Math Interlude*: Lagrange Interpolation as Self-Checking Algebra Practice

My kids started algebra last year, and, while they seem to get the basic concepts, they need practice – practice using the distributive law correctly to simplify algebraic expressions, practice dealing correctly with all those negative signs, etc.

Of course, I could just give them a huge number of polynomials to multiply, but, aside from being boring, that would require me to work out the answers myself in order to check their answers!

I’ve found a simple alternative that goes back to the great eighteenth-century mathematician Lagrange. “Lagrange interpolation” is actually interesting and useful in itself (although even well-educated technical people seem often to be ignorant of it nowadays), it happens to require quite a lot of multiplying of polynomials, checking of signs, etc. so that it is good algebra practice, and, best of all, it is automatically self-checking.

Here in a nutshell is how it works:

You are given a table of values for the variables x and y, and you want to find a polynomial that gives exactly the correct values of y when you plug in the values for x.

Suppose, for example, you are given the following values:

x y
1 1
2 4
3 9
5 9

and you want to find a polynomial

y = a x³ + b x² + c x + d

that goes through those points.

There are various ways to solve this problem -- for example, you can use linear algebra if you view (a,b,c,d) as a vector in a four-dimensional space.

The method published by Lagrange uses a much simpler idea.

What we do is find four separate polynomial, each of which vanishes at all but one of the values of x.

For example, expression A:

(x - 2) * (x - 3) * (x - 5)

obviously vanishes when x is 2, 3, or 5, but obviously does not vanish when x is 1.

What is the value of expression A when x equals 1? Well, just plug 1 in for x and you find that the value is -8.

Now, when x is 1, according to our table, we need y to have a value of 1, not -8. So, we will just divide expression A by -8 and multiply it by 1, getting expression B:

1 * (x - 2) * (x - 3) * (x - 5) / (-8)

If you do the same thing for the case where x is 2, you get expression C:

4 * (x - 1) * (x - 3) * (x - 5) / (3)

Run the same trick for x equal to 3, and you get expression D:

9 * (x - 1) * (x - 2) * (x - 5) / (-4)

Finally, for x equals 5, you get expression E:

9 * (x - 1) * (x - 2) * (x - 3) / (24)

Now, expression B gives the right value for y when x is 1, and, it is created so that it will vanish at the other three values of x, so it will not mess up the values of y there. Similarly, expression C is created so that it gives the right value for y when x is 2, and it is zero at the other three values of x.

So, if we simplify expressions B, C, D, and E by multiplying each one out, and then add them all together, combining like terms, we will get a polynomial that gives the right values of y for each of the four values of x.

I don't need practice on algebra, so I had my kids do this. Their answer is expression F:

y = (-2/3) x³ + 5 x² + (-22/3) x + 4

How do they (and I) know that they did the algebra right?

Simple – they plugged into expression F the values 1, 2, 3, and 5 for x. They should find that the values of y will then be 1, 4, 9, and 9 as planned. If they do not get the right values of y, they need to find their algebra error!

Note that Lagrange interpolation always works: you can choose any real numbers, positive, negative or zero, integral or fractional, for x and for y. (You can even use complex numbers if you wish.)

In particular, the values of x do not have to be evenly space: in my example, I “skipped” 4, and it worked fine. Nor do the values of y have to be in any pattern: 1, 4, and 9 in my example seemed to be starting a pattern, but I wrecked that pattern by using 9 twice instead of using the “obvious” choice of 16.

The values of x do have to all be different (although, you can play some interesting tricks by letting two values of x get “infinitely close” – basically, you can then control the slope at x as well as the value of y).

You can in fact prove that this is the only polynomial of degree three or lower that goes through our four points. (In general, if you have n points, with different values of x, there is always a unique polynomial of degree n-1 or lower that goes through those points.)

The more points you use, the more complicated the algebra gets. I'd start with only two or three points for someone who is just learning algebra.

Isn't this too hard for kids in first-year algebra? No, my kids have learned it without too much trouble: there is no division of polynomials here, no quadratic formula, no trig functions, etc. This really is just first-year algebra.

But, isn't it complicated?

A little.

But this is the kind of complication that you get in real math applied to real problems. No one in real life (not even in science or engineering) ever faces the problem of multiplying ( x - 2 ) times ( x- 3) just for the fun of it. And, it is very, very rare that anyone ever faces the familiar textbook sort of algebra problem: “A train leaves Albuquerque going towards Santa Fe at 70 mph and a train leaves Santa Fe...” or “Jane had five times as many dolls as Ginger, but after Jane got two more dolls....”

Algebra is an abstract science; traditional algebra is about understanding the abstract properties of the four basic arithmetic operations: what does and can happen when you use the operations of addition, subtraction, multiplication and division in a systematic way? (Modern university algebra is about the properties of more general systems of mathematical operations that can operate on elements very different from ordinary real numbers.)

Mathematics is not really about balancing your checkbook or calculating the amount of tile you need to re-tile the kitchen – we have electronic calculators to do that for us.

Mathematics is about the possible abstract structures that can logically exist.

Those structures are often based on arithmetic and geometry, so you do need to know traditional math to understand modern mathematics.

But really learning math means trying not just to learn to get the right answer but actually exploring the universe of mathematics much as traditional explorers explored newly-discovered continents.

Lagrange interpolation is a very simple example of such exploration. Actually graphing the polynomials you get through Lagrange interpolation can also be enlightening: while the method always works, it can give some pretty “snaky” curves if you fit more than three points (for three points, the resulting curve is much nicer).

Is it too complicated? Well, someone unwilling to tackle topics such as Lagrange interpolation is not really learning math. If you want to know what math is really about, rather than just working you way through the watered-down, over-simplified picture of mathematics portrayed in American public-school textbooks, you need to try to wrap you mind around ideas such as Lagrange interpolation.

____________
* From time to time, I plan on posting a “Math Interlude,” in which I’ll try to explain some significant idea in math not known to most educated American adults, that I have in fact taught to my own kids in grade school or middle school, and that a bright middle-school student should be able to grasp.

Socialization and the Carnival of Homeschooling

The current Carnival of Homeschooling links to an insightful (and truly hilarious!) post on the homeschooling socializtion issue.

It is mentioned in the same paragraph as my recent post on a Chinese-American perspective on the issue of whether homeschooled kids are not independent enough. I had thought my post was pretty decent, but the post from The Learning Curve wins out. Check it out.

Are Homeschooled Kids Too Dependent? Chinese vs. American Perspectives

The question of whether homeschooled children are too “dependent” on their parents is related to the oft-encountered “socialization” question, but the “dependency” question seems to get less attention among homeschoolers.

