Archive for February, 2011

Ancient Mound Builder Controversy: Did They Interact with and Memorialize Allosaurus, T-Rex or Carnotaurus? Plus, Champ or Ogopogo?

Church of Darwin, Crypto, Dinosaurs in Literature, s8int.com, Sophistication of Ancestors, Unexplained Artifact | Posted by Chris Parker
Feb 28 2011

by Chris Parker

Drawing: Mound Builder Artifact from the book: “Records of Ancient Races in the Mississippi Valley, 1887

It’s amazing how little we as a supposedly educated culture actually know about the past. Even if we’ve studied the past, much of what is being taught or that is commonly believed is untrue. For instance, Columbus did not “discover” America. He was preceded here certainly by its inhabitants of the time, but also by the Norse, the Chinese, Africans, the Irish and the Vikings to name a few. Nero did not fiddle while Rome burned, Eve did not eat an apple, Egyptians were Africans. Dinosaurs and man Did interact

When in comes to the American continent, Native Americans supposedly came here from Asia 20,000 years ago across the Bering Straight land Bridge. However, a mysterious people, now called the Mound Builders whose works the Native Americans didn’t know were apparently here before them and their artifacts show that they interacted with the mastodon, dinosaurs and aquatic reptiles. Dinosaurs and aquatic reptiles supposedly went extinct more than 65 million years ago.

No one knows for sure who the Mound Builders were and where they came from but one thing is certain; no one believes that they lived with the dinosaurs 65 million years ago. The question is; did some dinosaurs and Mound builders live together here on this continent in “recent” times?

I present here some theories about the actual aimals being depicted here. Of course I could be wrong but I feel that if science believes that the ancient mound depicted on the left represents a Swan rather than a quadruped such as a sauropod that I’m on safe ground.

“The group of cultures collectively called Mound Builders were prehistoric inhabitants of North America who constructed various styles of earthen mounds for burial, residential and ceremonial purposes. These included the Pre-Columbian cultures of the Archaic period; Woodland period (Adena and Hopewell cultures); and Mississippian period; dating from roughly 3000 BCE to the 16th century CE, and living in regions of the Great Lakes, the Ohio River valley, and the Mississippi River valley and its tributaries.

As a comparison, beginning with the construction of Watson Brake about 3500 BC in present-day Louisiana, indigenous peoples started building earthwork mounds in North America nearly 1000 years before the pyramids were constructed in Egypt. Since the 19th century, the prevailing scholarly consensus has been that the mounds were constructed by Indigenous peoples of the Americas, early cultures distinctly separate from the historical Native American tribes extant at the time of European colonization of North America. The historical Native Americans were generally not knowledgeable about the civilizations that produced the mounds. Research and study of these cultures and peoples has been based on archaeology and anthropology.” Wikipedia

Mound Builder Theropod Dinosaur


This artifact (also pictured at the top of the page) is a drawing from the book; Records of Ancient Races in the Mississippi Valley: Written By William McAdams published in 1887 Page 14.

The author and the experts refer to the depiction as a “dragon” and it might seem presumptuous of me to disagree (particularly since we have only the head of the creature) but I see something else entirely. The Mound Builder creatures surrounding this piece are for the most part recognizable and are sculpted in a realistic style. There is no reason to suppose that this is a mythological creature other than because one has an A Priori scientific belief that these creatures did not exist within the lifetime of man.

Here is the author’s discussion of the piece in question and the other pieces found with it in the mound.

Graphic:Left, T-rex, Right Carnotaurus

“In our collection of pottery from the ancient mounds we have several pieces ornamented with dragon-like devices. We give an illustration of two of these; burial vases, with a most pronounced dragon-head standing up from the rim of the vessel. There is the great mouth with the teeth revealed, and protruding tongue, with fierce eyes, and the general aspect, not only of the Piasa, but of those mythological representations of the dragon so frequently found in Asia. We present a sketch of another.

Graphic:Left, Allosaurus, Right Carnotaurus


It is all the more interesting since we found with it a magnificent collection of pottery, of more than a hundred pieces, at the base of the great Cahokia mound, (pictured above, right) in the American Bottom, in Madison County, Ills.

This is the largest artificial mound in the United States, and perhaps in the world, being one-hundred feet in height, and covering with its base sixteen acres of ground. It is the centre of a group of seventy-two others, which surround it, and of which a description will be given farther on in this work. They are situated on a level plain, miles from any natural elevation. For a complete description and survey of them, see “The Antiquities of Cahokia, or Monk’s Mound’

Upon taking these curious old burial vases from the place where they had rested for ages, it was like exhuming a museum of natural history in ceramics ; for these were the shapes of animals, birds, reptiles, fishes, and aflmost all animated nature, together with the shapes of From Cahoki Mound. ithe human form. Among them were several vases adorned with the dragon heads.”

This is a realistically sculpted creature, posed withn a frog, it is certainly not a lizard and its teeth and head indicate that it is a carnivorous reptile. Quite frequently, such a creature is laabeled a dragon, and if not then a crocodile or alligator. This creature is clearly not one of those. It appears to be a lifelike depiction of a theropod dinosaur; a species similar to the allosaurus, t-rex or Carnotaurus. Are those horns on the top of its head or are those prominent eye ridges as depicted in a number of t-rex renditions. Keep this in mind; the piece you see here may have been done by an eyewitness.

We have compared here the Mound Builder creature with several examples of meat eating theropod dinosaurs. In our minds, the Carnotaurus, to date found in South America, most closely matches the Mound Builder artrifact.

Carnotaurus (pronounced /?k?rn??t?r?s/; meaning “meat-eating bull”, referring to its distinct bull-like horns (Latin carne = flesh + Greek tauros = bull) was a large predatory dinosaur, with horns vaguely resembling a bull’s. Only one species, Carnotaurus sastrei has been described so far. Carnotaurus lived in Patagonia, Argentina (La Colonia Formation) during the Campanian to the Maastrichtian stage of the Late Cretaceous, and was discovered by José Bonaparte, who has uncovered many other South American dinosaurs. Wikipedia

Allosaurus and T-rex are North American dinosaurs according to science.

