Soon the assholes become unpersons, axiomatically and automatically unworthy of compassion.
|
|
There is, axiomatically, no disputing taste, and also no accounting for it.
|
|
If Trump pursues a policy, it cannot axiomatically be wrong, evil and dangerous.
|
|
So, it's logical to say there will axiomatically always be fat people who are not healthy.
|
|
"With precision" is the key phrase, of course, and it renders the statement almost axiomatically true.
|
|
Much as JFK picked up where Ike left off, the next foreign policy president might be axiomatically Trumpian.
|
|
For any given emission reduction goal, there is, almost axiomatically, some level of carbon tax that can achieve it.
|
|
That human numbers are, axiomatically, part of the story of human impact does not mean that human numbers have to take center stage.
|
|
That's the takeaway from Tuesday's special election, which demonstrated anew that Trump's considerable weakness doesn't translate automatically and axiomatically into a Democratic advantage.
|
|
The biggest subsidies go to those with the highest levels of education, which axiomatically means they will have the highest average lifetime earnings.
|
|
In this latter view, the US is simply the strongest power in the world, and any agreement with another country axiomatically reduces that power.
|
|
Once upon a time, every web developer knew, axiomatically, that you never mixed HTML and JavaScript, because of separation of concerns; but then came React.
|
|
More broadly, Sessions emphasized a number of times that a conversation between a president and the head of the FBI would not axiomatically be inappropriate.
|
|
Shaft says outrageous things, but he's fundamentally harmless, and also axiomatically irresistible — "a sex machine to all the chicks," as the ancient version of the title song has it.
|
|
Although this March will mark the 26th annual Silicon Valley Prayer Breakfast (recently renamed Silicon Valley Connect), Big Tech is still considered, almost axiomatically, allergic to expressions of faith.
|
|
It is the brilliance of Mr. Day — and let's generalize to say of British men's fashion — that any accepted version of good taste is axiomatically considered something to be flouted.
|
|
The bottom line is that while functional finance has a lot going for it, it's not the kind of axiomatically true doctrine that Lerner – and, I think, modern MMTers – imagined it to be.
|
|
Even if the F-16 can completely dominate the F-35 in a dogfight (and if it can't, it's very likely that other brand-new fighters designed for that one job can), it doesn't axiomatically make the F-16 a "better" plane.
|
|
"First, that ungoverned spaces in the Middle East and Central Asia will likely be exploited by Islamic extremists; second, that in dealing with such situations, U.S. leadership is not only invaluable, it is indispensable — though we obviously want as many partners with us as possible; and third, that we need to lead a comprehensive approach, not just a narrow [counterterrorism] approach — though that does not axiomatically mean that the U.S. has to provide the ground combat forces or perform a number of the other tasks that comprise a comprehensive approach," he told The Hill.
|
|
Edited by Francine Diener, Marc Diener. Springer, 1995.Nonstandard Analysis, Axiomatically. By V. Vladimir Grigorevich Kanovei, Michael Reeken.
|
|
According to Bailyn's survey of thermodynamics, Carathéodory's approach is called "mechanical," rather than "thermodynamic."Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics, Woodbury NY, . Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein.
|
|
SyGuS-Comp (Syntax-Guided Synthesis Competition) Still, the available algorithms are only able to synthesize small programs. A 2015 paper demonstrated synthesis of PHP programs axiomatically proven to meet a given specification, for purposes such as checking a number for being prime or listing the factors of a number.
|
|
They optimise sequence of decision rules, mappings of the available knowledge on possible actions. This sequence is called strategy or policy. Among various theories, Bayesian DM is broadly accepted axiomatically based theory that solves the design of optimal decision strategy. It describes random, uncertain or incompletely known quantities as random variables, i.e.
|
|
If one defines a homology theory axiomatically (via the Eilenberg–Steenrod axioms), and then relaxes one of the axioms (the dimension axiom), one obtains a generalized theory, called an extraordinary homology theory. These originally arose in the form of extraordinary cohomology theories, namely K-theory and cobordism theory. In this context, singular homology is referred to as ordinary homology.
|
|
As a result, some see the rise of right-libertarianism as popular political philosophy as partially responsible for climate change. Right-libertarians are also criticised for ignoring observation and historical fact and instead focusing on an abstract ideal. Imperfection is not accounted for and they are axiomatically opposed to government initiatives to counter the effects of climate change.
|
|
A Muslim woman's mahramss form the group of allowable escorts when she travels. An adopted brother who suckled from the mother of the woman is axiomatically a mahram. For a spouse, being mahram is a permanent condition. That means, for example, that a man will remain mahram to his ex-mother-in-law after divorcing her daughter.
