Math and science and other stuff

Combined with a head start in reading, I also have a mind that is nimble with numbers. I can remember, from an early age, mentally playing with the numbers on the hymnal board, finding mathematical relationships among them. As a result, in school, I was accelerated in math as well as in reading.

Our school had about ninety children in each grade; the ninety were divided into three classrooms. We were sorted by academic competence, and every student knew which group was smart, which was dumb, and which was in the middle. In fourth grade they resorted us for math class. I remember one girl being teased because she was in the smart class for everything else, but for math she was in the dumb class. I do not think the school district meant for us to be aware of the difference, let alone judgmental about it, but children will be children.

I was one of the few who was promoted a year in math (but not in other classes). As a fourth-grader, I went to the fifth grade class for math. In fifth grade I was with the sixth-graders, then in sixth grade with the seventh-graders, and in seventh-grade I was with the eighth graders. No one had a plan about what to do with me next, so I repeated eighth grade math in the eighth grade. If someone at school or at home had troubled to get a high school book and guide me through it, I might have taken more interest in math or science as a career, but no one bothered to think that far outside the box.

On the other hand, my parents surrounded me with scientific toys. I had a telescope and a microscope, a chemistry kit, and a kit of electronic projects from Radio Shack. I enjoyed those toys, and I’m sure they helped me to well in school, but math and science remained fun hobbies. I got easy As in those subjects all through school, but I never considered career opportunities in those fields.

I followed the space program on TV, watched the Apollo missions to the moon, and dreamed of being an astronaut. I dissected frogs in seventh grade and learned the parts of the body, even the Latin names for all the bones. I took to algebra, to trigonometry, and to symbolic logic for geometric proofs like a duck to water. My senior year of high school, I took calculus, which was as far as the high school math program could take us. By the end of the year, my friend Pete and I were go-to resources, along with the teacher, to help the rest of the class understand the calculus lessons. (Pete was high school valedictorian; he is now a family physician in the Chicago area.) Some high school teachers, and some other adults I knew, were disappointed that I was not pursuing further education and a career in science or mathematics.  

When I was in the fourth grade, a student teacher working on a paper for her school took me and a few other students from assorted grades out of class for testing that went beyond the standardized tests all students took. I was never told results of those tests. But, after I graduated college, I once took the privilege of visiting the school and seeing my “permanent file.” I learned that, in the earliest grades, my standardized test scores were only slightly above average. I scored well on the individualized tests conducted by this student teacher. Afterward, my test scores increased each year. In high school, I scored As in every class except some Physical Education classes and one typing class. (I graduated seventh in my class.) In college I managed straight As and was at the top of my class. On the SAT and ACT and later on the GRE, I landed in the highest percentile. But skill at test-taking does not translate to skill in all areas. I am capable of only the most basic household and car repair skills—I can change a light bulb or even replace a switch or electrical outlet; I can change a tire or a car battery. Beyond those basic tasks, I rely on professionals. I’ve done a smattering of learning in other languages, but I’ve never become fluent in a second language, and geniuses who can sense and describe nuances in the grammar of vocabulary of Biblical Hebrew and Greek or in more recent Latin and German texts blow me out of the water. I can read music and play several instruments but am proficient at none of them. It took me many years to begin to appreciate the complexity of classical music, let alone modern jazz. No one does all things well. At best, we do well in the things that matter most to us and to those who rely upon us. J.

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Beyond reason in creation and in redemption

I am thankful for fellow blogger Clyde Herrin for two reasons. First, he has been kind enough to repost several of my recent posts on his blog, thus expanding my potential audience. Second, he has given me food for thought in his comment on my recent “Summer Solstice” post. You may recall that I suggested that an Obsessive-Compulsive Creator would have given us thirty-day months and a 360-day year, allowing day and month and year to match mathematically. Clyde suggested that, in the beginning, the solar system operated in sync according to simple math, but that sin and the consequences of sin threw the system into a more chaotic set of relationships. He pointed me to a post of his ten years ago (which I had already read and liked some time in the past) in which he suggests that the turmoil of the Flood threw the earth’s day off from its previous length by about twenty-one minutes, resulting in the mismatch of days to years that complicates our calendars today.