Mound Builder Aquatic Reptile: Champ or Ogopogo

The Tusayan people are named after a ruin located in the Grand canyon National park in Arizona. These ancestral Puebloan people lived in the area until 1,000 years ago archaeologists believe. In American Anthropologist, Volume 5, January 1892, in an article entitled “ A Few Tusayan Pictographs” is a very interesting pictograph showing four long necked creatures.


Aquatic reptiles became extinct more than 65 million years ago as did the long necked sauropods according to science. The pictographs were thought to be a little as a thousand years old, possibly several thousand but certainly not millions of years old. Could science be wrong or are the more benign “nothing to see here, move it along” explanations of the archeologists satisfactory?

Here we compare the Mound Builder pictographs with; an ancient Roman “sea monster” and with several other classic sea or lake monsters.

Here’s what the author wrote about the pictograph in question:

“It will be noticed in examining the cut of these four pictographs of the great serpent that one is about horizontal and the other three erect. It will also be noticed that the horizontal specimen has a zigzag outline, as if in motion, which the others do not have. They are undoubtedly, however, figures of the same mythological personage”.

Ogopogo and Champ


“Ogopogo or Naitaka (Salish: n’ha-a-itk, “lake demon”) is the name given to a cryptid lake monster reported to live in Okanagan Lake, in British Columbia, Canada. Ogopogo has been allegedly seen by First Nations people since the 19th century. The most common description of Ogopogo is a forty- to fifty-foot-long (12 to 15 m) sea serpent. It has supposedly been photographed and even been caught on tape.” …Wikipedia


“Champ, or Champy is the name given to a reputed lake monster living in Lake Champlain, a natural freshwater lake in North America, partially situated across the U.S.-Canada border in the Canadian province of Quebec and partially situated across the Vermont-New York border. While there is no scientific evidence for the cryptid’s existence, there have been over 300 reported sightings. The legend of the monster is considered a draw for tourism in the Burlington, Vermont area.”… Wikipedia

The creature appears to be drawn with a beard which is typical of some “sea monsters” and drawn with a type of headcrest or ears.

I’m not suggesting that this creature is either Ogopogo or Champ but that it could represent a tyoe of sea or lake monster that has been seen and described by even the Romans and Greeks and by ancient and recent North and South Americans.

It appears that neither theropod dinosaurs or aquatic reptiles died out millons of years ago and were instead seen and memorialized by ancient peoples.

In Search of the Death Valley Cave System: Giants in Those Days? On The Ground in Death Valley- An Expedition to the Panamint Scorpions

Church of Darwin, Giants in Those Days, s8int.com, Uncategorized | Posted by Chris Parker
Feb 17 2011

Photo:Jef and team on the ground in Death Valley in search of the ancient cave system.

Several years ago, we reprinted an article originally printed in the San Diego Union on August 4, 1947 entitled;” Ancient Civilization Beneath Death Valley? EXPEDITION REPORTS NINE-FOOT SKELETONS.

This was a rather prominent article of the time concerning the alleged discovery of an ancient cave complex located in Death Valley which contained remnants of an ancient civilization and nine foot giants. The story eventually disappeared from the papers but we were able to find some possible collaborative evidence in the form of a mention of the discovery in a government archeologist’s papers.

In recounting the discovery, one of the principals; Hill claimed: “These giants,” said Hill, “are clothed in garments consisting of a medium length jacket and trouser extending slightly below the knees. The texture of the material is said to resemble gray dyed sheepskin, but obviously it was taken from an animal unknown today.”

S8int.com also included a number of other articles concerning the same cave system including descriptions of the area. With the advent of Google Maps and satellite photos of the area, those interested in the story have tried to find the location of the cave system. We’ve since had a number of updates about these aerial, virtual searches. Since our original article s8int.com has added nothing to the story but has only served to print comments and additional stories by others interested in the Death Valley Caves, if they exist.


The “Scorpions”: January 14, 2009 at 18:11 Google Earth:
36 31 23.67 N, 117 03 54.90 W

The search has recently focused on an area known as the scorpions, near Panamint Springs in Death Valley. There is a belief by some that the aerial maps of the area have been “airbrushed”. Scott S. who has supplied some updated material is a guy who has spent some of his vacations searching for the caves for up to a week at a time. He says he only lives 1,000 miles away. Disneyland is closer, I’m sure.

Recently, Jef Anderson, another adventurer and some of his compatriots took the possible coordinates of the cave system and set out for Death Valley. Here is an account of their journey.

Panamint Mountain Scorpions
Who says there’s no Intelligent Design?

Dateline: 2/14/11 – by Jef Anderson
All Photos and Text Copyright Jef Anderson, 2011

Right:Yahoo maps
36 31 23.67 N, 117 03 54.90 W

Below, Left: Bing Maps
36 31 23.67 N, 117 03 54.90 W
As near as I can tell… these images started appearing on all three major satellite mapping websites at approximately the same time – On January 14, 2009 Wouter W. of The Hague, Netherlands found this Google Earth image (the first image), which has since been removed from the Google sites, but can still be found by inputting the Long/Lat coordinates into the other map services.

It’s an extremely curious image, seemingly a massive geoglyph of a pair of Scorpions (or maybe Lobsters) positioned in the heart of the Panamint Mountains, in Death Valley California, and only visible from satellites or high flying aircraft, which would make the size on par with the Nazca geoglyphs.

On February 12, 2011 a group of 7 men set out to reach those GPS coordinates. The demographics of our group was comprised mainly of Educators, a few of them being former History teachers, all with Master’s degrees. Most of the members are experienced High Sierra outdoorsmen, having conquered several 14k peaks, with years of experience heading into extreme conditions.

This experience led us to choose February as the time to attempt this trek. We had great weather, temperatures ranging from 32º to 78º. Sudden thunder storms are really the only danger at this time of year, but we had blue skies and little to no wind. This window of temperate weather only occurs a few weeks every year before the legendary heat returns.

We decided to seize the opportunity.

Armed only with satellite maps and handheld GPS units, we set out from Salt Creek. There are no trails or paths heading toward the Scorpions, so we struck out across the one of the most inhospitable deserts in the America’s. The trackless journey took us across many types of terrain, rolling sand dunes, suddenly deep ravines, rocks ranging from gravel to boulders, and the ground covers with all manner of small cactus and nettled brush. The trek toward the coordinates was a steady climb of about 15º beginning under 200 feet below sea level.

