Wednesday, September 20, 2017

Religion & Politics: Cognitive Style vs. Cognitive Ability

Religion and politics appear to be related to different aspects of cognition, according to new psychological research. Religion is more related to quick, intuitive thinking while politics is more related to intelligence.

The study, which was published in the scientific journal Personality and Individual Differences, found evidence that religious people tend to be less reflective while social conservatives tend to have lower cognitive ability.

“We have been doing research on how certain cognitive (thinking) styles (i.e., tendency to think analytically vs. intuitively) may be associated with or even lead to different social attitudes for a couple of years,” explained the study’s corresponding author, S. Adil Saribay of Boğaziçi University. “This research is partly motivated by the observation that the growing religiosity, anti-secularism, and anti-science sentiment across the world seemed to go along with the spread of simple ideas about the world that have intuitive appeal.

“Empirical findings generally suggest that an intuitive thinking style and a lower IQ level (cognitive ability) are associated with both religiosity and conservatism. Thinking style and cognitive ability are positively associated; and so are religiosity and conservatism. We noticed that there are reasons to believe that religiosity and social conservatism may be differentially predicted by cognitive style and cognitive ability, respectively.”

The study examined 426 American adults. Among the sample were 225 Christians, 59 Agnostics, 37 Atheists, 9 Buddhists, 8 Jews, 5 Pagans, 3 Muslims , 30 “others”, and 50 with no affiliation.

Saribay and his colleague, Onurcan Yilmaz, found that an intuitive thinking style independently predicted religious belief while low cognitive ability independently predicted social but not economic conservatism. In other words, people who tended to think intuitively rather than analytically were more likely to believe in a variety of religious concepts. People with lower cognitive ability were more likely to endorse socially conservative views.

“We would like to warn readers to resist the temptation to draw conclusions that suit their ideological worldviews,” Saribay told PsyPost. “One must not think in terms of profiles or categories of people and also not draw simple causal conclusions as our data do not speak to causality. Instead, it’s better to focus on how certain ideological tendencies may serve psychological needs, such as the need to simplify the world and conserve cognitive energy.”

“Our findings suggest that intuitive thinking serves the upholding of religious beliefs and by extension, growing less religious has more to do with overcoming one’s intuitions, if one has received religious upbringing. On the other hand, adopting socially progressive ideas may have more to do with intelligence compared to cognitive style. Note that these relations are not so clear cut and effect sizes are small.”

The small effect sizes mean that there were large overlaps between the groups studied. “But if any differential relation between these constructs exists, our survey of the literature and our data both suggest that it is in this direction,” Saribay said.

“By extension and more practically, resolving different types of ideological conflict may require different approaches,” he continued. “If conflict involves religious beliefs, it may be best to invite the religious party to reason slowly and more carefully in a calmer atmosphere (to enable analytic scrutiny), rather than to attack them and generate heated emotions (which would only bolster their tendency to rely on intuitions).”

“Resolution of conflict that involves social conservatism, on the other hand, may benefit more from breaking down progressive ideas into pieces that are easier to comprehend and reason about. This, of course, requires more direct evidence, but is suggested by our findings.”

The findings dovetail with previous research that found liberals tend to use analytic thinking when processing moral judgments while conservatives tend to use intuitive thinking.

“This is a difficult area because of its political nature and people often assume we are ideologically motivated in the research. It is of course likely that various biases influence research outcomes and this has been a topic of discussion (e.g., the liberal bias in social psychology),” Saribay added.
“However, a more balanced understanding can only be reached via continued empirical research. Human beings may sometimes benefit from cognitive simplification of a complex and at times scary world of constant change and uncertainty. It does seem that certain aspects of religion and conservative ideology serve to deal with this, in slightly different ways. This is the direction that evidence points to thus far. However, researchers of course must resist this very need to simplify the world beyond a certain level.”

“Our field treated culture as a relatively static entity and made simple, sweeping distinctions such as individualism vs. collectivism; but as research continued, it moved on to a more nuanced person by situation by culture type of understanding,” Saribay said. “Same happened earlier with the construct of personality. Thus, we hope that our understanding of the current variables and their relations will grow more complex.”

The study, titled “Analytic cognitive style and cognitive ability differentially predict religiosity and social conservatism“, was published online March 30, 2017.

Monday, September 4, 2017

August 21, 2017: Total Solar Eclipse

"In ancient times, every culture had a sun god, and it was usually one of the chief gods of their whole pantheon," explains Bradley Schaefer, astronomy professor at Louisiana State University. "Humans couldn't touch what's in the sky, so they believed it must be where the gods are. When you have a total solar eclipse, it looks like the death of a god, and to them, that couldn't be a good thing."

In many cultures throughout human history, the sun was seen as an entity of supreme importance, crucial to their very existence. It was regularly worshipped as a god – Amun-Ra to the Egyptians and Helios to the Greeks – or as a goddess, such as Amaterasu for the Japanese and Saule for many Baltic cultures.

One reason the sun served as a god or goddess in so many cultures was its awesome power: Looking directly at it would severely damages the eyes, a sign of the sun deity’s wrath.

For thousands of years people learned about the sun through careful observation. Understanding the sun and seasons was critical to survival. As early as 4,000 years ago, ancient astronomers tried to predict solar eclipses in China and Greece.
Ancients texts from China, Mesopotamia and Greece that mention solar eclipses all suggest the phenomena “were just trouble,” says Ed Krupp, director of the Griffith Observatory in Los Angeles. “They represented a serious disturbance in the natural order of things.

"The eclipse always seems to coincide with some sort of panic,” adds astronomy historian Steve Ruskin, author of the book America's First Great Eclipse.

Different cultures have different ways of explaining why eclipses happen, but the stories generally share a theme of the sun being “devoured.” The Chinese word for eclipse, rishi, is composed to the characters for “sun/day” and “eat.” The word eclipse itself derives from the the Ancient Greek root for "abandonment," ékleipsis, which makes sense since the Greeks viewed the eclipse as the sun “abandoning” the earth, Krupp explains.

Several East Asian cultures believed the eclipse was caused by a giant frog eating the sun, and in China, myths tell of a dragon doing the devouring, Ruskin says. In Norse mythology, the eclipse was the result of two sky wolves, Sköll and Hati, chasing and finally eating the sun, leading to its temporary disappearance, Krupp adds. (Some scholars doubt the veracity of that interpretation, however.)

Many cultures thought that such a disastrous event required their immediate action to help restore order. Ancient Chinese and Mesopotamians made loud noises to scare away the spirits or creatures doing the devouring. Hugh Lenox Scott, who at the time was a member of the U.S. Cavalry and later a superintendent of West Point, recorded his observations of the Cheyenne tribe during the solar eclipse of 1878. “They became very much excited when the eclipse began, shooting off guns and making every sort of noise they could to frighten away the evil medicine which they thought was destroying the sun,” he wrote.

The eclipse was a bad omen for many ancient civilizations, but how the omen was interpreted varied greatly by culture. “An eclipse of either the sun or moon is looked upon as a terribly calamity, being sure to be the forerunner of disease or death,” wrote J.G. Wood on the beliefs of Australian Aborigines in his 1870 tome The Natural History of Man. In Ancient China, solar eclipses were a sign that the emperor, considered partially divine, had done something wrong. In Mesoamerican cultures, they were occasions for human sacrifice to ward off evil, Krupp explains.

Sometimes, however, the event was explained away. In several references from China and Mesopotamia, royal astronomers interpreted the eclipse “as bad news for somebody else... some other king or country,” says Krupp.

The Conversation:
Eskimos thought an eclipse meant that the sun and moon had become temporarily diseased. In response, they’d cover up everything of importance – themselves included – lest they be infected by the “diseased” rays of the eclipsed sun.

For the Ojibwe tribe of the Great Lakes, the onset of total eclipse represented an extinguished sun. To prevent permanent darkness, they proceeded to fire flaming arrows at the darkened sun in an attempt to rekindkle it.

Amidst the plethora of the myths and legends and interpretations of this strange event, there are seeds of understanding about their true nature.

For example, the famed total solar eclipse of May 28, 585 B.C., occurred in the middle of a battle between the Medes and the Lydians in what is now the northeast region of modern-day Turkey. The eclipse actually ended the conflict on the spot, with both sides interpreting the event as a sign of the displeasure from the gods. But based on the writings of the ancient Greek historian Heroditus, it’s thought that the great Greek philosopher-mathematician Thales of Miletus had, coincidentally, predicted its occurrence.

Chinese, Alexandrian and Babylonian astronomers were also said to be sophisticated enough to not only understand the true nature of solar eclipses, but also to roughly predict when the “dragon” would come to devour the sun. (As with much knowledge back then, however, astronomical and astrological findings were relayed only to the ruling elites, while myths and legends continued to percolate among the general population.)

A Hindu myth tells the story of the demon Rahu's disembodied head attempting to swallow the sun whole... but because he was lacking in a throat, it would fall out through the hole in his neck shortly thereafter.

In other branches of Hindu culture, the “sun eater” took the more traditional form of a dragon. To fight this beast, certain Hindu sects in India immersed themselves up to the neck in water in an act of worship, believing that the adulation would aid the sun in fighting off the dragon.

More recently, scientists planned experiments during eclipses to test theories and equipment. With the sun blocked, other atmospheric features become visible. Scientists proved Einstein’s theory of relativity, and they searched for a theoretical planet Vulcan but it was proven not to exist.

Albert Einstein wrote a paper on special relativity in 1905 and formed his theory of general relativity, or the relationship between gravity and the curvature of space and time, just before World War I. He believed that an event like an eclipse would showcase this idea of light bending as it nears a massive object.

Arthur Eddington, an English astronomer, read his paper on relativity and set out to test it during the May 1919 eclipse. Eddington's photographs of the eclipse verified the theory by capturing the bending of starlight passing near the sun. Both Eddington's achievement and Einstein's brilliance were trumpeted in the media.

More information:
» Why eclipses have inspired terror and awe
» Conspiracy theorist claims this month's solar eclipse will signal the end

Sunday, August 6, 2017

"The arc of the moral universe is long, but it bends toward justice."

“Evil may so shape events that Caesar will occupy a palace and Christ a cross,” Dr. King wrote, “but that same Christ will rise up and split history into A.D. and B.C., so that even the life of Caesar must be dated by his name. Yes, ‘the arc of the moral universe is long, but it bends toward justice.’” (Note: King has those words in quotes because he was actually citing 19th century clergyman Theodore Parker, who first coined the phrase.)
Quote Investigator:
Theodore Parker was a Unitarian minister and prominent American Transcendentalist born in 1810 who called for the abolition of slavery. In 1853 a collection of “Ten Sermons of Religion” by Parker was published and the third sermon titled “Of Justice and the Conscience” included figurative language about the arc of the moral universe: 1
Look at the facts of the world. You see a continual and progressive triumph of the right. I do not pretend to understand the moral universe, the arc is a long one, my eye reaches but little ways. I cannot calculate the curve and complete the figure by the experience of sight; I can divine it by conscience. But from what I see I am sure it bends towards justice.

Things refuse to be mismanaged long. Jefferson trembled when he thought of slavery and remembered that God is just. Ere long all America will tremble.
The words of Parker’s sermon above foreshadowed the Civil War fought in the 1860s. The passage was reprinted in later collections of Parker’s works. A similar statement using the same metaphor was printed in a book called “Morals and Dogma of the Ancient and Accepted Scottish Rite of Freemasonry” with a copyright date of 1871 and publication date of 1905. The author was not identified: 2
We cannot understand the moral Universe. The arc is a long one, and our eyes reach but a little way; we cannot calculate the curve and complete the figure by the experience of sight; but we can divine it by conscience, and we surely know that it bends toward justice. Justice will not fail, though wickedness appears strong, and has on its side the armies and thrones of power, the riches and the glory of the world, and though poor men crouch down in despair. Justice will not fail and perish out from the world of men, nor will what is really wrong and contrary to God’s real law of justice continually endure.
In 1918 a concise instance of the expression similar to the modern version was printed in a book titled “Readings from Great Authors” in a section listing statements attributed to Theodore Parker: 3
The arc of the moral universe is long, but it bends toward justice.
Here are additional selected citations in chronological order.

In 1932 a columnist for the Cleveland Plain Dealer in Ohio reported on an adage that he saw posted at a church. This saying nearly matched the 1918 expression, but the word “moral” was omitted. Boldface has been added to excerpts: 4
A Euclid Avenue church which displays weekly epigrams on its bulletin board, has this current offering: “The arc of the universe is long, but it bends toward justice.” I don’t know where the quotation is from—it sounds like Emerson—but I have tried in vain for three days now to puzzle out a meaning for it. It sounds so good that it really ought to mean something. Perhaps it is a vague and poetic way of claiming that nature is eventually just.
A newspaper reader recognized that the statement was based on the words of Theodore Parker and notified the columnist. Fourteen days after the original article was published the columnist reprinted the relevant excerpt from Parker’s sermon as published in 1853 and then praised his words. 5
In 1934 a version of the phrase was used in a sermon by Rev. Seth Brooks, pastor of the First Parish church of Malden, Massachusetts as reported in the Lowell Sun newspaper. No attribution was given: 6
“We must believe that the arc of the universe is long, but that it bends toward justice, toward one Divine end towards which creation moves onward and onward, forever.”
In 1940 a version was included in a New Year’s message by Rabbi Jacob Kohn in Los Angeles. No attribution was given: 7
“Our faith is kept alive by the knowledge, founded on long experience, that the arc of history is long and bends toward justice,” Rabbi Jacob Kohn told his audience at Temple Sinai. “We have seen so many ancient tyrannies pass from earth since Egypt and Rome held dominion that our eyes are directed not to the tragic present, but to the beyond, wherein the arc of history will be found bending toward justice, victory and freedom.”
In 1958 an article by Martin Luther King, Jr. was printed in “The Gospel Messenger” periodical. King employed the saying, and he placed it between quotation marks which signaled that it was a pre-existing aphorism: 8
Evil may so shape events that Caesar will occupy a palace and Christ a cross, but that same Christ arose and split history into A.D. and B.C., so that even the life of Caesar must be dated by his name. Yes, “the arc of the moral universe is long, but it bends toward justice.” There is something in the universe which justifies William Cullen Bryant in saying, “Truth crushed to earth will rise again.”
In 1964 King delivered the Baccalaureate sermon at the commencement exercises for Wesleyan University in Middletown, Connecticut, and he included the saying: 9
“The arc of the moral universe is long,” Dr. King said in closing, “but it bends toward justice.”
In 2009 Time magazine published an article by President Barack Obama that included the distinctive subphrase about history: “bends toward justice.” Obama credited the words to King: 10
But as I learned in the shadow of an empty steel plant more than two decades ago, while you can’t necessarily bend history to your will, you can do your part to see that, in the words of Dr. King, it “bends toward justice.” So I hope that you will stand up and do what you can to serve your community, shape our history and enrich both your own life and the lives of others across this country.
In 2010 the quote appeared in the pages of Time magazine again, and the words were credited to King: 11
Martin Luther King Jr. once said, “Let us realize the arc of the moral universe is long, but it bends toward justice.”
In conclusion, QI believes that Theodore Parker should be credited with formulating this metaphor about historical progress which was published in a collection of his sermons in 1853. By 1918 a concise version of the saying was being credited to Parker. In 1958 Martin Luther King, Jr. included the expression in an article, but he placed the words in quotation marks to indicate that the adage was already in circulation. King found the phrase attractive and included it in several of his speeches.
(In Memoriam: Thanks to my brother Stephen who asked about this saying. Special thanks to David Weinberger who pointed out that “Ten Sermons of Religion” was published in 1853.)

Update History: On June 28, 2015 the date of “Ten Sermons of Religion” was specified as 1853. The bibliographical note had the correct date of 1853, but within the main body of the article the date previously indicated was 1857. Please note that there were multiple editions and collections that included the quotation, and 1853 was the earliest edition located by QI.

Sunday, July 23, 2017

- Best Way to Repel Mosquitos


Experts recommend applying sunscreen first, then bug spray, not the other way around. And even if you wear bug repellent clothing, Consumer Reports recommends also using bug spray for greater protection.

In addition, the Centers for Disease Control and Prevention recommends wearing long-sleeve shirts and long pants, and staying in places with air conditioning and window and door screens to keep mosquitoes outside.

An experimental vaccine for the Zika virus is due to begin human testing soon, after getting the green light from U.S. health officials.


According to Consumer Reports, a number of insect repellents on the market can stop the mosquitoes that transmit the dangerous virus, as well as another mosquito-borne illness, West Nile virus. They put 16 insect repellents to the test to see which ones are the most effective.

"In a laboratory, very brave volunteers put their arms into a cage full of 200 disease-free mosquitoes and we use two types of mosquitoes," Trisha Calvo of Consumer Reports told CBS New York. "One is the Culex mosquito that carries West Nile and the other is the Aedes mosquito that carries Zika."

The results? "We found three ingredients that are safe for everyone to use, even pregnant women, and are very effective. And those are Deet, oil of lemon eucalyptus, and Picaridin," Calvo said.

Sawyer's Fisherman's Formula Picaridin got top results in Consumer Reports testing, keeping mosquitoes away for up to eight hours.

Other top performing repellents on the list include:

  • Ben's 30% DEET Tick & Insect Wilderness Formula
  • Natrapel 8 Hour, with 20 percent picaridin
  • Off! Deepwoods VIII, with 25 percent DEET
  • Repel Lemon Eucalyptus, with 30 percent lemon eucalyptus
  • Repel Scented Family, with 15 percent DEET

Consumer Reports advises skipping most repellents made with natural plant oils such as lemongrass or citronella, which were not very effective in their testing.


More information:
» The Scientist: "Plan to Fight Zika with GM Mosquitos in Florida Faces Opposition"
» Paste: "Gear Geek: Seven Ways to Ward Off Mosquitos"

Saturday, July 22, 2017

Saturday, June 24, 2017

Testosterone Supplements: Overconfidence and Bad Judgment

New York Times:
“Does being over 40 make you feel like half the man you used to be?”

Ads like that have led to a surge in the number of men seeking to boost their testosterone. The Food and Drug Administration reports that prescriptions for testosterone supplements have risen to 2.3 million from 1.3 million in just four years.

There is such a condition as “low-T,” or hypogonadism, which can cause fatigue and diminished sex drive, and it becomes more common as men age. But according to a study published in JAMA Internal Medicine, half of the men taking prescription testosterone don’t have a deficiency. Many are just tired and want a lift. But they may not be doing themselves any favors. It turns out that the supplement isn’t entirely harmless: Neuroscientists are uncovering evidence suggesting that when men take testosterone, they make more impulsive — and often faulty — decisions.

Researchers have shown for years that men tend to be more confident about their intelligence and judgments than women, believing that solutions they’ve generated are better than they actually are. This hubris could be tied to testosterone levels, and new research by Gideon Nave, a cognitive neuroscientist at the University of Pennsylvania, along with Amos Nadler at Western University in Ontario, reveals that high testosterone can make it harder to see the flaws in one’s reasoning.

How might heightened testosterone lead to overconfidence? One possible explanation lies in the orbitofrontal cortex, a region just behind the eyes that’s essential for self-evaluation, decision making and impulse control. The neuroscientists Pranjal Mehta at the University of Oregon and Jennifer Beer at the University of Texas, Austin, have found that people with higher levels of testosterone have less activity in their orbitofrontal cortex. Studies show that when that part of the brain is less active, people tend to be overconfident in their reasoning abilities. It’s as though the orbitofrontal cortex is your internal editor, speaking up when there’s a potential problem with your work. Boost your testosterone and your editor goes reassuringly (but misleadingly) silent.

Men are also more likely to overestimate how well they’ll perform compared with their peers. Researchers at Kiel University in Germany and at Oxford gave a group of adults a test that assesses judgment and reasoning called the Cognitive Reflection Test, or C.R.T.

To see what the C.R.T. looks like, try answering this question: A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

If you’re like most people, your first thought is that the ball costs 10 cents. But that is incorrect. If the ball costs $0.10, and the bat costs $1.00 more (or $1.10), then the total would be $1.20. So the ball costs 5 cents and the bat costs $1.05.

If you got this wrong, you’re not alone. Even at Ivy League schools such as Harvard and Princeton, less than 30 percent of students answer all the questions correctly. This is how the clever questions are designed. There’s an immediate, obvious answer that feels right but is actually wrong.

In the Kiel University study, both genders thought they’d scored higher on the test than they actually had. When asked to predict how others would fare, however, women expected other women to earn comparably high scores, but men thought they’d significantly outperform other men.

People don’t like to believe that they’re average. But compared with women, men tend to think they’re much better than average.

If you feel your judgment is right, are you interested in how others see the problem? Probably not. Nicholas D. Wright, a neuroscientist at the University of Birmingham in Britain, studies how fluctuations in testosterone shape one’s willingness to collaborate. Most testosterone researchers study men, for obvious reasons, but Dr. Wright and his team focus on women. They asked women to perform a challenging perceptual task: detecting where a fuzzy pattern had appeared on a busy computer screen. When women took oral testosterone, they were more likely to ignore the input of others, compared with women in the placebo condition. Amped up on testosterone, they relied more heavily on their own judgment, even when they were wrong.

The findings of the latest study, which have been presented at conferences and will be published in Psychological Science in January, offer more reasons to worry about testosterone supplements.

Dr. Nave and Dr. Nadler’s team asked 243 men in Southern California to slather gel onto their shoulders, arms and chest. Half of the men rubbed in a testosterone gel, and the rest rubbed in a placebo. Once the gel dried, they put on their shirts and went about their day.

Four and a half hours later, enough time for their testosterone levels to peak and stabilize, the men returned to the lab. They sat down at a computer and took several tests — a math test, a mood questionnaire and the C.R.T.

For the men with extra testosterone, their moods hadn’t changed much, but their ability to analyze carefully had. They were, on average, 35 percent more likely to make the intuitive mistake on the bat and ball question. They were also rushed in their bad judgment and gave incorrect answers faster than the men with normal testosterone levels, while taking longer to generate correct answers.

Some will shrug and say that making a mistake on a sneaky word problem isn’t a concern in daily life, but researchers are discovering that these reasoning errors could affect financial markets. A team of neuroeconomists, led by Dr. Nadler, along with Paul J. Zak at Claremont Graduate University, gave 140 male traders either testosterone gel or a placebo. The next day, the traders came back into the lab and participated in an asset trading simulation.

The results are disturbing. Men with boosted testosterone significantly overpriced assets compared with men who got the placebo, and they were slower to incorporate data about falling values into their trading decisions. In other words, they created a trading bubble that was slow to pop. (Fortunately, Dr. Nadler didn’t have these men participate in a real stock market, out of concern for what a single dose of this drug could do.)

History has long labeled women as unreliable and hysterical because of their hormones. Maybe now it’s time to start saying, “He’s just being hormonal.”

The research has its limitations. On average, men in these studies were in their early 20s, and a surge in testosterone might not impair older men’s reasoning in quite the same way. And of course this research doesn’t prove that all men are bad decision makers because of their testosterone or that they’re worse decision makers than women. Confidence can spur a person to action, to take risks. But we should all be more aware of when confidence tips into overconfidence, and testosterone supplements could encourage that. Ironically, these supplements might make someone feel bold enough to lead but probably reduce his ability to lead well.

The television ads promise youth and vigor, but they’ve left out the catch: Testosterone enhancement doesn’t just make you feel like an invincible 18-year-old. It makes you think like one, too.

More information:
» Testosterone Testing and Testosterone Replacement Therapy
» Should the Modern Man Be Taking Testosterone?
» Male Hormone Molds Women, Too, In Mind and Body

Friday, June 23, 2017

The Cognitive-Theoretic Model of the Universe

Christopher Michael Langan:
Those interested in serious theories include just about everyone, from engineers and stockbrokers to doctors, automobile mechanics and police detectives.  Practically anyone who gives advice, solves problems or builds things that function needs a serious theory from which to work.   But three groups who are especially interested in serious theories are scientists, mathematicians and philosophers.  These are the groups which place the strictest requirements on the theories they use and construct. 

While there are important similarities among the kinds of theories dealt with by scientists, mathematicians and philosophers, there are important differences as well.  The most important differences involve the subject matter of the theories.  Scientists like to base their theories on experiment and observation of the real world…not on perceptions themselves, but on what they regard as concrete “objects of the senses”.  That is, they like their theories to be empirical.  Mathematicians, on the other hand, like their theories to be essentially rational…to be based on logical inference regarding abstract mathematical objects existing in the mind, independently of the senses.  And philosophers like to pursue broad theories of reality aimed at relating these two kinds of object.  (This actually mandates a third kind of object, the infocognitive syntactic operator…but another time.)     

Of the three kinds of theory, by far the lion’s share of popular reportage is commanded by theories of science.  Unfortunately, this presents a problem.  For while science owes a huge debt to philosophy and mathematics – it can be characterized as the child of the former and the sibling of the latter - it does not even treat them as its equals.  It treats its parent, philosophy, as unworthy of consideration.  And although it tolerates and uses mathematics at its convenience, relying on mathematical reasoning at almost every turn, it acknowledges the remarkable obedience of objective reality to mathematical principles as little more than a cosmic “lucky break”.  

Science is able to enjoy its meretricious relationship with mathematics precisely because of its queenly dismissal of philosophy.  By refusing to consider the philosophical relationship between the abstract and the concrete on the supposed grounds that philosophy is inherently impractical and unproductive, it reserves the right to ignore that relationship even while exploiting it in the construction of scientific theories.  And exploit the relationship it certainly does!  There is a scientific platitude stating that if one cannot put a number to one's data, then one can prove nothing at all.  But insofar as numbers are arithmetically and algebraically related by various mathematical structures, the platitude amounts to a thinly veiled affirmation of the mathematical basis of knowledge. 

Although scientists like to think that everything is open to scientific investigation, they have a rule that explicitly allows them to screen out certain facts.  This rule is called the scientific method.  Essentially, the scientific method says that every scientist’s job is to (1) observe something in the world, (2) invent a theory to fit the observations, (3) use the theory to make predictions, (4) experimentally or observationally test the predictions, (5) modify the theory in light of any new findings, and (6) repeat the cycle from step 3 onward.  But while this method is very effective for gathering facts that match its underlying assumptions, it is worthless for gathering those that do not. 

In fact, if we regard the scientific method as a theory about the nature and acquisition of scientific knowledge (and we can), it is not a theory of knowledge in general.  It is only a theory of things accessible to the senses.  Worse yet, it is a theory only of sensible things that have two further attributes: they are non-universal and can therefore be distinguished from the rest of sensory reality, and they can be seen by multiple observers who are able to “replicate” each other’s observations under like conditions.  Needless to say, there is no reason to assume that these attributes are necessary even in the sensory realm.  The first describes nothing general enough to coincide with reality as a whole – for example, the homogeneous medium of which reality consists, or an abstract mathematical principle that is everywhere true - and the second describes nothing that is either subjective, like human consciousness, or objective but rare and unpredictable…e.g. ghosts, UFOs and yetis, of which jokes are made but which may, given the number of individual witnesses reporting them, correspond to real phenomena. 

