Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every integer greater than 2 can be expressed as the sum of two primes; the conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. A Goldbach number is a positive integer that can be expressed as the sum of two odd primes. Since four is the only number greater than two that requires the prime 2 in order to be written as the sum of two primes, another form of the statement of Goldbach's conjecture is that all integers greater than 4 are Goldbach numbers; the expression of a given number as a sum of two primes is called a Goldbach partition of that number. The following are examples of Goldbach partitions for some numbers: 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7 = 5 + 5 12 = 7 + 5... 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53... The number of ways in which 2n can be written as the sum of two primes is: 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3....
On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler in which he proposed the following conjecture: Every integer which can be written as the sum of two primes, can be written as the sum of as many primes as one wishes, until all terms are units. He proposed a second conjecture in the margin of his letter: Every integer greater than 2 can be written as the sum of three primes, he considered 1 to be a prime number, a convention subsequently abandoned. The two conjectures are now known to be equivalent, but this did not seem to be an issue at the time. A modern version of Goldbach's marginal conjecture is: Every integer greater than 5 can be written as the sum of three primes. Euler replied in a letter dated 30 June 1742, reminded Goldbach of an earlier conversation they had, in which Goldbach remarked his original conjecture followed from the following statement Every integer greater than 2 can be written as the sum of two primes,which is, thus a conjecture of Goldbach.
In the letter dated 30 June 1742, Euler stated: "Dass … ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann." Goldbach's third version is the form in which the conjecture is expressed today. It is known as the "strong", "even", or "binary" Goldbach conjecture, to distinguish it from a weaker conjecture, known today variously as the Goldbach's weak conjecture, the "odd" Goldbach conjecture, or the "ternary" Goldbach conjecture; this weak conjecture asserts that all odd numbers greater than 7 are the sum of three odd primes, appears to have been proved in 2013. The weak conjecture is a corollary of the strong conjecture, as, if n – 3 is a sum of two primes n is a sum of three primes; the converse implication, the strong Goldbach conjecture remain unproven. For small values of n, the strong Goldbach conjecture can be verified directly. For instance, Nils Pipping in 1938 laboriously verified the conjecture up to n ≤ 105.
With the advent of computers, many more values of n have been checked. One record from this search is that 3,325,581,707,333,960,528 is the smallest number that has no Goldbach partition with a prime below 9781. Statistical considerations that focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, the more "likely" it becomes that at least one of these representations consists of primes. A crude version of the heuristic probabilistic argument is as follows; the prime number theorem asserts that an integer m selected at random has a 1 / ln m chance of being prime. Thus if n is a large integer and m is a number between 3 and n/2 one might expect the probability of m and n − m being prime to be 1 /. If one pursues this heuristic, one might expect the total number of ways to write a large integer n as the sum of two odd primes to be ∑ m = 3 n / 2 1 ln m 1 ln ≈ n 2 2.
Since this quantity goes to infinity as n increases, we expect that every large integer
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, applies to concepts of chance and information entropy; the fields of mathematics and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space; this association facilitates the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions.
These and other constructs are useful in probability theory and the various applications of randomness. Randomness is most used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input, are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators. Random selection, when narrowly associated with a simple random sample, is a method of selecting items from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen.
That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen we can say the selection process is random. In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, this evolved into games of chance. Most ancient cultures used various methods of divination to attempt to circumvent randomness and fate; the Chinese of 3000 years ago were the earliest people to formalize odds and chance. The Greek philosophers discussed randomness at length, but only in non-quantitative forms, it was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance John Venn wrote a chapter on The conception of randomness that included his view of the randomness of the digits of the number pi by using them to construct a random walk in two dimensions.
The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid- to late-20th century, ideas of algorithmic information theory introduced new dimensions to the field via the concept of algorithmic randomness. Although randomness had been viewed as an obstacle and a nuisance for many centuries, in the 20th century computer scientists began to realize that the deliberate introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases such randomized algorithms outperform the best deterministic methods. Many scientific fields are concerned with randomness: In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics to explain phenomena in thermodynamics and the properties of gases. According to several standard interpretations of quantum mechanics, microscopic phenomena are objectively random.