Our base camp was Furnace Creek campground and we tented it, finding the National Parks service campground a nice one (although dealing with the Rangers is always a pain, and 12 Sax was not exception).

At about 6:45am and just before the sun shone down into the Panamint Valley we drove north up Hwy 190 toward the Salt Creek Trail parking lot, arriving around 7:10am. After quickly donning our daypacks, we set the compass and struck out at a heading of 220º. Our first waypoint was Trellis Canyon, though the way was obscured by a sharp, rising ridgeline bordering the salt creek.

The map below shows our starting point at Salt Creek, Trellis Canyon, and finally the Scorpions.

What follows is the log of the expedition.

07:15: Salt Creek Interpretive Trail Parking lot: 36 º 35’ 26.8N, 116 º 59’ 24.3W
- Sunrise, approx. 50º temp
- We climbed a small, steep ridgeline due West, then followed rolling desert
- Heading: 220 º
- Elevation: -200 ft.

07:45: 36 º 34’ 45.7N, 116 º 59 49.3W
- approx 55 º temp
- Now we encountered steep broken ravines cut into the rising desert floor, we were forced to pick our way down into the ravines, often traveling along them for a time before climbing out again, trying to maintain our bearing. Will’s foot issues force him to turn back – very bummed.

- Heading 200 º
- Elevation: – 200 ft.

08:15: 36 º 34’ 22.9N, 117 00’ 6.9W
- Approx 60 º temp

- We are now through the ravines and face a slope rising North by Northwest. It consists of softball to bowling ball size rocks, making the footing difficult. They’re too big to get flat footing and too small to “boulder” across, very annoying. In the distance we see two possible canyons that may be Trellis, so we split the difference between the two, heading toward the center.

- Heading: 200 º
- Elevation: -150 ft.

08:45: 36 º 33’ 43.9N, 117 00’ 46.8W
- Approx 60 º
- Still on the long climbing slope of rocks, we pass power lines and talk about what a miserable job installing those must have been
- Heading 220 º
- Elevation: -50 ft.

09:15: 36 º 33’ 17.3N, 117 01’ 33.4W
- Approx 60 º
- The angle of the slope has significantly increased, though the rocks become smaller, with many places very gravelly. We begin to think that the Northern canyon is indeed Trellis, so adjust out bearing and head that way.
- Heading: 240 º
- Elevation: +283

09:45: 36 º 33’ 7.7N, 117 02’ 34.0W
- Approx 65 º

- The bugs appear. These are little gnats, virtually invisible and SUPER annoying. What could they possible live on when they can’t get humans? We’ve seen no wildlife or water. We take a break in the shade of first set of uprising rocks that will eventually form the canyons, but the gnats are everywhere and we don’t stay long.

- Heading 310 º
- Elevation: +938

10:15: 36 º 32’ 53.8N, 117 03’ 20.0W
- Approx 65 º

- We enter Trellis Canyon. We’ve been shooting for 36 32 55.1N, 117 03 19.11W coordinates and pretty much navigated to the spot, once we decided the Northern canyon was our mark. The bugs are still present but as long as we keep moving they don’t bother too badly. The mouth of the canyon is about 200 yards across with several large rock formations that are evident in the satellite maps.
- Heading: 190 º
- Elevation: +1212 ft.

10:45: 36 º 32’ 13.2N, 117 03’ 35.2W
- Approx 65 º
- We kept working our way back into the ever narrowing canyon. The walls squeeze in, emphasizing the height of the canyon. The outer canyon walls much reach 3000 to 4000 feet up, with the ones we can touch easily over a 1000 feet. At one place in the “Narrows” I can touch both sides of the walls at the same time and yet the canyon towers about 2000 feet straight up – It’s an awesome place, and by the gps coordinates we know we’re getting close to the Scorpions.

All along this part of the canyon, we can see “hanging valleys” open up above us in the rocks. All of them are at least 50 to 100 feet up from the riverbed that we’re walking on. The evidence of water erosion is significant, and it’s easy to see how this canyon was formed, being cut so sharply by the flash floods that race down the narrow canyon. One of these hanging valleys is easy to spot when we climbed a rock wall across from it.

There is now a shear rock wall that must be climbed in order to really see into the valley, but from our view it’s clearly evident that it has a lot of square acreage. It’s now absolutely inaccessible without a 3 or 4 rated climb of over 100 feet. And it’s not alone, we see several places were areas open up, but they’re all very high up in the canyon walls, some 1000 to 3000 feet up. We didn’t plan on climbing, knowing that we wouldn’t have the time. But climbing there is the only way to really explore the mysteries of the place. We believe the coolest stuff is now high up in the canyon walls, and can only be reached by climbing.

- Heading: 190 º
- Elevation: +1524 ft.

11:36: 36 º 31’ 23.67N, 117 03’ 54.90W
- Approx 65 º
- THE SCORPIONS, we reach the coordinates a little after 11:30am.

The narrow sliver of sky we see is still bright blue, with golden sunlight reflecting hard off the towering canyon walls. We however are covered in dark shadow. It’s evident that some places within the canyon never get sunlight because the extremely high walls. I’m not just spouting poetic imagery… this is the point –
The Scorpions are created by shadow. There are so many twisting, turning corners to the canyon that it’s easy to see how the Scorpions are formed.

I’ve been so mystified by the darkness of the Scorpions, when every other shadow on the satellite images are a lot softer. But now, standing in these impressively high and narrow canyons, the darkness of the images is obvious. The contrast of the light and darkness is extreme and the blackness of the shadow in such close proximity to that of the brightly lit rocks, no doubt creates the deep blackness of the satellite photo images.


We’ve found no caves or caverns at ground level, but as I’ve stated, we’ve seen the potential for both high up in many places along the canyon. We also haven’t seen any geoglyphs or any evidence that humans have ever been in this place. BUT, it’s also very obvious that the canyon get scoured by flood waters and has (relatively) quickly sunk down many dozens if not hundreds of feet since the time people may have lived here.

