Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.[1][2]
The precise
In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (A and B) implies A), while non-logical axioms are substantive assertions about the elements of the domain of a specific mathematical theory, for example a + 0 = a in integer arithmetic.
Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). 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), and there are typically many ways to axiomatize a given mathematical domain.
Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.[5]
Etymology
The word axiom comes from the
The root meaning of the word postulate is to "demand"; for instance, Euclid demands that one agree that some things can be done (e.g., any two points can be joined by a straight line).[7]
Ancient geometers maintained some distinction between axioms and postulates. While commenting on 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."[8] Boethius translated 'postulate' as petitio and called the axioms notiones communes but in later manuscripts this usage was not always strictly kept.[citation needed]
Historical development
Early Greeks
The logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge) through the application of sound arguments (
The ancient Greeks considered
An "axiom", in classical terminology, referred to a
When an equal amount is taken from equals, an equal amount results.
At the foundation of the various sciences lay certain additional hypotheses that 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. Aristotle warns that the content of a science cannot be successfully communicated if the learner is in doubt about the truth of the postulates.[9]
The classical approach is well-illustrated[a] 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).
- Postulates
- It is possible to draw a straight linefrom 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.
- ("intersect on that side on which are the anglesless 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.
Modern development
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,
Structuralist mathematics goes further, and develops theories and axioms (e.g.
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 mathematics itself can be regarded as a branch of logic. Frege, Russell, Poincaré, Hilbert, and Gödel are some of the key figures in this development.
Another lesson learned in modern mathematics is to examine purported proofs carefully for hidden assumptions.
In the modern understanding, a set of axioms is any
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[b] of Euclidean geometry,[10] and the related demonstration of the consistency of those axioms.
In a wider context, there was an attempt to base all of mathematics on
The formalist project suffered a setback a century ago, when
It is reasonable to believe in the consistency of Peano arithmetic because it is satisfied by the system of
Other sciences
Experimental sciences - as opposed to mathematics and logic - also have general founding assertions from which a deductive reasoning can be built so as to express propositions that predict properties - either still general or much more specialized to a specific experimental context. For instance,
As a matter of facts, the role of axioms in mathematics and postulates in experimental sciences is different. In mathematics one neither "proves" nor "disproves" an axiom. A set of mathematical axioms gives a set of rules that fix a conceptual realm, in which the theorems logically follow. In contrast, in experimental sciences, a set of postulates shall allow deducing results that match or do not match experimental results. If postulates do not allow deducing experimental predictions, they do not set a scientific conceptual framework and have to be completed or made more accurate. If the postulates allow deducing predictions of experimental results, the comparison with experiments allows falsifying (falsified) the theory that the postulates install. A theory is considered valid as long as it has not been falsified.
Now, the transition between the mathematical axioms and scientific postulates is always slightly blurred, especially in physics. This is due to the heavy use of mathematical tools to support the physical theories. For instance, the introduction of Newton's laws rarely establishes as a prerequisite neither Euclidean geometry or differential calculus that they imply. It became more apparent when Albert Einstein first introduced special relativity where the invariant quantity is no more the Euclidean length (defined as ) > but the Minkowski spacetime interval (defined as ), and then
In quantum physics, two sets of postulates have coexisted for some time, which provide a very nice example of falsification. The '
Mathematical logic
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical and non-logical (somewhat similar to the ancient distinction between "axioms" and "postulates" respectively).
Logical axioms
These are certain
Examples
Propositional logic
In
Each of these patterns is an axiom schema, a rule for generating an infinite number of axioms. For example, if , , and are propositional variables, then and are both instances of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of the propositional calculus. It can also be shown that no pair of these schemata is sufficient for proving all tautologies with modus ponens.
Other axiom schemata involving the same or different sets of primitive connectives can be alternatively constructed.[13]
These axiom schemata are also used in the
First-order logic
Axiom of Equality.
