorphic even though they are constitutionally different. Some philosophers refer to this as
4845-491: The continuously changeable aspects of physical phenomena such as electrical , mechanical , or hydraulic quantities to model the problem being solved, in contrast to digital computers that represented varying quantities symbolically, as their numerical values change. As an analog computer does not use discrete values, but rather continuous values, processes cannot be reliably repeated with exact equivalence, as they can with Turing machines . The first modern analog computer
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#17327982411044940-431: The contrary, it is likely that creatures that cannot be in identical physical states, due to their different composition and structure, can nevertheless be in identical psychological states. Some philosophers refer to this argument — that multiple realizability is much more likely than reductionism — as the likelihood argument . Putnam also formulates a complementary argument based on functional isomorphism . He defines
5035-520: The course of Allied bombing campaigns. Apparently his work remained largely unknown to engineers in the UK and US until much later, although at least IBM was aware of it as it financed his post-war startup company in 1946 in return for an option on Zuse's patents. In 1944, the Harvard Mark I was constructed at IBM's Endicott laboratories. It was a similar general purpose electro-mechanical computer to
5130-572: The decimal system. Around 1820, Charles Xavier Thomas de Colmar created what would over the rest of the century become the first successful, mass-produced mechanical calculator, the Thomas Arithmometer . It could be used to add and subtract, and with a moveable carriage the operator could also multiply, and divide by a process of long multiplication and long division. It utilised a stepped drum similar in conception to that invented by Leibniz. Mechanical calculators remained in use until
5225-414: The derivability of all terms in the theory to be reduced from terms in the reducing theory and the bridging laws, as the canonical theory of reduction. Given generalized multiple realizability, the physical science part of these psychophysical bridge laws will end up being a (possibly infinite) disjunction of all the terms referring to possible physical realizations of a mental kind. This disjunction cannot be
5320-478: The developments from early devices used for simple calculations to today's complex computers, encompassing advancements in both analog and digital technology. The first aids to computation were purely mechanical devices which required the operator to set up the initial values of an elementary arithmetic operation, then manipulate the device to obtain the result. In later stages, computing devices began representing numbers in continuous forms, such as by distance along
5415-407: The early 20th century. Torres Quevedo designed an electromechanical machine with floating-point arithmetic, while Bush's later work explored electronic digital computing. By the mid-20th century, these innovations paved the way for the first fully electronic computers. In the first half of the 20th century, analog computers were considered by many to be the future of computing. These devices used
5510-436: The evolution of computing hardware, as the era's rapid advancements in machinery and manufacturing laid the groundwork for mechanized and automated computing. Industrial needs for precise, large-scale calculations—especially in fields such as navigation, engineering, and finance—prompted innovations in both design and function, setting the stage for devices like Charles Babbage's Difference Engine (1822). This mechanical device
5605-543: The first binary adder . Typically signals have two states – low (usually representing 0) and high (usually representing 1), but sometimes three-valued logic is used, especially in high-density memory. Modern computers generally use binary logic , but many early machines were decimal computers . In these machines, the basic unit of data was the decimal digit, encoded in one of several schemes, including binary-coded decimal or BCD, bi-quinary , excess-3 , and two-out-of-five code . The mathematical basis of digital computing
5700-496: The first binary electronic digital calculating device. This design was semi-electronic (electro-mechanical control and electronic calculations), and used about 300 vacuum tubes, with capacitors fixed in a mechanically rotating drum for memory. However, its paper card writer/reader was unreliable and the regenerative drum contact system was mechanical. The machine's special-purpose nature and lack of changeable, stored program distinguish it from modern computers. Computers whose logic
5795-421: The flight times of the shells. Various spotters on board the ship would relay distance measures and observations to a central plotting station. There the fire direction teams fed in the location, speed and direction of the ship and its target, as well as various adjustments for Coriolis effect , weather effects on the air, and other adjustments; the computer would then output a firing solution, which would be fed to