The fact that the scientific method does not permit the investigation of abstract mathematical principles is especially embarrassing in light of one of its more crucial steps: “invent a theory to fit the observations.”  A theory happens to be a logical and/or mathematical construct whose basic elements of description are mathematical units and relationships.  If the scientific method were interpreted as a blanket description of reality, which is all too often the case, the result would go something like this: “Reality consists of all and only that to which we can apply a protocol which cannot be applied to its own (mathematical) ingredients and is therefore unreal.”  Mandating the use of “unreality” to describe “reality” is rather questionable in anyone’s protocol.  

What about mathematics itself?  The fact is, science is not the only walled city in the intellectual landscape.  With equal and opposite prejudice, the mutually exclusionary methods of mathematics and science guarantee their continued separation despite the (erstwhile) best efforts of philosophy.  While science hides behind the scientific method, which effectively excludes from investigation its own mathematical ingredients, mathematics divides itself into “pure” and “applied” branches and explicitly divorces the “pure” branch from the real world.  Notice that this makes “applied” synonymous with “impure”.  Although the field of applied mathematics by definition contains every practical use to which mathematics has ever been put, it is viewed as “not quite mathematics” and therefore beneath the consideration of any “pure” mathematician.   

In place of the scientific method, pure mathematics relies on a principle called the axiomatic method.  The axiomatic method begins with a small number of self-evident statements called axioms and a few rules of inference through which new statements, called theorems, can be derived from existing statements.  In a way parallel to the scientific method, the axiomatic method says that every mathematician’s job is to (1) conceptualize a class of mathematical objects; (2) isolate its basic elements, its most general and self-evident principles, and the rules by which its truths can be derived from those principles; (3) use those principles and rules to derive theorems, define new objects, and formulate new propositions about the extended set of theorems and objects; (4) prove or disprove those propositions; (5) where the proposition is true, make it a theorem and add it to the theory; and (6) repeat from step 3 onwards. 

The scientific and axiomatic methods are like mirror images of each other, but located in opposite domains.  Just replace “observe” with “conceptualize” and “part of the world” with “class of mathematical objects”, and the analogy practically completes itself.  Little wonder, then, that scientists and mathematicians often profess mutual respect.  However, this conceals an imbalance.  For while the activity of the mathematician is integral to the scientific method, that of the scientist is irrelevant to mathematics (except for the kind of scientist called a “computer scientist”, who plays the role of ambassador between the two realms).  At least in principle, the mathematician is more necessary to science than the scientist is to mathematics. 

As a philosopher might put it, the scientist and the mathematician work on opposite sides of the Cartesian divider between mental and physical reality.  If the scientist stays on his own side of the divider and merely accepts what the mathematician chooses to throw across, the mathematician does just fine.  On the other hand, if the mathematician does not throw across what the scientist needs, then the scientist is in trouble.  Without the mathematician’s functions and equations from which to build scientific theories, the scientist would be confined to little more than taxonomy.  As far as making quantitative predictions were concerned, he or she might as well be guessing the number of jellybeans in a candy jar.  

From this, one might be tempted to theorize that the axiomatic method does not suffer from the same kind of inadequacy as does the scientific method…that it, and it alone, is sufficient to discover all of the abstract truths rightfully claimed as “mathematical”.  But alas, that would be too convenient.  In 1931, an Austrian mathematical logician named Kurt Gödel proved that there are true mathematical statements that cannot be proven by means of the axiomatic method.  Such statements are called “undecidable”.  Gödel’s finding rocked the intellectual world to such an extent that even today, mathematicians, scientists and philosophers alike are struggling to figure out how best to weave the loose thread of undecidability into the seamless fabric of reality. 

To demonstrate the existence of undecidability, Gödel used a simple trick called self-reference.  Consider the statement “this sentence is false.”  It is easy to dress this statement up as a logical formula.  Aside from being true or false, what else could such a formula say about itself?  Could it pronounce itself, say, unprovable?  Let’s try it: "This formula is unprovable".  If the given formula is in fact unprovable, then it is true and therefore a theorem.  Unfortunately, the axiomatic method cannot recognize it as such without a proof.  On the other hand, suppose it is provable.  Then it is self-apparently false (because its provability belies what it says of itself) and yet true (because provable without respect to content)!  It seems that we still have the makings of a paradox…a statement that is "unprovably provable" and therefore absurd.  

But what if we now introduce a distinction between levels of proof?  For example, what if we define a metalanguage as a language used to talk about, analyze or prove things regarding statements in a lower-level object language, and call the base level of Gödel’s formula the "object" level and the higher (proof) level the "metalanguage" level?  Now we have one of two things: a statement that can be metalinguistically proven to be linguistically unprovable, and thus recognized as a theorem conveying valuable information about the limitations of the object language, or a statement that cannot be metalinguistically proven to be linguistically unprovable, which, though uninformative, is at least no paradox.  Voilà: self-reference without paradox!  It turns out that "this formula is unprovable" can be translated into a generic example of an undecidable mathematical truth.  Because the associated reasoning involves a metalanguage of mathematics, it is called “metamathematical”.