Let be a
This means that, for any variable symbol , the formula can be regarded as an axiom. Also, in this example, for this not to fall into vagueness and a never-ending series of "primitive notions", either a precise notion of what we mean by (or, for that matter, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol has to be enforced, only regarding it as a string and only a string of symbols, and mathematical logic does indeed do that.
Another, more interesting example
Axiom scheme for Universal Instantiation.
Given a formula in a first-order language , a variable and a
Where the symbol stands for the formula with the term substituted for . (See
Axiom scheme for Existential Generalization. Given a formula in a first-order language , a variable and a term that is substitutable for in , the below formula is universally valid.
Non-logical axioms
Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example, the
Almost every modern
Non-logical axioms are often simply referred to as axioms in mathematical
Thus, an axiom is an elementary basis for a
Examples
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
The study of topology in mathematics extends all over through
This list could be expanded to include most fields of mathematics, including
.Arithmetic
The
We have a language where is a constant symbol and is a unary function and the following axioms:
- for any formula with one free variable.
The standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0.
Euclidean geometry
Probably the oldest, and most famous, list of axioms are the 4 + 1
Real analysis
The objectives of the study are within the domain of
Role in mathematical logic
Deductive systems and completeness
A
that is, for any statement that is a logical consequence of there actually exists a deduction of the statement from . This is sometimes expressed as "everything that is true is provable", but it must be understood that "true" here means "made true by the set of axioms", and not, for example, "true in the intended interpretation". Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.
Note that "completeness" has a different meaning here than it does in the context of
There is thus, on the one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of non-logical axioms. The completeness theorem and the incompleteness theorem, despite their names, do not contradict one another.
Further discussion
Early
See also
- Axiomatic system
- Dogma
- First principle, axiom in science and philosophy
- List of axioms
- Model theory
- Regulæ Juris
- Theorem
- Presupposition
- Principle
Notes
References
- ^ Cf. axiom, n., etymology. Oxford English Dictionary, accessed 2012-04-28.
- ISBN 9780199891535.
a statement or proposition that is regarded as being established, accepted, or self-evidently true
- ^ "A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, Posterior Analytics I.2.72a18-b4.
- ^ "A proposition (whether true or false)" axiom, n., definition 2. Oxford English Dictionary Online, accessed 2012-04-28.
- realistview.
- ^ a b "Axiom — Powszechna Encyklopedia Filozofii" (PDF). Polskie Towarzystwo Tomasza z Akwinu. Archived (PDF) from the original on 9 October 2022.
- ^ Wolff, P. Breakthroughs in Mathematics, 1963, New York: New American Library, pp 47–48
- Heath, T. L.(1956). The Thirteen Books of Euclid's Elements. New York: Dover. p. 200.
- ^ Aristotle, Metaphysics Bk IV, Chapter 3, 1005b "Physics also is a kind of Wisdom, but it is not the first kind. – And the attempts of some of those who discuss the terms on which truth should be accepted, are due to want of training in logic; for they should know these things already when they come to a special study, and not be inquiring into them while they are listening to lectures on it." W.D. Ross translation, in The Basic Works of Aristotle, ed. Richard McKeon, (Random House, New York, 1941)
- ^ For more, see Hilbert's axioms.
- ^ Raatikainen, Panu (2018), "Gödel's Incompleteness Theorems", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Fall 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 19 October 2019
- ^ Koellner, Peter (2019), "The Continuum Hypothesis", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 19 October 2019
- ^ Mendelson, "6. Other Axiomatizations" of Ch. 1
- ^ Mendelson, "3. First-Order Theories" of Ch. 2
- ^ Mendelson, "3. First-Order Theories: Proper Axioms" of Ch. 2
- ^ Mendelson, "5. The Fixed Point Theorem. Gödel's Incompleteness Theorem" of Ch. 2
Further reading
- Mendelson, Elliot (1987). Introduction to mathematical logic. Belmont, California: Wadsworth & Brooks. ISBN 0-534-06624-0
- )