Douglas Richard Hofstadter ( born February 15, 1945 ) is an American professor of cognitive science whose research focuses on consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics.

In response to confusion over the book's theme, Hofstadter has emphasized that GEB is not about mathematics, art, and music but rather about how cognition and thinking emerge from well-hidden neurological mechanisms.

Dewdney followed Martin Gardner and Douglas Hofstadter in authoring Scientific American's recreational mathematics column, which he renamed to " Computer Recreations ", then " Mathematical Recreations ", from 1984 to 1993 ( with the last few appearing in Algorithm ).

He then worked for two years in the Geological Survey of New York, and shortly after he had reached his majority he became an instructor in mathematics and the sciences at the Albany Female Academy, where he taught for four years, after which he resumed his studies in Germany with Justus von Liebig.

There are various and quite different notions of sequences in mathematics, some of which ( e. g., exact sequence ) are not covered by the notations introduced below.

Numbering sequences starting at 0 is quite common in mathematics, in particular in combinatorics.

In mathematics, several specific infinite sequences of bits have been studied for their mathematical properties ; these include the Baum – Sweet sequence, Ehrenfeucht – Mycielski sequence, Fibonacci word, Kolakoski sequence, regular paperfolding sequence, Rudin – Shapiro sequence, and Thue – Morse sequence.

In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

* Elliptic divisibility sequence, a class of integer sequences in mathematics

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences.

In mathematics, the dot product, or scalar product ( or sometimes inner product in the context of Euclidean space ), is an algebraic operation that takes two equal-length sequences of numbers ( usually coordinate vectors ) and returns a single number obtained by multiplying corresponding entries and then summing those products.

Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or ( especially ) estimation of certain types of sums.

Different levels of mathematics are taught at different ages and in somewhat different sequences in different countries.

* Convergence ( mathematics ), refers to the notion that some functions and sequences approach a limit under certain conditions

Hurewicz is best remembered for two remarkable contributions to mathematics, his discovery of the higher homotopy groups in 1935-36, and his discovery of exact sequences in 1941.

In mathematics, lexicographical order is a means of ordering sequences in a manner analogous to that used to produce alphabetical order.

In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f < sub > n </ sub > = f < sub > n − 1 </ sub > ± f < sub > n − 2 </ sub >, where the signs + or − are chosen at random with equal probability 1 / 2, independently for different n. By a theorem of Harry Kesten and Hillel Fürstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly.

Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences ( though mathematics does speak of ordered sets, and these can be seen to include the musical kind in some sense, they are far more involved ).

* The space ℓ < sup > 2 </ sup > of square-summable sequences, in mathematics

In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups.

In mathematics, the Lucas sequences U < sub > n </ sub >( P, Q ) and V < sub > n </ sub >( P, Q ) are certain integer sequences that satisfy the recurrence relation

In mathematics, the Cauchy product, named after Augustin Louis Cauchy, of two sequences,, is the discrete convolution of the two sequences, the sequence whose general term is given by

In mathematics, and in particular in combinatorics, the combinatorial number system of degree k ( for some positive integer k ), also referred to as combinadics, is a correspondence between natural numbers ( taken to include 0 ) N and k-combinations, represented as strictly decreasing sequences c < sub > k </ sub > > ... > c < sub > 2 </ sub > > c < sub > 1 </ sub > ≥ 0.

Scientists say that the world and everything in it are based on mathematics.

They involve only simple mathematics that are taught in grammar school arithmetic classes.

These keys are the working principles of physics, mathematics and astronomy, principles which are then extrapolated, or projected, to explain phenomena of which we have little or no direct knowledge.

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

Associative operations are abundant in mathematics ; in fact, many algebraic structures ( such as semigroups and categories ) explicitly require their binary operations to be associative.

Two aspects of this attitude deserve to be mentioned: 1 ) he did not only study science from books, as other academics did in his day, but actually observed and experimented with nature ( the rumours starting by those who did not understand this are probably at the source of Albert's supposed connections with alchemy and witchcraft ), 2 ) he took from Aristotle the view that scientific method had to be appropriate to the objects of the scientific discipline at hand ( in discussions with Roger Bacon, who, like many 20th century academics, thought that all science should be based on mathematics ).

Overall, his contributions are considered the most important in advancing chemistry to the level reached in physics and mathematics during the 18th century.

His criticisms of the scientific community, and especially of several mathematics circles, are also contained in a letter, written in 1988, in which he states the reasons for his refusal of the Crafoord Prize.

In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1.

The anthropic principle is often criticized for lacking falsifiability and therefore critics of the anthropic principle may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, " conditions that are observed in the universe must allow the observer to exist ", is " easy " to support in mathematics and philosophy, i. e. it is a tautology or truism

The advantages of abstraction in mathematics are:

These ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

Other examples are readily found in different areas of mathematics, for example, vector addition, matrix multiplication and conjugation in groups.

In mathematics terminology, the vector space of bras is the dual space to the vector space of kets, and corresponding bras and kets are related by the Riesz representation theorem.

In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem.

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

Sets are of great importance in mathematics ; in fact, in modern formal treatments, most mathematical objects ( numbers, relations, functions, etc.

It is ' naive ' in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system.

In mathematics, the Bernoulli numbers B < sub > n </ sub > are a sequence of rational numbers with deep connections to number theory.

The graduate instructional programs emphasize doctoral studies and are dominated by science, technology, engineering, and mathematics fields.

Although a " bijection " seems a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets.

Some branches of mathematics such as algebraic geometry, typically influenced by the French school of Bourbaki, use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact.