So I have a friend who wants to become an engineer who is overtly obsessed with his mathematical foundations at the moment. He has confessed recently that he didn't understand the definition of the derivative, and asked me to elaborate. And so I did. However, what he asked next kind of confused me, and it took me a while to understand what exactly he was asking. He asked me "Why is this definition the way it is", to which I said that it is simply the most convenient way to formulate it. He was still confused, so I tried to explain that definitions simply served as shorthand for concepts that we want to refer to. I emphasized the need for definitions by showing him a non-recursive definition for ordinal arithmetic and showing him how tedious it would be to use that definition without the shorthand ## \alpha + \beta## in everyday life. He followed that part, but then he asked me to "prove" that this definition "works". I wasn't sure exactly what he means by this, but I assumed that he wanted me to show that the definition was logically consistent. I told him that we derive certain concepts from the axioms, and we give those concepts a name to make it easier to work with them. As long as the deductions are valid, so are our definitions. He doesn't seem to follow that explanation though. Is there any better way I can explain this to him? Or is it simply a matter of mathematical maturity? His highest math is BC Calculus, getting a 5 on the AP exam, if that is of any relevance.