INTPs are pensive, analytical folks. They may venture so deeply into thought as to seem detached and often actually are oblivious to the world around them.
Precise about their descriptions, INTPs will often correct others (or at least be sorely tempted to) if the shade of meaning is a bit off. This sometimes annoying habit is at once also a strength in grammarians, a perfect arena for the literal INTP.
INTPs are relatively easy-going and amenable to almost anything until their principles are violated, about which they may become outspoken and inflexible. But they usually return to a reserved albeit benign ambiance, not wishing to make spectacles of themselves.
A major concern for INTPs is the haunting sense of impending failure. They spend considerable time second-guessing themselves. The open-endedness (from Perceiving) conjoined with the need for competence (NT) is expressed in a sense that one's conclusion may well be met by an equally plausible alternative solution, and that, after all, one may very well have overlooked some critical bit of data. An INTP arguing a point may very well be trying to convince himself as much as his opposition. In this way INTPs are markedly different from INTJs, who are much more confident in their competence and willing to act on their convictions.
Mathematics is a system where many INTPs love to play, similarly languages, computer systems--potentially any complex system. INTPs thrive on systems. Understanding, exploring, mastering, and manipulating systems can overtake the INTP's conscious thought. This fascination for logical wholes and their inner workings is often expressed in a detachment from the environment, a concentration where time is forgotten and extraneous stimuli are held at bay.
INTPs and Logic -- One of the tipoffs that a person is an INTP is her obsession with logical correctness. Errors are not often due to poor logic -- apparent faux pas in reasoning are usually a result of 'overlooking details' or of incorrect context.