Because I married into a Chinese family (my parents-in-law were born in mainland China), I try to think about such issues from a “multicultural” perspective: how does our American idea of children’s “independence” compare to Chinese ideas of “independence”?

Chuansheng Chen, a professor at the University of California at Irvine, has done research in the area of Chinese vs. Western adolescence, and has noted:

Feldman and Rosenthal (1991) found that U.S. and Australian adolescents had earlier expectations for autonomy than did Hong Kong adolescents. The largest cultural differences were found for behaviors that would fall into the category of misconduct (e.g., smoking and drinking alcohol) and those related to peers (e.g., "attending boy-girl parties," "dating," and "preferring to do things with friends than with family")…

For example, peer factors play a less important role in Chinese adolescents' misconduct than in American adolescents' misconduct because Chinese adolescents spend less time with their peers (Chen et al., 1998).

Does the fact that Chinese have “less expectations of autonomy” than Americans imply that Chinese kids are more “dependent” than American kids?

My wife has explained to me that it is more subtle than that.

Consider which of the following sorts of children should be considered truly dependent:

• Those kids who spend a lot of time with their parents and whose parents work hard to instill mature values and an understanding of the consequences of one’s decisions, so that, when the children finally become adults, they can independently make intelligent and informed decisions

or

• Those kids who lack an adult’s perspective on values and consequences and are therefore, in reality, heavily dependent on the ill-formed judgments of their adolescent peers.

The American concept of a child’s independence tends to be that the independent child makes decisions on his own, even though those decisions may show no regard for his parents' values or judgments or what the parents have tried to teach him.

The Chinese concept of independence is that a child shows independence when he has properly internalized the parents’ teaching and is able to make a judgment similar to the judgment that the parents would have made had the parents been present and privy to all the relevant information.

The American concept of independence means ignoring the value of adult (especially parental) judgment. The Chinese attitude is that true independence comes when the child has acquired and accepted adult standards of judgment.

Most importantly, when a child makes an “independent” decision as a result of bowing to peer pressure, Americans still see this as a sign of the child’s independence, even if the decision is obviously unwise.

To Chinese, this is a sign of a very unhealthy form of dependence: dependence on the ill-informed and immature opinions of other children.

To Americans, a child’s spending more time with his peers is therefore a sign of his growing independence. Chinese have a different perspective.

I think this helps illuminate the “dependency” question we homeschoolers face.

If “independence” means kids’ making decisions without having internalized an adult’s understanding of values and consequences, then, yes, our homeschooled kids are less “independent” than many American kids.

At least, I hope so.

But, if “independence” means that a child is not dependent on peer pressure and can make mature decisions because he has acquired and internalized an adult perspective on decision-making, then I think that homeschooled kids may often be more independent than typical American adolescents.

Carnival of Homeschooling

My post on “Should Homeschooled Kids Study Philosophy?” is mentioned in the September 7, 2009 Carnival of Homeschooling.

I did not submit anything to this week’s Carnival , and I will probably only submit something once every month or two. I do think the Carnival is a good way for homeschoolers to hear about other homeschoolers’ blogs.

I notice that my blog seems to be less personal, less about family activities, than most of the homeschooler blogs. I’m not sure if this is because I am a male and most homeschool bloggers are females, or, perhaps, as a physicist, I am less inclined to focus on personal matters.

One blog that is a bit more similar to mine is Kitchen Table Math : this is not specifically limited to homeschoolers, but does have a large homeschooler presence. Kitchen Table Math also has a lot of nice links that are good for learning about math education.

Tyranny vs. Chaos: The False Dichotomy of “Progressive” vs. “Traditional” Education

Alfie Kohn has a revealing essay* posted on Education Week, currently available for free to the public.

Kohn is a well-known advocate of “progressive” education and his article exhibits nicely the basic error that “progressives” make, as well as the opposite error made by far too many advocates of “traditional” education.

On the one hand, Kohn criticizes the traditional teacher-as-lord-and-master approach as consisting of “mandates handed down from on high… where test scores drive the instruction and students are essentially bullied into doing whatever they’re told.”

I think that is a fair rap.

“To divide fractions, invert and multiply.” (Why?)

“Democracy is the best system of government.” (Then why not carry out open-heart surgery democratically?)

“Humans are descended from fish.” (How do we know this?)

To simply order children to believe the “right” answer causes them to accept that one can only learn the truth from authority, and that they, and humans in general, lack the ability to reliably determine the truth for themselves.

And, if kids ever start to wonder how the “authorities” learned the right answer, since the authorities themselves also are mere humans, the kids may fall into a naïve skepticism, thinking that no human can ever really know any truths at all.

But, sadly, Kohn offers, as a false alternative, the old “progressive” solution of giving kids the “opportunities to discover answers to their own questions,” i.e., the “constructivist” approach where kids have to create knowledge for themselves rather than systematically being taught what humans have discovered, at enormous effort, during the last three thousand years.

That really would be a swell approach – if kids had a spare three thousand years to work out everything, and if all kids were as bright as Euclid, Einstein, etc.

So, how to avoid the false choice of “progressive” vs. “traditional” education?

The answer should really be obvious from everyday life.

In real life, we explain to kids that you need to brush your teeth because you will otherwise get cavities, you need to wash your hands because there are germs on your hands that can make you sick, etc. We give explanations.

The idea of giving rational explanations, as opposed to the false dichotomy of either issuing irrational commands or forcing the students to discover everything for themselves, is really not that complicated!

E.g., why “invert and multiply” to divide fractions?

Well, division is the inverse operation to multiplication: if you multiply by some fraction and then wish to undo the multiplication, inverting and multiplying will indeed undo the original multiplication.

Liping Ma, in her brilliant Knowing and Teaching Elementary Mathematics, a “must-read” for all homeschooling parents, goes into much greater depth on this issue of dividing fractions: this is one of the toughest things to explain clearly in elementary mathematics.

But it can be explained. It is unrealistic to expect kids to discover for themselves how to divide fractions, or to fully understand on their own why it works, even if they do stumble upon it. However, it is also not necessary to teach the standard algorithm as an arbitrary rule imposed, for some mysterious reason, by adults.

As Ma explains, if one wishes to use division as the inverse of multiplication, if one wants division to be a means of carrying out repeated subtractions, if one wants the “cancellation law” to apply to division, one has no choice: there is only one right answer, the one given by the standard algorithm.

Ma advocates a “profound understanding of fundamental mathematics”: i.e., both a serious conceptual understanding of elementary math, as well as a practical mastery of the elementary math facts and the standard algorithms.