It would be bad enough if undecidability were the only thing inaccessible to the scientific and axiomatic methods together. But the problem does not end there.  As we noted above, mathematical truth is only one of the things that the scientific method cannot touch.  The others include not only rare and unpredictable phenomena that cannot be easily captured by microscopes, telescopes and other scientific instruments, but things that are too large or too small to be captured, like the whole universe and the tiniest of subatomic particles; things that are “too universal” and therefore indiscernable, like the homogeneous medium of which reality consists; and things that are “too subjective”, like human consciousness, human emotions, and so-called “pure qualities” or qualia.  Because mathematics has thus far offered no means of compensating for these scientific blind spots, they continue to mark holes in our picture of scientific and mathematical reality. 

But mathematics has its own problems.  Whereas science suffers from the problems just described – those of indiscernability and induction, nonreplicability and subjectivity - mathematics suffers from undecidability.  It therefore seems natural to ask whether there might be any other inherent weaknesses in the combined methodology of math and science.  There are indeed.  Known as the Lowenheim-Skolem theorem and the Duhem-Quine thesis, they are the respective stock-in-trade of disciplines called model theory and the philosophy of science (like any parent, philosophy always gets the last word).  These weaknesses have to do with ambiguity…with the difficulty of telling whether a given theory applies to one thing or another, or whether one theory is “truer” than another with respect to what both theories purport to describe.  

But before giving an account of Lowenheim-Skolem and Duhem-Quine, we need a brief introduction to model theory.  Model theory is part of the logic of “formalized theories”, a branch of mathematics dealing rather self-referentially with the structure and interpretation of theories that have been couched in the symbolic notation of mathematical logic…that is, in the kind of mind-numbing chicken-scratches that everyone but a mathematician loves to hate.  Since any worthwhile theory can be formalized, model theory is a sine qua non of meaningful theorization.  

Let’s make this short and punchy. We start with propositional logic, which consists of nothing but tautological, always-true relationships among sentences represented by single variables.  Then we move to predicate logic, which considers the content of these sentential variables…what the sentences actually say.  In general, these sentences use symbols called quantifiers to assign attributes to variables semantically representing mathematical or real-world objects.  Such assignments are called “predicates”.  Next, we consider theories, which are complex predicates that break down into systems of related predicates; the universes of theories, which are the mathematical or real-world systems described by the theories; and the descriptive correspondences themselves, which are called interpretations.  A model of a theory is any interpretation under which all of the theory’s statements are true.  If we refer to a theory as an object language and to its referent as an object universe, the intervening model can only be described and validated in a metalanguage of the language-universe complex. 

Though formulated in the mathematical and scientific realms respectively, Lowenheim-Skolem and Duhem-Quine can be thought of as opposite sides of the same model-theoretic coin.  Lowenheim-Skolem says that a theory cannot in general distinguish between two different models; for example, any true theory about the numeric relationship of points on a continuous line segment can also be interpreted as a theory of the integers (counting numbers).  On the other hand, Duhem-Quine says that two theories cannot in general be distinguished on the basis of any observation statement regarding the universe. 

Just to get a rudimentary feel for the subject, let’s take a closer look at the Duhem-Quine Thesis.  Observation statements, the raw data of science, are statements that can be proven true or false by observation or experiment.  But observation is not independent of theory; an observation is always interpreted in some theoretical context. So an experiment in physics is not merely an observation, but the interpretation of an observation.  This leads to the Duhem Thesis, which states that scientific observations and experiments cannot invalidate isolated hypotheses, but only whole sets of theoretical statements at once.  This is because a theory T composed of various laws {Li}, i=1,2,3,… almost never entails an observation statement except in conjunction with various auxiliary hypotheses {Aj}, j=1,2,3,… .  Thus, an observation statement at most disproves the complex {Li+Aj}.   

To take a well-known historical example, let T = {L1,L2,L3} be Newton’s three laws of motion, and suppose that these laws seem to entail the observable consequence that the orbit of the planet Uranus is O.  But in fact, Newton’s laws alone do not determine the orbit of Uranus.  We must also consider things like the presence or absence of other forces, other nearby bodies that might exert appreciable gravitational influence on Uranus, and so on.  Accordingly, determining the orbit of Uranus requires auxiliary hypotheses like A1 = “only gravitational forces act on the planets”, A2 = “the total number of solar planets, including Uranus, is 7,” et cetera.  So if the orbit in question is found to differ from the predicted value O, then instead of simply invalidating the theory T of Newtonian mechanics, this observation invalidates the entire complex of laws and auxiliary hypotheses {L1,L2,L3;A1,A2,…}.  It would follow that at least one element of this complex is false, but which one?  Is there any 100% sure way to decide? 

As it turned out, the weak link in this example was the hypothesis A2 = “the total number of solar planets, including Uranus, is 7”.  In fact, there turned out to be an additional large planet, Neptune, which was subsequently sought and located precisely because this hypothesis (A2) seemed open to doubt.  But unfortunately, there is no general rule for making such decisions.  Suppose we have two theories T1 and T2 that predict observations O and not-O respectively.  Then an experiment is crucial with respect to T1 and T2 if it generates exactly one of the two observation statements O or not-O.  Duhem’s arguments show that in general, one cannot count on finding such an experiment or observation.  In place of crucial observations, Duhem cites le bon sens (good sense), a non-logical faculty by means of which scientists supposedly decide such issues.  Regarding the nature of this faculty, there is in principle nothing that rules out personal taste and cultural bias.  That scientists prefer lofty appeals to Occam’s razor, while mathematicians employ justificative terms like beauty and elegance, does not exclude less savory influences.
So much for Duhem; now what about Quine?  The Quine thesis breaks down into two related theses.  The first says that there is no distinction between analytic statements (e.g. definitions) and synthetic statements (e.g. empirical claims), and thus that the Duhem thesis applies equally to the so-called a priori disciplines.  To make sense of this, we need to know the difference between analytic and synthetic statements.  Analytic statements are supposed to be true by their meanings alone, matters of empirical fact notwithstanding, while synthetic statements amount to empirical facts themselves.  Since analytic statements are necessarily true statements of the kind found in logic and mathematics, while synthetic statements are contingently true statements of the kind found in science, Quine’s first thesis posits a kind of equivalence between mathematics and science.  In particular, it says that epistemological claims about the sciences should apply to mathematics as well, and that Duhem’s thesis should thus apply to both. 