That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case; the modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection. The latter retains some random mutations in the gene pool due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them.
Several authors claim that evolution and sometimes development require a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities; the characteristics of an organism arise to some extent deterministically and to som
Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, loosely speaking, 0 indicates impossibility and 1 indicates certainty; the higher the probability of an event, the more it is that the event will occur. A simple example is the tossing of a fair coin. Since the coin is fair, the two outcomes are both probable; these concepts have been given an axiomatic mathematical formalization in probability theory, used in such areas of study as mathematics, finance, science, artificial intelligence/machine learning, computer science, game theory, philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is used to describe the underlying mechanics and regularities of complex systems; when dealing with experiments that are random and well-defined in a purely theoretical setting, probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes.
For example, tossing a fair coin twice will yield "head-head", "head-tail", "tail-head", "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents possess different views about the fundamental nature of probability: Objectivists assign numbers to describe some objective or physical state of affairs; the most popular version of objective probability is frequentist probability, which claims that the probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. This interpretation considers probability to be the relative frequency "in the long run" of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome if it is performed only once.
Subjectivists assign numbers per subjective probability. The degree of belief has been interpreted as, "the price at which you would buy or sell a bet that pays 1 unit of utility if E, 0 if not E." The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some prior probability distribution; these data are incorporated in a likelihood function. The product of the prior and the likelihood, results in a posterior probability distribution that incorporates all the information known to date. By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions regardless of how much information the agents share; the word probability derives from the Latin probabilitas, which can mean "probity", a measure of the authority of a witness in a legal case in Europe, correlated with the witness's nobility.
In a sense, this differs much from the modern meaning of probability, which, in contrast, is a measure of the weight of empirical evidence, is arrived at from inductive reasoning and statistical inference. The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers. According to Richard Jeffrey, "Before the middle of the seventeenth century, the term'probable' meant approvable, was applied in that sense, unequivocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially,'probable' could apply to propositions for which there was good evidence.
The sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal. Christiaan Huygens gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi and Abraham de Moivre's Doctrine of Chances treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture for histories of the early development of the concept of mathematical probability; the theory of errors may be traced back to Roger Cotes's Opera Miscellanea, but a memoir prepared by Thomas Simpson in 1755 first applied the theory to the discussion of errors of observation. The reprint of this memoir lays down the axioms that positive and negative errors are probable, that certain assignable limits define the range of all errors.
Simpson discusses c
In algorithmic information theory, the Kolmogorov complexity of an object, such as a piece of text, is the length of the shortest computer program that produces the object as output. It is a measure of the computational resources needed to specify the object, is known as descriptive complexity, Kolmogorov–Chaitin complexity, algorithmic complexity, algorithmic entropy, or program-size complexity, it is named after Andrey Kolmogorov, who first published on the subject in 1963. The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, Turing's halting problem. In particular, for all objects, it is not possible to compute a lower bound for its Kolmogorov complexity, let alone its exact value. Consider the following two strings of 32 lowercase letters and digits. Example 1: abababababababababababababababab Example 2: 4c1j5b2p0cv4w1x8rx2y39umgw5q85s7The first string has a short English-language description, namely "ab 16 times", which consists of 11 characters.
The second one has no obvious simple description other than writing down the string itself, which has 32 characters. More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language, it can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings like the abab example above, whose Kolmogorov complexity is small relative to the string's size, are not considered to be complex; the Kolmogorov complexity can be defined for any mathematical object, but for simplicity the scope of this article is restricted to strings. We must first specify a description language for strings; such a description language can be based on any computer programming language, such as Lisp, Pascal, or Java virtual machine bytecode. If P is a program which outputs a string x P is a description of x; the length of the description is just the length of P as a character string, multiplied by the number of bits in a character.