That indeed should be the goal in all academic (and non-academic) subjects: a conceptual understanding of American history combined with detailed factual knowledge of dates and historical events; an understanding of the experimental bases for scientific theories as well as detailed knowledge of the important scientific facts; etc.

As a practical matter, it is sometimes necessary to say to a student, “We will see the justification for this next month or next year.” Sometimes, one cannot fully understand the evidence for a theory until one has grasped exactly what the theory is. And, it would be foolish to slavishly imitate all the false starts and errors made in the historical development of scientific theories, in the historical creation of various concepts in economics, in the historical discovery of various methods in mathematics, etc.

A student’s learning need not and should not recapitulate the historical process by which knowledge was originally discovered. The whole point is to make it easier for the student than it was for the original discoverer.

Often, the historical experiments or reasoning that led to a discovery are relevant: this is true, for example, of Rutherford’s discovery of the nucleus. But the important thing is to present the best proof and justification we possess today for a particular piece of knowledge.

The other error to avoid, which tends to be shared by both “progressive” and traditional approaches to education, is the false belief that kids have to wait until they are mature to be told of discoveries that the human race only stumbled upon in the last century or so.

It may have taken humans a long time to discover the Big Bang, the fact that humans are descended from fish, etc. But a six-year-old can grasp those ideas – they are not that complex. Even the basic evidence for those facts – e.g., the fossil record, the fact that the galaxies are expanding outward – can be explained at a simple level to six-year-olds.

That the human race had to be “mature” to discover such things does not mean that young kids cannot understand those discoveries.

An educational approach based on giving rational explanations, as opposed to the false dichotomy of either issuing irrational commands or expecting the students to discover everything for themselves, is really not that hard to grasp. Both progressivism and the traditional approach to education are wrong.

We need to teach our kids that humans can and do have a rational understanding of reality. Neither the “progressive” nor the “traditional” approach to schooling really achieves that.

Our kids deserve a "content-rich" approach that teaches them, at an early age, the marvelous and amazing facts that human beings have discovered about reality.

____________

* Thanks to Barry Garelick at Kitchen Table Math for bringing Kohn's essay to my attention.

Science vs. Religion: Teaching the Controversy

Scientific American has just published “a new column that examines the intersection between science and society,” an essay by my fellow physicist Lawrence Krauss that argues that science is fundamentally inconsistent with traditional religion. As Krauss’s conclusion states:

Until we are willing to accept the world the way it is, without miracles that all empirical evidence argues against, without myths that distort our comprehension of nature, we are unlikely to bridge the divide between science and culture and, more important, we are unlikely to be fully ready to address the urgent technical challenges facing humanity.

So, what does all this have to do with homeschooling? Like most scientists, I disagree with so-called "young earth creationists” as well as the more subtle creationism of “Intelligent Design.” But, on one point, I think the creationists are absolutely right: the “tolerant,” “moderate” position that holds that of course religion and mainstream, established science are compatible, and that, if they aren’t, at least we should be quiet about it, leads to an educational disaster.

Religion has been of crucial importance in human history; it is still extremely important to a large number of Americans.

Whether religion and mainstream science are compatible is not just a sophomoric question that can and should be quietly ignored by mature people. It is a question that is central to our understanding of ourselves, our civilization, and our future.

Krauss points out, quite correctly, that science does not disprove the existence of “a God that does not directly intervene in the daily operations of the cosmos.”

However, science does have a perspective and orientation that differs quite radically from traditional, organized religion.

The central ethic of science is that scientists should actively work to undermine existing theories, to find little details that do not fit into accepted theories and that push us onwards to new and better theories.

I know of none of the major religions that similarly urges its practitioners to do their best to undermine the beliefs of that religion!

Furthermore, science as it now exists is based on a mechanistic perspective that is antithetical to both common sense and to religious sensibilities.

Common sense thinks of flowers as striving to grow upwards towards the air and sun. Modern science thinks of a flower as a strange, squishy little machine, “designed” only by the long process of evolution, in which the banging about of atoms, the pushing and pulling due to the quantum electronic interactions of molecules, is all that is happening. To modern science, there is no real purpose, no real goals, exhibited by a growing plant – just the large-scale result of all those atoms banging into each other.

Similarly, common sense thinks of the difference between a peach and a zucchini as consisting of all the properties that a peach has that a zucchini lacks and vice versa: a peach has the property of being reddish orange, sweet, and so on; zucchinis have contrasting properties.

To a scientist, peaches and zucchinis are simply slightly different ways of arranging carbon, hydrogen, oxygen, and nitrogen atoms (with a smattering of other atoms): all the apparent differences between peaches and zucchinis are due to how these different arrangements of atoms reflect light, interact with various molecules in our taste buds, etc.

Just atoms (and photons) banging into each other – the peachiness and zucchininess have no separate existence.

I think it is hard for most non-scientists to grasp how certain scientists are of the correctness of this perspective: the shining of the sun, the eruption of a volcano, the growing of a rosebush, the internal operation of our own nervous system – nothing but tiny particles of matter (and force fields) pushing and pulling on each other.

I hope it is obvious how radically this view conflicts not only with common sense and traditional philosophy but also with traditional religion.

This conflict is central to contemporary human civilization. Science cannot simply be ignored – its stunning successes not only in creating material comforts and clever gadgets but also in explaining everything from the interior of a neutron star to the functioning of our own genes makes science an overwhelming intellectual and cultural force.

Any education that ignores this conflict between science and the traditional perspectives embodied in religion and philosophy (and “common sense”) is failing horribly either to teach about traditional religion, or about the full scope of the current scientific viewpoint, or both.

So, as a scientist, am I claiming that the scientific perspective is the final word, and everyone must meekly surrender to it?

Well… the perspective of science has been stunningly successful in understanding nature, and I think that needs to be acknowledged.

But, in the interests of full disclosure, let me present here the three little clouds on the horizon that suggest that maybe science as we know it has not grasped all aspects of reality.

First, in physics, we do not know how to combine quantum mechanics with Einstein’s theory of gravity (general relativity). For a long time, it was thought that this was merely a technical, mathematical difficulty that we would soon overcome (superstring theory has promised to do this, for example). But, in recent decades, an increasing number of physicists have come to wonder if maybe this is a sign that we are missing something more fundamental than we realize.

The second cloud on the horizon is quantum mechanics itself. Anyone who has learned anything about quantum mechanics, even from really bad popular books (and there are lots of those!), knows that quantum mechanics is really weird. In a nutshell, quantum mechanics seems to say that everything that could have happened affects the future, not just those things that actually did happen.

There are various ways of trying to escape this ghostly effect of unrealized possibilities. None has yet convinced the majority of physicists.

The third cloud is the problem of consciousness. Physics has nothing to say about what it feels like to be an electron. But we all know that it does feel like something to be a human being. How can electrons, protons, and neutrons, whirling around inside our head become aware of themselves?

Lots of suggestions have been made: they all ignore the fact that electrons, protons, and neutrons are a part of physics, and that the idea of an “internal perspective” is utterly alien to physics.

Quite a few philosophers and physicists have mulled over this problem: an increasing number have the humility to admit that we cannot see what an answer could even look like (for one of my favorite discussions, see the philosopher Colin McGinn’s readable and informed book The Mysterious Flame: Conscious Minds in a Material World).

So, how will these disturbing clouds be resolved? What will the angry dance among science, religion, and philosophy look like a hundred years from now?

I don’t know. But I am confident that it is one of the great questions of the twenty-first century.

An education that tries to sweep such questions under the rug, that ignores how radically the modern scientific view of reality differs both from common sense and from traditional religion, is really no education at all.

A Japanese Musician on Selling Our Kids Short

While googling for information on piano equipment (pedal extenders for the kids), I accidentally stumbled on an essay that criticizes the modern tendency to underestimate the mental potential of our children, written by Haruko Kataoka, founder of the Suzuki piano method.

Dr. Kataoka wrote:

I am always dismayed to see the content of so-called "children's entertainment" and educational materials. They are so simplistic, as if children cannot understand anything….

Making only childish materials available to children is based on a huge misconception…. adults become convinced that children need simplistic materials for their entertainment and education, and they arbitrarily provide children with only these things. The great majority of children's toys and activity books are based on this basic misunderstanding….

The result is that society itself makes children look below themselves in their studies. It is not the children's fault that they are looking down. The adults are forcing them to look down….

The beauty of nature and the splendid fragrance of the arts are necessary for people from childhood. Whether they are exposed to such things daily for ten or twenty years or whether they have been exposed only to lowly, common things will determine that individual's sensibility for the rest of his or her life.

Indeed.

That is one of the core messages that I hope I am communicating in this blog: kids are capable of far, far more intellectually than most adults give them credit for. We are routinely denying children access to the best knowledge of nature and history that humans possess, and to our greatest cultural creations such as classical music, and instead feeding them "simplistic" pabulum considered “developmentally appropriate.”

That is not only an insult to our kids' intelligence. It also cripples our kids in developing the primary means of survival possessed by human beings – their minds.

While Kataoka’s specific interest was in music and music teaching, she makes clear that she intends her point to apply not just to music but rather to all aspects of a child’s development.

The entire essay is a bit over a page in length, and well worth reading: I hope you’ll take a minute to click and read it.

What Barack Should Tell the Schoolkids

Here’s what President Obama should tell the schoolkids on Tuesday if he really wants to help solve the problems in American education:

Since I’m supposed to be the leader of this country, I’m gonna give you guys some straight talk. The grown-ups in this country are ruining you kids’ lives. Your teachers have been trained to make sure you don’t learn anything interesting or challenging; half of your teachers should never have been allowed to graduate from grade school themselves. Your parents want you to end up just as stupid and ignorant as they are: they’re really afraid you might actually end up smart, which is why they make fun of smart folks by calling them “nerds.”

So, here’s what you gotta do: fight back! Hard. Force your parents to take you a couple times a week to the biggest library in your town. Go through the books on science and history and math. You’ll find that most of them are just as full of empty words as your schoolbooks. But, if you look carefully, you’ll find a few that tell you the truth. When you find one of those books that tells you the truth, read it, and work hard to understand it.

It’s not gonna be easy, because no one ever taught you to understand books that tell you real stuff. But there are books that tell you the truth about how people create computers and bridges and medicines and airplanes and skyscrapers. The grown-ups don’t want you to learn that stuff, because then you’ll be smarter than most of the grown-ups and then the grown-ups will feel like fools. You just show ‘em.

And you can find books that tell you the truth about all the grown-up leaders, too – how Presidents fought wars for no reason, lied just to keep power, and stole as much money as they could get their hands on. The grown-ups don’t want you to know that either. They want you to grow up to be suckers, just like they have been.

Knowledge is power, guys. You wanna be poor, ignorant fools like your parents, or you wanna fight back?

Your choice.

Good luck

Kids would be pretty startled to hear someone telling them the truth about education for the first time. But they'd listen. And they might even believe him.

Of course, that kind of courage would require that Barack have a backbone. And there is no evidence that he does.

Teaching Science the Harry Potter Way

Seeing the most recent Harry Potter movie raised an interesting question: why do so many kids find the idea of attending Hogwarts so enticing?

After all, J. K. Rowling makes clear that the students at Hogwarts actually study hard.

And, notoriously, American kids do not like hard schoolwork.

The answer of course is obvious: at Hogwarts, you get to learn magic.

At Hogwarts, you get to acquire knowledge not possessed by mere “muggles,” knowledge that lets you in on arcane secrets about how the universe really works, knowledge that gives you vast and amazing powers not possessed by ordinary humans.

Such magic actually exists in the real world: it is called science.

For example, who would have thought, prior to the twentieth century, that a rather mundane material, so-called “yellowcake,” had within it a mysterious substance that, when properly separated out through arcane and laborious methods, could be used to make some of the most dangerous weapons in history, far more dangerous than Rowling’s “death-eaters”?

(Yellowcake is a substance produced from uranium ore, and uranium is of course the original raw material for nuclear weapons.)

Who would have thought, before the late nineteenth century, that all ordinary matter had within it tiny little particles, making up less than a thousandth of the weight of matter, that could be used not only to control each other’s motion but also to control everything from a cell phone to a jet plane?

(J. J. Thomson discovered the electron in the 1890s; electronics is the art of using a small flow of electrons to control a much larger, more powerful flow of electrons.)

Who would have imagined, prior to the twentieth century, that the entire structure, growth, and daily functioning of our bodies is controlled by a tiny stringy molecule, hidden in the nucleus of nearly every cell of our body, that embodies a sort of computer program that builds a human being from a single cell?

(The stringy molecule is DNA, of course.)

As the example of nuclear weapons illustrates, science is not an entirely benign form of magic. But that fact should not make it less interesting to children: “black” magic is at least as interesting as “white”!

I could continue at great length in this vein: almost all of the great discoveries in science, from plate tectonics to relativity theory, from quantum theory to the theory of evolution, amount to showing that the universe is a radically different, much more mysterious, magical place than “common sense” would ever have suggested.