If people did once inhabit this place, the evidence for them wouldn’t be down at the current ground level, but up much higher in the canyon walls and in the suspended/hanging valleys that would have been much closer to ground level long ago.
- Elevation: +2169 ft.

This picture (top article photo) was taken at precisely 36 31 23.67 N, 117 03 54.90 W – the coordinates given by all three satellite map services, showing the Scorpions


The fact that the Scorpions exist in the satellite images is, in my mind, evidence of the awesome creativity that God shows in his creation. No one can deny there are images on the satellite photos, images that are created by shadow and light, yet clearly form recognizable, symmetrical shapes; shapes that don’t happen randomly in nature. It seems this place is marked specially for some reason and though evidence wasn’t observable at ground level, who can guess what kinds of things may be found higher up in the canyon walls.

We’ll be going back, with climbing gear ;)

Prior Articles on this Topic

Ancient Civilizations, Giants, Tunnels Beneath Southern California?
Update on:Ancient Civilizations, Giants, Tunnels Beneath Southern California? Tuesday, December 23rd, 2008

More on Death Valley Cave Complex–Giants Etc. Thursday, January 1st, 2009

Updating the Update of the Death Valley Giants Cave Complex-with Coordinates Monday, January 12th, 2009

Death Valley Giants Cave Complex Satellite Photo Blurred Out? Wednesday, April 28th, 2010

New Update on the Death Valley Cave Complex-Quest for Death Valley Giants and the Cover Up? Scott S. June 16 2010

New Update on the Death Valley Cave Complex Scott S. June 17 2010

How Darwinism Deals with Human Giants: Go Lieth?

Church of Darwin, Giants in Those Days, Science, Unexplained Artifact | Posted by Chris Parker
Feb 10 2011

Numbers 13:33
And there we saw the giants, the sons of Anak, which come of the giants: and we were in our own sight as grasshoppers, and so we were in their sight.

The Quote Box following contains an Email sent to s8int.com by Jim S. which itself contains quotes and references to an article on the website; The Naked Scientists, which is an interview with a South African anthropologist.

Photo: Sellards: giant of early 20th century Florida anthropology and geology.
“As just one example of the “deception” that Jim S. references, Elias Howard Sellards, who became the first state geologist of the State of Florida, found human bones near Vero Beach Florida of what he admitted were “fully modern” individuals he estimated to be between 10 and 12 feet tall. Today, Vero Beach man remains controversial because of his “modernity” but no refererence is ever found concerning his alleged giant stature. A link to that article is at the end of this one.

Following Jim’s email is a news article we’re presenting that appears to be somewhat related….s8int.com

Hello,

Photo: Femur of Ancient Giant compared to normal human

“This is Jim, the guy who found the Castelnau giant report. I recently stumbled across this interesting new bit of information concerning a race of giants in Africa:

Although you will never hear about it in the National Geographic or Nature Magazine, Evolutionist scientists have known all along that races of giants have existed. We all know about Meganthropus and Gigantopithecus, but many skeletons of other giants have been found by anthropologists for decades, and in this rare 2007 interview by “The Naked Scientist” we can get a quick glimpse of some of what has been found.

A 2007 interview with Professor Lee Burger, University of Witwatersrand South Africa reveals some startling clues:”
Prof. Lee Burger -
“One of the most interesting things that the fossil record reveals is that we went through a period of extreme giantism. These were people routinely over 7ft tall, they were huge. This was before we turned into the modern humans of today. ”

” What we’re looking at is the most enormous femur: the bit that forms your hip joint. That’s huge. As a doctor I know how big they normally are, that’s huge.

They are huge. That’s so big we can’t even calculate how big this individual was.”

“….we found a lot of them. Everywhere we find them we find them enormous. These are what we call archaic Homo sapiens. Some people refer to them as Homo heidelbergensis. These individuals are extraordinary, they are giants”

thenakedscientists

Jim’s Remarks Continued:
“This is just another example of the establishment finding races of giants, and trying to reconcile them into their pet ape ancestry agenda. It’s also very remarkable that this race of giants has never made the headlines in Nature Magazine or Nat Geo, whereas the discovery of the 3 1/2 foot tall pygmies on the Island of Flores in 2004 created an Academic sensation with headlines all over the press announcing the discovery of tiny ape men, who were probably just smaller versions of us.

Perhaps they are concerned that if they were to report the discovery of a race of 8 foot tall ancient giants, they would somehow be vindicating scripture and they simply cannot allow this.

Dwarfs and small ape sized men are OK to report, but giants twice our size cannot be reported for fear that Bibles might start thumping and accountability and God might actually exist. Well maybe I am going a bit too far, but you get my point.

God bless,”
Jim S.

United Press Feb. 9, 1929
Drawing by Dan Smith, 1929

Ancient Battle:
Capetown, South Africa. Feb. 9. An important anthropological discovery consisting of the fossil of a gigantic prenegroid man and a huge extinct type of buffalo fossil was announced last night by the Transvaal Museum authorities.

The positions of the fossils indicated the buffalo had trampled the man in a battle.
The skull of the man had been crushed.

The fossil of the buffalo showed that the animal measured 12 feet between the horn tips. The fossils were found near Springbok Flats, Transvaal.

See Also: Giant Men? Evidence; Amen! Giant Tracks? Science Attacks! 12 Feet Tall? Darwinists; Not at All! Historical Evidence That There Were Giants in Those Days-As Alleged by Scientists of the Recent Past

The Splay-Footed Cricket’s Giving Darwinists a Spot of Bother
For 100 Million Years Baby Looked Just Like Mother and Father?
So How Have These Crickets Stayed in Stasis For So Long?
Does This Evolving, Non Evolving Problem Show Darwin Was Wrong?

Church of Darwin, s8int.com, Science, Uncategorized, Unexplained Artifact | Posted by Chris Parker
Feb 05 2011


Rare insect fossil reveals 100 million years of evolutionary stasis
February 3, 2011
Physorg.com

Photo: A fossil found in northeastern Brazil confirmed that the splay-footed cricket of today has at least a 100-million-year-old pedigree. Credit: Hwaja Goetz

Researchers have discovered the 100 million-year-old ancestor of a group of large, carnivorous, cricket-like insects that still live today in southern Asia, northern Indochina and Africa.

