Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of <+,ℤ²>. To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.
Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.
This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.
To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.
Algebraically, 12 is not the same as 10+2. 12 is an element of, say, ℤ, while 10+2 is one of <+,ℤ²>. To make them interchangeable, we need to establish an equivalence relation. Given that relation, we then have the opportunity to express useful, non-obvious equivalences using transitivity.
Now that we have X=10+2=12, we need to choose which one represent the equivalence class of X. 12 is certainly shorter, but a seemingly magic constant. 10+2 implies that in other number systems, X=b+1+1 may be also true. If the scribe subscribes to the principle of MDL, we can speculate that this is the reason he chose the longer version, and if that is accurate, we have gained more information. If we chose 9+3, we would arguably lose information, since this expression is (hypothetically) misleading.
This is all to say that expressions are more informative than their equivalence classes, since they have been hand-picked to be representative.
To represent the equivalence class of 6.28... with 2*3.14... implies that the equivalence class of 3.14... is more important, and that the prototype likely involves two separate instances of the concept of π. This is misleading.