Pertanyaan: Atas dasar apa anda mendefinisikan dogma dan axioma. Saya khawatir definisa anda akan menyebabkan salah tafsir terhadap axioma dan dogma, sehingga bertentangan dengan pengertian secara umum.
Saya mengikuti tulisan anda, anda menggunakan istilah-istilah yang tidak sesuai dengan pengertian umum, misalnya: dark energy, microcosmos, non energy, energy metafisika dll.
Jawaban: Ini saya kutipkan penjelasan mengenai axioma dan dogma dari Wikipedia sebagai pembanding dengan yang saya jelaskan di Teori Minimalis. Jika ada bahan pembanding lain silakan kirimkan pada saya.
A.Dogma is a principle or set of principles laid down by an authority as incontrovertibly true. It serves as part of the primary basis of an ideology or belief system, and it cannot be changed or discarded without affecting the very system’s paradigm, or the ideology itself. They can refer to acceptable opinions of philosophers or philosophical schools, public decrees, religion, or issued decisions of political authorities.
The term derives from Greek δόγμα “that which seems to one, opinion or belief” and that from δοκέω (dokeo), “to think, to suppose, to imagine”. Dogma came to signify laws or ordinances adjudged and imposed upon others by the First Century. The plural is either dogmas or dogmata, from Greek δόγματα. The term “dogmatics” is used as a synonym for systematic theology, as in Karl Barth‘s defining textbook of neo-orthodoxy, the 14-volume Church Dogmatics.
Dogmata are found in religions such as Christianity, Judaism, Buddhism, Pastafarianism, and Islam, where they are considered core principles that must be upheld by all believers of that religion. As a fundamental element of religion, the term “dogma” is assigned to those theological tenets which are considered to be well demonstrated, such that their proposed disputation or revision effectively means that a person no longer accepts the given religion as his or her own, or has entered into a period of personal doubt. Dogma is distinguished from theological opinion regarding those things considered less well-known. Dogmata may be clarified and elaborated but not contradicted in novel teachings (e.g., Galatians 1:6-9). Rejection of dogma may lead to expulsion from a religious group.
In Christianity, religious beliefs are defined by the Church. It is usually on scripture or communicated by church authority. It is believed that these dogmas will lead human beings towards redemption and thus the “paths which lead to God”.
For Catholicism and Eastern and Oriental Orthodox Christianity, the dogmata are contained in the Nicene Creed and the canon laws of two, three, seven, or twenty ecumenical councils (depending on whether one is Nestorian, Oriental Orthodox, Eastern Orthodox, or Roman Catholic). These tenets are summarized by St. John of Damascus in his Exact Exposition of the Orthodox Faith, which is the third book of his main work, titled The Fount of Knowledge. In this book he takes a dual approach in explaining each article of the faith: one, for Christians, where he uses quotes from the Bible and, occasionally, from works of other Fathers of the Church, and the second, directed both at non-Christians (but who, nevertheless, hold some sort of religious belief) and atatheists, for whom he employs Aristotelian logic and dialectics.
The decisions of fourteen later councils that Catholics hold as dogmatic and numerous decrees promulgated by Popes‘ exercising papal infallibility (for examples, see Immaculate Conception and Assumption of Mary) are considered as being a part of the Church’s sacred body of doctrine.
Roman Catholic dogmata are a distinct form of doctrine taught by the Church.
As a possible reaction to skepticism, dogmatism is a set of beliefs or doctrines that are established as undoubtedly in truth. They are regarded as (religious) truths relating closely to the nature of faith.
A notable use of the term can be found in the Central Dogma of Molecular Biology. In his autobiography, What Mad Pursuit, Francis Crick wrote about his choice of the word dogma and some of the problems it caused him:
I called this idea the central dogma, for two reasons, I suspect. I had already used the obvious word hypothesis in the sequence hypothesis, and in addition I wanted to suggest that this new assumption was more central and more powerful. … As it turned out, the use of the word dogma caused almost more trouble than it was worth…. Many years later Jacques Monod pointed out to me that I did not appear to understand the correct use of the word dogma, which is a belief that cannot be doubted… I used the word the way I myself thought about it, not as most of the world does, and simply applied it to a grand hypothesis that, however plausible, had little direct experimental support.
An axiom, or postulate, is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy. The word comes from the Greek ἀξίωμα (āxīoma) ‘that which is thought worthy or fit’ or ‘that which commends itself as evident.’ As used in modern logic, an axiom is simply a premise or starting point for reasoning. Axioms define and delimit the realm of analysis; the relative truth of an axiom is taken for granted within the particular domain of analysis, and serves as a starting point for deducing and inferring other relative truths. No explicit view regarding the absolute truth of axioms is ever taken in the context of modern mathematics, as such a thing is considered to be an irrelevant and impossible contradiction in terms.
In mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”. Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually defining properties for the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, “axiom,” “postulate”, and “assumption” may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally “true” systems of logic, precisely the same thing must be said for logical axioms – they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.
In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
he word “axiom” comes from the Greek word ἀξίωμα (axioma), a verbal noun from the verb ἀξιόειν (axioein), meaning “to deem worthy”, but also “to require”, which in turn comes from ἄξιος (axios), meaning “being in balance”, and hence “having (the same) value (as)”, “worthy”, “proper”. Among the ancient Greekphilosophers an axiom was a claim which could be seen to be true without any need for proof.
