Here is an example of what I mean by physical significance:
When we use some geometric or trigonometric identity, let us say Pythagoras' theorem to calculate the length of the diagonal of a field, according to my understanding what we are doing is using the property of space (and/or the structure of our mind which perceives space) to use the consistent behaviour of space to predict the length of the diagonal without directly measuring it.
If that is not a wrong way of looking at mathematics, my question is, what exactly are we doing when we perform calculations such as 941 * 67 = 63047 on paper?