|
|
As opposed to the above formula one may define the formal derivative axiomatically as the map (\ast)^\prime\colon R[x] \to R[x] satisfying the following properties. 1) r'=0 for all r\in R\subset R[x]. 2) The normalization axiom, x' = 1. 3) The map commutes with the addition operation in the polynomial ring, (a+b)' = a'+b'.
|
|
Schmeidler has made many other contributions, ranging from conceptual issues in implementation theory, to mathematical results in measure theory. But his most influential contribution is probably in decision theory. Schmeidler was the first to propose a general-purpose, axiomatically- based decision theoretic model that deviated from the Bayesian dictum, according to which any uncertainty can and should be quantified by probabilities.
|
|
The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x0:x1: ... :xn) is the underlying group of the geometry.This is therefore PGL(n + 1, F) for projective space of dimension n Stated differently, the natural action of GL(V) on V descends to an action of PGL(V) on the projective space P(V). The projective linear groups therefore generalise the case PGL(2, C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line. Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure".
|
|
The first clear definition of an abstract field is due to .. See also . In particular, Heinrich Martin Weber's notion included the field Fp. Giuseppe Veronese (1891) studied the field of formal power series, which led to introduce the field of p-adic numbers. synthesized the knowledge of abstract field theory accumulated so far. He axiomatically studied the properties of fields and defined many important field-theoretic concepts.
|
|
The voting system that returns all maximal lotteries is axiomatically characterized as the only one satisfying probabilistic versions of population-consistency (a weakening of reinforcement) and composition-consistency (a strengthening of independence of clones). A social welfare function that top-ranks maximal lotteries is characterized using Arrow's independence of irrelevant alternatives and Pareto efficiency.F. Brandl and F. Brandt. Arrovian Aggregation of Convex Preferences. Econometrica. Forthcoming.
|
|
IL provides an axiomatically formulated theory of language, which currently covers, in particular, phonology, morphology, syntax, semantics, and language variability. IL is a non- generative and non-transformational approach in linguistics: it assumes neither "deep structures" nor transformational relations between sequentially ordered structures. Rather, IL conceives linguistic entities as interrelated, "multidimensional" objects, which are typically modelled as set-theoretic constructs.Nolda, Andreas; and Oliver Teuber (2011).
|
|
Weizsäcker developed the theory of ur- alternatives (archetypal objects), publicized in his book Einheit der Natur (literal translation Oneness of Nature, 1971) and further developed through the 1990s. The theory axiomatically constructs quantum physics from the distinction between empirically observable, binary alternatives. Weizsäcker used his theory, a form of digital physics, to derive the 3-dimensionality of space and to estimate the entropy of a proton falling into a black hole.
|
|
Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded. Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically. At the time, it was thought that utility was quantifiable, in units known as utils.
|
|
Nambooripad's basic contributions are in the structure theory of regular semigroups. A semigroup is a set S together with an associative binary operation in S. A semigroup S is said to be regular if for every a in S there is an element b in S such that aba = a. Nambooripad axiomatically characterised the structure of the set of idempotents in a regular semigroup. He called a set having this structure a biordered set.
|
|
The study of categories is an attempt to axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure-preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category. Consider the following example. The class Grp of groups consists of all objects having a "group structure".
|
|
A projective plane may be defined axiomatically as an incidence structure, in terms of a set of points, a set of lines, and an incidence relation that determines which points lie on which lines. These sets can be used to define a plane dual structure. Interchange the role of "points" and "lines" in : to obtain the dual structure :, where is the converse relation of . is also a projective plane, called the dual plane of .
|
|
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set- theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.
|
|
The hyperreals can be developed either axiomatically or by more constructively oriented methods. The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle. In the following subsection we give a detailed outline of a more constructive approach. This method allows one to construct the hyperreals if given a set-theoretic object called an ultrafilter, but the ultrafilter itself cannot be explicitly constructed.
|
|
Physicist Carl Friedrich von Weizsäcker's theory of ur-alternatives (theory of archetypal objects), first publicized in his book The Unity of Nature (1971), further developed through the 1990s, is a kind of digital physics as it axiomatically constructs quantum physics from the distinction between empirically observable, binary alternatives. Weizsäcker used his theory to derive the 3-dimensionality of space and to estimate the entropy of a proton. In 1988 Görnitz has shown that Weizsäcker's assumption can be connected with the Bekenstein–Hawking entropy.
