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🔗 Biography 🔗 Mathematics 🔗 Physics 🔗 Telecommunications 🔗 Biography/science and academia 🔗 Energy 🔗 Electrical engineering 🔗 Physics/Biographies 🔗 Devon

Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of telecommunications, mathematics, and science.

🔗 Mathematics

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0. Laurence Kirby and Jeff Paris showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε₀-induction in Peano arithmetic. The Paris–Harrington theorem gave another example.

Kirby and Paris introduced a graph-theoretic hydra game with behavior similar to that of Goodstein sequences: the "Hydra" (named for the mythological multi-headed Hydra of Lerna) is a rooted tree, and a move consists of cutting off one of its "heads" (a branch of the tree), to which the hydra responds by growing a finite number of new heads according to certain rules. Kirby and Paris proved that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very long time. Just like for Goodstein sequences, Kirby and Paris showed that it cannot be proven in Peano arithmetic alone.

🔗 Mathematics 🔗 Philosophy 🔗 Philosophy/Philosophical literature 🔗 Books 🔗 Philosophy/Logic 🔗 Philosophy/Contemporary philosophy 🔗 Linguistics 🔗 Philosophy/Philosophy of language 🔗 Linguistics/Philosophy of language 🔗 Philosophy/Analytic philosophy

The Tractatus Logico-Philosophicus (widely abbreviated and cited as TLP) is the only book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein that was published during his lifetime. The project had a broad goal: to identify the relationship between language and reality and to define the limits of science. Wittgenstein wrote the notes for the Tractatus while he was a soldier during World War I and completed it during a military leave in the summer of 1918. It was originally published in German in 1921 as Logisch-Philosophische Abhandlung (Logical-Philosophical Treatise). In 1922 it was published together with an English translation and a Latin title, which was suggested by G. E. Moore as homage to Baruch Spinoza's Tractatus Theologico-Politicus (1670).

The Tractatus is written in an austere and succinct literary style, containing almost no arguments as such, but consists of altogether 525 declarative statements, which are hierarchically numbered.

The Tractatus is recognized by philosophers as one of the most significant philosophical works of the twentieth century and was influential chiefly amongst the logical positivist philosophers of the Vienna Circle, such as Rudolf Carnap and Friedrich Waismann and Bertrand Russell's article "The Philosophy of Logical Atomism".

Wittgenstein's later works, notably the posthumously published Philosophical Investigations, criticised many of his ideas in the Tractatus. There are, however, elements to see a common thread in Wittgenstein's thinking, in spite of those criticisms of the Tractatus in later writings. Indeed, the legendary contrast between ‘early’ and ‘late’ Wittgenstein has been countered by such scholars as Pears (1987) and Hilmy (1987). For example, a relevant, yet neglected aspect of continuity in Wittgenstein’s central issues concerns ‘meaning’ as ‘use’. Connecting his early and later writings on ‘meaning as use’ is his appeal to direct consequences of a term or phrase, reflected e.g. in his speaking of language as a ‘calculus’. These passages are rather crucial to Wittgenstein’s view of ‘meaning as use’, though they have been widely neglected in scholarly literature. The centrality and importance of these passages are corroborated and augmented by renewed examination of Wittgenstein’s Nachlaß, as is done in "From Tractatus to Later Writings and Back - New Implications from the Nachlass" (de Queiroz 2023).

🔗 Mathematics

The gömböc (Hungarian: [ˈɡømbøt͡s]) is a convex three-dimensional homogeneous body that when resting on a flat surface has just one stable and one unstable point of equilibrium. Its existence was conjectured by the Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by the Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all with a very strict shape tolerance (about 0.1 mm per 100 mm).

The most famous solution, capitalized as Gömböc to distinguish it from the generic gömböc, has a sharpened top, as shown in the photo. Its shape helped to explain the body structure of some tortoises in relation to their ability to return to equilibrium position after being placed upside down. Copies of the gömböc have been donated to institutions and museums, and the largest one was presented at the World Expo 2010 in Shanghai in China. In December 2017, a 4.5 m gömböc statue was installed in the Corvin Quarter in Budapest.

🔗 Mathematics 🔗 Economics 🔗 Game theory

A Vickrey auction or sealed-bid second-price auction (SBSPA) is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. This type of auction is strategically similar to an English auction and gives bidders an incentive to bid their true value. The auction was first described academically by Columbia University professor William Vickrey in 1961 though it had been used by stamp collectors since 1893. In 1797 Johann Wolfgang von Goethe sold a manuscript using a sealed-bid, second-price auction.

Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms Vickrey auction and second-price sealed-bid auction are, in this case only, equivalent and used interchangeably. In the case of multiple identical goods, the bidders submit inverse demand curves and pay the opportunity cost.

Vickrey auctions are much studied in economic literature but uncommon in practice. Generalized variants of the Vickrey auction for multiunit auctions exist, such as the generalized second-price auction used in Google's and Yahoo!'s online advertisement programmes (not incentive compatible) and the Vickrey–Clarke–Groves auction (incentive compatible).

🔗 Computer science 🔗 Mathematics 🔗 Systems

An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial "axiom" string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals.

🔗 Mathematics 🔗 Philosophy 🔗 Statistics

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1 both inclusive. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. By contrast, in Boolean logic, the truth values of variables may only be the integer values 0 or 1.

The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic had, however, been studied since the 1920s, as infinite-valued logic—notably by Łukasiewicz and Tarski.

Fuzzy logic is based on the observation that people make decisions based on imprecise and non-numerical information. Fuzzy models or sets are mathematical means of representing vagueness and imprecise information (hence the term fuzzy). These models have the capability of recognising, representing, manipulating, interpreting, and utilising data and information that are vague and lack certainty.

Fuzzy logic has been applied to many fields, from control theory to artificial intelligence.

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🔗 Mathematics

In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated packing density, η, of an arrangement is the proportion of the surface covered by the circles. Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.

While the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible. The "worst" shape to pack onto a plane is not known, but the smoothed octagon has a packing density of about 0.902414, which is the lowest maximum packing density known of any centrally-symmetric convex shape. Packing densities of concave shapes such as star polygons can be arbitrarily small.

The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.

🔗 Mathematics

In a foundational controversy in twentieth-century mathematics, L. E. J. Brouwer, a supporter of intuitionism, opposed David Hilbert, the founder of formalism.