I replied to Clyde that, in my opinion, God delights in complexity within creation and does not limit himself to simple relations. I mentioned complexity in biology and in subatomic physics, and I then offered the thought that God purposely put the sun, moon, and planets (including our earth) into a complex dance that does not simplify to easy mathematics. Continuing to ponder the possibilities after posting that comment, I have arrived at even more evidence that the patterns in our solar system are intended to be complex.

The evidence has been known for a very long time. Two thousand years ago, Greek mathematicians used geometry to study the world and even to comprehend complex ideas in number theory. Reality frustrated these mathematical geniuses. They wanted every number in the universe to be a fraction, a ratio, a balance of two other numbers. But these students of nature discovered that the relationship of the diameter of a circle to its circumference is not a rational number. It cannot be expressed as a fraction of two other numbers. That relationship of the trip around a circle to a trip across a circle is called “pi,” a number about (but not exactly) one-seventh more than three. Likewise, the relationship of the diagonal of a square to the side of a square is another irrational number, which happens to be the square root of two. Every square in the world, no matter how big or how small, has the same relationship of diagonal to side, and the number that describes that relationship is never a fraction or ratio of two other numbers.

It is no coincidence that we call those numbers irrational. Not only are “pi” and “the square root of two” not expressed by fractions, or ratios of two numbers; they also do not make sense to people who want mathematical simplicity in their world. It seems that God delights in complexity and does not settle for simple relationships in his creation. For people like Clyde and me, who believe in an Almighty God who created heaven and earth and all that exists, that raises interesting questions. Is the Almighty God limited by rules of geometry, so that circles and squares could not exist apart from the irrational numbers that describe them? Or could God have created a world with different mathematical rules and different geometric proportions, a world that was fully rational even to ancient Greeks who studied the world and the things it contains?

Such questions go beyond science and mathematics and geometry. Identical questions can be raised about ethics. Is the Almighty God answerable to rules about good and evil, or does he get to write all the rules? Those who call Him Almighty define “good” as “whatever God likes” and “evil” as “whatever God does not like.” Our debates about good and evil, then, come down to God’s statements to us about what he likes and what he hates, the behavior of which he approves and the behavior of which he disapproves. Yet some people feel qualified to judge God, to apply their own rules to the Creator and decide whether he meets with their approval. To such people, God speaks as he spoke to Job: “Where were you when I created the world?”

Imagining a world with different rules for mathematics and geometry goes beyond our comprehension. Imagining a world with different rules for right and wrong goes beyond our imagination. God, at his essence, is love; for love flows among the Persons of the Father and the Son and the Holy Spirit. We are created in God’s image. The most important commandments God gave us are that we love him and that we love one another. God’s other commandments teach us how to love. Sometimes, what seems loving to us attacks others and harms others rather than truly loving them. God’s love sometimes is “tough love,” discouraging us from harmful behavior we might characterize as love and guiding us into true love for God and for one another.

But, because God is love, he also rescues us from the consequences of disobeying his rules. We cannot disobey some rules: we cannot defy gravity, and we cannot cause the relationship of the diagonal of a square to its side to become a rational number. In cases where we have broken God’s commandments telling us how to love, God rescues us from the consequences of our failure. Jesus, on the cross, bore the burden for our sins to reconcile us to God. Jesus defeated our enemies—even our own sins—and shares his victory with us. In a sense, God breaks the rules of justice, of power and authority, to establish grace and mercy and peace in our lives.

And he supports that message about his love and his grace by leaving in his creation other mysteries that defy reason and logic and the way we would do things—including quantum mechanics, including irrational numbers, and including the complex dance of the sun, the moon, and the planets. J.

Let’s get small, part two

If you tear a sheet of paper into tiny pieces of paper, you will not be able to determine if you have discovered the smallest possible piece of paper. Therefore, you cannot prove in this way that whether paper is made of nothing but paper. Remember: there are three possibilities: every piece of paper might be divisible into smaller pieces of paper, continuing endlessly to smaller and smaller pieces; or there might be a smallest piece of paper that cannot be divided; or paper might be made out of small pieces of something else, small pieces which together have the properties of paper. We will test the third possibility. Weigh the piece of paper; then set it on fire. When it has burned, weigh the ashes that remain. In this way, you prove that paper consists of at least two ingredients. One ingredient is the ash that is left; the other ingredient somehow disappeared in the fire.

