Some thoughts on 2021 Part 1

     This post is going to be highly speculative and unusual.  

    I predict that 2021 will be even stranger than 2020.  The reason has to do with Rudolf Steiner's ideas.  I believe Steiner had genuine insight, but also his lectures are full of ideas which it is difficult to either confirm or reject.  So, the best way I have found to read or listen to Steiner is to view his writings not as an essay, where each statement is meant to support the main point, but a collection of ideas.  If any statement or idea proves memorable or seems intuitively plausible, then I will think more about it and see where it leads.  And that is what this post is an attempt to do.

    Steiner has a vast schema concerning past and future ages in the evolution of the earth and human beings.  Currently we are in the Fifth of the Post-Atlantean Epochs, of which there are supposed to be seven in total.  Each of these Epochs is said to last 2160 years.  Further, each of the Post-Atlantean Epochs are supposed to have a particular quality and characteristic which it is the task of the people in that epoch to develop.  After thinking over the matter, I believe it is better to regard the epochs as spiritual qualities that will manifest in the world and in the souls that live in that era rather than as fixed periods of time.  In other words, the spiritual quality is more fundamental than the length of time of the epoch.  

    If we view the Epochs this way, then they could be either lengthened or brought to a close depending on how they are going.  For instance, some textbooks have bonus chapters so that a particularly adept class can learn additional material.  Likewise, an epoch that is going well could be extended so that its qualities can be developed even further.  On the other hand, if an epoch is going badly, it may either be brought to a close or the next epoch may be introduced early.  

      The overarching task of the Fifth Epoch, which was supposed to have begun in 1413 and is supposed to last until 3573, was for human beings to develop their individuality.  The Fifth Epoch was also supposed to prepare the way for the Sixth Epoch where a new form of social organization is supposed to be brought forth.  Also, the Slavic peoples are predicted to play a special role in the Sixth Epoch.  Terry Boardman in one of his lectures said it is almost unimaginable at our current time.  

    But, I think we have some clues.  Bruce Charlton has written extensively about the importance of the family in the present day and age.  One reason is that as other forms of social cohesion have diminished, the family is what we have left.  But, the other reason is that he believes (and I agree) that the destiny of human beings is to move towards a form of social organization based on the family but in a new way.  

    That is what I believe the Sixth Epoch to be about, the family.  It makes sense.  In earlier times (but not the earliest), the family was something that existed within the overarching structure of the nation or the culture.  They carved out space so that the family could then exist.  However, in the Sixth Epoch, nations will not go away, they will move within so that countries will become more like extended families.  Further, people will view having a family as their duty.  In fact, this makes sense of the industrial revolution and the vast abundance that we in the West have experienced over the past two centuries.  The intended purpose was not for people to live luxurious lives, but rather for almost everyone to be able to raise families.  

Part 2

    

A Guardian Angel of Earth?

    The guardian angels of countries are known as principalities.  There is a good post on this about the guardian angel of England by William Wildblood.  The idea is that for a country to be a real thing, rather than an arbitrary creation of laws, there must be a spiritual force which binds the people of the country together.  But not just the people; the guardian angel of a country would also be concerned with the land and the physical nature of the nation as well.  

    But why stop at countries?  Is there a guardian angel for the whole earth as well, who organizes the principalities and is concerned with the physical environment of the planet?  

    The first time this thought occurred to me was reading this passage from the second installment of John Fitzgerald's excellent story "That Hideous Strength Unveiled"

"Because, before they can be handed to the deep King of Britain, they must first be blessed by the greatest king of all - God's vice-regent on Earth - the silent watcher, the still point, the secret overlord of the planet. He has many names. Some call him Melchizedeck, others the Chakravatin, others the King of the World. And you and I will receive his blessing too before we travel back to return the Jewels to Arthur. Then my work will be complete. "

In this story, Melchizedek is the guardian angel of Earth.  And then I remembered that this notion also exists in C.S. Lewis's Space Trilogy, where each planet has its own guardian spirit, called an Oyarsa.  Lewis in turn borrowed this from older notions, such as the idea that the planets were moved by "planetary intelligences." 