And, the mysteries unveiled by science not only tell us amazing things about reality; the knowledge provided by science is also enormously powerful.

So, how can American public-school educators manage to transmute science education from the unveiling of deep and powerful mysteries to a numbingly boring subject that most kids shun?

A big part of the answer lies in the dogma of “developmental appropriateness” that plagues American elementary schools. It is “developmentally inappropriate” for young grade-schoolers to learn about black holes or the Big Bang or mutants or nuclear chain reactions – although the X-Men comic series has proven for decades that kids are interested in “mutants” and although it is hard not to be interested in things that make very big bangs (which include nuclear reactions and black holes, and, of course, the granddaddy of them all, the Big Bang itself).

Instead, in early grade school, kids are taught that plants have roots and leaves, that seeds sprout and turn into plants, etc. – as if any normal kid did not already know this.

If anyone thinks I am being unfair, glance through the “Science Content Standards for California Public Schools: Kindergarten through Grade 12”, adopted October 1998 . The terms “DNA,” “Big Bang,” and “mutation” do not occur in the grade one through six standards at all. The terms “black hole” and “relativity” occur nowhere, not even in the high school physics standards.

Another problem is the over-emphasis in American science education on experiments.

It’s true, of course, that natural science is based on detailed observations and experiments. But real scientific experiments are not just random fooling around to see what happens. Real experiments are the result of careful thought and study and mastery of all that is already known about the subject, before one decides on an experiment that will give us insights into nature that we do not yet possess.

Real science is an obsessive search for secrets that nature is carefully hiding from common sense.

Grade-school students and, by and large, even high-school students cannot do that sort of experiment. The result is that the experiments that can be done by schoolchildren tend to be exceptionally boring and uninformative.

(The one major exception is experiments that blow things up: understandably, both teachers and parents tend to be wary of that sort of experiment!)

The emphasis on pointless experiments is connected to the educratic dogma of “constructivism,” the idea that kids can and should “construct” knowledge from their own experience, rather than learn it from a book.

The problem, of course, is that no amount of experience can cause an ordinary person to “construct” the theory of relativity for himself: we needed a genius, Einstein, to figure out how to “construct” relativity.

Prior to the university level, experiments should be put in the same category as field trips or science specials on television: perhaps an entertaining break from the daily grind, but no substitute for actually learning science.

Really learning science means reading books (not just a single textbook, but a variety of books on the subject), working problems, and, above all, trying hard to think about the concepts of science and trying to understand how science has shown that so many of our “common-sense” beliefs about the world are in fact radically wrong.

Still another problem is political correctness from both the Right and the Left.

Science is revolutionary: it proves that many of our common-sense ideas are false.

Most people have finally adjusted to the idea that the earth moves around the sun, and so the schools can safely teach that as a fact.

But many people have not yet adjusted to the idea that we are descended from fish, and so the schools must tread lightly in presenting that idea, especially in early grade school. (Needless to say, the words “evolve” and “evolution” do not occur anywhere in the grade one through six California science standards.)

Similarly, the political Left is uncomfortable with the fact that our genes have a large influence on our intelligence, our personality, etc. So, don’t expect to learn much about evolutionary psychology or behavioral genetics in American schools, either.

A final problem is the American emphasis on pragmatism: how will it help my kid practically to know more than a bare minimal amount of science?

Well, actually, kids who are engaged and excited by discussions about mutants and the Big Bang and relativity and black holes are more likely to stick with science, so that they are willing to slog through the tough advanced science courses needed to become an engineer, a scientist, or a physician.

So, yes, teaching the deep and amazing aspects of science may well help your kids practically.

But, more than that, science is the first organized body of knowledge in human history that gives us systematic, verifiable, non-obvious knowledge about reality, knowledge that is the same whether you live in New Delhi or New York, Nairobi or Shanghai.

That is a remarkable change in human history. When my great grandmother was born in 1883, no one knew what atoms were made of or how old the universe was or what made the stars shine or that there were other galaxies besides the Milky Way.

We now know all of that and more – that there are planets around other stars, how the continents have moved during the earth’s long history, how stars can end their lives as neutron stars or black holes, etc.

Before the rise of science, humans lived in, as Carl Sagan put it, a “demon-haunted world.” “Truth” was a matter of the arbitrary beliefs enforced in your native land: as Pascal sardonically suggested, what was true on one side of the Pyrenees was false on the other.

Science changed that – it banished the imaginary demons, it discovered real, objective truth.

So, kids need to know real science – exciting, revolutionary, disturbing science – not simply because it will motivate them to study enough science to get into dentistry school but also because science frees humans from the lies, myths, and dogmas of the past.

“You shall know the truth, and the truth will make you free.”

Science is magic made real. Real education in real science should be more exciting than learning magic at J. K. Rowling’s imaginary Hogwarts.

Books are available nowadays for even young kids to learn real science in a way that explains the excitement and mystery of science without sacrificing accuracy – e.g., Mahlon Hoagland’s The Way Life Works or Jenny Morgan’s The Universe Tells Our Cosmic Story trilogy. Get books like these and let your kids learn real magic.

Should Homeschooled Kids Study Philosophy?

Should homeschooled kids study philosophy and, more specifically, should homeschooled kids of grade-school age study philosophy?

The answer to both questions is “Yes, but…”

Educated people know something about the history of philosophy, just as they know something about the history of literature, music, architecture, etc. Ideas created or elaborated by philosophers hundreds of years ago are still floating around today, and kids need to learn about those ideas because they affect us all today. Kids need especially to learn about those ideas that happen to be utter nonsense, so that they are not sucked into believing in the nonsense themselves.

Furthermore, philosophy is, in fact, less complicated than most people imagine, and even grade-school kids can seriously learn some things about the history of philosophy.

When my kids were in third-grade, we read together, out loud, Jeremy Weate’s A Young Person’s Guide to Philosophy: like all DK books, it is engaging and nicely illustrated. It is of course necessarily superficial, but it does give kids a thumbnail sketch of some of the major figures in the history of philosophy, and it offers a good basis for further discussion.

When our kids were fourth-graders we read together Lloyd Spencer’s Introducing the Enlightenment, which, in the format of a graphic novel, does a nice job of introducing the leading characters and ideas of the Enlightenment. I found especially interesting his emphasis on the importance of Locke as a key figure behind the Enlightenment, and also his contrasting of Rousseau vs. Voltaire.

So, yes, even grade-school kids should learn something about the history of philosophy, if nothing else to be immunized against philosophical nonsense.

