Only small tidbits of math remain unresolved for Rubik’s Cube. While God’s number is 20, it’s unknown exactly how many of the 43,252,003,274,489,856,000 combinations require a whole 20 moves to be solved.

In particular Erwin Schrödinger is said (Wigner (1981)) to have spoken of the Gruppenpest (German for “plague of group theory”) which ought to be abandoned. In his autobiography John Slater, an MIT physicist, claimed: It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their “Gruppenpest”: the pest of the group theory… The authors of the “Gruppenpest” wrote papers which were incomprehensible to those like me who had not studied group theory, in which they applied these theoretical results to the study of the many electron problem. The practical consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. Yet there were no good texts from which one could learn group theory. It was a frustrating experience, worthy of the name of a pest. I had what I can only describe as a feeling of outrage at the turn which the subject had taken… As soon as this [Slaters] paper became known, it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as “Slater has slain the ‘Gruppenpest’”. I believe that no other piece of work I have done was so universally popular.

This is the theory of the extended mind, introduced more than two decades ago by the philosophers Andy Clark and David Chalmers. A 1998 article of theirs published in the journal Analysis began by posing a question that would seem to have an obvious answer: “Where does the mind stop and the rest of the world begin?” They went on to offer an unconventional response. The mind does not stop at the usual “boundaries of skin and skull,” they maintained. Rather, the mind extends into the world and augments the capacities of the biological brain with outside-the-brain resources.

One last resource for augmenting our minds can be found in other people’s minds. We are fundamentally social creatures, oriented toward thinking with others. Problems arise when we do our thinking alone — for example, the well-documented phenomenon of confirmation bias, which leads us to preferentially attend to information that supports the beliefs we already hold. According to the argumentative theory of reasoning, advanced by the cognitive scientists Hugo Mercier and Dan Sperber, this bias is accentuated when we reason in solitude. Humans’ evolved faculty for reasoning is not aimed at arriving at objective truth, Mercier and Sperber point out; it is aimed at defending our arguments and scrutinizing others’. It makes sense, they write, “for a cognitive mechanism aimed at justifying oneself and convincing others to be biased and lazy. The failures of the solitary reasoner follow from the use of reason in an ‘abnormal’ context’” — that is, a nonsocial one. Vigorous debates, engaged with an open mind, are the solution. “When people who disagree but have a common interest in finding the truth or the solution to a problem exchange arguments with each other, the best idea tends to win,” they write, citing evidence from studies of students, forecasters and jury members.

First, according to Trumbull, Olson underestimates diffuse groups’ ability to develop compelling narratives about how they serve the public interest. In fact, weak, diffuse groups have a paradoxical political advantage: precisely because they are weak and diffuse, the public sees them as less self-interested and thus comparatively trustworthy. Second, Olson also underestimates the power of ideological motivation, rather than just money and concentration, to spur activism. Third, “diffuse interests can be represented without mobilization,” thanks to activism by politicians and government officials who take up their cause. (FDR started a federal pension program at a time when “retirees,” as a self-identified social class, did not yet exist. The program created the constituency, rather than the other way around.) Fourth, weak or diffuse interests can link up with concentrated groups to amplify their effectiveness, as when consumers align with exporters to oppose trade protections or when free-speech advocates join with political parties to oppose campaign-finance limits.