Heather has shared some thoughts on the significance of the statement that 2+2=4 with reference to a statement by Will Smith.
Smith is clearly talking rubbish, I have no idea whether he thought about what he was saying but I see an opportunity to advise film producers who offer to pay Will $5 million to appear in a film, and offer it in two instalments of $2 million, Will would quickly revise his beliefs.
The statement "two plus two is four" is often used to close down discussion. I associate it with a rejection of the possibility that mathematics is socially constructed: "two plus two was four at the time of the dinosaurs". Of course it was not because (as far as I know) dinosaurs did not speak English. A lot of mathematics is tautology and if we define two as "one plus one" and four as "one plus one plus one plus one" then two plus two is "(one plus one) plus (one plus one)" which is identical to "one plus one plus one plus one". If I define five as "one plus one plus one plus one" then two plus two is five. If I am working in base three, 2+2=10. But I doubt this was what Will Smith was thinking about in the interview, rather I think he is using mathematics metaphorically, a better analogue would have been Euclidean geometry, if we constrain ourselves to the parallel postulate then the sum of the internal angles of a triangle is 180 degrees, if we abandon the postulate we discover more interesting geometries. I like to think Will is presenting a metaphor.
Michael brings in Descartes and Leibnitz. One of the talks that I need to follow up on was Norma's because of what she said about seventeenth century scientists seeing nature as a performance. This is interesting to me because of the problem Pragmatists have with the idea of scientific objectivity, i.e. the scientist as uninvolved spectator rather than participant - this is important in understanding economics. It is normal to see the statement 2+2=4 as 'god given' in some sense (it was true at the time of dinosaurs). An alternative explanation is 2+2=4 works practically (this is why we can disabuse Will of his beliefs), and we reject such statements when they prove to be practically untrue, as when Gauss realised that in surveying the internal angles of a triangle do not always sum to 180 degrees.
My recollection of Orwell's discussion of 2+2=4 in 1984 is centred on empiricism, that the Party in denying 2+2=4 it was asking you to deny the evidence of your own experience.
There were a number of mentions of Holmes-Moriarty at the workshop. Will's comments connect to this via economics. The Austrian economist Oskar Morgenstern became involved in the Vienna Circle of mathematicians and philosophers, never as an active participant but as a bridge between them and economics. In 1935 he presented a paper to the mathematicians associated with the Vienna Circle on the problem of perfect foresight. Morgenstern often referred to an episode in Conan Doyle's story 'The Final Problem', which describes the 'final' intellectual battle between Sherlock Holmes and the mathematician, Professor Moriarty which results in them both falling to apparent death in the Reichenbach Falls. At the start of the adventure Holmes and Watson are trying to flee to the Continent, pursued by the murderous Moriarty. Watson and Holmes are sat on the train to the Channel ports
For Morgenstern this captured the fundamental problem of economics. While Frank Knight had realised that profit was impossible without unquantifiable uncertainty, Mogernstern came to think that perfect foresight was pointless in economics. If the world was full of Laplacian demons making rational decisions then everything would, in effect, grind to a halt with the economy reaching its equilibrium where it would remain forever. There is a story in the Zen tradition of two Go masters playing a game. The first player places their stone on the board, and after considering the implications of this first move, the second player concludes that they have lost and resigns. This is the consequence of perfect foresight.
Morgenstern offers a more individualistic alternative, writing
The imp of the perverse would confound Laplace's demon by doing something unexpected, irrational and inspired. It was this feature of the social science that makes it fundamentally different from the natural sciences, since physics is never perverse. It is also a justification for the Austrian economic schools interest with individuality
Heather argues this is a neoliberal position, but I cannot see the link based on how I understand neoliberalism. From the economic perspective (where the term originates, I believe) neoliberalism is a synthesis of classical liberalism and strong states, it was a "middle way" between free-markets and state-intervention. Pinochet's Chile characterises neoliberalism; Reagan, Thatcher, Clinton, Blair and Bush are neoliberals because they head strong centralised governments; UKIP are neoliberals, Ayn Rand and the Tea Party are not because they look to restrict the role of governments and so are classical liberals. Will Smith is taking a liberal position, not a neoliberal position, and that tradition has its roots in Locke's politics and empiricism.
There is a link to my research here. The Scholastics followed Aristotle in recognising the vital role markets played in keeping societies together (this is illiberal-see this paper that coined the term neoliberal) and they began to realise that markets were places were communities determined an 'objective' market price based on collective, not personal, need. This diverged from Aristotle who had only considered individual exchange in Nicomachean Ethics. In addition, the 'market price' was impersonal, it did not matter whether you were an Emperor, king, lord, knight, peasant or priest, the price was the price and could not be controlled. This had theological significance for Aquinas, the order in markets, what Adam Smith would term "the hidden hand'' five hundred years later, implied to the Dominican that there was an 'active' intelligence guiding the market. Not only was there order in nature but there was order in the cities, an order independent of the will of Kings and Emperors, evidence for God's existence. This was starkly demonstrated in 1304-1305 when economic instability and a market failure led the French King, Philip the Fair, to issue decrees fixing the price of bread. His decrees failed spectacularly, clear evidence to the scholars that it was 'God' and not the King, who governed nature. This episode can be seen as one of the streams that flowed into Enlightenment conceptions of democracy.