In addition, William James Tychonievich has written a post entitled: "Each continent and region has its own biological 'style' " that ends with this interesting idea:

"Just as Rembrandt and Titian, and even Picasso and Chagall, are all part of a larger style called 'European art,' which is quite distinct from, say, Chinese art -- so I sometimes feel that I can sense a common "Earth" style underlying all the animals in the world, even though I obviously have no other set of animals to compare them too. The planetary spirit of Earth may be harder to see, for lack of contrast, than the regional spirits of various countries and continents, but it is no less real. " 

Which suggests that if we were to see an animal or plant from another planet, we would immediately recognize it.  Such an animal or plant would be different in some big, difficult to imagine way from any of those we are familiar with.  

    After thinking through these considerations, I do believe in a guardian angel of the Earth.  And in these times, I think such thoughts are well worth thinking about. 

No, they can't imagine it either

    Edward Feser has a good series of posts about eliminativism (toward the end of this list).  This is the idea in the philosophy of mind that no one thinks; people only think they think.  In other words, the mind and all of conscious experience, from thinking to the senses is an illusion.  Three of the most prominent defenders of this idea are Daniel Dennet and Paul and Patricia Churchland.  Feser has a great metaphor to describe why this does not work: 

"Here I want to focus on the presupposition of Bakker’s question, and on another kind of fallacious reasoning I’ve called attention to many times over the years.  The presupposition is that science really has falsified our commonsense understanding of the rest of the world, and the fallacy behind this presupposition is what I call the “lump under the rug” fallacy.

Suppose the wood floors of your house are filthy and that the dirt is pretty evenly spread throughout the house.  Suppose also that there is a rug in one of the hallways.  You thoroughly sweep out one of the bedrooms and form a nice little pile of dirt at the doorway.  It occurs to you that you could effectively “get rid” of this pile by sweeping it under the nearby rug in the hallway, so you do so.  The lump under the rug thereby formed is barely noticeable, so you are pleased.  You proceed to sweep the rest of the bedrooms, the bathroom, the kitchen, etc., and in each case you sweep the resulting piles under the same rug.  When you’re done, however, the lump under the rug has become quite large and something of an eyesore.  Someone asks you how you are going to get rid of it.  “Easy!” you answer.  “The same way I got rid of the dirt everywhere else!  After all, the ‘sweep it under the rug’ method has worked everywhere else in the house.  How could this little rug in the hallway be the one place where it wouldn’t work?  What are the odds of that?”

This answer, of course, is completely absurd.  Naturally, the same method will not work in this case, and it is precisely because it worked everywhere else that it cannot work in this case.  You can get rid of dirt outside the rug by sweeping it under the rug.  You cannot get of the dirt under the rug by sweeping it under the rug.  You will only make a fool of yourself if you try, especially if you confidently insist that the method must work here because it has worked so well elsewhere. 

 

...

 

Now, the 'Science has explained everything else, so how could the human mind be the one exception?' move is, of course, standard scientistic and materialist shtick.  But it is no less fallacious than our imagined 'lump under the rug' argument.  

 

...

 

In short, the scientific method 'explains everything else' in the world in something like the way the 'sweep it under the rug' method gets rid of dirt -- by taking the irreducibly qualitative and teleological features of the world, which don’t fit the quantitative methods of science, and sweeping them under the rug of the mind.  And just as the literal 'sweep it under the rug' method generates under the rug a bigger and bigger pile of dirt which cannot in principle be gotten rid of using the 'sweep it under the rug' method, so too does modern science’s method of treating irreducibly qualitative, semantic, and teleological features as mere projections of the mind generate in the mind a bigger and bigger 'pile' of features which cannot be explained using the same method."   

    In other words, science presupposes the existence of the mind because scientific reasoning and observation takes place within the mind of the scientist.  Thus, for science to eliminate the mind would eliminate science itself.  

     

    But one thing that always struck me about the eliminativists was how confident they are.  They boldly assert that consciousness is an illusion.  I cannot imagine consciousness being an illusion because if there is no one there, how can you fool them?  But, for a long time I assumed that the eliminativists had some way of imagining consciousness being an illusion.  Then, about a year and a half ago, I was reading about eliminativism and it hit me, "No, they can't imagine it."  They can't imagine consciousness being an illusion because no one can.  

