That about sums it up. It's made worse by professors who have been trained to teach things in an unmotivated fashion.
I had a graduate probability teacher who said (paraphrased) "one of the beautiful things about math is that you start with a definition, and you prove some very simple things with the definition... so simple that you feel you're just using circular reasoning. Then you find that your definition is equivalent to a useful property!"
Actually, what happened when the stuff was developed is that someone wanted the useful property, and managed to rationalize a definition which fit. (example: group theory was done for hundreds of years before we had a definition of group. Noether noticed that a lot of folks were writing the same sorts of things in different contexts, so she distilled it down to three axioms.)
For a really concrete example, consider ordered pairs. A person being coy about their intent would say that "we define an ordered pair (x,y) on a set S as a set of the form {{x},{x,y}}, where x and y are in S."
It's safe to say that no one thinks of an ordered pair in that fashion. The intent was to have two ordered pairs be equal if and only if each of the two coordinates are equal. You can prove this from the above definition, but it's far more helpful to tell the audience in advance that we really want a structure with this property.
I had a graduate probability teacher who said (paraphrased) "one of the beautiful things about math is that you start with a definition, and you prove some very simple things with the definition... so simple that you feel you're just using circular reasoning. Then you find that your definition is equivalent to a useful property!"
Actually, what happened when the stuff was developed is that someone wanted the useful property, and managed to rationalize a definition which fit. (example: group theory was done for hundreds of years before we had a definition of group. Noether noticed that a lot of folks were writing the same sorts of things in different contexts, so she distilled it down to three axioms.)
For a really concrete example, consider ordered pairs. A person being coy about their intent would say that "we define an ordered pair (x,y) on a set S as a set of the form {{x},{x,y}}, where x and y are in S."
It's safe to say that no one thinks of an ordered pair in that fashion. The intent was to have two ordered pairs be equal if and only if each of the two coordinates are equal. You can prove this from the above definition, but it's far more helpful to tell the audience in advance that we really want a structure with this property.