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Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century

2 CommentsPosted by on May 23, 2011

In 1931, the young mathematician Kurt Gödel made a landmark discovery, as powerful as anything Albert Einstein developed.

In one salvo, he completely demolished an entire class of scientific theories.

Gödel’s discovery not only applies to mathematics but literally all branches of science, logic and human knowledge. It has earth-shattering implications.

Oddly, few people know anything about it.

Allow me to tell you the story.

Mathematicians love proofs. They were hot and bothered for centuries, because they were unable to PROVE some of the things they knew were true.

So for example if you studied high school Geometry, you’ve done the exercises where you prove all kinds of things about triangles based on a set of theorems.

That high school geometry book is built on Euclid’s five postulates. Everyone knows the postulates are true, but in 2500 years nobody’s figured out a way to prove them.

Yes, it does seem perfectly “obvious” that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that Euclid’s postulates are a reasonable, and in fact necessary, set of 5 assumptions.

Towering mathematical geniuses were frustrated for 2000+ years because they couldn’t prove all their theorems. There were so many things that were “obviously true,” but nobody could find a way to prove them.

In the early 1900′s, however, a tremendous wave of optimism swept through mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) became convinced that they were rapidly closing in on a final synthesis.

A unifying “Theory of Everything” that would finally nail down all the loose ends. Mathematics would be complete, bulletproof, airtight, triumphant.

In 1931 this young Austrian mathematician, Kurt Gödel, published a paper that once and for all PROVED that a single Theory Of Everything is actually impossible. He proved they would never prove everything. (Yes it sounds a little odd, but is exactly what Kurt Gödel did)

Gödel’s discovery was called “The Incompleteness Theorem.”

If you’ll give me just a few minutes, I’ll explain what it says, how Gödel proved it, and what it means – in plain, simple English that anyone can understand.

Gödel’s Incompleteness Theorem says:

“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”

You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.

Stated in Formal Language:Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.

Syllogism:

1. All non-trivial computational systems are incomplete

2. The universe is a non-trivial computational system

3. Therefore the universe is incomplete



You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.


You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.


Gödel proved that there are ALWAYS more things that are true than you can prove. Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.


Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it’s equally true in science or language and philosophy.


Gödel created his proof by starting with “The Liar’s Paradox” — which is the statement

“I am lying.”

“I am lying” is self-contradictory, since if it’s true, I’m not a liar, and it’s false; and if it’s false, I am a liar, so it’s true.

Gödel, in one of the most ingenious moves in the history of math, converted this Liar’s Paradox into a mathematical formula. He proved that no statement can prove its own truth.

You always need an outside reference point.

The Incompleteness Theorem was a devastating blow to the “positivists” of the time. They insisted that literally anything you could not measure or prove was nonsense. He showed that their positivism was nonsense.

Gödel proved his theorem in black and white and nobody could argue with his logic. Yet some of his fellow mathematicians went to their graves in denial, believing that somehow or another Gödel must surely be wrong.

He wasn’t wrong. It was really true. There are more things that are true than you can prove.

A “theory of everything” – whether in math, or physics, or philosophy – will never be found.  Because it is mathematically impossible.

OK, so what does this really mean? Why is this super-important, and not just an interesting geek factoid?

Here’s what it means:

  • Faith and Reason are not enemies. In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.
  • All closed systems depend on something outside the system.
  • You can always draw a bigger circle but there will still be something outside the circle.

Reasoning inward from a larger circle to a smaller circle (from “all things” to “some things”) is deductive reasoning.

Example of a deductive reasoning:

1.    All men are mortal
2.    Socrates is a man
3.    Therefore Socrates is mortal

Reasoning outward from a smaller circle to a larger circle (from “some things” to “all things”) is inductive reasoning.

Examples of inductive reasoning:

1. All the men I know are mortal
2. Therefore all men are mortal

1. When I let go of objects, they fall
2. Therefore there is a law of gravity that governs all falling objects

Notice than when you move from the smaller circle to the larger circle, you have to make assumptions that you cannot 100% prove.

For example you cannot PROVE gravity will always be consistent at all times. You can only observe that it’s consistently true every time.

Nearly all scientific laws are based on inductive reasoning. All of science rests on an assumption that the universe is orderly, logical and mathematical based on fixed discoverable laws.

You cannot PROVE this. (You can’t prove that the sun will come up tomorrow morning either.) You literally have to take it on faith. In fact most people don’t know that outside the science circle is a philosophy circle. Science is based on philosophical assumptions that you cannot scientifically prove. Actually, the scientific method cannot prove, it can only infer.

(Science originally came from the idea that God made an orderly universe which obeys fixed, discoverable laws – and because of those laws, He would not have to constantly tinker with it in order for it to operate.)

Now please consider what happens when we draw the biggest circle possibly can – around the whole universe. (If there are multiple universes, we’re drawing a circle around all of them too):

  • There has to be something outside that circle. Something which we have to assume but cannot prove
  • The universe as we know it is finite – finite matter, finite energy, finite space and 13.8 billion years time
  • The universe (all matter, energy, space and time) cannot explain itself
  • Whatever is outside the biggest circle is boundless. So by definition it is not possible to draw a circle around it.
  • If we draw a circle around all matter, energy, space and time and apply Gödel’s theorem, then we know what is outside that circle is not matter, is not energy, is not space and is not time. Because all the matter and energy are inside the circle. It’s immaterial.
  • Whatever is outside the biggest circle is not a system – i.e. is not an assemblage of parts. Otherwise we could draw a circle around them. The thing outside the biggest circle is indivisible.
  • Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect.

We can apply the same inductive reasoning to the origin of information:

  • In the history of the universe we also see the introduction of information, some 3.8 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.
  • The information had to come from the outside, since information is not known to be an inherent property of matter, energy, space or time.
  • All codes we know the origin of are designed by conscious beings.
  • Therefore whatever is outside the largest circle is a conscious being.

When we add information to the equation, we conclude that not only is the thing outside the biggest circle infinite and immaterial, it is also self-aware.

Isn’t it interesting how all these conclusions sound suspiciously similar to how theologians have described God for thousands of years?

Maybe that’s why it’s hardly surprising that 80-90% of the people in the world believe in some concept of God. Yes, it’s intuitive to most folks. But Gödel’s theorem indicates it’s also supremely logical. In fact it’s the only position one can take and stay in the realm of reason and logic.

The person who proudly proclaims, “You’re a man of faith, but I’m a man of science” doesn’t understand the roots of science or the nature of knowledge!

Interesting aside…

If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:

“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”

If you know Gödel’s theorem, you know all systems rely on something outside the system. So according to Gödel’s Incompleteness theorem, the folks at Infidels cannot be correct. Because the universe is a system, it has to have an outside cause.

Therefore Atheism violates the laws mathematics.

The Incompleteness of the universe isn’t proof that God exists. But… it IS proof that in order to construct a consistent model of the universe, belief in God is not just 100% logical… it’s necessary.

Euclid’s 5 postulates aren’t formally provable and God is not formally provable either. But… just as you cannot build a coherent system of geometry without Euclid’s 5 postulates, neither can you build a coherent description of the universe without a First Cause and a Source of order.

Thus faith and science are not enemies, but allies. They are two sides of the same coin. It had been true for hundreds of years, but in 1931 this skinny young Austrian mathematician named Kurt Gödel proved it.

No time in the history of mankind has faith in God been more reasonable, more logical, or more thoroughly supported by rational thought, science and mathematics.

“Math is the language God wrote the universe in.” –Galileo Galile, 1623

Always Yours — As Usual — Saurabh Singh

Source: http://www.cosmicfingerprints.com/incompleteness/

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I Keep Thinking About:

Comments Off on I Keep Thinking About:Posted by on December 17, 2009

“Why majority of social scientists often try to turn mathematicians, which they are not and neither it’s expected of them…Gentlemen you are expected to have Mastered humanities …society won’t take mathematical equations in exchange…equations will be taken as…adding salt to injury …if you do so…You are the one who destabilized the fine balance of Psycho-Socio-Economic elements of Humanity…we can hire a lot of Mathematics graduates…But Philosophers, Thinkers, Social Scientists are Rare Creations of Almighty…We are Searching them Pan Disciplines identified till Date.”

“Want to keep Yourself Updated about Saurabh’s Every move ! — Then Click The Google Button Just Below. It would do the Job, So Now You can Relax.”

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Finance in History

1 CommentPosted by on December 10, 2009

Dear Learned Audiences,

History is not just a forte of Kings, Emperors, Social Workers, Leaders and so on only. It keeps on silently recording the numerous developments happening in the various spheres of learning too. Sometimes, it may be in the form of thought other day it may be principles and following day may be for practices.