But… many introductory books on philosophy take the tack that “philosophy is not so much a set of answers as a way of asking questions: the important thing about philosophy is not specific answers, but rather the philosophical way of thinking”

Yeah – that is because the answers that philosophers have come up with over the centuries have been almost uniformly bad!

I have in mind, for example, Kant’s claim that the structure of the human mind forces us to think in terms of Euclidean geometry, just a few years before non-Euclidean geometry was discovered, or Comte’s “positivistic” claim that we would never know the composition of the stars, just a few years before scientists discovered how to analyze stellar composition using spectral lines.

The problem is not simply that these guys goofed up. The problem is that, for centuries, philosophers have supposed that they could gain insight into the inner nature of reality by thinking about how we use words, by studying carefully how we think about the world, and by utilizing the basic common-sense knowledge that we all already possess – without doing complex experiments, without learning advanced math, without making painfully detailed observations of nature, etc.

The “method” of philosophy, in short, is that of a literate, articulate gentleman, someone who is very skilled at using words but who does not want to soil his hands with actually doing detailed observations or experiments, or bother his mind with doing high-level math or learning about the detailed observations and experiments carried out by others.

(A few philosophers today are actually going to the trouble to learn real, advanced math, science, etc., but they are very few in number. People who can really handle tough math and science, naturally, tend to become mathematicians, scientists, engineers, etc., rather than philosophers.)

This philosophical method is the exact opposite of the scientific method: science has progressed by assuming that nature hides its secrets in the hard, difficult to come by, details.

Kepler discovered that planets move in ellipses by trying to understand why Tycho Brahe’s actual observations differed in tiny details from what any existing model predicted. Darwin proved the fact of natural selection through his bizarre obsession with Galapagos finches, and many other varieties of animals. The second law of thermodynamics was developed by thinking in great detail about how heat engines worked, and trying to figure out what the performance of the most ideal possible heat engine would be.

Science is the result of an obsessive-compulsive disorder directed to making unnaturally detailed observations of nature in the paranoid belief that nature is hiding deep secrets in those absurd, tiny little details.

And it has worked.

The method of philosophy has been to assume that the inner nature of reality is inherently accessible and transparent and can be understood by thinking calmly and carefully in the comfort of one's armchair, as if reality were simply another well-spoken gentleman who can be understood if one courteously and attentively listens to his words.

That has failed.

So, while kids should learn about the history of philosophy, they most emphatically should not be encouraged to think “philosophically,” i.e., in the way that professional philosophers have thought for the last couple centuries. That method of thinking is a proven loser.

Of course, some areas that are usually considered part of “philosophy” are subjects that humans simply must deal with. The obvious example is ethics: you do have ethical views, whether you admit it or not, and it would be best if you consciously investigated and understood those views.

But… the fact that “ethics” is classified by libraries and by universities under the subject of philosophy should not compel us to think that professional “philosophers” are actually experts on ethics or that the “philosophical way of thinking” is necessarily the right way to think about ethics.

Ethics is too important to be left to the philosophers.

Ethics has to do with the practical issue of how we should live our daily lives: on the face of it, numerous different subjects have something interesting to say about that question – history, anthropology, economics, biology, religion, etc.

And, surely, the accumulated common-sense wisdom of human beings through the generations has some relevance, too.

Yes, kids certainly need to learn about the difference between right and wrong, and from a very early age. But it is an error simply to assume that philosophers are better able to think about ethics in a careful, systematic way than experts in other intellectual disciplines.

In thinking about ethics, the quality of someone’s thoughts must be judged on its own merits, not by whether or not he is officially a “philosopher.” And, based on the failures of philosophy in so many other areas, one should not be optimistic about philosophers’ contributions to thinking about morality, either.

So, yes, homeschooled kids should be taught about Socrates, Plato, and Aristotle, about Locke, Hume, Kant, and Hegel – indeed, they need to be intellectually immunized against Kant and Hegel.

But children should also be taught not to think “philosophically,” in the manner of current and recent academic and professional philosophers. On the contrary, they should be explicitly told that, for at least the last two centuries, the philosophical enterprise as carried out by professional philosophers has been an obvious failure and that the vast increase in our knowledge of reality during the last several centuries has been due not to philosophy but to natural science.

The Solution to Global Warming?

“A Sunshade for Planet Earth,” an article* by Robert Kunzig that discusses serious scientific proposals to avoid global warming without the need to cut back on human CO₂ emissions, was published in the November 2008 issue of Scientific American.

But I have not been able to find any discussion of this in the “mainstream” news media.**

Why?

I think it is fair to say that the “mainstream” news media is obsessed with the “global warming” issue. And, indeed, if the worst fears of global warming become reality, it is going to be a very serious problem.

The possibility that there may be a simple, cheap technological fix should therefore be great news.

Why didn’t the “mainstream” media discuss this?

Let’s be blunt – over the next few decades, human beings are going to continue to dump huge amounts of CO₂ into the atmosphere. It is politically impossible for Europe, Japan, and the US to cut back dramatically on CO₂ emissions in a way that will seriously impact our standard of living.

And, China, India, and other developing countries will dramatically increase their CO₂ emissions in the next several decades.

Human CO₂ emissions will in fact be higher, probably a great deal higher, in 2030 than they are today.

Now, it is possible that those increased CO₂ emissions might not end up being a problem.

There are in fact compelling scientific reasons to conclude that human CO₂ emissions cause the globe to be warmer than it otherwise would be.

But the devil is in the details. Without human CO₂ emissions,would the earth naturally be in a warming or a cooling period? Is it possible that, without our CO₂ emissions, the globe would actually be cooling, and that therefore we need those emissions to stabilize the global temperature?

Yes, it’s possible. Our current geological epoch is one of intermittent Ice Ages: for example, the so-called “Little Ice Age” started in the late Middle Ages and only ended in the 1800s. Perhaps, we would soon enter another "Little Ice Age," except for the protection provided by human CO₂ emissions.

The other key question is: if anthropogenic CO₂ is indeed causing warming, exactly how much warming will end up occurring?

That is much harder to determine than the “mainstream” media have admitted.

I have been interested in global climate modeling since the late ‘60s, long before “global warming” became a big political issue, and, as a physicist, I have some concept of the scientific and computational difficulties involved in modeling the global climate.

The scientific reasoning that indicates that human CO₂ emissions will make the earth warmer than it otherwise would be is quite straightforward. But to calculate the actual value of that warming is fiendishly difficult.

It’s the clouds.

A major part of the problem is clouds: as the globe starts to warm, more moisture should go into the air. Water vapor is itself a “greenhouse gas,” and thus will tend to increase global warming. However, more water vapor also tends to mean more clouds, and clouds tend to reflect sunlight back into space and therefore reduce the effect of global warming.