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#17327982411045890-399: The grounds that it violates the causal closure of the physical , which assumes that physics provides a full explanation of physical events. If mental properties are causally efficacious, they must either be identical to physical properties or there must be widespread overdetermination. The latter is often held to be either unlikely or even impossible on conceptual grounds. If Kim is right, then
5985-400: The halting problem and the busy beaver game . It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. Some examples of mathematical statements that are computable include: Some examples of mathematical statements that are not computable include: Computation can be seen as
6080-407: The idea that the physical realization of a particular mental state is not a particular physical state but the disjunction of the physical states that realize that mental state — creates problems for multiple realizability. Putnam also argued against this "disjunctive" possibility in earlier work. Block and Fodor had also argued against it. Jaegwon Kim has argued against non-reductive physicalism on
6175-544: The machine proposed by Basile Bouchon . These bands would inspire information recording for automatic pianos and more recently numerical control machine tools. In the late 1880s, the American Herman Hollerith invented data storage on punched cards that could then be read by a machine. To process these punched cards, he invented the tabulator and the keypunch machine. His machines used electromechanical relays and counters . Hollerith's method
6270-611: The machines in 1624 and it is believed Schickard was too disheartened to build another. In 1642, while still a teenager, Blaise Pascal started some pioneering work on calculating machines and after three years of effort and 50 prototypes he invented a mechanical calculator . He built twenty of these machines (called Pascal's calculator or Pascaline) in the following ten years. Nine Pascalines have survived, most of which are on display in European museums. A continuing debate exists over whether Schickard or Pascal should be regarded as
6365-528: The main (non-volatile) storage medium. Engineer Tommy Flowers joined the telecommunications branch of the General Post Office in 1926. While working at the research station in Dollis Hill in the 1930s, he began to explore the possible use of electronics for the telephone exchange . Experimental equipment that he built in 1934 went into operation 5 years later, converting a portion of
6460-411: The mapping account of pancomputationalism , the idea that everything can be said to be computing everything. Gualtiero Piccinini proposes an account of computation based on mechanical philosophy . It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or the manipulation (by a functional mechanism) of a "medium-independent" vehicle according to
6555-400: The mental state of "being in pain." Mental states have been claimed to be multiply realizable not only across species and between individuals but also within individuals. At different times, the same individual may realize the same mental states in physically different forms. Neural plasticity — the fact that areas of the brain can assume the functions of other parts that have been damaged as
6650-541: The modern slide rule in 1632, essentially a combination of two Gunter rules , held together with the hands. Slide rules were used by generations of engineers and other mathematically involved professional workers, until the invention of the pocket calculator . In 1609 Guidobaldo del Monte made a mechanical multiplier to calculate fractions of a degree. Based on a system of four gears, the rotation of an index on one quadrant corresponds to 60 rotations of another index on an opposite quadrant. Thanks to this machine, errors in
6745-471: The most powerful was constructed at the University of Pennsylvania 's Moore School of Electrical Engineering , where the ENIAC was built. A fully electronic analog computer was built by Helmut Hölzer in 1942 at Peenemünde Army Research Center . By the 1950s the success of digital electronic computers had spelled the end for most analog computing machines, but hybrid analog computers , controlled by digital electronics, remained in substantial use into
6840-483: The multiplication and division of numbers could be performed by the addition and subtraction, respectively, of the logarithms of those numbers. While producing the first logarithmic tables, Napier needed to perform many tedious multiplications. It was at this point that he designed his ' Napier's bones ', an abacus-like device that greatly simplified calculations that involved multiplication and division. Since real numbers can be represented as distances or intervals on
6935-475: The neurosciences requires that some universal consistencies in brain structures must either exist or be assumed to exist. The similarity (produced by homology or convergent evolution) of brain structures allows us to generalize across species. If multiple realizability (especially the generalized form) were an empirical fact, then results from experiments conducted on one species of animal (or one organism) would not be meaningful or useful when generalized to explain
7030-405: The options seem to be either reduction or elimination. One criticism of multiple realizability is that any theory that attempts to address the possibility of generalized multiple realizability must necessarily be so local and context-specific in nature (referring exclusively to a certain token system of a certain structure-type at a certain time) that its reductions would be incompatible with even