Finance in History

If you are doomed to repeat history, let’s hope you can pick your era. Once upon a time, business bankruptcies resulted in jail time (if you were lucky), treasurers defended their funds with a sword, and financial planning was tested by plagues and fire. Things improved during the American Revolution, when the father of our country also proved to be one of its best bookkeepers. But accounting couldn’t keep up during the Industrial Revolution, with disastrous consequences for workers. If you tend to think of history as the third quarter of the last fiscal year, it may be time to learn a little bit more about your profession’s checkered past.

The 17th-century business world revealed in Samuel Pepys’s famous diary is not so far removed from our own.

“Most happy in the keeping of all my accounts, for that after all the changings and turnings necessary in such an account, I find myself right to a farthing in an account of 127,000 pounds.” — Samuel Pepys’s diary entry, August 20, 1666

Public officials in 17th-century England had not yet refined the notion that one has to pay to play; that is, pony up political contributions to obtain government contracts or favors. But when Samuel Pepys was an important naval administrator in London during the mid-1660s, the basic idea was well understood. Like others similarly situated, Pepys gladly accepted gifts, and he recognized the debt he incurred in accepting them.

We know this from reading Pepys’s diary, regarded by many as the greatest in the English language. Between January 1, 1660, and May 31, 1669, Pepys (rhymes with “keeps”) chronicled his everyday life, from his professional concerns to his sexual escapades, from the state of the financial accounts he kept to the painful progress of his kidney stone. The practice of diary keeping began to catch on during the 17th century, according to Pepys biographer Claire Tomalin. But his is prized for its confessional insights, large cast of characters, accounts of significant events, and entertaining narrative, combining to reveal a singular sensibility.

“What is extraordinary is that he went into areas no one else considered recording, looked at himself with as much curiosity as he looked at the exterior world, weighing himself and the world equally in the balance,” observes Tomalin in Samuel Pepys: The Unequalled Self (2002). Writing for his eyes only, Pepys used a private shorthand and, in especially delicate passages, French. His six-volume diary was only deciphered and published in the 1820s, more than 100 years after his death.

To historians, Pepys was an invaluable chronicler of a period when the press was censored by the government of Charles II. From him we have poignant accounts of the Great Plague, which decimated England in 1665, and of the Great Fire of London, which destroyed half the city in 1666. On a more personal scale, Pepys supplied entertaining accounts of his financial wheelings and dealings as a government administrator.

“The Diary sends a beam of light into the way in which government officers and businessmen worked together, through clubs, through hospitality, through trips that mixed business and pleasure, through well-chosen and discreetly given presents and through cultivating the friendship of those in a position to be helpful in giving contracts or licenses,” observes Tomalin. “The circumstances were different, but there is something eerily familiar about it too: today’s arms and building contracts, entertainment of clients, quiet words at the club, conferences in luxury hotels, boardroom rivalries and contributions to favourite charities are all in the same tradition. Pepys was, among other things, mapping a recognizably modern world.”

Accounting for the Royal Navy

As one learns from the diary, Pepys was ambitious, intelligent, and well connected. Born in 1633, he never became a sailor, but gained an accounting post in the British Navy and turned it to steady profit. Pepys had the good fortune to capitalize on his family’s one political connection: he was a distant cousin to Sir Edward Montagu, later the Earl of Sandwich. Oliver Cromwell put Montagu in joint command of the British fleet, and the 27-year-old Pepys sailed in on Montagu’s coattails. In 1660 Pepys was appointed Clerk of the Acts to the Navy Board, and as such was responsible for requesting funds from Parliament and dispensing them to build the navy and keep it afloat.

Pepys advanced steadily during the next 13 years, eventually becoming Secretary of the Admiralty. Anyone who wanted a government contract to supply the Royal Navy had to go through his office. Shipbuilders, victuallers, slopsellers, and many others did their best to curry favor with the young finance minister.

On August 16, 1660, in the first year of his diary, Pepys recorded a telling conversation he had with Lord Sandwich. Riding across town in a coach, Sandwich told Pepys that he hopes the Clerk of the Acts position will be good to him, saying “it was not the salary of any place that did make a man rich, but the opportunity of getting money while he is in the place.”

Pepys took this advice to heart. Once sworn in as Clerk of the Acts, he almost immediately found himself on the receiving end of a steady stream of gifts, from barrels of oysters, wine, and brandy to gold coins and silver plate. In 17th-century London, merchants clearly considered these donations to be money well spent, just another cost of doing business.