    

     The eliminativists believe that their position is a counterintuitive truth based on scientific argument, like curved spacetime or quantum mechanics.  But it is not like that at all because curved spacetime is at least imaginable, if difficult to understand.  Eliminativism is not counter-intuitive, it's anti-intuitive.  Imagine a group of people who are convinced that the color red is actually blue.  That seems crazy enough, but eliminativism is even worse than that: even if these people could not stop seeing red, they could at least imagine everything red being blue.  You cannot even imagine eliminativism being true.  

   

Tiers of Ability, Part 2

Continued from Part 1

 

    In his book Hereditary Genius, Francis Galton provides an excellent illustration of this phenomenon with data from the Cambridge Mathematical Tripos.  There were two degrees in the old Cambridge system, an honours degree and a pass degree.  In the original Tripos, which ran from 1748 - 1909, the honours students were ranked based on how many points they scored on the exam.  The highest scorers were called Wranglers and those below them Optimes, while the very highest scorer was called the Senior Wrangler.  The candidate who scored lowest, but still managed to perform at the honours level was called the Wooden Spoon.  Here is what Galton has to say about the matter:       

 "There can hardly be a surer evidence of the enormous difference between the intellectual capacity of men, than the prodigious differences in the numbers of marks obtained by those who gain mathematical honours at Cambridge. I therefore crave permission to speak at some length upon this subject, although the details are dry and of little general interest. There are between 400 and 450 students who take their degrees in each year, and of these, about 100 succeed in gaining honours in mathematics, and are ranged by the examiners in strict order of merit.

About the first forty of those who take mathematical honours are distinguished by the title of wranglers, and it is a decidedly creditable thing to be even a low wrangler; it will secure a fellowship in a small college. It must be carefully borne in mind that the distinction of being the first in this list of honours, or what is called the senior wrangler of the year, means a vast deal more than being the foremost mathematician of 400 or 450 men taken at hap-hazard. No doubt the large bulk of Cambridge men are taken almost at hap-hazard. A boy is intended by his parents for some profession; if that profession be either the Church or the Bar, it used to be almost requisite, and it is still important, that he should be sent to Cambridge or Oxford. These youths may justly be considered as having been taken at hap-hazard. But there are many others who have fairly won their way to the Universities, and are therefore selected from an enormous area. Fully one-half of the wranglers have been boys of note at their respective schools, and, conversely, almost all boys of note at schools find their way to the Universities. Hence it is that among their comparatively small number of students, the Universities include the highest youthful scholastic ability of all England. The senior wrangler, in each successive year, is the chief of these as regards mathematics, and this, the highest distinction, is, or was, continually won by youths who had no mathematical training of importance before they went to Cambridge.

... 

The examination lasts five and a half hours a day for eight days. All the answers are carefully marked by the examiners, who add up the marks at the end and range the candidates in strict order of merit. The fairness and thoroughness of Cambridge examinations have never had a breath of suspicion cast upon them.

Unfortunately for my purposes, the marks are not published. They are not even assigned on a uniform system, since each examiner is permitted to employ his own scale of marks; but whatever scale he uses, the results as to proportional merit are the same. I am indebted to a Cambridge examiner for a copy of his marks in respect to two examinations, in which the scales of marks were so alike as to make it easy, by a slight proportional adjustment, to compare the two together. This was, to a certain degree, a confidential communication, so that it would be improper for me to publish anything that would identify the years to which these marks refer. I simply give them as groups of figures, sufficient to show the enormous differences of merit. The lowest man in the list of honours gains less than 300 marks; the lowest wrangler gains about 1,500 marks; and the senior wrangler, in one of the lists now before me, gained more than 7,500 marks. Consequently, the lowest wrangler has more than five times the merit of the lowest junior optime, and less than one-fifth the merit of the senior wrangler.

The results of two years are thrown into a single table.

The total number of marks obtainable in each year was 17,000.

 

The precise number of marks obtained by the senior wrangler in the more remarkable of these two years was 7,634; by the second wrangler in the same year, 4,123; and by the lowest man in the list of honours, only 237. Consequently, the senior wrangler obtained nearly twice as many marks as the second wrangler, and more than thirty-two times as many as the lowest man. I have received from another examiner the marks of a year in which the senior wrangler was conspicuously eminent.