7125-487: The philosophy of mind posits that the same mental state can be realized by different physical states; another way of putting it is that there is a many-to-one mapping from physical states to mental states. Multiple realizability in general is not restricted to the multiple realizability of mental states. Many kinds of things can be realized by numerous physical devices. A wide variety of physical devices can serve as corkscrews, for example. Mental states can also be realized in
7220-500: The relation between higher-level sciences and physics. Fodor also uses multiple realizability to argue against reductionism not only of psychology but of any special sciences (that is, any sciences that are "higher level" than physics). In his characterization of reductionism, all mental kind predicates in an ideal and completed psychology must correspond with physical kind predicates in an ideal and completed physics. He suggests taking Ernest Nagel 's theory of reduction, which insists on
7315-737: The result of traumatic injury, pathology, natural biological development, or other processes — has long been considered to be an example. But so are more mundane facts about neurophysiology, such as the fact that neurons die and connections between them are rewired. The argument that neural plasticity supports multiple realizability has also been contested. Gualtiero Piccinini differentiates three related properties: variable realizability, multiple realizability, and medium independence. Multiple realizability has been used as an argument against type-identity theory, against reductionist theories of mind in general, for functionalist theories of mind, and even against functionalist theories of mind. Starting in
7410-566: The thesis of multiple realizability would be to examine brain structures and determine whether some homoplasious "psychological processes or functions might be 'constructed' from different material" and supported by different brain structures just as the flight capacities of bats and birds emerge from different morphophysiologies. The emergence of similar behavioral outputs or psychological functions brought about by similar or identical brain structures in convergent evolutionary lineages would provide some evidence against multiple realizability, since it
7505-614: The turrets for laying. In 1912, British engineer Arthur Pollen developed the first electrically powered mechanical analogue computer (called at the time the Argo Clock). It was used by the Imperial Russian Navy in World War I . The alternative Dreyer Table fire control system was fitted to British capital ships by mid-1916. Mechanical devices were also used to aid the accuracy of aerial bombing . Drift Sight
7600-508: Was a tide-predicting machine , invented by Sir William Thomson , later Lord Kelvin, in 1872. It used a system of pulleys and wires to automatically calculate predicted tide levels for a set period at a particular location and was of great utility to navigation in shallow waters. His device was the foundation for further developments in analog computing. The differential analyser , a mechanical analog computer designed to solve differential equations by integration using wheel-and-disc mechanisms,
7695-572: Was a job title assigned to primarily women who used these calculators to perform mathematical calculations. By the 1920s, British scientist Lewis Fry Richardson 's interest in weather prediction led him to propose human computers and numerical analysis to model the weather; to this day, the most powerful computers on Earth are needed to adequately model its weather using the Navier–Stokes equations . Companies like Friden , Marchant Calculator and Monroe made desktop mechanical calculators from
7790-419: Was able to compute the roots of arbitrary polynomials of order eight, including the complex ones, with a precision down to thousandths. An important advance in analog computing was the development of the first fire-control systems for long range ship gunlaying . When gunnery ranges increased dramatically in the late 19th century it was no longer a simple matter of calculating the proper aim point, given
7885-476: Was conceptualized in 1876 by James Thomson , the brother of the more famous Lord Kelvin. He explored the possible construction of such calculators, but was stymied by the limited output torque of the ball-and-disk integrators . In a differential analyzer, the output of one integrator drove the input of the next integrator, or a graphing output. A notable series of analog calculating machines were developed by Leonardo Torres Quevedo since 1895, including one that
7980-607: Was created in 1939 at the UK Government Code and Cypher School (GC&CS) at Bletchley Park by Alan Turing , with an important refinement devised in 1940 by Gordon Welchman . The engineering design and construction was the work of Harold Keen of the British Tabulating Machine Company . It was a substantial development from a device that had been designed in 1938 by Polish Cipher Bureau cryptologist Marian Rejewski , and known as
8075-461: Was intended to automate the calculation of polynomial functions and represented one of the earliest applications of computational logic. Babbage, often regarded as the "father of the computer," envisioned a fully mechanical system of gears and wheels, powered by steam, capable of handling complex calculations that previously required intensive manual labor. His Difference Engine, designed to aid navigational calculations, ultimately led him to conceive