In the ancient world, many philosopher/scientists concluded that the material world and everything in it consists of four elements: earth, water, air, and fire. From this experiment, they would say that paper must consist of earth (the ashes left when the paper was burned) and fire (the missing weight that disappeared when the paper was burned). For centuries, philosopher/scientists called alchemists proposed theories and conducted experiments to learn more about the material world and the substances in this world. Often alchemists are portrayed as magicians trying to turn lead into gold. They did, in fact, attempt to make that change. However, they also performed many other investigations which led to the modern discovery of the science called chemistry.

Modern science would determine that most of the ash produced by burning the paper is an element called carbon. Carbon is one of the elements found in paper. Modern science also reports that water is not an element. Each molecule of water contains three atoms—two atoms of hydrogen and one atom of oxygen. Water molecules can be broken. Connect wires to the terminals of a battery and put the wires into a glass of water. Bubbles will form at each wire—hydrogen molecules at the cathode (attached to the negative pole of the battery) and oxygen molecules at the anode (attached to the positive pole).

If you could see a molecule of water, it would look like a Mickey Mouse head—two little atoms of hydrogen set sixty degrees apart on a larger atom of oxygen. Remember that these molecules are very tiny. A large number of them are required for the water to have any properties that our senses can detect. But another interesting fact about a glass of water is that –in addition to chemicals contained in the water—even a full glass of water contains much empty space.

To prove this, try the following experiment. Take a measuring cup and carefully add half a cup (four ounces) of water. Now carefully add a tablespoon of water and notice that the water level is above the half-cup (or four ounce) mark. Add a second tablespoon of water, and you now have five ounces. Add two more tablespoons of water, and you have six ounces, or three-quarters of a cup of water.

Now add a tablespoon of sugar and gently stir until all the sugar is dissolved. Notice that the water level has not increased above the six-ounce mark. Do so again, and you still have only three-quarters of a cup of water. A third time will not have the same results, because all the sugar cannot dissolve. But, even with three tablespoons of sugar in six ounces of water, you will still be far closer to the six ounce line than you are to the full cup of water. The dissolved sugar has found empty spaces between the molecules of water in your cup.

Matter contains atoms, but it also contains much empty space. Empty space exists between electrons and nuclei in each atom (and none of the sugar was able to fit into that empty space in the water). Empty space exists between molecules in even the most seemingly solid substances. Empty space exists beyond the atmosphere of the earth. Except for the brief times when the Moon, Mercury, Venus, or some asteroid crosses between the Earth and the Sun, that distance is more than ninety million miles of empty space. Even with one of those objects in the way, the bulk of that ninety million miles is empty space. More empty space separates the Sun from other stars, and yet more empty space lies between the galaxies. Most of the universe is empty space.

Empty space is not nothing. People who confuse emptiness, or the void, with nothing make the same mistake that the Cyclops named Polyphemus made in Homer’s Odyssey. Clever Odysseus introduced himself to the Cyclops as “No one.” Later, when Odysseus poked a sharpened stick into Polyphemus’ eye, the Cyclops roared out, “No one is attacking me! No one has blinded me!” None of his friends came to help him; they thought that, if no one was attacking him, everything was fine. Likewise, in Lewis Carroll’s Through the Looking Glass, the king asks his messenger who he passed on the road and the messenger answers, “nobody.” The king remarks that nobody is slower than his messenger. Indignant, the messenger says that he believes nobody is faster than he is. “He can’t do that,” the king said, “or he’d have been here first.”

In the same way, even experienced philosophers sometimes confuse themselves, mistaking emptiness or the void with nothing. Earlier philosophers felt that empty space was impossible. For anything to move, they figured, it must displace something else. When you walk into a room, you displace some air. When you lower yourself into a bathtub, you displace some water. (When Archimedes realized the significance of that displacement, he was so invited that he invented streaking.) Those early philosophers were certain that, from the very smallest pieces of the world to the very largest, everything must displace something else as it moved. They were the ones who coined the expression, “Nature abhors a vacuum,” which has nothing to do with housecleaning.

But Nature cannot abhor a vacuum; nature is almost entirely vacuum. From the empty space inside each atom to the empty space between galaxies, most of the universe is empty space. But emptiness, or void, can be measured. The tiny space between electrons of an atom and its nucleus can be measured (and that empty space is much bigger than the nucleus of the atom, let alone the electrons). The empty space between planets and the sun, or between stars, or between galaxies, can be measured. Because it can be measured, it is not nothing.