He obtained 9,422 marks, whilst the second in the same year—whose merits were by no means inferior to those of second wranglers in general—obtained only 5,642. The man at the bottom of the same honour list had only 309 marks, or one-thirtieth the number of the senior wrangler.

...

The mathematical powers of the last man on the list of honours, which are so low when compared with those of a senior wrangler, are mediocre, or even above mediocrity, when compared with the gifts of Englishmen generally. Though the examination places 100 honour men above him, it puts no less than 300 “poll men” below him. Even if we go so far as to allow that 200 out of the 300 refuse to work hard enough to get honours, there will remain 100 who, even if they worked hard, could not get them. "

    From the table, we see that the scores are indeed positively skewed.  One amusing story about the Tripos is that William Thompson, later Lord Kelvin was universally acknowledged as the best in his year at Cambridge, so when the Tripos results were posted, he asked one of the college servants, "Go see who is the second Wrangler."  The servant did so and then responded, "You are, sir."  Someone else had beaten him. 

 

Tiers of Ability, Part 1

     Bruce Charlton made an astute observation in a comment on this post:

        "mathematical and musical abilities seem more varied than anything else I know of.

(And I think there is a surprising positive between music composition and maths.)

Most people are pretty close to zero! - and then the people who are good are So much better - and then there are some who are So much better than the people who are good...

But the distribution doesn't seem anything like a bell curve - more like something very positively skewed."

   

The famous Bell Curve

A negatively and positively skewed distribution

 

    The Bell Curve is symmetric.  Whatever population we are examining, if a trait is distributed normally within the population, then there will be as many members of the population who have the trait a given distance above the average as there are who have the trait a given distance below average.  However, with positively and negatively skewed distributions, more members of the population are either above the mean or below the mean respectively.  

    So, Bruce Charlton makes the point that because the range of mathematical and musical ability is so large most people are at the low end of the spectrum with a very small number of people at the high end.  But not only that, there are large gaps between each of the levels, so the best way we can conceptualize this is by tiers of ability. 

	Part 2

Idea for a proof of William James Tychonievich's Assumption about Dice

     In a previous post, I used a metaphor about rolling all 5 Platonic solid dice, mistakenly believing that the distribution of the sum of the dice rolls would not follow a Bell Curve.  William James Tychonievich mentioned in a comment that he thought it would and has demonstrated this by plotting the data.  

    Here is an idea for how one might prove why this works.  I suspect that as the number of dice we use increases, the distribution converges to the normal distribution.  However, I have not shown this, but I have shown that the distribution is unimodal (meaning that the probability of the lowest roll is the lowest, then the probability starts to increase until it gets to the highest point, then continues to decrease).  The Bell Curve is a unimodal distribution.  

    If we roll fair dice, each number on the die has an equal probability.  If we roll multiple dice, then to find the probability of any given roll we multiply the probabilities of each roll.  So, for example, the chance of simultaneously rolling a 1 on a four-sided die (d4), a 3 on a six-sided die (d6), and a 7 on an 8-sided die (d8) is (1/4) x (1/6) x (1/8), which is 1/192.  The reason we multiply the probabilities is because each roll is independent - what we roll on the d4 does not affect what we roll on the d6 or d8.  

    Likewise, since there are four possible rolls for the first die, 6 for the second, and 8 for the third, there are 192 total possible rolls for the three dice.  Since each roll has an equal probability, we can dispense with the probability and only focus on the rolls themselves.  We can view each die as a list of consecutive integers: (1,2,3,4), (1,2,3,4,5,6), and (1,2,3,4,5,6,7,8).  A roll is then the same as selecting one number from the first list, one from the second, and one from the third.  

