Logic - 3 Quarks Daily

by Ashutosh Jogalekar

*Gödel and Einstein in Princeton (Source: Emilio Segre Visual Archives)*

Two men walking in Princeton, New Jersey on a stuffy day. One shaggy-looking with unkempt hair, avuncular, wearing a hat and suspenders, looking like an old farmer. The other an elfin man, trim, owl-like, also wearing a fedora and a slim white suit, looking like a banker. The elfin man and the shaggy man used to make their way home from work every day. Passersby and motorists would strain their heads to look. Everyone knew who the shaggy man was; almost nobody knew who his elfin companion was. And yet when asked, the shaggy man would say that his own work no longer meant much to him, and the only reason he came to work was to have the privilege of walking home with the elfin man. The shaggy man was Albert Einstein. His walking companion was Kurt Gödel.

What made Gödel, a figure unknown to the public, so revered among his colleagues? The superlatives kept coming. Einstein called him the greatest logician since Aristotle. The legendary mathematician John von Neumann who was his colleague argued for his extraction from fascism-riddled Europe, writing a letter to the director of his institute saying that “Gödel is absolutely irreplaceable; he is the only mathematician about whom I dare make this assertion.” And when I made a pilgrimage to Gödel’s house during a trip to his native Vienna a few years ago, the plaque in front of the house made his claim to posterity clear: “In this house lived from 1930-1937, the great mathematician and logician Kurt Gödel. Here he discovered his famous incompleteness theorem, the most significant mathematical discovery of the twentieth century.”

*The author in front of the house in Vienna where Gödel was living with his mother and brother when he proved his Incompleteness Theorems*

The reason Gödel drew gasps of awe from colleagues as brilliant as Einstein and von Neumann was because he revealed a seismic fissure in the foundations of that most perfect, rational and crystal-clear of all creations – mathematics. Of all the fields of human inquiry, mathematics is considered the most exact. Unlike politics or economics, or even the more quantifiable disciplines of chemistry and physics, every question in mathematics has a definite yes or no answer. The answer to a question such as whether there is an infinitude of prime numbers leaves absolutely no room for ambiguity or error – it’s a simple yes or no (yes in this case). Not surprisingly, mathematicians around the beginning of the 20th century started thinking that every mathematical question that can be posed should have a definite yes or no answer. In addition, no mathematical question should have both answers. The first requirement was called completeness, the second one was called consistency. Read more »

by Scott F. Aikin and Robert B. Talisse

Plato In last month's post, we contrasted a formal conception of argument with a dialectical one. We claimed that a dialectical model must be developed in order to capture the breadth not only of the good arguments we give, but also the bad. To review, the formal conception takes arguments as products, specifically as sets of claims with subsets of premises and conclusions. These arguments are understood as abstract objects, and they are, as one might say, purely logical entities. By contrast, the dialectical perspective sees arguments as more like processes; they happen, unfold, emerge, and they take various twists and turns. They erupt, get heated, go nowhere or cover ground. In short, on the dialectical conception, arguments are events of reason-exchange between people. And just as there are rules for argument-construction as formal entities, there are rules for good argument-performance as interpersonal processes.

The first thing to note is that argument-as-process is a turn-taking game. Alfred and Betty may disagree – perhaps Alfred accepts some proposition, p, and Betty rejects p. They aim to resolve their disagreement through argument. They could, of course, resolve the disagreement through other means – Alfred could threaten Betty, or bribe her – but they, instead, decide to enact a means of deciding the matter according to their shared reasons. That's argument, and the point is to share and jointly weigh the reasons. That's where the turn-taking is important. Alfred presents his reasons, and Betty presents hers. They respond to each other's reasons in turn.

A few things about the turn-taking are worth noting. When the sides present their respective cases, they present arguments in the formal sense – they articulate sets of claims comprised by premises and conclusions. The other side, then, may accept the premises but hold they don't support the conclusions, or they may hold that the premises themselves are false or unacceptable. Or they may change their minds and accept the conclusion. In that case, the argument concludes: Dispute resolved. Otherwise, what the two sides do is give each other reasons and then take turns giving each other reasoned feedback about how to change their arguments so they can rationally be better, or how they can change their views to fit with the rationally better reasons. When it's well-run, argument is a cooperative enterprise. Hence it's not uncommon to use the term argumentation in discussions of the dialectical conception of argument.

The turn-taking element of argumentation makes the feedback process possible. And this feedback process is what separates argumentation from simple speechmaking or sermonizing. But there's no guarantee that things will work out like they should. Sometimes, there are misfires in argumentation. One common misfire involves misrepresenting the other side in providing critical feedback. The straw man fallacy occurs specifically when one side strategically misrepresents the other side's arguments as weaker than those they actually gave. Straw-manning is a dialectical fallacy par excellence – it is a failure of the turn-taking element of proper argumentative exchange; it's a turn that doesn't properly respond to the contents of the previous turns.

Tag: Logic

Kurt Gödel’s Open World

Monday, November 11, 2013

Logic and Dialogue