Yes, the famous dichotomy of the equality through intension and equality through extension. The easiest way to explain this dichotomy is thus
- equality intension refers to the meaning of the things you equate, 4/2 is equal to 2 because you want them to be equal in meaning
- equality in extension is equality through computation, it is also called Propositional equality
Be careful because there is also the dichotomy equality-identity. The best part is that these notions differ with the mathematical sub-field wherein you work...
You clearly see that 1+1 is not the same thing at 4/2 which differs equally from 2. The work involved in each of the illustration of 2 is not the same, the reasoning is not the same.
Of course, a few people cannot stand that 2, 1+1 and 4/2 differ, because they hate to contextualize the result, through the method of getting the result. These people who hate this situation crave some absolute from which they claim that the method of obtaining the result is irrelevant in obtaining the result.
What matters to them is the result and nothing else.
Now, let's look at what happens once you take into account the method.
Once you say that the method matters, you attach to the result 2 the method of getting this 2. This brings you the concept of equivalence of proof (of the number 2): the easiest proof of the number is the data of the number 2; you show to yourself 2 and this is the easiest way to prove 2. But you have plenty of other algorithm, other method to obtain the number 2 and these methods lead you to the notion of equivalence of proofs.
I let you dwell in this mess of extension and intension for equality and identity.