A modern physicist says that the universe is expanding. Ask, “into what is it expanding?” and the physicist answers, “Nothing.” They physicist does not mean empty space or a void; the nothing that surrounds the known universe is not measurable empty space or void. Christians say that God created the universe out of nothing. They do not mean that He created out of void or empty space. Before God created, according to Christian teachings, nothing but God existed—not even empty space, not even a void. The difference is very important to Christian teachers and to modern physicists. J.

Let’s get small, part one

You have before you a piece of paper. Being a philosopher and a scientist (for the present, we will take those words to be synonyms), you use your senses to analyze this piece of paper. You see that it is white. You see that it is flat. Seeing a measuring device, you use it to determine that the paper is eight and one half inches long and eleven inches wide—or, if you prefer, 261 mm (millimeters) x 279 mm. Measuring the thickness of the paper is not so easy, but you are a clever philosopher and scientist. You stack one hundred sheets of paper and find that the stack is one-half inch tall, or 1.2 mm. Therefore, you know that one sheet of paper is one-two-hundredth of an inch thick, or 0.012 mm—about the smallest size your eyes can see or your fingers can feel.

Scent and taste do not reveal useful information about the paper, but your hands tell you that the paper is smooth, but the edges and corners feel sharp. Left alone, the paper makes no noise. With your hands, though, you can cause it to make noises when you flap it in the air, when you crumple it, or when you tear it.

Tearing the piece of paper gives you a new thought. You now have two pieces of paper, both remaining white and flat, with the same thickness and width, but neither as long as the original. How long can this process continue? Can you continue to tear the paper into smaller and smaller pieces? And will each piece remain a piece of paper? Modern science teaches us about molecules and atoms while we are young, but for centuries philosophers and scientists were lacking that information. For centuries they wondered how small a piece of paper could remain paper, and what it might be if it was no longer paper.

With a microscope, you can see that paper consists of fibers. Perhaps with tiny, delicate tools, you could isolate one fiber from the paper and chop it into shorter and shorter lengths. Even this experiment will not answer the age-old question about what tiny parts might make up a piece of paper. Logically, three possibilities exist. Perhaps the process can continue forever—however small a piece of paper you have, you can divide it into two smaller pieces. Perhaps the process reaches a limit—a small particle of paper exists that cannot be divided into smaller pieces. Or perhaps at some point we will find tiny pieces that are no longer paper, but are something else, ingredients, elements of which paper is made. By tearing and shredding the paper, we will never determine which of these results is real.

We set the paper aside for a little while, and instead we consider a drop of water. How big is a drop? For convenience, we will define one drop as one twentieth of a milliliter (0.2 mL) or one one-hundredth of a teaspoon. Modern science tells us that one drop of water contains 1.5 sextillion molecules of water. Sextillion is a real number, unlike jillion or zillion. It is written as a one followed by twenty-one zeroes. When dealing with huge numbers or tiny numbers, scientists prefer to use “scientific notation”—in the case of one sextillion, writing the number as 1 x 10²¹. Another shortcut is to use special measurements, such as nanometers or Angstroms. To try to put this number into perspective, though, let’s take that drop of water and divide it in half. Then divide the half-drop in half again, and do so a third time, a fourth time, a fifth time, and on to a tenth time. Now we have a speck of water that is about as high and wide and thick as the thickness of one piece of paper—and it still contains 1.5 quintillion molecules of water.

With special instruments, we continue dividing that one speck of water—slightly less than one thousandth the original drop—in half, and divide that half in half, until we have done that process another ten times. The invisible bit of water we have now is one millionth the size of the original drop, and it contains 1.5 quadrillion molecules of water. Repeat the process another ten times, and what we have is one billionth the size of the original drop and contains 1.5 trillion molecules of water. Now we are getting to numbers we recognize—at least if we pay attention to the national budget. Billions and trillions are somewhat familiar. Along the way, we may begin to appreciate just how tiny one molecule of water happens to be.

But another thing has happened. By the eleventh or twelfth division of that drop of water, what we had left was not really water. It still contained water molecules—an unimaginarily huge number of molecules—but that water was no longer wet. Drop it on your skin, and you would not feel it. Drop it into a glass of water, and you would not hear it land or see the ripples. It takes an enormous number of molecules of water to be sensed as water, just as it takes an enormous number of molecules of chlorophyl before we can see any green in a leaf.