And understanding clouds is very hard.

Among other things, cloud formation depends on the amount of dust and kind of dust in the atmosphere, and understanding how much and what kind of dust will be kicked up into the atmosphere under global warming is also very hard.

The amount of global warming also depends on how vegetation (and bacteria) respond to the CO₂ increase and to the increase in temperature.

Bacteria are complicated.

The moral here is that when the “mainstream” media report that the “scientific consensus” expects a global warming of between x and y degrees, do not believe them.

This is not a problem that can yet be definitively settled scientifically: I am not confident that it can ever be settled.

It is hard.

So, I do not know if global warming will be a real problem or not. My “gut feeling,” for what it is worth, is that we may be on the verge of another natural cooling period, and that there may therefore not be much of a problem. But that may simply be “wishful thinking” due to my naturally optimistic disposition.

“Global warming” may indeed turn out to be a very serious problem.

So, what solution does the Scientific American article suggest?

The most promising solution seems to be to dump sulfur dioxide (about one-and-a-half million tons, in terms of the weight of the sulfur, per year) into the stratosphere. The estimated cost is between twenty-five and fifty billion dollars a year, a trivial amount when spread out over all the citizens of the industrialized nations. (Incidentally, we are already dumping much more sulfur dioxide into the lower atmosphere each year: this would be a very modest increase in our sulfur dioxide emissions into the atmosphere.)

This works by causing increased cloud formation in the stratosphere, which reflects sunlight and cools the planet. As the article notes, we have reason to be confident that this will work: the 1991 eruption of Mount Pinatubo in the Philippines dumped a lot of sulfur dioxide into the stratosphere and did indeed have a cooling effect.

Is it a perfect solution?

No – as I said above, climate modeling is hard.

There are questions about whether this will have an effect on ozone holes over the arctic and antarctic, how it will affect different regional climates, etc.

And, the article also discusses other possible methods of counteracting global warming, although I myself found the sulfur dioxide approach to be the most promising. (The sea salt scheme bears further looking into, but the idea of a literal sunshade in space, while theoretically possible, is, in my judgment, not practicable.)

Go to your library and read the article yourself.

And then ask yourself: why did you not hear about this in the “mainstream” media?

The “mainstream” media are, after all, obsessed with global warming: they should find this welcome news.

Exactly what game are the “mainstream” media playing?
__________

* The article is behind a wall; only the introduction, Geoengineering: How to Cool Earth-- At a Price is available for free online. Fortunately, any decent library has Scientific American.

**A Google search on the title, restricted to the last year (i.e., all the time since it was published) brings up less than three hundred hits (many to other pages on the Scientific American site that link to the article) – none in “mainstream” US news media, except of course for Scientific American itself.

Physicists Dissing Philosophy

The feisty Czech physicist Luboš Motl recently weighed in on the “what’s gone wrong with philosophy” issue. I think it is fair to say that Luboš specializes in controversy, but I also think his views on the issue of philosophy are very widespread among practicing scientists.

Luboš also links to an interesting essay by an old professor of mine, the Nobel laureate Steve Weinberg, in which Steve addresses the “unreasonable ineffectiveness of philosophy”: why has all the effort of philosophers during the last couple centuries borne so little fruit?

Steve’s basic conclusion is that the best that good philosophy can do is simply to serves as an antidote to bad philosophy: as he states at one point, “But here again the service of philosophy was a negative one; it helped only to free science from the constraints of philosophy itself.”

I do think that Steve is a bit too harsh towards the philosophy of “mechanism,” i.e., the idea that all of reality consists fundamentally of simple entities that do nothing but push and pull on each other. He is of course correct that the rise of the concepts of fields (e.g., the magnetic field) in the nineteenth century disproved the most naïve versions of mechanism – the world is indeed more than just billiard balls bouncing off each other, which is the underlying picture behind the most primitive version of mechanism.

However, as much as we physicists may admire ourselves for our new, more sophisticated picture of reality, which Steve discusses – quantum fields, superstrings, etc. – still, to an ordinary layperson the scientific world-view remains pretty mechanistic. Quantum fields, superstrings, etc. are still simple, mindless little things that push and pull on each other in simple, mindless ways.

If the modern physicist’s view of reality is not quite the simple “clockwork universe” of the nineteenth-century physicist, it is still, as Steve himself has noted elsewhere, basically a view of the universe bereft of meaning, purpose, or feelings.

In short, a pretty mechanistic universe, from the perspective of most human beings (and traditional philosophy).

I think the underlying issue here is the essential three-way conflict among science, philosophy, and religion.

Science, philosophy, and religion all make claims to have a broad, integrated view of reality. But, the views of reality they arrive at differ dramatically.

The scientific view of reality is based on actively trying to disprove one’s hypotheses (believe me – I, and any good scientist, would dearly love to show that we have a wonderful new theory that overturns all the existing theories in our field!) and on only retaining those theories that survive the most vigorous attempts at disproof. Science also embodies the rather paranoid concept that nature is hiding its secrets from us and that we can uncover those secrets only through obsessively detailed observation and experiment. And, science rests on a broadly mechanistic picture of reality, a universe lacking in objective purpose, meaning, or feelings.

Philosophy is radically different: it supposes that the secrets of reality can be uncovered by concentrated thought alone, that reality is naturally open to human understanding. The “testing” of philosophical theories consists basically of verbal assaults by other philosophers. And, historically, most philosophers seem somehow to have uncovered a reality in which human feelings and concerns have a natural home.

Religion differs dramatically from both philosophy and science, most notably in the fact that very few religions welcome attempts to prove that they are wrong: indeed, in practice, the primary warrant for religious belief is that those who deny or seriously question the core beliefs are encouraged or compelled to leave the religious community. And, of course, religions commonly claim that the driving Spirit behind reality has made an active effort to explain Himself to us. Needless to say, religion generally offers a universe full of purpose, meaning, and feelings.

It would be quite surprising if three such radically different approaches to confronting reality were to give compatible pictures of reality.

Of course, they do not.

This is a deeper cultural conflict than often acknowledged: these three views of reality cannot all be right.

It is easy for us scientists to deride creationists who claim that their religious beliefs disprove modern biology and paleontology. It is also easy for us to dismiss “postmodernists” who claim that their verbal musings trump the findings of modern science.

But, in some ways, both the creationists and the postmodernists deserve credit for seeing something that more sensible, moderate folks try to evade: in the long-term, science, philosophy, and religion cannot co-exist.

Choices have to be made.

