An Overview of Axioms

Mohamed Y. Syed^*
*Correspondence: Mohamed Y. Syed, Department of P.G. Studies and Research in Mathematics, Kuvempu University, Shankaraghatta, Shivamogga, Karnataka, India, Email:

Introduction

A statement that is assumed to be true is referred to as an axiom, postulate, or assumption. It is used as the foundation for future deductive reasoning and arguments. The word is derived from the Ancient Greek word axma, which means "that which commends itself as evident" or "that which is judged worthy or fit." When employed in relation to several academic disciplines, the phrase has nuances in its definition. A statement that is so obvious or well-established that it is accepted without debate or question is called an axiom, according to the definition of the term used in classical philosophy. Axioms are premises or the basis for reasoning in contemporary logic. The terms "logical axioms" and "non-logical axioms" are two closely related but distinct uses of the word "axiom" in mathematics. Non-logical axioms are genuine substantive declarations regarding the constituents of the domain of a particular mathematical theory, while logical axioms are typically statements that are taken to be true within the system of logic they describe and are sometimes shown in symbolic form [1].

Description

One technique to axiomatize a given mathematical subject is to demonstrate that a system of knowledge's claims may be deduced from a condensed, understandable set of statements. A statement that serves as the foundation from which subsequent truths can be logically deduced is known as an axiom. The philosophical discussion of mathematics centres on the question of whether it is meaningful for an axiom to be "true [2].

"A remark that may be considered as self-evidently true without the requirement for justification was known as an axiom among the ancient Greek philosophers. The word postulate's basic meaning is "demand," as in Euclid's demand that one concur that some things can be done. Early geometers kept an axioms vs postulates split. Proclus makes the following statement on Euclid's works: "Geminus held that this Postulate should not be categorised as a postulate but as an axiom, since it does not, like the previous three Postulates, indicate the possibility of some construction but states a fundamental quality. Boethius termed the axioms notiones communes and translated "postulate" as "petitio," but this usage was not always fully followed in later editions [3].

The ancient Greeks created the logico-deductive method, which is the fundamental idea behind contemporary mathematics, whereby conclusions follow from premises by the use of sound arguments. Nothing can be inferred from nothing, tautologies aside. Thus, the fundamental presumptions underlying a specific corpus of deductive knowledge are axioms and postulates. They are acknowledged without need for proof. With the aid of these fundamental hypotheses, all additional statements must be supported. Axiom and postulate, however, now have slightly different meanings for current mathematicians than they did for Aristotle and Euclid due to changes in how mathematics is understood between ancient and modern times. The theorems of geometry were regarded by the ancient Greeks as being on par with scientific facts and as one of several disciplines. As a result, they created and applied the logico-deductive technique, which they utilised to structure knowledge, communicate it, and prevent errors. The classical idea is clearly explained in Aristotle's posterior analytics. In classical jargon, a "axiom" was a self-evident presumption shared by numerous scientific disciplines. The adage "When an equal quantity is removed from equals, an equal amount results" is a nice illustration of this [4,5].

Conclusion

A few further theories that were accepted without evidence were at the core of the various disciplines. A postulate was used to describe such a hypothesis. Although numerous sciences shared the same axioms, each unique field had its own set of postulates. It was necessary to use actual experience to validate them. Aristotle cautions that if the learner has any doubts regarding the veracity of the postulates, it will be impossible to effectively explain the subject matter of a science. Euclid's Elements serves as a good example of the classical method in which a series of postulates is offered, followed by a list of "common ideas".

Conflict of Interest

None.

References

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