Unless you're dealing with business executive types (that wouldn't notice if you were lying anyway), most of the time, to most of the people, a Java programmer and a C programmer and a Ruby programmer are all really just programmers.
They may have different ways of dealing with problems (the C programmer will lie awake trying to remember if he dropped a free(), while the Ruby programmer will stay up nights trying to figure out how to stop copying that list O(n) times), but the point is that they're still dealing with the same problems (it's always memory, isn't it?). The techniques may be different, but the questions and basic concepts never change.
In math, this is far from the case. In geometry, you never have to deal with infinity the same way you do in set theory. In fact, you don't even have to understand the idea of infinity the same way as a set theorist, and because of this, you _can't_ become a set theorist (unless you want to go back to undergrad and disappoint your parents _again_). It's like saying that a Java programmer doesn't need to understand that memory exists. Maybe they don't have to directly allocate and free it, but they still need to know how much they have and what happens when they use too much of it, and they can certainly recognize the same issues in any other language, even if they don't know how to fix them.
Mathematicians just don't have the same amount of common ground, and it's not even that they can't (because there's too much or something), it's that they don't really even want to (because they see it as boring or a waste of time (and in the case of analysis, they'd be right ;-)).
Unless you're dealing with business executive types (that wouldn't notice if you were lying anyway), most of the time, to most of the people, a Java programmer and a C programmer and a Ruby programmer are all really just programmers.
They may have different ways of dealing with problems (the C programmer will lie awake trying to remember if he dropped a free(), while the Ruby programmer will stay up nights trying to figure out how to stop copying that list O(n) times), but the point is that they're still dealing with the same problems (it's always memory, isn't it?). The techniques may be different, but the questions and basic concepts never change.
In math, this is far from the case. In geometry, you never have to deal with infinity the same way you do in set theory. In fact, you don't even have to understand the idea of infinity the same way as a set theorist, and because of this, you _can't_ become a set theorist (unless you want to go back to undergrad and disappoint your parents _again_). It's like saying that a Java programmer doesn't need to understand that memory exists. Maybe they don't have to directly allocate and free it, but they still need to know how much they have and what happens when they use too much of it, and they can certainly recognize the same issues in any other language, even if they don't know how to fix them.
Mathematicians just don't have the same amount of common ground, and it's not even that they can't (because there's too much or something), it's that they don't really even want to (because they see it as boring or a waste of time (and in the case of analysis, they'd be right ;-)).