Feynman gave a prescription for calculating the amplitude for any given diagram from a field theory Lagrangian-the Feynman rules.

In this route, theory ( description ) and prescription were non-overlapping formal discussions of a subject.

Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

According to bundle theory, an object consists of its properties and nothing more: thus neither can there be an object without properties nor can one even conceive of such an object ; for example, bundle theory claims that thinking of an apple compels one also to think of its color, its shape, the fact that it is a kind of fruit, its cells, its taste, or at least one other of its properties.

The difficulty in conceiving of or describing an object without also conceiving of or describing its properties is a common justification for bundle theory, especially among current philosophers in the Anglo-American tradition.

The inability to comprehend any aspect of the thing other than its properties implies, this argument maintains, that one cannot conceive of a bare particular ( a substance without properties ), an implication that directly opposes substance theory.

Bundle theory maintains that properties are bundled together in a collection without describing how they are tied together.

Critics question how bundle theory accounts for the properties ' compresence ( the togetherness relation between those properties ) without an underlying substance.

This extension is needed for, among other things, modeling the concepts of " is an element of " or " is a subset of " in set theory, without running into logical inconsistencies such as Russell's paradox.

Inflation and baryogenesis remain more speculative features of current Big Bang models: they explain important features of the early universe, but could be replaced by alternative ideas without affecting the rest of the theory.

The flatness and horizon problems are naturally solved in the Einstein-Cartan-Sciama-Kibble theory of gravity, without needing an exotic form of matter and introducing free parameters.

Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

The deficiencies of valence bond theory became apparent when hypervalent molecules ( e. g. PF < sub > 5 </ sub >) were explained without the use of d orbitals that were crucial to the bonding hybridisation scheme proposed for such molecules by Pauling.

The theory is said to assume many aspects of how the universe came to be without scientific analysis, and assumes a monotheistic religious outlook.

As such, critical theory was left, in Jürgen Habermas ’ words, without " anything in reserve to which it might appeal ; and when the forces of production enter into a baneful symbiosis with the relations of production that they were supposed to blow wide open, there is no longer any dynamism upon which critique could base its hope.

In theory, speakers of the Western languages would understand written or spoken Interlingua immediately, without prior study, since their own languages were its dialects.

A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ ( a ) holds in the standard model of set theory ( see Kunen 1980: 153 ).

Either there are some basic beliefs that we can be justified for holding, without being able to justify them on the basis of any other belief, or else for each justified belief there is an infinite regress of ( potential ) justification nebula theory.

This theory has the advantage of avoiding the infinite regress without claiming special, possibly arbitrary status for some particular class of beliefs.

In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.

Analogously, anyone who argues that Einstein's theory of gravitation is simpler than Newton's, must say rather more to explain how it is that the latter is mastered by student-physicists, while the former can be managed ( with difficulty ) only by accomplished experts.

The strongest appeal of the Copernican formulation consisted in just this: ideally, the justification for dealing with special problems in particular ways is completely set out in the basic ' rules ' of the theory.

My reply is that I associate myself with all those who affirm that Gentile-Jewish relations should contribute to the theory and practice of human dignity.

The point is that an ethical critic, with an assist from Freud, can seize on this theory to argue that tragedy provides us with a harmless outlet for our hostile urges.

But beginning, for all practical purposes, with Frederick Seebohm's English Village Community scholars have had to reckon with a theory involving institutional and agrarian continuity between Roman and Anglo-Saxon times which is completely at odds with the reigning concept of the Anglo-Saxon invasions.

As you have doubtless guessed already, the plot is plastered with Freudian, Jungian, and Meinckian theory.

The experimental second moment Af conforms to the relation Af in agreement with theory.

We then outline the theory of the interaction of paramagnetic dipoles with nuclei and show that the theory is in excellent agreement with experiment.

Chromium analysis gave 58.8% Cr as compared with 61.2% theory.

We have been using the word `` public '' in quotation marks, that is, in its vernacular connotation with reference to the odd-lot index theory.

What with traders trading for so many different objectives, and what with there being so many unique and individualized market theories and trading techniques in use, and more coming into use all the time, it is hard to imagine how any particular theory or technique could acquire enough `` fans '' to invalidate itself.

Perhaps one way to sharpen our sense of the modernity of Utopian communism is to contrast it with the principal earlier types of communistic theory.

Other theories of origin are compatible with the formulaic theory: Beowulf may contain a design for terror, and The Iliad may have a vast hysteron-proteron pattern answering to a ceramic pattern produced during the Geometric Period in pottery.

Dreams that display events of the future with photographic detail call for a theory explaining their basic mystery and all its components, including that weird feeling of deja vue, inevitably fantastic though that theory must seem.

Dr. Bonnor begins with a discussion of the relativistic theories of the universe, based on the general theory of relativity.

Professor Bondi disagrees with the expansion-contraction theory.

Dr. Lyttleton backs the theory that we live in an electric universe and this theory starts with the behavior of protons and electrons.