All the Ways of Intuiting 1729

Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning the number 1729. The idea of running through the cubes of all integers from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition of 1729 as the smallest positive integer that can be written in two distinct ways as the sum of two integral cubes is inappropriate and obscures the workings of the naive mathematical mind. To be sure, a computer-mind could come up with that list at a wink. But what would induce it ...

 Beginning to Reason

Beginning to reason is like stepping onto an escalator that leads upward and out of sight. Once we take the first step, the distance to be traveled is independent of our will and we cannot know in advance where we shall end.... If we do not understand what an escalator is, we might get on it intending to go a few meters, only to find that once we are on, it is difficult to avoid going all the way to the end. Similarly, once reasoning has got started it is hard to tell where it will stop. The...

 Human Combinatoric Reasoning

Humans, of course, were not created in a state of Original Reason. We descended from apes, spent hundreds of millennia in small bands, and evolved our cognitive processes in the service of hunting, gathering, and socializing. Only gradually, with the appearance of literacy, cities, and long-distance travel and communication, could our ancestors cultivate the faculty of reason and apply it to a broader range of concerns, a process that is still ongoing. One would expect that as collective rati...

 Humans Are Abstraction Masters

What distinguishes us from cavemen is the level of abstraction we can reach. Abstraction enabled humans to move from barter to money, and from gold coins to plastic cards. These days, what's left of "money" is often just an account record we read on a computer screen, and soon it could just be a line of code in a bitcoin ledger. Today, abstraction is all around us — and math is the language of abstraction. In the words of the great mathematician Henri Poincare, mathematics is valuable beca...

 Analytical Thinking Reduces Religiosity

Some have argued that belief in God is intuitive, a natural (by-)product of the human mind given its cognitive structure and social context. If this is true, the extent to which one believes in God may be influenced by one’s more general tendency to rely on intuition versus reflection. Three studies support this hypothesis, linking intuitive cognitive style to belief in God. Study 1 showed that individual differences in cognitive style predict belief in God. Participants completed the Cogni...

 Associative vs. Analogous Reasoning

There are many different types of reasoning, but not too many have been successfully automated beyond deductive linear reasoning and various statistical methods. What alternative methods has the Web facilitated? One obvious candidate is associative reasoning, where reasoning on the basis of associations - which can be extremely unpredictable and personalized - takes one down a train of thought [202]. So, for example, the classic case of associative reasoning is given in Proust's novel Remembr...