We could, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring <M>. If M is a Turing Machine which, on input w, outputs string x the concatenated string <M> w is a description of x. For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is preferred in the research literature. In this article, an informal approach is discussed. Any string s has at least one description. For example, the second string above is output by the program: function GenerateExample2String return "4c1j5b2p0cv4w1x8rx2y39umgw5q85s7" where the first string is output by the pseudo-code: function GenerateExample1String return "ab" * 16 If a description d of a string s is of minimal length, it is called a minimal description of s. Thus, the length of d is the Kolmogorov complexity of s, written K. Symbolically, K = |d|; the length of the shortest description will depend on the choice of description language. There are some description languages which are optimal, in the following sense: given any description of an object in a description language, said description may be used in the optimal description language with a constant overhead.
The constant depends only on the languages involved, not on the description of the object, nor the object being described. Here is an example of an optimal description language. A description will have two parts: The first part describes another description language; the second part is a description of the object in that language. In more technical terms, the first part of a description is a computer program, with the second part being the input to that computer program which produces the object as output; the invariance theorem follows: Given any description language L, the optimal description language is at least as efficient as L, with some constant overhead. Proof: Any description D in L can be converted into a description in the optimal language by first describing L as a computer program P, using the original description D as input to that program; the total length of this new description D′ is: |D′| = |P| + |D|The length of P is a constant that doesn't depend on D. So, there is at most a constant overhead, regardless of the object described.
Therefore, the optimal language is universal up to this additive constant. Theorem: If K1 and K2 are the complexity functions relative to Turing complete description languages L1 and L2 there is a constant c – which depends only on the languages L1 and L2 chosen – such that ∀s. −c ≤ K1 − K2 ≤ c. Proof: By symmetry, it suffices to prove that there is some constant c such that for all strings s K1 ≤ K2 + c. Now, suppose there is a program in the language L1 which acts as an interpreter for L2: function InterpretLanguage where p is a program in L2; the interpreter is characterized by the following property: Running InterpretLanguage on input p returns the result of running p. Thus, if P is a program in L2, a minimal description of s InterpretLanguage returns the string s; the length of this description of s is the sum of The length of the program InterpretLanguage, which we can take to be the constant c. The length of P which
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 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, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a
Jürgen Schmidhuber is a computer scientist most noted for his work in the field of artificial intelligence, deep learning and artificial neural networks. He is a co-director of the Dalle Molle Institute for Artificial Intelligence Research in Manno, in the district of Lugano, in Ticino in southern Switzerland, he is sometimes called the "father of AI" or "father of deep learning."Schmidhuber did his undergraduate studies at the Technische Universität München in Munich, Germany. He taught there from 2004 until 2009 when he became a professor of artificial intelligence at the Università della Svizzera Italiana in Lugano, Switzerland. In 1997, Schmidhuber and Sepp Hochreiter published a paper on a type of recurrent neural network which they called Long short-term memory or LSTM. In 2015, LSTM was used in a new implementation of speech recognition in Google's software for smartphones. Google used LSTM for the smart assistant Allo and for Google Translate. Apple used LSTM for Siri. Amazon used LSTM for Amazon Alexa.
In 2017, Facebook performed some 4.5 billion automatic translations every day using LSTM networks. Bloomberg Business Week wrote: "These powers make LSTM arguably the most commercial AI achievement, used for everything from predicting diseases to composing music."In 2011, Schmidhuber's team at IDSIA with his postdoc Dan Ciresan achieved dramatic speedups of convolutional neural networks on fast parallel computers called GPUs. An earlier CNN on GPU by Chellapilla et al. was 4 times faster than an equivalent implementation on CPU. The deep CNN of Dan Ciresan et al. at IDSIA was 60 times faster and achieved the first superhuman performance in a computer vision contest in August 2011. Between May 15, 2011 and September 10, 2012, their fast and deep CNNs won no less than four image competitions, they significantly improved on the best performance in the literature for multiple image databases. The approach has become central to the field of computer vision, it is based on CNN designs introduced much earlier by Yann LeCun et al. who applied the backpropagation algorithm to a variant of Kunihiko Fukushima's original CNN architecture called neocognitron modified by J. Weng's method called max-pooling.
