What is mathematical
intelligence? 
Definition* 
Mathematical and logical
intelligence is the ability to quickly understand
mathematical and logical problems. People with mathematical
intelligence can easily translate abstract problems into
mathematical notation ("writing in Greek"). They like to "play"
with numbers and detect (numeric and visual) patterns. It is
easy for them to mentally carry out complicated calculations. 
Mathematically gifted people have
uncovered general laws from empirical observations and have
formulated these laws in mathematical equations. The law of
gravity, or the famous relation between energy and mass have
been specified in elegant mathematical equations by Newton and
Einstein. 
Some people with mathematical
intelligence can do astounding numeric calculations in their
head, such as carrying out multiplication of large numbers or
identifying large prime numbers. People with mathematical and
logical intelligence are usually very well organized, precise in
their use of language, and pedantic  sometimes to the extent
that others find pathological. There are numerous examples of
mathematical geniuses that have rather limited social, emotional
and practical intelligence. However, mathematical intelligence
is often related to musical talent. Johann Sebastian Bach could
intuitively compose music (fugue), which strictly follows certain
mathematical patterns. 
Mathematical and logical
intelligence include the following mental abilities: 
a) 
the
mental ability to comprehend abstract concepts, 
b) 
the
mental ability to quickly make logical deductions, 
c) 
the
ability to imagine general solutions for specific problems, 
d) 
the
ability to recognize (numeric, visual, or logical) patterns, 
e) 
the
mental ability to develop rational strategies for games
("prisoner's dilemma") 
f) 
the
mental ability to create efficient (numeric) algorithms or
analytical solutions for very complex (procedural) problems
("traveling salesman", optimization theory) 

* I
am very uncomfortable writing something about mathematics, because
I know very little about it. You have to read this description
as the impressions of an outsider, who has observed for a long
time colleagues with advanced mathematical abilities. 
