It is interesting to note that although Alan Turing believed humans were merely machines, much like the computers he had envisioned, failed to realize that his idea for computers came to him suddenly, 'in a vision', thus confirming Godel's contention that humans had access to the 'divine spark of intuition'. A divine spark which enables humans to transcend the limits he, and Turing, had found in his incompleteness theorem for computers, mathematics, and even for all material reality generally:
Gifted people being able to instantaneously know answers to complex problems, as Turing himself did with his vision of a computer, is something that argues forcefully against the notion that our minds are merely the 'emergent' products of molecules in motion in our brain;
Electrical genius Nicola Tesla was born in Serbia in 1856,,, his father was a clergyman.
Excerpt: While walking in Budapest Park, Hungary, Nikola Tesla had seen a vision of a functioning alternating current (AC) electric induction motor. This was one of the most revolutionary inventions in the entire history of the world.
The boy in this following video rivals, or surpasses, Nikola Tesla as an example of innovative ideas coming fully formed to the mind without any need for trial and error:
Bluejay: The Mind of a Child Prodigy – video
At the 11:50 minute mark of this following video 21 year old world Chess champion Magnus Carlsen explains that he does not know how he knows his next move of Chess instantaneously, that ‘it just comes natural’ to him to know the answer instantaneouly.
Mozart of Chess: Magnus Carlsen – video
A chess prodigy explains how his mind works – video
Excerpt: What’s the secret to Magnus’ magic? Once an opponent makes a move, Magnus instantaneously knows his own next move.
This ability to 'instantaneously' know answers to complex problems has long been a very intriguing characteristic of some autistic savants;
Is Integer Arithmetic Fundamental to Mental Processing?: The mind's secret arithmetic
Excerpt: Because normal children struggle to learn multiplication and division, it is surprising that some savants perform integer arithmetic calculations mentally at "lightning" speeds (Treffert 1989, Myers 1903, Hill 1978, Smith 1983, Sacks 1985, Hermelin and O'Connor 1990, Welling 1994, Sullivan 1992). They do so unconsciously, without any apparent training, typically without being able to report on their methods, and often at an age when the normal child is struggling with elementary arithmetic concepts (O'Connor 1989). Examples include multiplying, factoring, dividing and identifying primes of six (and more) digits in a matter of seconds as well as specifying the number of objects (more than one hundred) at a glance. For example, one savant (Hill 1978) could give the cube root of a six figure number in 5 seconds and he could double 8,388,628 twenty four times to obtain 140,737,488,355,328 in several seconds. Joseph (Sullivan 1992), the inspiration for the film "Rain Man" about an autistic savant, could spontaneously answer "what number times what number gives 1234567890" by stating "9 times 137,174,210". Sacks (1985) observed autistic twins who could exchange prime numbers in excess of eight figures, possibly even 20 figures, and who could "see" the number of many objects at a glance. When a box of 111 matches fell to the floor the twins cried out 111 and 37, 37, 37.
The following video is fairly direct in establishing the 'spiritual' link to man's ability to learn new information, in that it shows that the SAT (Scholastic Aptitude Test) scores for students showed a steady decline, for seventeen years from the top spot or near the top spot in the world, after the removal of prayer from the public classroom by the Supreme Court, not by public decree, in 1963. Whereas the SAT scores for private Christian schools have consistently remained at the top, or near the top, spot in the world:
The Real Reason American Education Has Slipped – David Barton – video
You can see that dramatic difference, of the SAT scores for private Christian schools compared to public schools, at this following site;
Aliso Viejo Christian School – SAT 10 Comparison Report
Of related interest:
Bruce Charlton's Miscellany - October 2011
Excerpt: I had discovered that over the same period of the twentieth century that the US had risen to scientific eminence it had undergone a significant Christian revival. ,,,The point I put to (Richard) Dawkins was that the USA was simultaneously by-far the most dominant scientific nation in the world (I knew this from various scientometic studies I was doing at the time) and by-far the most religious (Christian) nation in the world. How, I asked, could this be - if Christianity was culturally inimical to science?
