Final answer:
The name of the mathematician is likely 'Fermat', following the cryptic instructions of changing specified letters to their corresponding positions in the alphabet.
Step-by-step explanation:
The question asks for the name of a famous mathematician that starts with B, the second letter is E, the fifth letter is R. The name has six letters in the first name, and ten letters in the last name. We can apply a cryptic clue provided: Change the first consonant in the word to the fifth consonant in the alphabet and change the second to the fourteenth letter in the alphabet.
Following the clues, we know the first name should start with the fifth consonant 'F', since 'B' is the second consonant, and changing 'B' to 'F' fulfills that condition. The second letter is given as 'E'. No change is mentioned here. The fifth letter should be 'R'. For the second 'e' in the name, we need to change it to the fourteenth letter in the alphabet, which is 'N'.
Piecing these clues together, a likely first name that fits all these criteria is 'Fermat'. The last name with ten letters is not provided by the clues, but knowing a famous mathematician with this first name is Pierre de Fermat, we can be quite confident in this solution.