Thinking Complete

In yesterday's post I briefly mentioned Claude Shannon, founder of information theory. It occurred to me that even most Oxford students wouldn't know who he was, despite his work being crucial for the subsequent development of the "Information Age". How many other intellectual revolutions in obscure fields haven't become common knowledge yet? Perhaps the best way of finding out is just to start listing the greatest and most influential intellectual achievements ever, and seeing if anyone else has more to add - particularly in the humanities and social sciences, where my knowledge is a little shallower. Also, is there anything you expect to deserve a place on this list in the next few years? I'm focusing on achievements which either founded or greatly influenced an entire field (also, although there are many amazing artistic and engineering achievements, to keep things relatively concise I'm not going to include either here). Of course this will be necessa

I want to investigate the nature of and limits of mathematics. To do that, I'll explore logic, set theory, computability, information theory and philosophy. Be warned that this essay is rather long; it starts relatively easy and gets significantly harder. I hope everyone will get something out of it (including my tutors!) but if you're new to mathematics, that might not be much. If you're philosophically inclined, definitely look at the last section on Truth and Induction. Any discussion or errata are of course very welcome. Let's start with some historical background. Although the Babylonians were able to do some interesting calculations (particularly astronomical), mathematics in the sense of deductive arguments from premises to conclusions began with Thales, Pythagoras and their disciples. Greek maths was focused on geometry, which was formalised by Euclid into 5 axioms - foundational assertions - from which he proved many theorems. Euclidean geometry (as formalis

I recently took part in a speech competition at the European Forum Alpbach (a conference, by the way, which I highly recommend; if you're interested, several hundred scholarships are available for students and recent graduates). The topic of the competition was "Tolerance and Conflict"; my argument, inspired by ideas in Effective Altruism , was that a societal focus on the "passive" virtue of tolerance has made modern Westerners associate moral goodness with "doing no harm", which leaves many important global issues - global warming, extreme poverty, disease pandemics - under-addressed (video here ). In hindsight, though, that isn't quite the right way to frame this point. Even if creating an ideally tolerant society would only require each individual to avoid discriminating against each other, in practice the fight against discrimination is an active one which has given birth to great leaders like Mandela, Gandhi and King. The admiration we feel