The argument hinges on the idea that a satisfactory naturalistic account of thought processes in terms of brain processes can be given for mathematical reasoning along with everything else.

S Haldane's book, The Causes of Evolution, reestablished natural selection as the premier mechanism of evolution by explaining it in terms of the mathematical consequences of Mendelian genetics.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

It seeks to understand physical systems, using mathematical modeling, in terms of inputs, outputs and various components with different behaviors ; use control systems design tools to develop controllers for those systems ; and implement controllers in physical systems employing available technology.

The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system.

From a mathematical viewpoint, continuous-time IIR LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation.

In mathematical terms, the structure of a fullerene is a trivalent convex polyhedron with pentagonal and hexagonal faces.

In mathematical terms, the sequence F < sub > n </ sub > of Fibonacci numbers is defined by the recurrence relation

In terms of practice, mathematical finance also overlaps heavily with the field of computational finance ( also known as financial engineering ).

This new class of preferred motions, too, defines a geometry of space and time — in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential.

In mathematical terms, this defines a conformal structure.

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

Stated in mathematical terms, for the Cassie – Baxter state to exist, the following inequality must be true.

* Curtain accessories: mathematical and geometrical terms

Maynard Smith published a book entitled The Evolution of Sex which explored in mathematical terms, the notion of the " two-fold cost of sex ".

Restated in mathematical terms, on the surface of the Earth, the weight w of an object is related to its mass m by w

In mathematics and computer science, mutual recursion is a form of recursion where two mathematical or computational functions are defined in terms of each other.

The situation changed rapidly in the years 1925 – 1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas.

Tegmark writes that " abstract mathematics is so general that any Theory Of Everything ( TOE ) that is definable in purely formal terms ( independent of vague human terminology ) is also a mathematical structure.

By revealing ( in modern terms ) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history ; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect.

Oliver Heaviside FRS ( ( 18 May 1850 – 3 February 1925 ) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations ( later found to be 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.

The mathematical analysis originated in observations of the behaviour of game equipment such as playing cards and dice, which are designed specifically to introduce random and equalized elements ; in mathematical terms, they are subjects of indifference.

The modern mathematical understanding is to describe such a structural sequence in terms of an " abstract " polygon which is a partially ordered set ( poset ) of elements.

This is called an eigenstate of position — or, stated in mathematical terms, a generalized position eigenstate ( eigendistribution ).

Whereas the primary meanings of the Lo Shu diagram seemed to have been based on its inner mathematical properties -- and we shall see that even its secondary meanings rested on some mathematical bases -- the urgent desire to place everything into categories of fives led to other groupings based on other numbers, until an exaggerated emphasis on mere numerology pervaded Chinese thought.

This latter construal is sometimes expressed by saying " there is no fact of the matter as to whether or not P ." Thus, we may speak of anti-realism with respect to other minds, the past, the future, universals, mathematical entities ( such as natural numbers ), moral categories, the material world, or even thought.

Abstracting again, a category is itself a type of mathematical structure, so we can look for " processes " which preserve this structure in some sense ; such a process is called a functor.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them ; we are studying the relationships between various classes of mathematical structures.

:" Since a precise mathematical definition of the term effectively calculable ( effectively decidable ) has been wanting, we can take this thesis ... as a definition of it ..."

If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition.

The polyphonic organization of different melodies to sound at the same time was still a relatively new invention then, and it is understandable that the mathematical or physical relationships in frequency that give rise to the musical intervals as we hear them, should be foremost among the preoccupations of Medieval musicians.

Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we cannot measure.

" Kant stated that all mathematical and scientific statements are synthetic a priori propositions because they are necessarily true but our knowledge about the attributes of the mathematical or physical subjects we can only get by logical inference.

In mathematical logic, there are two quantifiers, " some " and " all ", though as Brentano ( 1838 – 1917 ) pointed out, we can make do with just one quantifier and negation.

This means that we should regard logical and mathematical laws as being independent of the human mind, and also as an autonomy of meanings.

In this way we get a proof of the Euler – Maclaurin summation formula by mathematical induction, in which the induction step relies on integration by parts and on the identities for periodic Bernoulli functions.

In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation ( in some sense ) to the desired response using a smaller N, as we shall now illustrate.

And in mathematical notation we can write these as

In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form ( they had already been recast as quaternions ) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations.

This commentary is one of the most valuable sources we have for the history of ancient mathematics, and its Platonic account of the status of mathematical objects was influential.

If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is arithmetically sound if all theorems of T are actually true about the standard mathematical integers.

Once we know what kind of mathematical structure we are concerned with, we should be able to pinpoint what mappings preserve the structure.