Pythagoras believed that behind the appearance of things, there was the permanent principle of mathematics, and that the forms were based on a transcendental mathematical relation.

Despite these facts, most mathematicians accept the axiom of choice as a valid principle for proving new results in mathematics.

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 originality of Descartes ' thinking, therefore, is not so much in expressing the cogito — a feat accomplished by other predecessors, as we shall see — but on using the cogito as demonstrating the most fundamental epistemological principle, that science and mathematics are justified by relying on clarity, distinctiveness, and self-evidence.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).

* Large deviation principle, the rate function in mathematics

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle.

Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism.

The Church – Turing thesis states that this is a law of mathematics — that a universal Turing machine can, in principle, perform any calculation that any other programmable computer can.

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

This means that man becomes immortal only if and to the extent that he acquires knowledge of what he can in principle know, e. g. mathematics and the natural sciences.

According to Lewis ( 1918 ), the " principle of these diagrams is that classes set ( mathematics ) | set s be represented by regions in such relation to one another that all the possible logical relations of these classes can be indicated in the same diagram.

Both Spinoza and Leibniz asserted that, in principle, all knowledge, including scientific knowledge, could be gained through the use of reason alone, though they both observed that this was not possible in practice for human beings except in specific areas such as mathematics.

Although the concepts of a law or principle in nature is borderline to philosophy, and presents the depth to which mathematics can describe nature, scientific laws are considered from a scientific perspective and follow the scientific method ; they " serve their purpose " rather than " questioning reality " ( philosophical ) or " statements of logical absolution " ( mathematical ).

Calculus, though allowed in solutions, is never required, as there is a principle at play that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge.

In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input ; a linear system fulfills these conditions.

* The superposition principle in physics, mathematics, and engineering, describes the overlapping of waves.

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a smallest element.

Jürgen Schmidhuber, however, says the " set of mathematical structures " is not even well-defined, and admits only universe representations describable by constructive mathematics, that is, computer programs.

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Key to the study is the realization that the mathematics of such jams, which the researchers call " jamitons ," are strikingly similar to the equations that describe detonation waves produced by explosions, says Aslan Kasimov, lecturer in MIT's Department of Mathematics.

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere.

Laurent says: What are mathematics helpful for?

He says certain advocates of the new topics " ignored completely the fact that mathematics is a cumulative development and that it is practically impossible to learn the newer creations if one does not know the older ones " ( p. 17 ).

In mathematics ( specifically linear algebra ), the Woodbury matrix identity, named after Max A. Woodbury says that the inverse of a rank-k correction of some matrix can be computed by doing a rank-k correction to the inverse of the original matrix.

In mathematics, a Galois extension is an algebraic field extension E / F satisfying certain conditions ( described below ); one also says that the extension is Galois.

In mathematics, the multinomial theorem says how to expand a power of a sum in terms of powers of the terms in that sum.

The theorems of mathematics, he says, apply absolutely to all things, from things divine to original matter.

In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves ( up to linear equivalence ) has a one-dimensional subspace on which it is positive definite ( not uniquely determined ), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.

In algebraic topology, a branch of mathematics, the excision theorem is a useful theorem about relative homology — given topological spaces X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out ( excise ) U from both spaces such that the relative homologies of the pairs ( X, A ) and ( X

In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, " fails badly to be invertible ".

In an example of an elementary mathematics in the Suàn shù shū, the square root is approximated by using an " excess and deficiency " method which says to " combine the excess and deficiency as the divisor ; ( taking ) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend.

He supports this with an analysis of the mathematics of ancient maps and of their accuracy, which he says surpassed instrumentation available at the time of the map's drafting.

Charlie's research often interferes with his relationships: as with Amita on their first date, for all they could talk about is mathematics ; Fleinhardt says that it is a common interest and they should not struggle to avoid the subject.

This leads to Charlie being confrontational, but she calms him when she says she just wants him to be " the Sean Connery of the mathematics department.

In mathematics, the Riemann series theorem ( also called the Riemann rearrangement theorem ), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges.

He states that his 6 or so ( famous ) examples of paradoxes ( antinomies ) are all examples of impredicative definition, and says that Poincaré ( 1905 – 6, 1908 ) and Russel ( 1906, 1910 ) " enunciated the cause of the paradoxes to lie in these impredicative definitions " ( p. 42 ), however, " parts of mathematics we want to retain, particularly analysis, also contain impredicative definitions.

Nemeth says that this skill allowed him to succeed in mathematics, during an era without much technology, during which even Braille was difficult to use in mathematics.

The madman says, while playing Beethoven's 9th Symphony, that mathematics are wrong, that 1 + 1 does not equal 2 but a bigger 1, and illustrates it with two drops of olive oil.

In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t and any vectors x and y in R < sup > n </ sup >, the following inequality holds: