Our molecular data, which were rigorously analyzed with phylogenetic methods, contradict species distinctions based upon non-rigorous intuition using highly variable morphological features, and we conclude that our rigorous analyses are more scientifically sound.
Yet, the theory has only become formal and general with the work by Ambrosio, Gigli and Savaré (which does not mean that proofs in [54] were not rigorous, but the intuition on the general structure still needed to be better understood).
As Tao notes, "It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.
"What we're doing is giving a rigorous analysis behind that intuition, and also saying that the estimate is a very good estimate, rather than some coarse estimate," MIT electrical engineering and computer science grad student Cameron Musco explains, previewing a new paper on the subject.
But perhaps intuition can be more rigorous than reason.
The numerical tools we have developed may be a way to build an intuition to guide a rigorous analysis of these phenomena.
Therefore, all of the rigorous theorems are merely guides to our intuition or expectations: empirical results from simulations are necessary to proceed further.
The first is that the engineer must solve the problems of integrating humans into machine systems by rigorous scientific methods and not rely on logic, intuition, or common sense.
Philosophically inclined thinkers have both historically and at present labored to shape the relevant intuition into a more formal, logically rigorous inference.
Despite many notable examples demonstrating the success of this iterative procedure, progress has been slow due to the ad hoc nature of its implementation: the iterations between the development of models and the production of data is still mostly guided by the intuition of the modellers, and no rigorous algorithms exist to render this process more systematic and less biased.
As such, though x-phi stresses intuitions be gathered in controlled, statistically rigorous, and systematic ways, conceptual analysis and the positive program as it is typically understood share a similar philosophical agenda and fate.