     List each of the numbers in increasing order and consider just the first two lists: (1,2,3,4) and (1,2,3,4,5,6).   Next, generate sums of rolls by starting with 1 and adding 1 to each of the numbers in the second list, then do the same with 2 and so on to generate the following lists: 

2, 3, 4, 5, 6, 7 

       3, 4, 5, 6, 7, 8

                4, 5, 6, 7, 8, 9

                           5, 6, 7, 8, 9, 10

    Look at what is happening.  We have 24 rolls and 8 possible sums (from 2 to 10).  Since these are consecutive integers, when we generate the sums in this way, moving fro 1 to 2 only increases by one, so the sums generated by adding 2 to every number of the second list (the first row) are only one greater than those generated by adding one to  the second list (the second row).  So, every number but the first and last overlap.  This pattern continues and at the end,  we see that we get one 2, one 10, two 3's, three 4's, three 8's, and four 5's, 6's and 7's.  

    For three dice, we do the same thing.  I will not write all 192 rolls, but instead consider the lists: (1,2,3), (4,5,6), (7,8,9).  Generate lists of sums of rolls by first selecting the smallest number from the first list and the smallest from the second and adding these to all the numbers in the third list to form 12, 13, and 14.  Then, add the smallest number in the first list and the second smallest in the second list to form 13, 14, and 15 and continue the process.  Once we exhaust the rolls with 1 from the first list, then use 2, and then 3, following the same process.  Altogether, we generate the following lists:  

12, 13, 14

              13, 14, 15 

                          14, 15, 16

                13, 14, 15 

                            14, 15, 16

                                         15, 16, 17

                              14, 15, 16

                                           15, 16, 17

                                                         16, 17, 18 

 Once again, when we line up the columns, we see that the sums in the middle occur more often than those on the ends and so this also forms a unimodal distribution.  Observe also that there is nothing special about dice - the important thing is that we are using lists of consecutive integers.  So, in general, if we have m lists of consecutive integers with n₁ integers in the first list, n₂ in the second, and so on until we have n_m in the final list, we can organize the lists in ascending order.  If we generate sums where each sum takes one member from each list in the same way as above, we will create a unimodal distribution. 

Leonhard Euler, John Von Neumann, and IQ Part 3

 Continued from Part 2

    I remember reading an blog post which said that Von Neumann's incredible intellect was produced by a large number of single-nucleotide polymorphisms.  The implication was that it was high IQ, high g alone which caused Von Neumann's intelligence.  In other words, like height, which is mostly thought to be caused by a small number of genes, each of which add a little bit more height, this post said that the same is true for intelligence.  This is true to some extent, but when we try to explain extreme outliers by this method, it fails.  It goes without saying that to be a world class intellect you need high general intelligence, but I believe that Euler and Von Neumann's unusual abilities can only be explained by interaction between general and special intelligence. 

    To see why, consider a list of the 10 documented oldest people who have lived: 

Jeanne Calment: 122 years, 164 days 

Sarah Knauss: 119 years, 97 days

Kane Tanaka: 117 years, 357 days 

Nabi Tajima: 117 years, 260 days 

Marie-Louise Meilleur: 117 years, 230 days 

Violet Brown: 117 years, 189 days 

Emma Morano: 117 years, 137 days 

Chiyo Miyako: 117 years, 81 days 

Misao Okawa: 117 years, 27 days 

Maria Capovilla: 116 years, 347 days 

    Notice that even though we have some slight outliers with the top two, the difference between the oldest and youngest is about 6 years and many of these ages are very close together.  Which suggests that the differences between these people can be explained by small differences in genetic and environmental influences.  Most likely by a lack of deleterious mutations causing unhealthiness.  So we can model lifespan as being caused by a large number of independent mutations which each have a similar degree of influence.  If we model lifespan in this way, we would expect to see what we actually do see - a fairly smooth gradation between the highest and lowest with most of the gaps filled in.  We can compare this to this distribution of sums of rolls caused by rolling five 6-sided dice. 

    On the other hand, if lifespan was caused by independent mutations each of which had significantly different degrees of influence, we would expect to see something like one person living 120 years, one 140, one 90 ... large gaps in between the ages.  We might compare this to rolling all 5 Platonic solids: one 4 sided die, one 6 sided, one 8 sided, one 12 sided, and one 20 sided, which would have a completely different distribution from rolling five 6-sided dice

    I suspect that speed of calculation for most people does follow a Bell Curve with few gaps.  In other words, the differences in speeds between ordinary people are not so great on an absolute scale.  However, when we compare Von Neumann and Euler to an ordinary person, they are like someone who has lived 150 years.  The gaps are too large.  