We will return to the water again and will look at its molecules and consider even smaller parts of the molecule. But, first, we will experiment again with the paper. J.

Reality starts getting weird

Our senses tell us of the world around us, the world in which we live. But how can we be sure that the information delivered by our senses is complete? What if other information lies outside our perception, realities we cannot comprehend because nature or its Creator have not equipped us to detect those realities?

My example of the singing refrigerator hints at such a possibility. My sister and I could hear the sounds the refrigerator made. Other family members could not hear them and refused to believe that such sounds existed. Human ears vary slightly regarding the pitches they can detect and report to the brain. Such a difference in hearing appears to be only the tip of the iceberg.

In the 1860s, at the height of the Victorian Era, scientists began to detect some sort of radiation associated with electricity and magnetism. Twenty years later, further research had provided a better understanding of that radiation. What we humans know as visible light—red, green, blue, and white—is only part of the spectrum of light waves in the world. Other wavelengths are longer or shorter than the wavelengths our eyes witness. Radio waves and microwaves had been found in the latter part of the nineteenth centuries; X-rays would not be discovered until 1895. Not only did science unveil the existence of these waves that have always been there; inventors swiftly found ways to harness this knowledge for the benefit of humankind.

Imagine telling a scientist from the year 1850 that in our time invisible waves are used to allow people to communicate across thousands of miles, to speak to one another and hear immediate replies. Imagine describing the way the same invisible waves convey not only sounds but also images—even moving pictures—all around the earth. Imagine adding to that fantastic tale the detail that bones and internal organs of a person can be observed without removing that person’s skin. These innovations would surely be as marvelous and unexpected as motorcars, airplanes, and other modern tools that we take for granted today.

A few people claim to believe that the Earth is flat, insisting that evidence of a spherical world is misinformation distributed to fool the general public. Perhaps somewhere a few people also insist that all light is visible light. They might claim that reports of radio waves and microwaves and X-rays are a trick and that such things do not exist. Cell phones, garage door openers, TV remotes, and medical and dental X-rays are all part of the trickery, clever illusions to persuade us to believe in unseen waves that constantly surround us and pass through us.

Because science stumbled upon these unseen versions of light, we must accept the possibility that other real things exist in the world, unobserved because we have not yet found a way to look for them. Meanwhile, further studies of the observable world bring us new and amazing bits of news. For everything we consider solid and reliable—the red apple in the refrigerator, and the square table in the middle of the room, and my foot, and my shoe, and the ant crawling on the floor next to my shoe—all these things are formed from an unimaginably large number of unimaginably tiny pieces. And those pieces follow rules that are far different from the rules of geometry and physics we have learned about the world our senses observe. Even the light that enables us to see those things follows a different set of rules. This is where things start becoming truly weird. J.

Our senses and our world, part three

If we agree that a tomato in the dark refrigerator is only potentially red—not truly red when no light is shining on it—then must we agree that the properties of objects do not exist when they are not perceived? Is sugar not sweet when it is not being tasted? And is salt not salty when it is not being tasted? Are they only potentially sweet and potentially salty? If that is the case, then we have abandoned dualism and are functioning in the realm of idealism. In that realm, minds and thoughts and ideas (and spirits) are real, but the material world is only in illusion formed by our minds and thoughts and ideas (and spirits).

Imagine a small pile, half a teaspoon, of white crystals on the kitchen counter. They might be sugar or salt, but you don’t know which. Clearly, by tasting a few of the crystals, you will know if the pile is sugar or salt. Does that mean that the crystals are neither sugar nor salt until they have been sampled?

Taste is the quickest way to discern sugar from salt, but a chemist could provide other tests that would identify the crystals apart from their taste. Sugar consists of hydrocarbon molecules, but table salt is a lattice of sodium and chlorine ions. These chemical facts remain true even if the crystals are not tasted. Therefore, we do not have to taste them for them to be either sweet or salty.

By the same token, the brown table in the center of the room is not brown in the dark, but it is still a table, hard and unyielding. If I walk into that table in the dark, it will bruise my shin and cause me to lose my balance. Even in the dark, when it is no longer brown, that table retains all its other physical properties as a material object.

