I consider myself a mathematician, since I have a PhD in math. I have always been able to do arithmetic pretty easily, and accurately, and so can all the other mathematicians I know. So I believe those stories about mathematicians not being able to do arithmetic are pretty much false. My wife balances the checkbook, but I could also do it. I might be more accurate, but suffer much more at it. In this task, patience is also a virtue.
Mathematicians make elementary mistakes like everyone else, but in general are far less likely to do so, rather than more likely. E.g. as a young man (less than 50 or 60), I never used a calculator to add up my grades because I could accurately add the scores in my head, faster than I could punch the buttons on the calculator.
Once at church when the calculator seemed to fail, I added up all the collection moneys and checks from all the plates in my head as they read them out, perfectly. The man following along on the adding machine made an error but I did not.
This is no longer true, as I have aged, I have become less quick, but am still much faster and more accurate at arithmetic than my students by and large. Probably this is not because I am a mathematician but rather because I was trained in high school this way as a child, before calculators were introduced. Calculator use is probably the main reason students today cannot calculate.
On the other hand it is true that when a mathematician makes a computational mistake he is usually not lost because he still knows what he is doing, so he does not worry about it so much. For example I assume the people on this site who go on and on about tensors being various kinds of matrices or symbols with indices, are non mathematicians who depend more on calculations for dealing with mathematics, and are not guided by the meaning of the concepts.
I would say a mathematician is someone who tries to understand, and add some unbderstanding to, number theory or geometry, or some such topic. Thus the person who pointed to the difference between whether 2+2=4 and why 2+2 =4 was on the right track, although perhaps not with such trivial examples.
Fermat asked himself which primes p could be expressed as n^2 + m^2 = p? for integers n,m. This is beautiful problem that can appeal to a junior high student, and whose most natural solution uses complex numbers (to show that if the equation above has a solution, then p = (n+mi)(n-mi) would no longer be prime over the complex numbers.)
I am not particularly advanced as mathematicians go, and I have recently been curious as to why the dimension of the space of sections h^0(L) of a line bundle L on a compact Riemann surface X is equal to
1-genus(X) + degree(L) + h^0(K-L), where K is the cotangent bundle of X.
and why generalizations of this to higher dimensional manifolds hold.
Mathematicians are more likely to wonder why that is true than why 2+2 = 4, but they still might remember it wrong as 1-genus(X) + degree(L) - h^0(K-L), for instance.
I was once amused while reading the works of a much stronger mathematician than myself who asserted that no one could be a grownup algebraic geometer until he/she had learned the riemann roch theorem for surfaces, and then proceeded to state it incorrectly.
I am also proud of my recently acquired familiarity with that theorem and why it is true. I have even worked out detailed proofs of it in many important special cases, but I am not sure I can correctly state it either. So it is definitely true I care more why it is true than exactly what the formula says.
For example, the key point of the previous formula for line bundles on "curves" is the fact that the difference h^0(L) - h^0(K-L), although defined in terms of the analytic structure of X, can be expressed merely in terms of the topology of X and K, i.e. as 1-g + degree(L).
Now simply knowing there is such a formula is already something, and then knowing the actual formula is a bit more. (for surfaces i will guess the formula says that h^0(L)+h^0(L) >= (1/2)(L.([L-K]) + (1/12)(K^2 + chi(top)).
but i am not sure and would have to go through the proof to be sure.
the point again is that a certain combination of h^0's on the left equals a number on the right which is purely topological.
In elementary mathematics the more difficult question of whether a problem does or does not have a solution is often ignored, and the explicit solution to a problem which does have one, is merely displayed without comment, to be memorized.
it is more enlightening to ask why one problem has a solution, but another similar one does not. E.g. why are there solution formulas for polynomial equaitons of degrees 1,2,3,4, but none in general for degree 5?
This is a much deeper and more interesting question than merely knowing what is the explicit formula for a 4th degree polynomial. or in calculus, not just what is the integral of this function, but are there any functions which do not have integrals? if so, which ones do have?
Mathematicians are not mindless memorizers, or calculating machines, they are curious scientists.
Still it is likely thay have more than the average skill in the simplest mechanical aspects of their subject. Julia Child probably knew how to break eggs too, even if she declined to do so for Martha Stewart.
peace
Real math (pure math anyway) is much more like philosophy since it's more about concepts than a science or calculuations, I think.
) said he will give me the power to know everything that is going on right now (including everything in all the journals 
), and all I have to do is lose the ability to remember names (including gf's name), I would.