Quine’s second thesis involves the concept of reductionism.  Reductionism is the claim that statements about some subject can be reduced to, or fully explained in terms of, statements about some (usually more basic) subject.  For example, to pursue chemical reductionism with respect to the mind is to claim that mental processes are really no more than biochemical interactions.  Specifically, Quine breaks from Duhem in holding that not all theoretical claims, i.e. theories, can be reduced to observation statements.  But then empirical observations “underdetermine” theories and cannot decide between them.  This leads to a concept known as Quine’s holism; because no observation can reveal which member(s) of a set of theoretical statements should be re-evaluated, the re-evaluation of some statements entails the re-evaluation of all. 

Quine combined his two theses as follows.  First, he noted that a reduction is essentially an analytic statement to the effect that one theory, e.g. a theory of mind, is defined on another theory, e.g. a theory of chemistry.  Next, he noted that if there are no analytic statements, then reductions are impossible.  From this, he concluded that his two theses were essentially identical.  But although the resulting unified thesis resembled Duhem’s, it differed in scope. For whereas Duhem had applied his own thesis only to physical theories, and perhaps only to theoretical hypothesis rather than theories with directly observable consequences, Quine applied his version to the entirety of human knowledge, including mathematics.  If we sweep this rather important distinction under the rug, we get the so-called “Duhem-Quine thesis”. 

Because the Duhem-Quine thesis implies that scientific theories are underdetermined by physical evidence, it is sometimes called the Underdetermination Thesis.  Specifically, it says that because the addition of new auxiliary hypotheses, e.g. conditionals involving “if…then” statements, would enable each of two distinct theories on the same scientific or mathematical topic to accommodate any new piece of evidence, no physical observation could ever decide between them.  

The messages of Duhem-Quine and Lowenheim-Skolem are as follows: universes do not uniquely determine theories according to empirical laws of scientific observation, and theories do not uniquely determine universes according to rational laws of mathematics.  The model-theoretic correspondence between theories and their universes is subject to ambiguity in both directions.  If we add this descriptive kind of ambiguity to ambiguities of measurement, e.g. the Heisenberg Uncertainty Principle that governs the subatomic scale of reality, and the internal theoretical ambiguity captured by undecidability, we see that ambiguity is an inescapable ingredient of our knowledge of the world.  It seems that math and science are…well, inexact sciences. 

How, then, can we ever form a true picture of reality?  There may be a way.  For example, we could begin with the premise that such a picture exists, if only as a “limit” of theorization (ignoring for now the matter of showing that such a limit exists).  Then we could educe categorical relationships involving the logical properties of this limit to arrive at a description of reality in terms of reality itself.  In other words, we could build a self-referential theory of reality whose variables represent reality itself, and whose relationships are logical tautologies.  Then we could add an instructive twist.  Since logic consists of the rules of thought, i.e. of mind, what we would really be doing is interpreting reality in a generic theory of mind based on logic.  By definition, the result would be a cognitive-theoretic model of the universe 

Gödel used the term incompleteness to describe that property of axiomatic systems due to which they contain undecidable statements.  Essentially, he showed that all sufficiently powerful axiomatic systems are incomplete by showing that if they were not, they would be inconsistent.  Saying that a theory is “inconsistent” amounts to saying that it contains one or more irresolvable paradoxes.  Unfortunately, since any such paradox destroys the distinction between true and false with respect to the theory, the entire theory is crippled by the inclusion of a single one.  This makes consistency a primary necessity in the construction of theories, giving it priority over proof and prediction.  A cognitive-theoretic model of the universe would place scientific and mathematical reality in a self-consistent logical environment, there to await resolutions for its most intractable paradoxes. 

For example, modern physics is bedeviled by paradoxes involving the origin and directionality of time, the collapse of the quantum wave function, quantum nonlocality, and the containment problem of cosmology.  Were someone to present a simple, elegant theory resolving these paradoxes without sacrificing the benefits of existing theories, the resolutions would carry more weight than any number of predictions.  Similarly, any theory and model conservatively resolving the self-inclusion paradoxes besetting the mathematical theory of sets, which underlies almost every other kind of mathematics, could demand acceptance on that basis alone.  Wherever there is an intractable scientific or mathematical paradox, there is dire need of a theory and model to resolve it. 

If such a theory and model exist – and for the sake of human knowledge, they had better exist – they use a logical metalanguage with sufficient expressive power to characterize and analyze the limitations of science and mathematics, and are therefore philosophical and metamathematical in nature.  This is because no lower level of discourse is capable of uniting two disciplines that exclude each other’s content as thoroughly as do science and mathematics.  

Now here’s the bottom line: such a theory and model do indeed exist.  But for now, let us satisfy ourselves with having glimpsed the rainbow under which this theoretic pot of gold awaits us.

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