If a tree falls in the forest and no one is there to hear it, does it make a sound? As it begins to tumble, it crashes into other trees, and the crackling of the branches sends vibrations through the air. When it finally hits the ground, it creates a thump that shakes the ground. That thump will be discernable for some distance in the ground, and it also will cause vibrations in the air. Now perhaps no person is in the forest to hear the crackling and the thump. If a scientist has left a listening/recording device in the forest—trying to gather evidence of a surviving ivory-billed woodpecker or of Bigfoot—that device will register the sound of the falling tree. Squirrels and sparrows will hear the crackle and the thump. But what if there are no squirrels, no sparrows, and no scientific listening device? Will the tree still make a sound? A Christian (or Muslim or Jew) is likely to say that God is still in the forest. God will hear the sound of the falling tree. If God is not present, then there is no tree and no forest, and (of course) no sound. On the one hand, this proposal lends itself to Berkeley’s brand of idealism—things we call material are ideas in the mind of God, and as a result they are real to all created beings that have senses and minds.

But a tree is big enough to make a sound. One leaf, falling from the tree, might not make a sound that is heard by any human being, squirrel, sparrow, or scientific device. Does God still hear the leaf when it lands on the floor of the forest? Perhaps. Philosophy alone cannot answer that question.

But substances in the material world must have a certain quantity to possess the qualities we apply to those substances. The half-teaspoon of sugar or salt was sweet or salty. One molecule of sugar, or one sodium ion linked to one chlorine ion, would have no flavor. Half a teaspoon of water is wet. One molecule of water is not wet. A steel knife is sharp. One iron molecule from that knife is not sharp.

I will address the atomic theory of material substances more completely a bit later in this writing. But we must concede right now that the smallest particles of matter lack the qualities that they attain when they gather in large numbers. A single molecule of chlorophyl is not green. It is too small to reflect any light. But millions of molecules of chlorophyl, gathered in the same leaf, are green. This fact forces us to reconsider our opinion about the reality of the material world, that world which is revealed to us by our senses. J.

More about philosophy

Philosophy is traditionally defined as the search for what is true, what is good, and what is beautiful. Technical terms for those topics are “metaphysics” (the search for what is true, or real—which is followed by “epistemology,” determining how we know what is real), “ethics” (the search for what is good), and “aesthetics” (the search for what is beautiful, and how we recognize what is beautiful). Some twentieth-century philosophers willingly surrendered these searches to other disciplines that had branched off from philosophy. They conceded the search for what is true to science, accepting that whatever scientists recognize as real should be considered real. They conceded the search for what is beautiful to the arts, accepting that whatever artists recognize as beautiful should be considered beautiful. The question of how we know what is real was bestowed upon psychologists, and the question of what is good was bestowed upon sociologists. After all, perception is done in the mind, and psychologists study the mind. Ethics are governed (if not formed) by groups of people, and sociologists study groups of people.

What, then, is left for philosophy to consider? With philosophy left only as a branch on the limb of “liberal arts,” much of the work of modern philosophers concerns language and communication. This is, indeed, a fertile field to plow and plant and tend. The signs and symbols used to communicate ideas fascinate philosophers. Take the idea of 2, which can also be represented as II. It can be called two or deux or dos or zwei. For English speakers, it must be distinguished from the preposition “to” or the synonym of “also,” “too.” Once considered or communicated, though, this sign or symbol represents a powerful idea, an idea that contains more than two apples or two triangles. Philosophers even ask whether the number two exists apart from two apples or two triangles. If it exists as a pure idea, what makes that idea real? Would the idea of “two” exist without a mind to consider the significance of “two” beyond its representation in any pair of objects in the universe?

These questions restore philosophy’s function as a search for truth. Thinkers trained in a scientific approach may fail to appreciate the importance of determining whether the idea of “two” exists apart from the observer or exists only in the mind of the observer. For that matter, philosophers should ask whether science can observe and measure and comprehend everything that is real. Science does a good job studying those things it is designed to study, but other existing things may retain their being outside the reach of science.

Likewise, sociology is not equipped to determine whether a rule or requirement is good. Observing groups of people all over the world, sociologists might report that nearly all groups of people frown upon murder and stealing. That, in itself, does not make those actions bad. A scientist might weigh each individual in a group of people, then establish an average, or normal, range of weights, with abnormal extremes at both ends of the spectrum, but that would not mean that the median weight was the healthiest weight for those people. A sociologist might closely observe a group of people and count the lies told by those people, then establish an average, or normal, number of lies told each day, with abnormal extremes at both ends of the spectrum, but that does not mean that the median honesty was the most ethical honesty for those people.

