Few savants can articulate their condition and explain how their minds work. Daniel Tammet can, and he's spoken with New Scientist to shed some light into his proficiency with numbers—a skill that allowed him to set a record by memorizing pi to the 22,514th digit.
Tammet says he thinks about numbers in much the same way other people think about language. Rather than being "separate" and "atomized" items, numbers are part of an interconnected web:
You wouldn't use a word like "giraffe" without understanding what the words "neck" or "tall" or "animal" mean. Words only make sense when they are in this web of interconnected meaning and I have the same thing with numbers. Numbers belong to a web. When somebody gives me a number, I immediately visualise it and how it relates to other numbers. I also see the patterns those relationships produce and manipulate them in my head to arrive at a solution, if it's a sum, or to identify if there is a prime.
For Tammet, numbers also have textures, which helps him build relationships and organize things in his mind:
For me, the ideal lumpy number is 37. It's like porridge. So 111, a very pretty number, which is 3 times 37, is lumpy but it is also round. It takes on the properties of both 37 and 3, which is round. It's an intuitive and visual way of doing maths and thinking about numbers.
Check out the rest of the interview for Tammet's insight into how he manages to learn languages quickly, and how he even constructed his own language to convey some eerily specific concepts.