|
|
Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes.
|
|
Encyclopedia of Philosophy, Vol.2, "Coherence Theory of Truth", auth: Alan R. White, p130-131 (Macmillan, 1969) However, formal reasoners are content to contemplate axiomatically independent and sometimes mutually contradictory systems side by side, for example, the various alternative geometries. On the whole, coherence theories have been rejected for lacking justification in their application to other areas of truth, especially with respect to assertions about the natural world, empirical data in general, assertions about practical matters of psychology and society, especially when used without support from the other major theories of truth.Encyclopedia of Philosophy, Vol.
|
|
Tönnies drew a sharp line between the realm of conceptualization (of sociological terms, including ‘normal types’) and the realm of reality (of social action). The first must be treated axiomatically and in a deductive way (pure sociology); the second, empirically and in an inductive way (applied sociology). Following Tönnies, reality (the second realm) cannot be explained without concepts, which belong to the first realm, or else you will fail because you try to define x by something derived from x. Tönnies’ Normaltyp was thus a conceptual tool created on a logical basis,P.
|
|
It is possible to treat different measures of algorithmic information as particular cases of axiomatically defined measures of algorithmic information. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to algorithmic information theory was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).
|
|
The radically performative character of the subject of non-philosophy would be meaningless without the concept of radical immanence. The philosophical doctrine of immanence is generally defined as any philosophical belief or argument which resists transcendent separation between the world and some other principle or force (such as a creator deity). According to Laruelle, the decisional character of philosophy makes immanence impossible for it, as some ungraspable splitting is always taking place within. By contrast, non-philosophy axiomatically deploys immanence as being endlessly conceptualizable by the subject of non-philosophy.
|
|
Thermodynamics had been a subject dear to Carathéodory since his time in Belgium. In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the second law of thermodynamics axiomatically, that is, without the use of Carnot engines and refrigerators and only by mathematical reasoning. This is yet another version of the second law, alongside the statements of Clausius, and of Kelvin and Planck. Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld.
|
|
A more satisfactory approach is to observe that the characteristic property of ordered pairs given above is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property. This was the approach taken by the N. Bourbaki group in its Theory of Sets, published in 1954. However, this approach also has its drawbacks as both the existence of ordered pairs and their characteristic property must be axiomatically assumed.
|
|
Techniques were supposed to be synthetic: in effect projective space as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the projective plane alone, the axiomatic approach can result in models not describable via linear algebra. This period in geometry was overtaken by research on the general algebraic curve by Clebsch, Riemann, Max Noether and others, which stretched existing techniques, and then by invariant theory.
|
|
In the 18th century, utilitarianism gave insight into the utility-maximizing versions of rationality, however, economists still have no single definition or understanding of what preferences and rational actors should be analyzed by. Since the pioneer efforts of Frisch in the 1920s, one of the major issues which has pervaded the theory of preferences is the representability of a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. Von Neumann and Morgenstern 1944 book "Games and Economic Behaviour" treated preferences as a formal relation whose properties can be stated axiomatically.
|
|
Positive political theory (PPT) or explanatory political theory is the study of politics using formal methods such as social choice theory, game theory, and statistical analysis. In particular, social choice theoretic methods are often used to describe and (axiomatically) analyze the performance of rules or institutions. The outcomes of the rules or institutions described are then analyzed by game theory, where the individuals/parties/nations involved in a given interaction are modeled as rational agents playing a game, guided by self-interest. Based on this assumption, the outcome of the interactions can be predicted as an equilibrium of the game.
|
|
In the 1940s Grigore Moisil introduced his Łukasiewicz–Moisil algebras (LMn-algebras) in the hope of giving algebraic semantics for the (finitely) n-valued Łukasiewicz logic. However, in 1956 Alan Rose discovered that for n ≥ 5, the Łukasiewicz–Moisil algebra does not model the Łukasiewicz n-valued logic. Although C. C. Chang published his MV-algebra in 1958, it is a faithful model only for the ℵ0-valued (infinitely-many-valued) Łukasiewicz–Tarski logic. For the axiomatically more complicated (finitely) n-valued Łukasiewicz logics, suitable algebras were published in 1977 by Revaz Grigolia and called MVn-algebras.
|
|
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields.
|
|
Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field).
