The axiom of equality can be expressed in three ways:
- Reflexivity: x = x
- Symmetry: if x = y, y = x
- Transitivity: if x = y and y = z, x = z
Reflexivity:
Any entity is equal to itself.
This is a fundamental idea that guarantees the internal consistency of mathematical systems. Many branches of mathematics including algebra and geometry use reflexivity. For example, in algebra, x = x is always true. Similarly, reflexivity is used in geometry to prove equality, as in the case of Angle A = Angle A, to make sure that the reasoning of the comparison between geometric objects is logically consistent.
The reflexivity axiom has philosophical implications, especially in metaphysics. Aristotle’s law of identity, which states that “everything is the same as itself”, mathematically stated as for all x: x = x, refers to the individuality of each entity and the possibility of telling one entity from another. But this is a highly paradoxical concept when applied to human nature. While mathematics is based on logic, human beings do not remain the same during their lifetime. Philosophers like Heraclitus argued that identity is constantly changing, whereas Locke believed that continuity of consciousness and memory is what makes a person. Ultimately, this is a question at the core of existentialist debate: what makes a person?
Symmetry:
Equality is mutual; the order of terms does not affect this equality.
The axiom of symmetry ensures that the relationship between variables is maintained, forwards and backwards. Algebraically, symmetry is used in group theory. In geometry, symmetry defines congruence and isometries, for example the isosceles triangle. The axiom of symmetry allows objects to be rotated, translated, and reflected whilst remaining unchanged.
The idea of moral reciprocity, the basic principle of treating others the way you would want to be treated, is undoubtedly influenced by the axiom of symmetry. According to Immanuel Kant’s categorical imperative, moral principles should be universally applied; a behaviour that is acceptable to one individual must be acceptable to everyone under comparable circumstances. This is the basis of the justice system. Furthermore, Rawls’ concept of the ‘veil of ignorance’, the theory that if no one knew their place in society, they would advocate for equality, is supported by the logic of the axiom of symmetry.
Transitivity:
If two entities are equal to a common third entity, the first and third entities must be equal to each other.
Transitivity in mathematics is required to ensure consistency between equations and logical frameworks. In algebra, equivalence relations rely on transitivity to create equalities between varying expressions. For instance, if a = b and b = c, it can be concluded that a = c, enabling substitution and simplification in equations. In geometry, transitivity makes sure that conditions between lengths, angles, and areas are logically consistent. If one triangle is the same as a second, and the second is the same as a third, then the first and third must be the same too. The philosophical implications of transitivity are relevant to rational argument and logical inference. The Socratic method relies on transitivity in argument—if a certain principle is true in one case and in another, then it must be true in similar cases. This comes into significant play in legal and political reasoning: if one group is granted equal rights, then a similar group should too.
According to the axiom of equality, things which are equal to the same thing are equal to one another. For mathematical systems to be logically consistent, this is essential. But its ramifications go much beyond mathematics, including philosophical understandings of rational reasoning and the nature of truth.