The new find, in a limestone fossil bed in northeastern Brazil, corrects the mistaken classification of another fossil of this type and reveals that the genus has undergone very little evolutionary change since the Early Cretaceous Period, a time of dinosaurs just before the breakup of the supercontinent Gondwana.

The findings are described in a paper in the open access journal ZooKeys.

“Schizodactylidae, or splay-footed crickets, are an unusual group of large, fearsome-looking predatory insects related to the true crickets, katydids and grasshoppers, in the order Orthoptera,” said University of Illinois entomologist and lead author Sam Heads, of the Illinois Natural History Survey.

“They get their common name from the large, paddle-like projections on their feet, which help support their large bodies as they move around their sandy habitats, hunting down prey.”

Although the fossil is distinct from today’s splay-footed crickets, its general features differ very little, Heads said, revealing that the genus has been in a period of “evolutionary stasis” for at least the last 100 million years.

Other studies have determined that the region where the fossil was found was most likely an arid or semi-arid monsoonal environment during the Early Cretaceous Period, Heads said, “suggesting that the habitat preferences of Schizodactylus have changed little in over 100 million years.”

More information: The paper, “On the Placement of the Cretaceous Orthopteran Brauckmanni groeningae From Brazil, With Notes on the Relationships of Schizodactylidae (Orthoptera, Ensifera),” is available online.

Provided by University of Illinois at Urbana-Champaign (news : web)

See Also: Pelican Fossil Poses Evolutionary Riddle; Why, They Haven’t Evolved, Not Even a Little

Thanks to Chris Z. and Scott S.

“The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Church of Darwin, Science, Uncategorized, Unexplained Artifact | Posted by Chris Parker
Feb 04 2011


by Eugene Wigner

Photo:Fibonacci Sequence; Left Milky way, Right, Hurricane

Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc. Copyright © 1960 by John Wiley & Sons, Inc.

Mathematics, rightly viewed, possesses not only truth, but supreme beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

–BERTRAND RUSSELL, Study of Mathematics

There is a story about two friends classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on.

His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. “How can you know that?” was his query. “And what is this symbol here?” “Oh,” said the statistician, “this is pi.” “What is that?” “The ratio of the circumference of the circle to its diameter.” “Well, now you are pushing your joke too far,” said the classmate, “surely the population has nothing to do with the circumference of the circle.”

Naturally, we are inclined to smile about the simplicity of the classmate’s approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense.

I was even more confused when, not many days later, someone came to me and expressed his bewilderment with the fact that we make a rather narrow selection when choosing the data on which we test our theories. “How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?” It has to be admitted that we have no definite evidence that there is no such theory.

The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.

Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories.

In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, “What is mathematics?”, then, “What is physics?”, then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.

WHAT IS MATHEMATICS?

Somebody once said that philosophy is the misuse of a terminology which was invented just for this purpose. This statement is quoted here from W. Dubislav’s Die Philosophie der Mathematik in der Gegenwart (Berlin: Junker and Dunnhaupt Verlag, 1932), p. 1.] In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts. Mathematics would soon run out of interesting theorems if these had to be formulated in terms of the concepts which already appear in the axioms.

Furthermore, whereas it is unquestionably true that the concepts of elementary mathematics and particularly elementary geometry were formulated to describe entities which are directly suggested by the actual world, the same does not seem to be true of the more advanced concepts, in particular the concepts which play such an important role in physics. Thus, the rules for operations with pairs of numbers are obviously designed to give the same results as the operations with fractions which we first learned without reference to “pairs of numbers.” The rules for the operations with sequences, that is, with irrational numbers, still belong to the category of rules which were determined so as to reproduce rules for the operations with quantities which were already known to us.

Most more advanced mathematical concepts, such as complex numbers, algebras, linear operators, Borel setsãand this list could be continued almost indefinitely were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty. In fact, the definition of these concepts, with a realization that interesting and ingenious considerations could be applied to them, is the first demonstration of the ingeniousness of the mathematician who defines them. The depth of thought which goes into the formulation of the mathematical concepts is later justified by the skill with which these concepts are used.

The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess. However, this is not our present subject. The principal point which will have to be recalled later is that the mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

M. Polanyi, in his Personal Knowledge (Chicago: University of Chicago Press, 1958), says: “All these difficulties are but consequences of our refusal to see that mathematics cannot be defined without acknowledging its most obvious feature: namely, that it is interesting” (p 188).]

The complex numbers provide a particularly striking example for the foregoing. Certainly, nothing in our experience suggests the introduction of these quantities. Indeed, if a mathematician is asked to justify his interest in complex numbers, he will point, with some indignation, to the many beautiful theorems in the theory of equations, of power series, and of analytic functions in general, which owe their origin to the introduction of complex numbers. The mathematician is not willing to give up his interest in these most beautiful accomplishments of his genius. The reader may be interested, in this connection, in Hilbert’s rather testy remarks about intuitionism which “seeks to break up and to disfigure mathematics,” Abh. Math. Sem., Univ. Hamburg, 157 (1922), or Gesammelte Werke (Berlin: Springer, 1935), p. 188.]

WHAT IS PHYSICS?

The physicist is interested in discovering the laws of inanimate nature. In order to understand this statement, it is necessary to analyze the concept, “law of nature.”

The world around us is of baffling complexity and the most obvious fact about it is that we cannot predict the future. Although the joke attributes only to the optimist the view that the future is uncertain, the optimist is right in this case: the future is unpredictable. It is, as Schrodinger has remarked, a miracle that in spite of the baffling complexity of the world, certain regularities in the events could be discovered. One such regularity, discovered by Galileo, is that two rocks, dropped at the same time from the same height, reach the ground at the same time. The laws of nature are concerned with such regularities. Galileo’s regularity is a prototype of a large class of regularities. It is a surprising regularity for three reasons.

The first reason that it is surprising is that it is true not only in Pisa, and in Galileo’s time, it is true everywhere on the Earth, was always true, and will always be true. This property of the regularity is a recognized invariance property and, as I had occasion to point out some time ago, without invariance principles similar to those implied in the preceding generalization of Galileo’s observation, physics would not be possible.