To illustrate this, here is a statement paraphrased from Wikipedia: 

Here is what Enrico Fermi said to Herbert Anderson: 

    "You know, Herb, Johnny can do calculations in his head ten times as fast as I can!  And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!"

    So, where are all the people who can calculate 9 times faster, 8 times faster, etc.  We are talking about a geometric increase, not merely an arithmetic increase.  So, I postulate that while obviously John Von Neumann and Euler had high general intelligence, their unusual traits must be explained by reference to special intelligence.  Further, it is possible that certain types of special intelligences, when they occur together, produce a "multiplier effect" where each one individually has an effect, but taken together their effects work synergistically to become even more powerful.  

    Update: William James Tychonievich has shown that rolling a mix of polyhedral dice does in fact make a normal distribution.  It turns out that was not the right metaphor to use.  I think the problem is that it does not take account of multiplier effects.

Leonhard Euler, John Von Neumann and IQ Part 2

Continued from Part 1 

    William Dunham has some descriptions about Leonhard Euler's incredible abilities in his book Journey through Genius: 

     "Euler's collected works fill over 70 large volumes, a testament to the genius of this unassuming Swiss citizen who changed the face of mathematics so profoundly.  Indeed, one's first inclination, upon encountering the volume and quality of his work, is to regard his story as an exaggerated piece of fiction rather than hard historical fact.  

... 

    Throughout his career, Euler was blessed with a memory that can only be called phenomenal.  His number-theoretic investigations were aided by the fact that he had memorized not only the first 100 prime numbers but also all of their squares, their cubes, and their fourth, fifth, and sixth powers.  While others were digging through tables or pulling out pencil and paper, Euler could simply recite from memory such quantities as 241⁴ or 337⁶.  But this was the least of his achievements.  He was able to do difficult calculations mentally, some of these requiring him to retain in his head up to 50 places of accuracy!  The Frenchman Francois Arago said that Euler calculated without apparent effort, 'just as men breathe, as eagles sustain themselves in the air.'  Yet this extraordinary mind still had room for a vast collection of memorized facts, orations, and poems, including the entire text of Virgil's Aeneid, which Euler had committed to memory as a boy and still could recite flawlessly half a century later.  No writer of fiction would dare to provide a character with a memory of this caliber.

... 

Incredible as it sounds, it has been estimated that, if one were to collect all publications in the mathematical sciences produced over the last three-quarters of the eighteenth century, roughly one-third of these were from the pen of Leonhard Euler! 

... 

Before he was done, Euler's number theory filled four large volumes of his Opera Omnia [Collected Works].  It has been observed that, had he done nothing else in his scientific career, these four volumes would place him among the greatest mathematicians of history."

    From the way Euler wrote, it also seems like he solved problems in the act of writing about them.  It should also be mentioned that Euler had 13 children and could work on mathematics with his children playing around him and Von Neumann could work well in loud, smoke-filled environments, so they didn't need quiet environments to concentrate.  

    Not only that, Euler became blind in one eye at age 31 and then completely blind at 59 and still he continued to churn out mathematics. 

Part 3

Leonhard Euler, John Von Neumann and IQ Part 1

    Leonhard Euler (1707 - 1783) and John Von Neumann (1903 - 1957) are, in terms of memory, calculation, and quick thinking, two of the most intelligent people to have ever lived.  There are numerous stories about Von Neumann's lightning calculation and extraordinary memory, but I will tell some lesser known stories that show he was also incredibly fast with higher level thinking: 

    The statistician David Blackwell told the following story in an interview: 

    "Also, I got a chance to meet Von Neumann that year.  He was a most impressive man.  Of course, everybody knows that.  Let me tell you a little story about him.  

    When I first went to the Institute, he greeted me, and we were talking and he invited me to come around and tell him about my thesis.  Well, of course I thought that was just his way of making a new young visitor feel at home, and I had no intention of telling him about my thesis.  He was a big, busy, important man.  But then a couple of months later, I saw him at tea and he said 'When are you coming around to tell me about your thesis?  Go in and make an appointment with my secretary.' So I did, and later I went in and started telling him about my thesis. 