Why We Are Homeschooling

Anyone familiar with the homeschooling universe knows of the wide diversity of “homeschooling philosophies” – e.g., the classical trivium, unschooling, Charlotte Mason, and, most of all, “eclectic.”

Rather than choosing from that smorgasbord of homeschooling philosophies, we have based our homeschooling approach on the idea that one’s homeschooling methods tend to be a reflection of one’s broader view of human life – including philosophy, politics, religion, etc.

We are homeschooling largely for academic reasons and also because we dissent from much of the prevailing values and attitudes that are typical of contemporary American life.

American society does not value intellectual achievement. We have “select” sports teams even for grade-school children who show unusual athletic talent. It does not surprise us that some kids can advance at twice the pace athletically of most other kids their age.

But, with very few exceptions, we do not have “select” schools for academically talented and motivated children. Very few Americans grasp that bright kids can advance intellectually at twice the speed of average kids.

Indeed, bright kids, as we all know, are disparaged as “nerds.”

Our own kids have been consistently testing at twice their grade level or higher (in tests administered by a local school district, not by me). I’d like to think that this shows that our kids are simply born geniuses, but I am pretty certain that this is not the case: I know lots of other kids who, at an early age, seemed to me as bright or brighter than our kids.

No, I think that our kids are excelling academically simply because our homeschooling approach is centered on the idea that learning is a very good thing, and that children are capable of learning much more, much faster, than most adults realize.

Aristotle said that man is the rational animal. Our mind is our primary tool for survival. To deny children the opportunity to develop that tool to its fullest is to cripple them.

Aristotle also said that long-term happiness is the result of developing our potential as rational beings to the fullest, in the pursuit of excellence.

As a modern scientist, I of course can find numerous points on which I differ with Aristotle’s philosophy. But on those two points, I agree with Aristotle.

Modern American society, especially the pop culture that so pervades most Americans’ lives, does not.

And, that is the central reason we are homeschooling: to enable our kids to develop their potential as rational beings to the fullest in an environment that values their intellectual efforts and achievements.

Is Philosophy Futile?

The historian Richard Carrier offers some on-target criticisms of present-day philosophy in this interview with Ben Dench.

Richard’s central point is that the original idea of philosophy as providing a grand integrated view of life and reality has been replaced by a concept of philosophy that involves dealing with isolated puzzles and riddles :

In academia, philosophy is almost dead. What passes for philosophy now is little more than a fancy system of games and puzzles. Even the few exceptions (Singer, Nussbaum, Haack) are mostly divorced from the original goal of philosophy, which was to bring a coherent worldview to the common man, based on facts and reason, that would tell us how to better live our lives and cope with the world as-it-is. But that requires building and defending systems of thought, not arguing isolated specialized problems in isolated specialized fields; it requires figuring out what actually counts as progress and then rolling up their collective sleeves and working on that progress, not just pontificating willy nilly and turning philosophy books and journals into what the rest of us call history of philosophy; and so on. There are no Humes or Ayers or Aristotles or even Ciceros or Senecas anymore….

The idea of a coherent worldview, every part as well thought out as the rest, has become an alien concept. Philosophy as it was, is no more. And as far as most people are concerned, what philosophy is now, is all but useless to anyone, and even what's useful, is so dense and jargonized as to be unintelligible. That's why decreasing numbers even bother studying it….

Richard also points out that philosophers are focused on the obsolete, discredited views of long-dead philosophers in a way that never occurs in the natural sciences:

History of philosophy can save people time, but only if you actually use it that way. Yes, by learning the blind alleys, you can avoid them yourself. But this has a pernicious tendency in two unfortunate directions. On the one hand, many philosophers dismiss new ideas by immediately labeling them as something that was already refuted even when in fact the new ideas are relevantly different. I have a hell of a time trying to get philosophers to understand that I am both a mathematical realist and a mathematical nominalist, and not a mathematical Platonist in any sense of the term--they can't fathom the synthesis, because all they hear are the past-and-dead categories "mathematical realist," "mathematical nominalist," "mathematical Platonist" and they can't get their minds out of the ruts of the way these things were characterized and debated in the past. History of philosophy has made them worse philosophers, not better ones. On the other hand, many philosophers keep trying to think in the same ruts as past debates--we're still dividing ethical theories into Utilitarian, Kantian, and Virtue Ethics, even though that very division is antiquated and confining….

History of philosophy also seems to be starting to replace actual philosophy outright… Pick up any philosophy journal today and see the amount of citing and referencing and discussing of "other" philosophers that occupies their pages, in ratio to anything that actually gets done as far as making progress in human understanding, and you might become as alarmed as I am.

Philosophers shouldn't be acting like historians. They should let historians do that, just as scientists do. You don't see articles in science journals laden with elaborate discussions of the history of science before attempting to establish a finding. They just present their findings. If there are other current findings and theories on the books (not past refuted findings and theories, but still unrefuted ones), they will survey them and respond to them, even if that involves them in historical reporting. But they don't waste time on inessentials. They build on established and agreed findings and results. No botanist would say you have to read up on the history of all the mistakes and dead ends in 19th century botanical science to make progress in botany today. Philosophy should operate the same way. It just doesn't…

Finally, he suggests that contemporary philosophy is woefully lacking even in basic intellectual standards:

Once at a dinner I stopped an editor of a philosophy journal and showed him that he had published an article with a glaringly obvious logical flaw, so obvious in fact it should have been embarrassing to his entire publication and certainly should never have passed any credible peer review. He actually responded by saying, "You actually expect philosophy journals to prevent the publication of fallacious arguments?" I was flabbergasted. Several of us at the same table answered in unison, "Uh, yes, we do." What the hell else is peer review for?

The interview goes into much more detail, and covers numerous other topics ranging from technology to religious fundamentalism (I don’t agree with all of Richard’s other points in the interview: I think he is way too optimistic about technologically induced “telepathy,” for example).

If anything, I think Richard may have understated the problems with current academic philosophy and the radical difference between modern academic “philosophy” and philosophy as practiced by Locke or Aristotle.

In later posts, I'll discuss in more detail how and why modern philosophy is in the doldrums and what I think this means for us homeschoolers.

The root problem, I think, lies in philosophers' failure fully to confront the fact that modern philosophy arose as an attempt to deal with the collision between modern science and traditional religion in seventeenth-century Europe. The philosopher/anthropologist Ernest Gellner wrote on this at great detail (e.g., in his The Legitimation of Belief , The Devil in Modern Philosophy, and Postmodernism, Reason, and Religion), and I agree with Gellner that you cannot grasp modern philosophy without addressing broader issues involving science and religion.