Difference between axioms and theorems
Axioms and theorems:
Axioms and theorems are terms that often get confused. While some may be well aware of the differences between the two terms, others may be completely bewildered by these which is why this articles seeks to define axioms and theorems at the same time, pointing out some of their differences as well.
What are axioms?
Based on logic, an axiom, which is also referred to as a postulate, or an assumption , can be defined as a statement which is viewed as being true. Since it is considered as self evident, it cannot be demonstrated or proven with evidence and therefore, anything that cannot be proven with due evidence nor a practical way of proving it, but has been declared as true is considered as an axiom. Thereby, an axiom completely disregards the truth.
In mathematics, axioms need to be accepted as true for the sole purpose of expressing or proving another statement. It is considered as the starting point and they pave the way, brick by brick for mathematical statements.
What are theorems?
Theorems are therefore, proven based on a set of logical connectives which sees light as a logical consequence of the axioms. They need to be proven through a thorough procedure based on one or several axioms. They are thus considered as expressed to be derived and therefore, these derivations ultimately become the proof of the expression. A theorem consists of a hypothesis and a conclusion which makes up the proof of a theorem.
What is the difference between theorems and axioms?
Axioms are considered to be the universal truth. It cannot be proved and is considered as self evident. And yet, theorems need to be proved with the aid of a lengthy and vigorous procedure using axioms and logical connectives. Thereby, a theorem is considered as the statement which had been derived from theorems while axioms remain as a building block which helped in thus providing the theorem.
Also, while axioms are considered as the universal truth with no need to provide proof for their veracity, only a few philosophers challenge its validity. And yet, theorems are often and more challenged than axioms which may or may not change as a result of various derivation methods which can be used in coming to that specific statement.
Although theorems are derived from axioms, theorems themselves are considered as axioms sometimes. And yet, they are considered to be more of theorems than axioms because of the fact that they can be changed and manipulated using other principles and methods of logical reasoning.