The second surprising feature is that the regularity which we are discussing is independent of so many conditions which could have an effect on it. It is valid no matter whether it rains or not, whether the experiment is carried out in a room or from the Leaning Tower, no matter whether the person who drops the rocks is a man or a woman.

It is valid even if the two rocks are dropped, simultaneously and from the same height, by two different people. There are, obviously, innumerable other conditions which are all immaterial from the point of view of the validity of Galileo’s regularity. The irrelevancy of so many circumstances which could play a role in the phenomenon observed has also been called an invariance. However, this invariance is of a different character from the preceding one since it cannot be formulated as a general principle.

The exploration of the conditions which do, and which do not, influence a phenomenon is part of the early experimental exploration of a field. It is the skill and ingenuity of the experimenter which show him phenomena which depend on a relatively narrow set of relatively easily realizable and reproducible conditions. See, in this connection, the graphic essay of M. Deutsch, Daedalus 87, 86 (1958).

A. Shimony has called my attention to a similar passage in C. S. Peirce’s Essays in the Philosophy of Science (New York: The Liberal Arts Press, 1957), p. 237.] In the present case, Galileo’s restriction of his observations to relatively heavy bodies was the most important step in this regard. Again, it is true that if there were no phenomena which are independent of all but a manageably small set of conditions, physics would be impossible.

The preceding two points, though highly significant from the point of view of the philosopher, are not the ones which surprised Galileo most, nor do they contain a specific law of nature. The law of nature is contained in the statement that the length of time which it takes for a heavy object to fall from a given height is independent of the size, material, and shape of the body which drops. In the framework of Newton’s second “law,” this amounts to the statement that the gravitational force which acts on the falling body is proportional to its mass but independent of the size, material, and shape of the body which falls.

The preceding discussion is intended to remind us, first, that it is not at all natural that “laws of nature” exist, much less that man is able to discover them.

E. Schrodinger, in his What Is Life? (Cambridge: Cambridge University Press, 1945), p. 31, says that this second miracle may well be beyond human understanding. The present writer had occasion, some time ago, to call attention to the succession of layers of “laws of nature,” each layer containing more general and more encompassing laws than the previous one and its discovery constituting a deeper penetration into the structure of the universe than the layers recognized before.

However, the point which is most significant in the present context is that all these laws of nature contain, in even their remotest consequences, only a small part of our knowledge of the inanimate world. All the laws of nature are conditional statements which permit a prediction of some future events on the basis of the knowledge of the present, except that some aspects of the present state of the world, in practice the overwhelming majority of the determinants of the present state of the world, are irrelevant from the point of view of the prediction. The irrelevancy is meant in the sense of the second point in the discussion of Galileo’s theorem. The writer feels sure that it is unnecessary to mention that Galileo’s theorem, as given in the text, does not exhaust the content of Galileo’s observations in connection with the laws of freely falling bodies.

As regards the present state of the world, such as the existence of the earth on which we live and on which Galileo’s experiments were performed, the existence of the sun and of all our surroundings, the laws of nature are entirely silent. It is in consonance with this, first, that the laws of nature can be used to predict future events only under exceptional circumstancesãwhen all the relevant determinants of the present state of the world are known. It is also in consonance with this that the construction of machines, the functioning of which he can foresee, constitutes the most spectacular accomplishment of the physicist. In these machines, the physicist creates a situation in which all the relevant coordinates are known so that the behavior of the machine can be predicted. Radars and nuclear reactors are examples of such machines.

The principal purpose of the preceding discussion is to point out that the laws of nature are all conditional statements and they relate only to a very small part of our knowledge of the world. Thus, classical mechanics, which is the best known prototype of a physical theory, gives the second derivatives of the positional coordinates of all bodies, on the basis of the knowledge of the positions, etc., of these bodies. It gives no information on the existence, the present positions, or velocities of these bodies. It should be mentioned, for the sake of accuracy, that we discovered about thirty years ago that even the conditional statements cannot be entirely precise: that the conditional statements are probability laws which enable us only to place intelligent bets on future properties of the inanimate world, based on the knowledge of the present state. They do not allow us to make categorical statements, not even categorical statements conditional on the present state of the world. The probabilistic nature of the “laws of nature” manifests itself in the case of machines also, and can be verified, at least in the case of nuclear reactors, if one runs them at very low power. However, the additional limitation of the scope of the laws of nature which follows from their probabilistic nature will play no role in the rest of the discussion.

THE ROLE OF MATHEMATICS IN PHYSICAL THEORIES

Having refreshed our minds as to the essence of mathematics and physics, we should be in a better position to review the role of mathematics in physical theories.

Naturally, we do use mathematics in everyday physics to evaluate the results of the laws of nature, to apply the conditional statements to the particular conditions which happen to prevail or happen to interest us. In order that this be possible, the laws of nature must already be formulated in mathematical language. However, the role of evaluating the consequences of already established theories is not the most important role of mathematics in physics. Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool.

Mathematics does play, however, also a more sovereign role in physics. This was already implied in the statement, made when discussing the role of applied mathematics, that the laws of nature must have been formulated in the language of mathematics to be an object for the use of applied mathematics. The statement that the laws of nature are written in the language of mathematics was properly made three hundred years ago; (It is attributed to Galileo) it is now more true than ever before.

In order to show the importance which mathematical concepts possess in the formulation of the laws of physics, let us recall, as an example, the axioms of quantum mechanics as formulated, explicitly, by the great physicist, Dirac. There are two basic concepts in quantum mechanics: states and observables. The states are vectors in Hilbert space, the observables self-adjoint operators on these vectors. The possible values of the observations are the characteristic values of the operators but we had better stop here lest we engage in a listing of the mathematical concepts developed in the theory of linear operators.

It is true, of course, that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics. It is true also that the concepts which were chosen were not selected arbitrarily from a listing of mathematical terms but were developed, in many if not most cases, independently by the physicist and recognized then as having been conceived before by the mathematician. It is not true, however, as is so often stated, that this had to happen because mathematics uses the simplest possible concepts and these were bound to occur in any formalism.

