# What exactly does 'abstract' signify in a course like Harvard's Math 55 (Honors 1st yr)?

Source: Logan R. Ury. Burden of Proof. Dec 6, 2006. Wikipedia on Math 55.

While Harrison ultimately chooses to remain in the class, such conferences motivate several more students to drop, including the only two females: [Elizabeth A] Cook ['10], who had looked forward to taking the class, and Laura P. Starkston ’10. “The problem was that I wasn’t prepared to think that abstractly,” Cook says. “Gaitsgory pointed it out to me in our private conference. Eventually I just got the picture.”

What exactly does 'abstract' signify here? What makes the thinking 'that abstract'?

• In the concrete context of Harvard's math 55, "think that abstractly" means thinking like a semi-formed mathematician - that is, able to handle rigorous proofs of formal statements in real and complex analysis and algebra, e.g. something like the Stone-Weierstrass theorem. To get an idea, look at Elkies's web page for the class: math.harvard.edu/~elkies/M55a.17 - they are given calculations of characters of finite groups as homework in the first semester, linear algebra includes things like exact sequences and fields of nonzero characteristic, etc. Feb 13, 2018 at 15:44

## 1 Answer

One definition of "abstract" is " disassociated from any specific instance".

In mathematics, we "abstract" by finding properties which underlie a class of examples. For instance, the concept of a "group" is an abstraction of many different concrete instances of groups which were important to mathematicians: composing symmetries of spaces (such as the isometries of the cube, or automorphisms of a vector space, or biholomorphisms of the unit disk), various bookkeeping devices ("homology classes" as bookkeeping device for domains of integration), etc. In all of these cases, we saw a common structure arising. It makes sense to study this structure "disassociated from any concrete instance", or "in the abstract", and so we began to study groups as their own abstract structure.