Ancient geometers maintained some distinction between axioms and postulates. While commenting Euclid’s books Proclus remarks that “Geminus held that this [4th] Postulate should not be classed as a postulate but as an axiom, since it does not, like the first three Postulates, assert the possibility of some construction but expresses an essential property”. Boethius translated ‘postulate’ as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (syllogisms, rules of inference), was developed by the ancient Greeks, and has become the core principle of modern mathematics. Tautologiesexcluded, nothing can be deduced if nothing is assumed. Axioms and postulates are the basic assumptions underlying a given body of deductive knowledge. They are accepted without demonstration. All other assertions (theorems, if we are talking about mathematics) must be proven with the aid of these basic assumptions. However, the interpretation of mathematical knowledge has changed from ancient times to the modern, and consequently the terms axiom andpostulate hold a slightly different meaning for the present day mathematician, than they did for Aristotle and Euclid.
The ancient Greeks considered geometry as just one of several sciences, and held the theorems of geometry on par with scientific facts. As such, they developed and used the logico-deductive method as a means of avoiding error, and for structuring and communicating knowledge. Aristotle’sposterior analytics is a definitive exposition of the classical view.
An “axiom”, in classical terminology, referred to a self-evident assumption common to many branches of science. A good example would be the assertion that
When an equal amount is taken from equals, an equal amount results.
At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. Their validity had to be established by means of real-world experience. Indeed, Aristotle warns that the content of a science cannot be successfully communicated, if the learner is in doubt about the truth of the postulates.
The classical approach is well-illustrated by Euclid’s Elements, where a list of postulates is given (common-sensical geometric facts drawn from our experience), followed by a list of “common notions” (very basic, self-evident assertions).
- It is possible to draw a straight line from any point to any other point.
- It is possible to extend a line segment continuously in both directions.
- It is possible to describe a circle with any center and any radius.
- It is true that all right angles are equal to one another.
- (“Parallel postulate“) It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
- Common notions
- Things which are equal to the same thing are also equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things which coincide with one another are equal to one another.
- The whole is greater than the part.
A lesson learned by mathematics in the last 150 years is that it is useful to strip the meaning away from the mathematical assertions (axioms, postulates,propositions, theorems) and definitions. One must concede the need for primitive notions, or undefined terms or concepts, in any study. Such abstraction or formalization makes mathematical knowledge more general, capable of multiple different meanings, and therefore useful in multiple contexts. Alessandro Padoa, Mario Pieri, and Giuseppe Peano were pioneers in this movement.
Structuralist mathematics goes further, and develops theories and axioms (e.g. field theory, group theory, topology, vector spaces) without any particular application in mind. The distinction between an “axiom” and a “postulate” disappears. The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid’s fifth postulate we get theories that have meaning in wider contexts, hyperbolic geometry for example. We must simply be prepared to use labels like “line” and “parallel” with greater flexibility. The development of hyperbolic geometry taught mathematicians that postulates should be regarded as purely formal statements, and not as facts based on experience.
When mathematicians employ the field axioms, the intentions are even more abstract. The propositions of field theory do not concern any one particular application; the mathematician now works in complete abstraction. There are many examples of fields; field theory gives correct knowledge about them all.
It is not correct to say that the axioms of field theory are “propositions that are regarded as true without proof.” Rather, the field axioms are a set of constraints. If any given system of addition and multiplication satisfies these constraints, then one is in a position to instantly know a great deal of extra information about this system.
Modern mathematics formalizes its foundations to such an extent that mathematical theories can be regarded as mathematical objects, and logic itself can be regarded as a branch of mathematics. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.
In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.
It was the early hope of modern logicians that various branches of mathematics, perhaps all of mathematics, could be derived from a consistent collection of basic axioms. An early success of the formalist program was Hilbert’s formalization of Euclidean geometry, and the related demonstration of the consistency of those axioms.
In a wider context, there was an attempt to base all of mathematics on Cantor’s set theory. Here the emergence of Russell’s paradox, and similar antinomies of naïve set theory raised the possibility that any such system could turn out to be inconsistent.
The formalist project suffered a decisive setback, when in 1931 Gödel showed that it is possible, for any sufficiently large set of axioms (Peano’s axioms, for example) to construct a statement whose truth is independent of that set of axioms. As a corollary, Gödel proved that the consistency of a theory like Peano arithmetic is an unprovable assertion within the scope of that theory.
It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of natural numbers, an infinite but intuitively accessible formal system. However, at present, there is no known way of demonstrating the consistency of the modern Zermelo–Fraenkel axioms for set theory. Theaxiom of choice, a key hypothesis of this theory, remains a very controversial assumption. Furthermore, using techniques of forcing (Cohen) one can show that the continuum hypothesis (Cantor) is independent of the Zermelo–Fraenkel axioms. Thus, even this very general set of axioms cannot be regarded as the definitive foundation for mathematics.
Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics. In particular, the monumental work of Isaac Newton is essentially based on Euclid‘s axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.
Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr, concerned the interpretation of quantum mechanics. This was in 1935. According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanical theory, i.e. the set of “theorems” derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics “hidden variables” to enforce determinism. However, thirty years later, in 1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yielded measurably different results using Einstein’s axioms compared to using Bohr’s axioms. And it took roughly another twenty years until an experiment of Alain Aspect got results in favour of Bohr’s axioms, not Einstein’s. (Bohr’s axioms are simply: The theory should be probabilistic in the sense of the Copenhagen interpretation.)
As a consequence, it is not necessary to explicitly cite Einstein’s axioms, the more so since they concern subtle points on the “reality” and “locality” of experiments.
Regardless, the role of axioms in mathematics and in the above-mentioned sciences is different. In mathematics one neither “proves” nor “disproves” an axiom for a set of theorems; the point is simply that in the conceptual realm identified by the axioms, the theorems logically follow. In contrast, in physics a comparison with experiments always makes sense, since a falsified physical theory needs modification.