On Paradoxes (2010-2019)

"The paradox of artificial intelligence is that any system simple enough to be understandable is not complicated enough to behave intelligently, and any system complicated enough to behave intelligently is not simple enough to understand. The path to artificial intelligence, suggested Turing, is to construct a machine with the curiosity of a child, and let intelligence evolve." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)

"Paradoxes often arise because theory routinely refuses to be subordinate to reality." (Lawrence K Samuels, "Defense of Chaos: The Chaology of Politics, Economics and Human Action", 2013)

"Political structures are excessively paternalistic, and to maintain them requires a high level of energy. The massive amounts of energy they consume are unsustainable and invite political meltdowns, bailouts, and fallout. On the other hand, proponents of complexity theory take the paradigm–shattering view that less is more. They understand that, paradoxically enough, the complexity of simplicity is the key to the emergence of systems, repeatable patterns and the social glue that holds community together and creates order. Anyone can make simplicity complicated; it takes a true genius to make the complicated simple." (Lawrence K Samuels, "Defense of Chaos", 2013)

"For group theoretic reasons the most impressive paradoxical decompositions occur in dimension at least three, but there are also interesting decompositions in lower dimensions. For example, one can partition a disc into finitely many subsets and rigidly rearrange these subsets to form a square of the same area as the original disc. Even without the Axiom of Choice it is possible to construct some counterintuitive subsets of the plane, so Choice cannot be held responsible for all that is counterintuitive in geometry. Some more radical alternatives to the Axiom of Choice obstruct such constructions more effectively." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Intuition is reliable only in the limited environment in which it has evolved. Unable to abandon its prejudices completely, we must constantly question what appears to be obvious, often revealing conceptual problems and hidden paradoxes. One intuitive notion which is ultimately paradoxical is that of arbitrary collections."  (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The basis of complex systems is actually quite simple (and this is not an attempt to be paradoxical, like an art critic who describes a sculpture as 'big yet small'. What makes a system unpredictable and thus nonlinear (which includes you and your perceptual process, or the process of making collective decisions) is that the components making up the system are interconnected." (Beau Lotto, "Deviate: The Science of Seeing Differently", 2017)

"The human mind isn’t a computer; it cannot progress in an orderly fashion down a list of candidate moves and rank them by a score down to the hundredth of a pawn the way a chess machine does. Even the most disciplined human mind wanders in the heat of competition. This is both a weakness and a strength of human cognition. Sometimes these undisciplined wanderings only weaken your analysis. Other times they lead to inspiration, to beautiful or paradoxical moves that were not on your initial list of candidates." (Garry Kasparov, "Deep Thinking", 2017)

"Zero is not a point of non-existence. Zero is always a balance point of existents. The human understanding of 'zero' must undergo the most radical of all transformations. Most people, especially scientists, associate it with absolute nothingness, with non-existence. This is absolutely untrue. Or, to put it another way, we can define it in two ways: 1) nothing as non-existence, in which case it has absolutely no consequences but leads to all manner of abstract paradoxes and contradictions, or 2) nothing as existence, in which case it is always a mathematical balance point for somethings. It is purely mathematical, not scientific, or religious, or spiritual, or emotional, or sensory, or mystical. It is analytic nothing and whenever you encounter it you have to establish the exact means by which it is maintaining its balance of zero." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)

"An infinitesimal is a hazy thing. It is supposed to be the tiniest number you can possibly imagine that isn’t actually zero. More succinctly, an infinitesimal is smaller than everything but greater than nothing. Even more paradoxically, infinitesimals come in different sizes. An infinitesimal part of an infinitesimal is incomparably smaller still. We could call it a second-order infinitesimal." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)