How Abstract Are You In Math?

[nightdove]

The question is, do people really have set preferences for different levels of abstraction?

You are as abstract as you need to be to solve a given interesting problem or to understand an insight with aesthetic features. I find it impossible to force myself to do ugly proofs (is there subjectivity here?), for instance, those in Optimisation Theory. Also Analysis only gets interesting once you move onto topology. Proofs regarding sequences of real numbers are absolutely banal.

But one of my favs in A-level was Group Theory while all the other students balked, so I vote for Option 5 (very abstract).

At that time, the Further Mathematics teacher was not able to give a single instance of a physical application of Group Theory (although many exist in the computer sciences), but this did not diminish my love for it. It seemed intrinsically logical and had a purity of spirit that set it apart from say, the concrete study of brutal objects like hyperbolic functions or complementary functions. I know I am not comparing like for like, but at A-level Further Math, Group Theory is the farthest you get into the abstract. The rest of it is methods based.

 

[andytoh]

So nightdove, it seems you are both concrete and abstract depending on your mood? Your opinions seem to go from one extreme to the other.

 

[nightdove]

Perhaps. I have already mentioned that I tend to be rather parsimonious with abstraction, too much conceals more than it reveals. So it isn't abstraction for its own sake, but abstraction for the sake of yielding an "einsicht", which you could describe as subjective (what is insight to one may not be insight to another), but probably not a function of mood.

Would you regard probability distribution theory (outside of measure theory) as abstract or concrete?