As we saw before, the concepts of mathematics are not chosen for their conceptual simplicity. Even sequences of pairs of numbers are far from being the simplest concepts but for their amenability to clever manipulations and to striking, brilliant arguments. Let us not forget that the Hilbert space of quantum mechanics is the complex Hilbert space, with a Hermitean scalar product. Surely to the unpreoccupied mind, complex numbers are far from natural or simple and they cannot be suggested by physical observations.

Furthermore, the use of complex numbers is in this case not a calculational trick of applied mathematics but comes close to being a necessity in the formulation of the laws of quantum mechanics. Finally, it now begins to appear that not only complex numbers but so-called analytic functions are destined to play a decisive role in the formulation of quantum theory. I am referring to the rapidly developing theory of dispersion relations.

It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind’s capacity to divine them. The observation which comes closest to an explanation for the mathematical concepts’ cropping up in physics which I know is Einstein’s statement that the only physical theories which we are willing to accept are the beautiful ones. It stands to argue that the concepts of mathematics, which invite the exercise of so much wit, have the quality of beauty. However, Einstein’s observation can at best explain properties of theories which we are willing to believe and has no reference to the intrinsic accuracy of the theory.

We shall, therefore, turn to this latter question.

IS THE SUCCESS OF PHYSICAL THEORIES TRULY SURPRISING?

A possible explanation of the physicist’s use of mathematics to formulate his laws of nature is that he is a somewhat irresponsible person. As a result, when he finds a connection between two quantities which resembles a connection well-known from mathematics, he will jump at the conclusion that the connection is that discussed in mathematics simply because he does not know of any other similar connection. It is not the intention of the present discussion to refute the charge that the physicist is a somewhat irresponsible person. Perhaps he is. However, it is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language. Let us consider a few examples.

The first example is the oft-quoted one of planetary motion. The laws of falling bodies became rather well established as a result of experiments carried out principally in Italy. These experiments could not be very accurate in the sense in which we understand accuracy today partly because of the effect of air resistance and partly because of the impossibility, at that time, to measure short time intervals. Nevertheless, it is not surprising that, as a result of their studies, the Italian natural scientists acquired a familiarity with the ways in which objects travel through the atmosphere. It was Newton who then brought the law of freely falling objects into relation with the motion of the moon, noted that the parabola of the thrown rock’s path on the earth and the circle of the moon’s path in the sky are particular cases of the same mathematical object of an ellipse, and postulated the universal law of gravitation on the basis of a single, and at that time very approximate, numerical coincidence. Philosophically, the law of gravitation as formulated by Newton was repugnant to his time and to himself. Empirically, it was based on very scanty observations.

The mathematical language in which it was formulated contained the concept of a second derivative and those of us who have tried to draw an osculating circle to a curve know that the second derivative is not a very immediate concept. The law of gravity which Newton reluctantly established and which he could verify with an accuracy of about 4% has proved to be accurate to less than a ten thousandth of a per cent and became so closely associated with the idea of absolute accuracy that only recently did physicists become again bold enough to inquire into the limitations of its accuracy. [ See, for instance, R. H. Dicke, Am. Sci., 25 (1959).]

Certainly, the example of Newton’s law, quoted over and over again, must be mentioned first as a monumental example of a law, formulated in terms which appear simple to the mathematician, which has proved accurate beyond all reasonable expectations. Let us just recapitulate our thesis on this example: first, the law, particularly since a second derivative appears in it, is simple only to the mathematician, not to common sense or to non-mathematically-minded freshmen; second, it is a conditional law of very limited scope. It explains nothing about the earth which attracts Galileo’s rocks, or about the circular form of the moon’s orbit, or about the planets of the sun. The explanation of these initial conditions is left to the geologist and the astronomer, and they have a hard time with them.

The second example is that of ordinary, elementary quantum mechanics. This originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say “if the mechanics as here proposed should already be correct in its essential traits.” As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg’s rules of calculation were abstracted from problems which included the old theory of the hydrogen atom.

The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg’s calculating rules were meaningless. Heisenberg’s rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg’s rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshita at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we “got something out” of the equations that we did not put in.

The same is true of the qualitative characteristics of the “complex spectra,” that is, the spectra of heavier atoms. I wish to recall a conversation with Jordan, who told me, when the qualitative features of the spectra were derived, that a disagreement of the rules derived from quantum mechanical theory and the rules established by empirical research would have provided the last opportunity to make a change in the framework of matrix mechanics. In other words, Jordan felt that we would have been, at least temporarily, helpless had an unexpected disagreement occurred in the theory of the helium atom. This was, at that time, developed by Kellner and by Hilleraas. The mathematical formalism was too dear and unchangeable so that, had the miracle of helium which was mentioned before not occurred, a true crisis would have arisen.

Surely, physics would have overcome that crisis in one way or another. It is true, on the other hand, that physics as we know it today would not be possible without a constant recurrence of miracles similar to the one of the helium atom, which is perhaps the most striking miracle that has occurred in the course of the development of elementary quantum mechanics, but by far not the only one. In fact, the number of analogous miracles is limited, in our view, only by our willingness to go after more similar ones. Quantum mechanics had, nevertheless, many almost equally striking successes which gave us the firm conviction that it is, what we call, correct.

The last example is that of quantum electrodynamics, or the theory of the Lamb shift. Whereas Newton’s theory of gravitation still had obvious connections with experience, experience entered the formulation of matrix mechanics only in the refined or sublimated form of Heisenberg’s prescriptions. The quantum theory of the Lamb shift, as conceived by Bethe and established by Schwinger, is a purely mathematical theory and the only direct contribution of experiment was to show the existence of a measurable effect. The agreement with calculation is better than one part in a thousand.

The preceding three examples, which could be multiplied almost indefinitely, should illustrate the appropriateness and accuracy of the mathematical formulation of the laws of nature in terms of concepts chosen for their manipulability, the “laws of nature” being of almost fantastic accuracy but of strictly limited scope. I propose to refer to the observation which these examples illustrate as the empirical law of epistemology. Together with the laws of invariance of physical theories, it is an indispensable foundation of these theories. Without the laws of invariance the physical theories could have been given no foundation of fact; if the empirical law of epistemology were not correct, we would lack the encouragement and reassurance which are emotional necessities, without which the “laws of nature” could not have been successfully explored.