|
|
Max Weber argued that sociology may be loosely described as a science as it is able to identify causal relationships of human "social action"—especially among "ideal types", or hypothetical simplifications of complex social phenomena. As a non- positivist, however, Weber sought relationships that are not as "historical, invariant, or generalisable" as those pursued by natural scientists. Fellow German sociologist, Ferdinand Tönnies, theorised on two crucial abstract concepts with his work on "gemeinschaft and gesellschaft" (). Tönnies marked a sharp line between the realm of concepts and the reality of social action: the first must be treated axiomatically and in a deductive way ("pure sociology"), whereas the second empirically and inductively ("applied sociology").
|
|
He also showed, in 1829, that the eigenvalues of symmetric matrices are real. Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric transformations at a local (or infinitesimal) level, see above; Kronecker's Vorlesungen über die Theorie der Determinanten and Weierstrass' Zur Determinantentheorie, both published in 1903, first treated determinants axiomatically, as opposed to previous more concrete approaches such as the mentioned formula of Cauchy. At that point, determinants were firmly established. Many theorems were first established for small matrices only, for example, the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices.
|
|
Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the proof of Kőnig's theorem for regular bipartite graphs, phrased in the language of configurations. In 1910 Steinitz published the very influential paper Algebraische Theorie der Körper (German: Algebraic Theory of Fields, Crelle's Journal (1910), 167–309). In this paper he axiomatically studies the properties of fields and defines important concepts like prime field, perfect field and the transcendence degree of a field extension.
|
|
The usual argument against Borel's "Law" is that if all possible outcomes of a natural process are highly improbable when taken individually, then a highly improbable outcome is certain. The true law being referenced is actually the Strong Law of large numbers, but creationists have taken a simple statement made by Borel in books written late in his life concerning probability theory and called this statement Borel's Law. This "Borel's Law" is actually the universal probability bound, which when applied to evolution is axiomatically incorrect. The universal probability bound assumes that the event one is trying to measure is completely random, and some use this argument to prove that evolution could not possibly occur, since its probability would be much less than that of the universal probability bound.
|
|
The condition which has only ones is thus dominated by any condition which has zeros in it [cf. pp. 367–71 in Being and Event].) Badiou reasons using these conditions that every discernible (nameable or constructible) set is dominated by the conditions which don't possess the property that makes it discernible as a set. (The property 'one' is always dominated by 'not one'.) These sets are, in line with constructible ontology, relative to one's being- in-the-world and one's being in language (where sets and concepts, such as the concept 'humanity', get their names). However, he continues, the dominations themselves are, whilst being relative concepts, not necessarily intrinsic to language and constructible thought; rather one can axiomatically define a domination – in the terms of mathematical ontology – as a set of conditions such that any condition outside the domination is dominated by at least one term inside the domination.
|
|
Further, REBT generally posits that disturbed evaluations to a large degree occur through over- generalization, wherein people exaggerate and globalize events or traits, usually unwanted events or traits or behavior, out of context, while almost always ignoring the positive events or traits or behaviors. For example, awfulizing is partly mental magnification of the importance of an unwanted situation to a catastrophe or horror, elevating the rating of something from bad to worse than it should be, to beyond totally bad, worse than bad to the intolerable and to a "holocaust". The same exaggeration and overgeneralizing occurs with human rating, wherein humans come to be arbitrarily and axiomatically defined by their perceived flaws or misdeeds. Frustration intolerance then occurs when a person perceives something to be too difficult, painful or tedious, and by doing so exaggerates these qualities beyond one's ability to cope with them.
|
|
Tönnies drew a sharp line between the realm of conceptuality and the reality of social action: the first must be treated axiomatically and in a deductive way ('pure' sociology), whereas the second empirically and in an inductive way ('applied' sociology). Both Weber and Georg Simmel pioneered the Verstehen (or 'interpretative') approach toward social science; a systematic process in which an outside observer attempts to relate to a particular cultural group, or indigenous people, on their own terms and from their own point-of-view. Through the work of Simmel, in particular, sociology acquired a possible character beyond positivist data-collection or grand, deterministic systems of structural law. Relatively isolated from the sociological academy throughout his lifetime, Simmel presented idiosyncratic analyses of modernity more reminiscent of the phenomenological and existential writers than of Comte or Durkheim, paying particular concern to the forms of, and possibilities for, social individuality.
|
|
This principle is rejected in minimal logic. This means the formula does not axiomatically hold for arbitrary A and B. As minimal logic represents only the positive fragment of intuitionistic logic, it is a subsystem of intuitionistic logic and is strictly weaker. Practically, this enables the disjunctive syllogism the intuitionistic context: :((A \lor B)\land (A\to \bot)) \to B. Given a constructive proof of A \lor B and constructive rejection of A, the principle of explosion unconditionally allows for the positive case choice of B. This is because if A \lor B was proven by proving B then B is already proven, while if A \lor B was proven by proving A, then B also follows if the system allows for explosion. Note that with \bot taken for B in the modus ponens expression, the law of non-contradiction :(A \land (A\to \bot))\to \bot, i.e.
