I have an exceedingly simple question, but something that has been eating at me for some time now. Imagine that we wish to perform subtraction of objects with the same units, say \$. I have a total of 10\$, composed of 5\$ of my own money, and 5\$ of my brother's. My brother then requests his money back (which I have identified by markings on the bills), and I have left $$ 10\$-5\$=5\$ $$
Imagine, however, that I had not identified my brother's portion of the money, and had jumbled it all up. My brother asks for his money back, and again, he receives $5\$,$but not necessarily his own money. I understand there is no difference in the quantity of money he receives- what is the property of arithmetic/objects that renders these two values the same? In which case is there an appreciable difference between these two values? If not, why is there no difference, even though we intuitively understand there to be one?
Many thanks!
There is no difference, because a dollar is a dollar, irrespective of where it comes from. If you want to make a difference you would have to add a label and you would have e.g. $(x,y)$ with $x$ the amount of dollars and $y$ and extra parameter, here $y=$ the money given by you or $y=$ the money given by your brother. If you call $V(x,y)$ the value of the money with parameters $(x,y)$ then we have generally accepted $V(x,y) = V(x,z)$ for all possible $x,y,z$. But you and your brother may decide that for your guys $V(x,y)\neq V(x,z)$ for whatever reason.