What is the idea behind 2-spinor calculus? In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor calculus. The books of course goes down in detail explaining this, but I'd like an overview of what the philosophy is.
 A: A $2$-spinor $\psi\in V\cong\mathbb{C}^2$ is here a (left) Weyl spinor, so Penrose & Rindler are e.g. exploring the fact that the complexified Minkowski spacetime $\mathbb{C}^{1,3}\cong V\otimes \bar{V}$ is a tensor product of a left (=un-dotted) and a right (=dotted) Weyl-spinor representation, cf. e.g. this Phys.SE post.
This e.g. provides a convenient way to classify irreducible representations of the Lorentz group, cf. e.g. this Phys.SE post.
A: May I suggest a somewhat intuitive way to try to answer the question, inspired by the late Sir Michael Atiyah's view that "spinors are the square root of geometry".
Atiyah argues in his lecture "What is a spinor ?"   that just as the really deep part of complex numbers lies in their role in complex analysis and Cauchy's theorem etc. (rather than just as solutions to polynomials), so the deeper aspect of spinors in his view relates not so much to representation theory and its associated algebra, but rather in their geometric meaning.*
For what it's worth, I therefore take the idea (the philosophy even?) behind 2-spinors, and their calculus, to be that spinors allow the notion of extension (whether in space, in time, or even otherwise) to have associated with it a hidden extra degree of freedom that can be useful in the full range of spinor applications (e.g. in accounting for the topologies of spaces), and not only in the description of the extra degree of freedom of quantum spin.
Historically, the idea of spinors can be traced back to classical mechanics "Is there any Classical Mechanics system which needs to be described by a spinor?" prior even to Elie Cartan’s introduction to them in geometry in 1913, or their naming as “spinors” by the theoretical physicist Paul Ehrehfest in 1929.
Nevertheless, it's still not obvious to Atiyah himself what "the square root of geometry" actually means. As he is quoted:

“Just as understanding the square root of -1 took centuries, the same might be true of spinors.”

However, in an extract from a private communication with Simon Altmann, author of Rotations, Quaternions and Double Groups (OUP, 1986; Dover, 2005), Altmann takes issue with Atiyah, expressing a view that is closer to the earlier answer here, from @Qmechanic:

"I have the greatest admiration for Michael Atiyah but he seems to me here to cloud the waters unnecessarily. Mathematical objects are whatever they are by virtue of what they DO, i.e. by their relations between themselves and other objects. To look for more than this is not only unnecessary but can lead to confusion, as was the case with imaginary numbers.
Having said this, I nevertheless sympathize with Michael’s ‘surprise’ about the relation of spinors and geometry.
Here my view is as follows. Spinors are the elementary objects from which by forming tensor products the bases of the representations of the rotation group are constructed. The bases of dimension three can be identified with vectors, which are formed by tensor product of a spinor with itself. Since vectors span euclidian space, it shows that the latter is generated from spinors."

*Atiyah tended to mumble in the lecture, so it's not always entirely clear what he's saying at this point - as I hear it, he says "geometric meaning".