On April 3, 1663, the diarist described a defense used by politicians to this day, which basically consists of sticking to an absurdly literal, and narrow, truth. After a certain Captain Grove gives him a letter that he can tell contains money, Pepys wrote: “But I did not open it till I came home to my office; and there I broke it open, not looking into it till all the money was out, that I might say I saw no money in the paper if ever I should be questioned about it.”

Another business associate gave him “a present for his wife,” a package said to contain a pair of gloves. On the evening of February 2, 1664, Pepys noted: “When I came home, Lord! in what pain I was to get my wife out of the room without bidding her go, that I might see what these gloves were; and by and by, she being gone, it proves a payre of white gloves for her and forty pieces in good gold, which did so cheer my heart that I could eat no victuals almost for dinner for joy to think how God do bless us every day more and more.”

Plague, Fire, and Fortune

Ironically, biographer Tomalin says the plague year of 1665 was one of Pepys’s happiest. During it his fortune quadrupled, thanks in part to two additional appointments: treasurer for Tangier and surveyor-general of victualling for the navy. Meanwhile, as his fortune grew, so did the plague. From June to September, deaths from the disease doubled nearly every week.

“But, Lord! to see how the plague spreads,” wrote Pepys on June 16. “It being now all over King’s Streete, at the Axe, and next door to it, and in other places.” At its height, in the last week of August 1665, the plague killed nearly 10,000 Londoners. “Thus this month ends with great sadness upon the publick, through the greatness of the plague every where through the kingdom almost,” wrote Pepys on August 31. “Every day sadder and sadder news of its encrease.”

The Great Fire of London, which began on September 2, 1666, and engulfed most of the central part of the city, helped quell the plague by killing the city’s disease-infected rats. As the fire raged toward his home, Pepys packed up his gold and silver and rode by cart in his nightshirt to a friend’s, safely outside the city. What he could not transport, he buried. Luck was on his side, however, and his neighborhood was spared.

As for the Lord of Sandwich, embezzlement was his downfall. While at war with the Dutch, Sandwich’s fleet captured several Dutch ships, including some loaded with goods from the East Indies. Instead of delivering these spoils of war to the King, Sandwich let the hatches be broken and divvied up the prizes with his fleet’s captains. His share’s worth came to 5,000 pounds. When news of this reached the King, Sandwich was stripped of his command. (He would later be reappointed and died in battle in 1672.)

Pepys’s assessment of the fall of “his Lord” is less forgiving. On December 31, 1665, he wrote: “The great evil of this year, and the only one indeed, is the fall of my Lord of Sandwich. The Duke of Albemarle goes with the Prince to sea this next year, and my Lord very meanly spoken of; and, indeed, his miscarriage about the prize goods is not to be excused, to suffer a company of rogues to go away with ten times as much as himself, and the blame of all to be deservedly laid upon him.”

Fearing for his eyesight, Pepys brought his diary to a close in 1669. He would later keep two other journals before his death in 1703, but Tomalin notes that they have “none of the qualities of the first Diary. Something essential was missing — some grit that had caused him to produce his pearl.” The luster of that pearl, and the qualities of the man, can be seen in the entry for Christmas day, December 25, 1666:

“To church in the morning, and there saw a wedding in the church, which I have not seen many a day; and the young people so merry one with another, and strange to see what delight we married people have to see these poor fools decoyed into our condition, every man and woman gazing and smiling at them. Here I saw again my beauty Lethulier. Thence to my Lord Bruncker’s by invitation and dined there, and so home to look over and settle my papers, both of my accounts private, and those of Tangier, which I have let go so long that it were impossible for any soul, had I died, to understand them, or ever come to any good end in them. I hope God will never suffer me to come to that disorder again.”

Observations from Samuel Pepys’s Diary On dog days:

“By coach to St. James’s and there did our business, which is mostly every day to complain of want of money.” (July 13, 1666)

On hard work: “How little merit do prevail in the world, but only favour; and for myself, chance without merit brought me in; and diligence only keeps me so, and will, living as I do among so many lazy people that the diligent man becomes necessary, that they cannot do anything without him.” (November 1, 1665)

On success: “But, Lord! to see what successe do, whether with or without reason, and making a man seem wise notwithstanding never so late demonstration of the profoundest folly in the world.” (August 15, 1666)

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