What is the meaning of equation?

Question

Do equations refer only to quantities or they can also refer to everything like physical objects , syllogisms etc ? I mean is legal to say that a equation doesnt only have meaning for numbers ?

Answer 1

An equation is an assertion that two things are equal. It has meaning as long as it is meaningful to assert that the objects on either side of the equals sign are or are not equal; for numbers we have a very natural way to do this. If you want to write an equation for something other than numbers, you need to be very careful to define what you mean by "equals".

Answer 2

This question does have a satisfying answer if you want your notion of equality to extend to mathematical objects, too, and are comfortable with accepting ZF as a basis for your discussion: It is a consequence of the axiom of extensionality that the mathematical cosmos exists of hereditary sets and hereditary sets only: Let's say we wanted to think about objects that are not sets, that is they wouldn't have any elements. By extensionality they would be equal to the empty set, that is you cannot introduce such objects in any meaningful way.

So, in mathematics, every object you will be speaking of will actually be some set hidden by notation (see e.g. the natural numbers). Now you have an exact meaning of equality: You can write "$=$" if and only if the objects on either side are equal as sets i.e. if and only if those sets have the same elements (again thanks to extensionality).

If you closely pay attention to your rigorous introduction to mathematics you will be able to trace back any equality to its interpretation in set theory and you will be finding yourself often identifying the exact sets things belong to in order to do so. If you progress further you will notice that notation and in particular the equality symbol will be used more and more loosely but any rigorous text will make sense of that too!