|
|
This section gives examples of mathematical theories that are developed entirely from a set of non-logical axioms (axioms, henceforth). A rigorous treatment of any of these topics begins with a specification of these axioms. Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo–Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann–Bernays–Gödel set theory, a conservative extension of ZFC. Sometimes slightly stronger theories such as Morse–Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.
|
|
Eventually logicians found that restricting Frege's logic in various ways—to what is now called first-order logic—eliminated this problem: sets and properties cannot be quantified over in first-order-logic alone. The now-standard hierarchy of orders of logics dates from this time. It was found that set theory could be formulated as an axiomatized system within the apparatus of first-order logic (at the cost of several kinds of completeness, but nothing so bad as Russell's paradox), and this was done (see Zermelo–Fraenkel set theory), as sets are vital for mathematics. Arithmetic, mereology, and a variety of other powerful logical theories could be formulated axiomatically without appeal to any more logical apparatus than first-order quantification, and this, along with Gödel and Skolem's adherence to first-order logic, led to a general decline in work in second (or any higher) order logic.
|
|
Conditional event algebras circumvent the obstacle identified by Lewis by using a nonstandard domain of objects. Instead of being members of a set F of subsets of some universe set Ω, the canonical objects are normally higher-level constructions of members of F. The most natural construction, and historically the first, uses ordered pairs of members of F. Other constructions use sets of members of F or infinite sequences of members of F. Specific types of CEA include the following (listed in order of discovery): : Shay algebras : Calabrese algebras : Goodman-Nguyen-van Fraassen algebras : Goodman-Nguyen-Walker algebras CEAs differ in their formal properties, so that they cannot be considered a single, axiomatically characterized class of algebra. Goodman-Nguyen-van Frassen algebras, for example, are Boolean while Calabrese algebras are non-distributive. The latter, however, support the intuitively appealing identity A → (B → C) = (A ∩ B) → C, while the former do not.
|
|
Breger points out a problem when one is approaching a notion "axiomatically", that is, an "axiomatic system" may have imbedded in it one or more tacit axioms that are unspoken when the axiom-set is presented. For example, an active agent with knowledge (and capability) may be a (potential) fundamental axiom in any axiomatic system: "the know-how of a human being is necessary – a know-how which is not formalized in the axioms. ¶ ... Mathematics as a purely formal system of symbols without a human being possessing the know-how with the symbols is impossible ..."Breger in (Groshoz and Breger 2002:221) He quotes Hilbert: : "In a university lecture given in 1905, Hilbert considered it "absolutely necessary" to have an "axiom of thought" or "an axiom of the existence of an intelligence" before stating the axioms in logic. In the margin of the script, Hilbert added later: "the a priori of the philosophers.
|
|
After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of the book concerns the authors' results. These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self- adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators. The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically-characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space. The authors are motivated in part by the computability of solutions to differential equations.
|
|
Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation."Preserving the incidence relation" means that if point is on line then is in ; formally, if then . Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the non-Desarguesian planes, and this definition permits one to define collineations in such projective planes. For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
|
|
Encyclopedia of Philosophy, Vol.2, "Coherence Theory of Truth", auth: Alan R. White, pp. 130–31 (Macmillan, 1969) However, formal reasoners are content to contemplate axiomatically independent and sometimes mutually contradictory systems side by side, for example, the various alternative geometries. On the whole, coherence theories have been rejected for lacking justification in their application to other areas of truth, especially with respect to assertions about the natural world, empirical data in general, assertions about practical matters of psychology and society, especially when used without support from the other major theories of truth.Encyclopedia of Philosophy, Vol.2, "Coherence Theory of Truth", auth: Alan R. White, pp. 131–33, see esp., section on "Epistemological assumptions" (Macmillan, 1969) Coherence theories distinguish the thought of rationalist philosophers, particularly of Baruch Spinoza, Gottfried Wilhelm Leibniz, and Georg Wilhelm Friedrich Hegel, along with the British philosopher F. H. Bradley.Encyclopedia of Philosophy, Vol.2, "Coherence Theory of Truth", auth: Alan R. White, p. 130 They have found a resurgence also among several proponents of logical positivism, notably Otto Neurath and Carl Hempel.
|
|