In mathematics and computer science, an algorithm is a finite sequence of well-defined instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. By making use of artificial intelligence, algorithms can perform automated deductions and use mathematical and logical tests to divert the code through various routes. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. The thesis is named after American mathematician Alonzo Church and the British mathematician Alan Turing. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the 1930s, several independent attempts were made to formalize the notion of computability:
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an infinity of infinities. He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.
In the philosophy of mathematics, intuitionism, or neointuitionism, is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
0 (zero) is a number, and the numerical digit used to represent that number in numerals. It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems. Names for the number 0 in English include zero, nought (UK), naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter "O"—oh or o. Informal or slang terms for zero include zilch and zip. Ought and aught, as well as cipher, have also been used historically.
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. As a result, potential infinity is often formalized using the concept of limit.
In the philosophy of mathematics, ultrafinitism is a form of finitism and intuitionism. There are various philosophies of mathematics that are called ultrafinitism. A major identifying property common among most of these philosophies is their objections to totality of number theoretic functions like exponentiation over natural numbers.
Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects are accepted as legitimate.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
Silvio Micali is an Italian computer scientist, professor at the Massachusetts Institute of Technology and the founder of Algorand. Micali's research centers on cryptography and information security.
Jeffrey David Ullman is an American computer scientist and the Stanford W. Ascherman Professor of Engineering, Emeritus, at Stanford University. His textbooks on compilers, theory of computation, data structures, and databases are regarded as standards in their fields. He and his long-time collaborator Alfred Aho are the recipients of the 2020 Turing Award, generally recognized as the highest distinction in computer science.
Leslie Gabriel Valiant is a British American computer scientist and computational theorist. He is currently the T. Jefferson Coolidge Professor of Computer Science and Applied Mathematics at Harvard University. Valiant was awarded the Turing Award in 2010, having been described by the A.C.M. as a heroic figure in theoretical computer science and a role model for his courage and creativity in addressing some of the deepest unsolved problems in science; in particular for his "striking combination of depth and breadth".
The history of computer science began long before our modern discipline of computer science, usually appearing in forms like mathematics or physics. Developments in previous centuries alluded to the discipline that we now know as computer science. This progression, from mechanical inventions and mathematical theories towards modern computer concepts and machines, led to the development of a major academic field, massive technological advancement across the Western world, and the basis of a massive worldwide trade and culture.
Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, with an Application to the Entscheidungsproblem." It was the second proof of the conjecture that some purely mathematical yes–no questions can never be answered by computation; more technically, that some decision problems are "undecidable" in the sense that there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In Turing's own words: "...what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]...".
Sir Adrian Frederick Melhuish Smith, PRS is a British statistician who is Chief Executive of the Alan Turing Institute and President of the Royal Society.
Patrick Colonel Suppes was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology and educational technology. He was the Lucie Stern Professor of Philosophy Emeritus at Stanford University and until January 2010 was the Director of the Education Program for Gifted Youth also at Stanford.
Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to number theory, abstract algebra, and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as Logicism.
Our Mathematical Universe: My Quest for the Ultimate Nature of Reality is a 2014 nonfiction book by the Swedish-American cosmologist Max Tegmark. Written in popular science format, the book interweaves what a New York Times reviewer called "an informative survey of exciting recent developments in astrophysics and quantum theory" with Tegmark's mathematical universe hypothesis, which posits that reality is a mathematical structure. This mathematical nature of the universe, Tegmark argues, has important consequences for the way researchers should approach many questions of physics.
Joel David Hamkins is an American mathematician and philosopher who is O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame. He has made contributions in mathematical and philosophical logic, set theory and philosophy of set theory, in computability theory, and in group theory.