In 2014, Schmidhuber formed a company, Nnaisense, to work on commercial applications of artificial intelligence in fields such as finance, heavy industry and self-driving cars. Sepp Hochreiter, Jaan Tallinn, Marcus Hutter are advisers to the company. Sales were under 11 million USD in 2016. Nnaisense raised its first round of capital funding in January 2017. Schmidhuber's overall goal is to create an all-purpose AI by training a single AI in sequence on a variety of narrow tasks. According to The Guardian, Schmidhuber complained in a "scathing 2015 article" that fellow deep learning researchers Geoffrey Hinton, Yann LeCun and Yoshua Bengio "heavily cite each other," but "fail to credit the pioneers of the field,” understating the contributions of Schmidhuber and other early machine learning pioneers including Alexey Grigorevich Ivakhnenko who published the first deep learning networks in 1965. LeCun denies the charge, stating instead that Schmidhuber "keeps claiming credit he doesn't deserve".
Schmidhuber received the Helmholtz Award of the International Neural Network Society in 2013, the Neural Networks Pioneer Award of the IEEE Computational Intelligence Society in 2016. He is a member of the European Academy of Sciences and Arts
Computer science is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems, its fields can be divided into practical disciplines. Computational complexity theory is abstract, while computer graphics emphasizes real-world applications. Programming language theory considers approaches to the description of computational processes, while computer programming itself involves the use of programming languages and complex systems. Human–computer interaction considers the challenges in making computers useful and accessible; the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, aiding in computations such as multiplication and division.
Algorithms for performing computations have existed since antiquity before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623. In 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner, he may be considered the first computer scientist and information theorist, among other reasons, documenting the binary number system. In 1820, Thomas de Colmar launched the mechanical calculator industry when he released his simplified arithmometer, the first calculating machine strong enough and reliable enough to be used daily in an office environment. Charles Babbage started the design of the first automatic mechanical calculator, his Difference Engine, in 1822, which gave him the idea of the first programmable mechanical calculator, his Analytical Engine, he started developing this machine in 1834, "in less than two years, he had sketched out many of the salient features of the modern computer".
"A crucial step was the adoption of a punched card system derived from the Jacquard loom" making it infinitely programmable. In 1843, during the translation of a French article on the Analytical Engine, Ada Lovelace wrote, in one of the many notes she included, an algorithm to compute the Bernoulli numbers, considered to be the first computer program. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information. In 1937, one hundred years after Babbage's impossible dream, Howard Aiken convinced IBM, making all kinds of punched card equipment and was in the calculator business to develop his giant programmable calculator, the ASCC/Harvard Mark I, based on Babbage's Analytical Engine, which itself used cards and a central computing unit; when the machine was finished, some hailed it as "Babbage's dream come true". During the 1940s, as new and more powerful computing machines were developed, the term computer came to refer to the machines rather than their human predecessors.
As it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. In 1945, IBM founded the Watson Scientific Computing Laboratory at Columbia University in New York City; the renovated fraternity house on Manhattan's West Side was IBM's first laboratory devoted to pure science. The lab is the forerunner of IBM's Research Division, which today operates research facilities around the world; the close relationship between IBM and the university was instrumental in the emergence of a new scientific discipline, with Columbia offering one of the first academic-credit courses in computer science in 1946. Computer science began to be established as a distinct academic discipline in the 1950s and early 1960s; the world's first computer science degree program, the Cambridge Diploma in Computer Science, began at the University of Cambridge Computer Laboratory in 1953. The first computer science degree program in the United States was formed at Purdue University in 1962.
Since practical computers became available, many applications of computing have become distinct areas of study in their own rights. Although many believed it was impossible that computers themselves could be a scientific field of study, in the late fifties it became accepted among the greater academic population, it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM 704 and the IBM 709 computers, which were used during the exploration period of such devices. "Still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, you would have to start the whole process over again". During the late 1950s, the computer science discipline was much in its developmental stages, such issues were commonplace. Time has seen significant improvements in the effectiveness of computing technology. Modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base.
Computers were quite costly, some degree of humanitarian aid was needed for efficient use—in part from professional computer operators. As computer adoption became more widespread and affordable, less human assistance was needed for common usage. Despite its short history as a formal academic discipline, computer science has made a number of fundamental contributions to science and society—in fact, along with electronics, it is