    He listened for about ten minutes and asked me a couple of questions and then he started telling me about my thesis.  What you could have really done is this, and probably this is true, and you could have done it in a somewhat simpler way, and so on.  He was a really remarkable man.  He listened to me talk about this rather obscure subject and in ten minutes knew more about it than I did.  He was extremely quick.  I think he may have wasted a certain amount of time, by the way, because he was so willing to listen to second- or third-rate people and think about their problems.  I saw him do that on many occasions."

  Eugene Wigner told this story in an interview: 

    "He [Von Neumann] wrote no articles on number theory.  But once I told him - this is a story which is perhaps of some interest - that I was much impressed by a new theorem about which I had read.  He said, 'Did you read the proof?'  I said, 'No, but the theorem itself is really amazing.'  He said, 'Well, would you like to have a proof?' I said, 'Yes, if you can give me one.' Then he asked me six questions: 'Do you know this theorem?', 'Do you know this theorem?' ... six theorems.  I knew three and I didn't know the other three.  And he gave me a wonderful proof, never mentioning the theorems which I did not know and using the theorems which I did know.  He was amazing in this respect."

    This story is ridiculous.  Constructing a mathematical proof is not an easy undertaking in general, but Von Neumann not only came up with a proof on the spot, he wrote Wigner a custom-made proof.  

    However, Von Neumann did have limitations.  Wigner also wrote: 

"I have known a great many intelligent people in my life. I knew Planck, von Laue and Heisenberg. Paul Dirac was my brother in law; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. But none of them had a mind as quick and acute as Jansci [John] von Neumann. I have often remarked this in the presence of those men and no one ever disputed.

But Einstein's understanding was deeper even than von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jansci's brilliance, he never produced anything as original."

 Paul Halmos wrote something similar in his memoir about Von Neumann ("The Legend of John Von Neumann): 

"He [Von Neumann] knew his own strengths and he admired, perhaps envied, people who had the complementary qualities, the flashes of irrational intuition that sometimes change the direction of scientific progress.  For Von Neumann it seemed impossible to be unclear in thought or in expression.  His insights were illuminating and his statements were precise."

Part 2

The Difference Between the 19th and 20th centuries in one picture

 Francis Galton and Karl Pearson (from 1909).  This picture is rather amusing because we have a man clearly of the 20th century (Karl Pearson) and then next to him Francis Galton is wearing gloves, a blanket, umbrella, and a hat: Galton knows he looks like a Dickens character and doesn't care.  Which tells us something about the personality of Francis Galton.

The Real 20th century, Part 2

 Continued from: The Real 20th century, Part 1  

    Another thing that Steiner writes about in his prophecy is that new impulses will come into play: 

    "The Angels form pictures in man's astral body and these pictures are accessible to thinking that has become clairvoyant. If we are able to scrutinise these pictures, it becomes evident that they are woven in accordance with quite definite impulses and principles. Forces for the future evolution of mankind are contained in them. If we watch the Angels carrying out this work of theirs — strange as it sounds, one has to express it in this way — it is clear that they have a very definite plan for the future configuration of social life on earth; their aim is to engender in the astral bodies of men such pictures as will bring about definite conditions in the social life of the future."

    I am not sure whether the means by which these impulses come into the world is as Steiner described it, but the point is that these new impulses will manifest themselves in human life, but the form they will take is up to human beings: we can respond to them well or badly.  

    If we combine this idea with the idea from the previous post, as well as Bruce Charlton, Owen Barfield and Rudolf Steiner's idea that from about 1750 human beings were supposed to develop a new form of consciousness, one thought that has occurred to me is that the entire 20th century, from 1914 onwards, was suboptimal.  All of the developments in this century that have been so lauded and for which so many panegyrics have been written and broadcast are all more or less second-best.  

    Not only that, taking the idea of the first part of this post, all the sociological and economic explanations ultimately miss the point because all they do is describe, they do not go deep enough to search for the true causes.  The only explanation is supernatural: how else could the communal ways of life that have endured for millenia go away so quickly?  

    So, it is an interesting thought experiment to ask the following question: If everything in the 20th century was a lesser manifestation of a good impulse, what would the real 20th century have looked like?