8170-507: Was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is undecidable : in general, it is not possible to decide algorithmically whether a given Turing machine will ever halt. He also introduced the notion of a "universal machine" (now known as a universal Turing machine ), with the idea that such a machine could perform the tasks of any other machine, or in other words, it
8265-636: Was one of the earliest examples of an electric operated digital computer built with electromechanical relays and was created by civil engineer Konrad Zuse in 1940 in Germany. It was an improvement on his earlier, mechanical Z1 ; although it used the same mechanical memory , it replaced the arithmetic and control logic with electrical relay circuits. In the same year, electro-mechanical devices called bombes were built by British cryptologists to help decipher German Enigma-machine -encrypted secret messages during World War II . The bombe's initial design
8360-622: Was primarily built using vacuum tubes are now known as first generation computers . Multiple realizability In the philosophy of mind , multiple realizability is the thesis that the same mental property, state, or event can be implemented by different physical properties, states, or events. Philosophers of mind have used multiple realizability to argue that mental states are not the same as — and cannot be reduced to — physical states. They have also used it to defend or criticize many versions of functionalism , especially machine-state functionalism . The multiple realizability thesis in
8455-586: Was probably a form of tally stick . The Lebombo bone from the mountains between Eswatini and South Africa may be the oldest known mathematical artifact. It dates from 35,000 BCE and consists of 29 distinct notches that were deliberately cut into a baboon 's fibula . Later record keeping aids throughout the Fertile Crescent included calculi (clay spheres, cones, etc.) which represented counts of items, probably livestock or grains, sealed in hollow unbaked clay containers. The use of counting rods
8550-579: Was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician Alan Turing , who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine . Other (mathematically equivalent) definitions include Alonzo Church 's lambda-definability , Herbrand - Gödel - Kleene 's general recursiveness and Emil Post 's 1-definability . Today, any formal statement or calculation that exhibits this quality of well-definedness
8645-467: Was quite similar to modern machines in some respects, pioneering numerous advances such as floating-point numbers . Replacement of the hard-to-implement decimal system (used in Charles Babbage 's earlier design) by the simpler binary system meant that Zuse's machines were easier to build and potentially more reliable, given the technologies available at that time. The Z3 was proven to have been
8740-496: Was the first programmable analog computer. Ramon Llull invented the Lullian Circle: a notional machine for calculating answers to philosophical questions (in this case, to do with Christianity) via logical combinatorics. This idea was taken up by Leibniz centuries later, and is thus one of the founding elements in computing and information science . Scottish mathematician and physicist John Napier discovered that
8835-553: Was the first such aid, developed by Harry Wimperis in 1916 for the Royal Naval Air Service ; it measured the wind speed from the air, and used that measurement to calculate the wind's effects on the trajectory of the bombs. The system was later improved with the Course Setting Bomb Sight , and reached a climax with World War II bomb sights, Mark XIV bomb sight ( RAF Bomber Command ) and
8930-471: Was the only electronic desktop calculator available, and was silent and quick. The tube technology was superseded in June 1963 by the U.S. manufactured Friden EC-130, which had an all-transistor design, a stack of four 13-digit numbers displayed on a 5-inch (13 cm) CRT , and introduced reverse Polish notation (RPN). The Industrial Revolution (late 18th to early 19th century) had a significant impact on
9025-1216: Was used in the 1890 United States Census . That census was processed two years faster than the prior census had been. Hollerith's company eventually became the core of IBM . By 1920, electromechanical tabulating machines could add, subtract, and print accumulated totals. Machine functions were directed by inserting dozens of wire jumpers into removable control panels . When the United States instituted Social Security in 1935, IBM punched-card systems were used to process records of 26 million workers. Punched cards became ubiquitous in industry and government for accounting and administration. Leslie Comrie 's articles on punched-card methods and W. J. Eckert 's publication of Punched Card Methods in Scientific Computation in 1940, described punched-card techniques sufficiently advanced to solve some differential equations or perform multiplication and division using floating-point representations, all on punched cards and unit record machines . Such machines were used during World War II for cryptographic statistical processing, as well as
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