The following video is also very suggestive to a 'spiritual' link in man's ability to learn new information in that the video shows that almost every, if not every, founder of each discipline of modern science was a devout Christian:
Christianity Gave Birth To Science - Dr. Henry Fritz Schaefer - video
As well Sir Isaac Newton stated this in regards to his own discoveries:
I have a fundamental belief in the Bible as the Word of God, written by men who were inspired. I study the Bible daily…. All my discoveries have been made in an answer to prayer. — Sir Isaac Newton (1642-1727), considered by many to be the greatest scientist of all time
These following studies, though of materialistic bent, offer strong support that Humans are extremely unique in 'advanced information capacity' when compared to animals:
Darwin’s mistake: Explaining the discontinuity between human and nonhuman minds:
Excerpt: There is a profound functional discontinuity between human and nonhuman minds. We argue that this discontinuity pervades nearly every domain of cognition and runs much deeper than even the spectacular scaffolding provided by language or culture can explain. We hypothesize that the cognitive discontinuity between human and nonhuman animals is largely due to the degree to which human and nonhuman minds are able to approximate the higher-order, systematic, relational capabilities of a physical symbol system.
In the beginning, the Word existed. The Word was with God, and the Word was God.
A very strong piece of suggestive evidence, which persuasively hints at a unique relationship that man has with 'The Word' of John 1:1, is found in these following articles which point out the fact that ‘coincidental scientific discoveries’ are far more prevalent than what should be expected from a materialistic perspective,:
In the Air – Who says big ideas are rare? by Malcolm Gladwell
Excerpt: This phenomenon of simultaneous discovery—what science historians call “multiples”—turns out to be extremely common. One of the first comprehensive lists of multiples was put together by William Ogburn and Dorothy Thomas, in 1922, and they found a hundred and forty-eight major scientific discoveries that fit the multiple pattern. Newton and Leibniz both discovered calculus. Charles Darwin and Alfred Russel Wallace both discovered evolution. Three mathematicians “invented” decimal fractions. Oxygen was discovered by Joseph Priestley, in Wiltshire, in 1774, and by Carl Wilhelm Scheele, in Uppsala, a year earlier. Color photography was invented at the same time by Charles Cros and by Louis Ducos du Hauron, in France. Logarithms were invented by John Napier and Henry Briggs in Britain, and by Joost Bürgi in Switzerland. ,,, For Ogburn and Thomas, the sheer number of multiples could mean only one thing: scientific discoveries must, in some sense, be inevitable.
List of multiple discoveries
Excerpt: Historians and sociologists have remarked on the occurrence, in science, of "multiple independent discovery". Robert K. Merton defined such "multiples" as instances in which similar discoveries are made by scientists working independently of each other.,,, Multiple independent discovery, however, is not limited to only a few historic instances involving giants of scientific research. Merton believed that it is multiple discoveries, rather than unique ones, that represent the common pattern in science.
Kurt Godel was well aware of the deep implications of his theorem as the following quotes from Godel make clear:
Quotes by Kurt Godel:
"The brain is a computing machine connected with a spirit." [6.1.19]
"Consciousness is connected with one unity. A machine is composed of parts." [6.1.21]
"I don’t think the brain came in the Darwinian manner. In fact, it is disprovable. Simple mechanism can’t yield the brain. I think the basic elements of the universe are simple. Life force is a primitive element of the universe and it obeys certain laws of action. These laws are not simple, and they are not mechanical." [6.2.12]
"The world in which we live is not the only one in which we shall live,,,."
"Materialism is false."
quotes taken from - Hao Wang’s supplemental biography of Gödel, A Logical Journey, MIT Press, 1996
Kurt Gödel (1906-1978) Germany, U.S.A.
Gödel, who had the nickname Herr Warum ("Mr. Why") as a child, was perhaps the foremost logic theorist ever, clarifying the relationships between various modes of logic.... He proved that first-order logic was indeed complete, but that the more powerful axiom systems needed for arithmetic (constructible set theory) were necessarily incomplete.
If Gödel's first theorem showed that systems of arithmetic cannot be complete, and that some truths of number theory can never be proved or disproved regardless of how much time and ingenuity we spend on them, then the second theorem shows that our confidence in the arithmetic we do have can never be perfect. It is not an exaggeration to say that the two theorems permanently destroyed two ideals of mathematics at the heart of its glory and prestige for millenia. Gödel published both theorems in 1931 at age 25.
No particular truth of number theory (or version of G) is absolutely unprovable, since any one you like can be added to the axiom set. But there will always be some unprovable truth for such systems. So every truth is provable, just not in the same system.
The theorem only applies to sufficiently powerful systems. Many weak and all inconsistent systems are immune to Gödel's theorem.
Similarly, the theorem is not provable in some non-standard logics specifically designed to block it.
Kurt Gödel - Incompleteness Theorem as it applies to material particles and the universe
Epistemology – Why should the human mind be able to comprehend reality so deeply? - referenced article
Intelligent Design - The Anthropic Hypothesis