see things differently

Posts tagged with logic

If you buy a loaf of bread from the supermarket both you and the supermarket (its shareholders, its employees, its bread suppliers) are made to some degree better off. How do I know? Because the supermarket offered the bread voluntarily and you accepted the offer voluntarily. Both of you must have been made better off, a little or a lot—or else you two wouldn’t have done the deal.

Economists have long been in love with this simple argument. They have since the eighteenth century taken the argument a crucial and dramatic step further: that is, they have deduced something from it, namely, Free trade is neat.

If each deal between you & the supermarket, and the supermarket & Smith, and Smith & Jones, and so forth is betterment-producing (a little or a lot: we’re not talking quantities here), then (note the “then”: we’re talking deduction here) free trade between the entire body of French people and the entire body of English people is betterment-producing. Therefore (note the “therefore”) free trade between any two groups is neat.

The economist notes that if all trades are voluntary they all have some gain. So free trade in all its forms is neat. For example, a law restricting who can get into the pharmacy business is a bad idea, not neat at all, because free trade is good, so non-free trade is bad. Protection of French workers is bad, because free trade is good. And so forth, to literally thousands of policy conclusions.

Deirdre McCloskey, Secret Sins of Economics

A wonderful essay. I’ll just add what I think are some common answers to common objections:

But it’s known that that stores loss-lead, that goods are bundled and tied. Models!
Choices are necessarily embedded. No one makes you shop at Wal-Mart. Sneer. So the government is going to tell us our true preferences?
People don’t always benefit from their voluntary choices. What’s wrong with you? Why don’t you behave more rationally?
Greedy! Envious! State interventionist!

The category of categories as a model for the Platonic World of Forms by David A Edwards & Marilyn L Edwards

Thales (7th cent. BC) made the first universal statement (proof w/o regard to the gods or mythology, just from pure reason)
pre-Greek mathematics was essentially engineering maths.
I owe ya a post on the illiterates in chapter 2 of James Gleick’s The Information. He tells the story of some illiterates in outer Soviet Union. According to the tale, they basically do not abstract at all. No abstract reasoning, no properties ascribed to members of a class, and so on.

It sounds kind of idyllic in the way of NYT tales of the Pirahã or Jill Bolte Taylor’s story of losing the logical half of her brain. I’m not sure if Thales set us on the path to Hell or Heaven.
“Plato set for himself the [goal] of extending geometry [beyond] triangles and circles and such, to all of human thought. He failed, but his vision has come to pass.”
Why did Lawvere succeed where Plato and Whitehead failed?
He had Descartes’ already-abstract notion of a function, along with
Eilenberg & Mac Lane’s notions of category and functor.
The definition of function for infinite sets is already implicit in the choice of “which set theory”.
Category theory, unlike earlier formalisations (think Peano arithmetic and Goedel’s proof), is stable to the “meta” step: you do 2-categories, you do n-categories … the abstraction is ultimately a k → k+1 kind of deal rather than a “And this is the ultimate finality!” kind of deal.

Primitives

I guess when most people hear the word “logic”, they think of

cold shoulders
loveless robots
a not-quite-rational preoccupation with principles, propositions, facts, and categorical truths over people, feelings, subjective impressions
making the wrong decision by thinking in straight lines

instead of reaction clouds and propagation networks and games and what-he-thinks-she-thinks-I-know and things that swirl or squish

But when I hear the word “logic”, I think of

the beautiful scrivenings of academics who have worked out the shapes of network relations, time & causality, Boolean algebra; untangled modality and self-reference;
the complete classification of all finite simple groups. This is a feat so massive that I have trouble describing its hugeness. It’s like, we pathetic worms called human beings have uncovered something real and unerring and definite and complete about the Universe. Not just our universe, but any possible universe. We’ve uncovered laws that constrain even G–d.
I remember looking out over a field of grasses wavering in the wind, a vector field billowing like waves of water, and thinking back on a talk by Stephen Wolfram.

Everything is computation, he said. I saw in my mind’s eye the bits of wind and of grass digitised—not necessarily in Q*bert blocks,

but in a dynamic and skeletal topology.

Conway’s Game of Life

playing out at a femto scale—the bits of air blowing on the bits of grain, the grains reacting back; the chromoclouds dynamically crunching their numbers in Douglas Noël Adams’ Monte Carlo simulation to compute the question whose answer is 42; the photo-rods −ct of timespace digitally passing into my retina, lighting off neural logics;

muscles contracting only at the micro scale because their chemical pathways each consist of thousands of nanoscale back-and-forths;

phospholipid cell-walls digitally repelling or allowing the chemicals for the micro breathers that make me up (and yet don’t—fuzzy logic? help?)
in short, Logic as the Hand of G-d, making everything in the Universe.

If God speaks to man, he undoubtedly uses the language of mathematics.

—Henri Poincaré

http://ej.iop.org/images/0295-5075/76/2/257/Full/img14.gif

Logicians on safari

Some cute applications of computability theory:

We now know that, if an alien with enormous computational powers came to Earth, it could prove to us whether White or Black has the winning strategy in chess. To be convinced of the proof, we would not have to trust the alien or its exotic technology, and we would not have to spend billions of years analyzing one move sequence after another. We’d simply have to engage in a short conversation with the alien about the sums of certain polynomials over finite fields.

There’s a finite (and not unimaginably-large) set of boxes, such that if we knew how to pack those boxes into the trunk of your car, then we’d also know a proof of the Riemann Hypothesis. Indeed, every formal proof of the Riemann Hypothesis with at most (say) a million symbols corresponds to some way of packing the boxes into your trunk, and vice versa. Furthermore, a list of the boxes and their dimensions can be feasibly written down.

Supposing you do prove the Riemann Hypothesis, it’s possible to convince someone of that fact, without revealing anything other than the fact that you proved it. It’s also possible to write the proof down in such a way that someone else could verify it, with very high confidence, having only seen 10 or 20 bits of the proof.