Dr. R. G. Sachs, with whom I discussed the empirical law of epistemology, called it an article of faith of the theoretical physicist, and it is surely that. However, what he called our article of faith can be well supported by actual examples many examples in addition to the three which have been mentioned.

THE UNIQUENESS OF THE THEORIES OF PHYSICS

The empirical nature of the preceding observation seems to me to be self-evident. It surely is not a “necessity of thought” and it should not be necessary, in order to prove this, to point to the fact that it applies only to a very small part of our knowledge of the inanimate world. It is absurd to believe that the existence of mathematically simple expressions for the second derivative of the position is self-evident, when no similar expressions for the position itself or for the velocity exist. It is therefore surprising how readily the wonderful gift contained in the empirical law of epistemology was taken for granted. The ability of the human mind to form a string of 1000 conclusions and still remain “right,” which was mentioned before, is a similar gift.

Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions. The question which presents itself is whether the different regularities, that is, the various laws of nature which will be discovered, will fuse into a single consistent unit, or at least asymptotically approach such a fusion.

Alternatively, it is possible that there always will be some laws of nature which have nothing in common with each other. At present, this is true, for instance, of the laws of heredity and of physics. It is even possible that some of the laws of nature will be in conflict with each other in their implications, but each convincing enough in its own domain so that we may not be willing to abandon any of them. We may resign ourselves to such a state of affairs or our interest in clearing up the conflict between the various theories may fade out. We may lose interest in the “ultimate truth,” that is, in a picture which is a consistent fusion into a single unit of the little pictures, formed on the various aspects of nature.

It may be useful to illustrate the alternatives by an example. We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity. These two theories have their roots in mutually exclusive groups of phenomena. Relativity theory applies to macroscopic bodies, such as stars. The event of coincidence, that is, in ultimate analysis of collision, is the primitive event in the theory of relativity and defines a point in space-time, or at least would define a point if the colliding panicles were infinitely small. Quantum theory has its roots in the microscopic world and, from its point of view, the event of coincidence, or of collision, even if it takes place between particles of no spatial extent, is not primitive and not at all sharply isolated in space-time. The two theories operate with different mathematical concepts the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively.

So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found. This example illustrates the two possibilities, of union and of conflict, mentioned before, both of which are conceivable.

In order to obtain an indication as to which alternative to expect ultimately, we can pretend to be a little more ignorant than we are and place ourselves at a lower level of knowledge than we actually possess. If we can find a fusion of our theories on this lower level of intelligence, we can confidently expect that we will find a fusion of our theories also at our real level of intelligence. On the other hand, if we would arrive at mutually contradictory theories at a somewhat lower level of knowledge, the possibility of the permanence of conflicting theories cannot be excluded for ourselves either. The level of knowledge and ingenuity is a continuous variable and it is unlikely that a relatively small variation of this continuous variable changes the attainable picture of the world from inconsistent to consistent.

This passage was written after a great deal of hesitation. The writer is convinced that it is useful, in epistemological discussions, to abandon the idealization that the level of human intelligence has a singular position on an absolute scale. In some cases it may even be useful to consider the attainment which is possible at the level of the intelligence of some other species. However, the writer also realizes that his thinking along the lines indicated in the text was too brief and not subject to sufficient critical appraisal to be reliable.

Considered from this point of view, the fact that some of the theories which we know to be false give such amazingly accurate results is an adverse factor. Had we somewhat less knowledge, the group of phenomena which these “false” theories explain would appear to us to be large enough to “prove” these theories. However, these theories are considered to be “false” by us just for the reason that they are, in ultimate analysis, incompatible with more encompassing pictures and, if sufficiently many such false theories are discovered, they are bound to prove also to be in conflict with each other. Similarly, it is possible that the theories, which we consider to be “proved” by a number of numerical agreements which appears to be large enough for us, are false because they are in conflict with a possible more encompassing theory which is beyond our means of discovery. If this were true, we would have to expect conflicts between our theories as soon as their number grows beyond a certain point and as soon as they cover a sufficiently large number of groups of phenomena. In contrast to the article of faith of the theoretical physicist mentioned before, this is the nightmare of the theorist.

Let us consider a few examples of “false” theories which give, in view of their falseness, alarmingly accurate descriptions of groups of phenomena. With some goodwill, one can dismiss some of the evidence which these examples provide. The success of Bohr’s early and pioneering ideas on the atom was always a rather narrow one and the same applies to Ptolemy’s epicycles. Our present vantage point gives an accurate description of all phenomena which these more primitive theories can describe. The same is not true any longer of the so-called free-electron theory, which gives a marvelously accurate picture of many, if not most, properties of metals, semiconductors, and insulators.

In particular, it explains the fact, never properly understood on the basis of the “real theory,” that insulators show a specific resistance to electricity which may be 1026 times greater than that of metals. In fact, there is no experimental evidence to show that the resistance is not infinite under the conditions under which the free-electron theory would lead us to expect an infinite resistance. Nevertheless, we are convinced that the free-electron theory is a crude approximation which should be replaced, in the description of all phenomena concerning solids, by a more accurate picture.

If viewed from our real vantage point, the situation presented by the free-electron theory is irritating but is not likely to forebode any inconsistencies which are unsurmountable for us. The free-electron theory raises doubts as to how much we should trust numerical agreement between theory and experiment as evidence for the correctness of the theory. We are used to such doubts.

A much more difficult and confusing situation would arise if we could, some day, establish a theory of the phenomena of consciousness, or of biology, which would be as coherent and convincing as our present theories of the inanimate world. Mendel’s laws of inheritance and the subsequent work on genes may well form the beginning of such a theory as far as biology is concerned. Furthermore, it is quite possible that an abstract argument can be found which shows that there is a conflict between such a theory and the accepted principles of physics. The argument could be of such abstract nature that it might not be possible to resolve the conflict, in favor of one or of the other theory, by an experiment. Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form.

It would give us a deep sense of frustration in our search for what I called “the ultimate truth.” The reason that such a situation is conceivable is that, fundamentally, we do not know why our theories work so well. Hence, their accuracy may not prove their truth and consistency. Indeed, it is this writer’s belief that something rather akin to the situation which was described above exists if the present laws of heredity and of physics are confronted.

Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.