In short, philosophers never should have limited themselves to studying language and communication, even though that topic is fascinating. Ancient Greeks made the same mistake when philosophy degenerated into sophistry, promising to teach speakers how to be convincing, no matter which side they took in a debate. The career of Socrates helped to correct that mistake. Philosophers need to keep asking the big questions: What is true? What is good? What is beautiful? Information from other specialties assists philosophers in their search for answers. Scientists and artists, though, cannot replace philosophers in the realm of human thinking. J.

Branches of philosophy

Although some early Christians rejected all secular philosophy, many other Christians found philosophy a useful tool to understand creation and to communicate with people living in the world. Church leaders came to regard philosophy as “the handmaiden of theology.” God and his revelation took first place, and the teachings of philosophy were not allowed to contradict the Word of God. Beyond that, philosophy had an honored place in the toolkit of Christian education, and also that of Jewish education and Muslim education.

In the Middle Ages, philosophy and education were expressed in what then were called the seven Liberal Arts. These began with the “trivium”—Grammar, Logic, and Rhetoric. These studies are in no way trivial: they remain the foundation of thinking and communication, including written composition and public speaking. The other Liberal Arts were Arithmetic, Geometry, Astronomy, and Music. A solid basis of knowledge in these areas prepared any student to specialize in other fields of knowledge, research, and understanding.

During the Enlightenment, appreciation of knowledge, understanding, and education underwent further revision. By modern times, two major boughs had grown on the tree of knowledge. They were called Science and the Arts. Even today, most colleges and universities grant degrees that are designated as either science or arts.

Branches on the bough of science begin with mathematics. This is the purest science, dealing only with numbers. Two is always two, whether it is represented by two apples, two triangles, or the two Natures of Christ. From the branch of mathematics grow further branches, including arithmetic, geometry, algebra, and calculus. Practical mathematics also are taught, such as accounting and statistics.

Physics is a second branch of science. Physics studies objects in the material world and analyzes their qualities and their movement. Astronomy was recognized as a branch of physics once Isaac Newton demonstrated that the heavenly bodies obey the same laws as earthly bodies. Nuclear, or subatomic, physics are another branch, one in which the geometry of Euclid and the physics of Newton no longer apply. Practical physics are found in the various departments of engineering.

Chemistry has become its own branch, although chemistry might be viewed as a more complicated field within physics. Alchemists began with a theory of four elements (air, earth, fire, and water). They eventually discovered a far more complicated table of elements. Theoretical chemistry and practical chemistry are not as easily distinguished as in mathematics and physics.

Biology is the next branch of the sciences. Biology studies living things, whether plants or animals or microscopic forms of life. The most important practical biology is medicine.

All these are pure sciences or natural sciences. On the same bough of sciences are the social sciences, beginning with psychology. Once uniquely identified with philosophy, the study of the psyche—or the self—was transferred to the sciences in the twentieth century, beginning with the work of Sigmund Freud. Along with psychology comes sociology. Psychology looks at human beings as individuals, while sociology studies people in groups. Among the practical branches of sociology are law, politics, and economics. Some schools even treat history as a social science, although most schools consider history one of the arts.

The bough of arts on the tree of knowledge divides into fine arts and liberal arts (also called humanities). The fine arts include visual arts, such as painting and sculpture, and music (which involves hearing rather than seeing). If fine arts involve the senses, then baking and cooking might also be listed as fine arts. But the fine arts also include literature—poetry and prose, fiction and non-fiction. History might also land here, as one of the literary arts, a branch of non-fiction. Drama also is a fine art, which branches further into the categories of theater and film

The liberal arts, or humanities, complete the major branches of the tree of knowledge. Most schools include history among the humanities. Other humanities involve the study of languages and the study of cultures. Religious studies are also included among the humanities. Ironically, the study of philosophy—once the essence of the entire tree—has now become a department within the humanities or liberal arts, merely one branch among the many branches of the tree.

Arguments for a third bough of the tree become increasingly common in the late twentieth century and twenty-first century. This third bough consists of vocational education, which focuses on neither science nor on the arts. Modern society needs plumbers, electricians, carpenters, auto mechanics, truck drivers, and hair stylists. Food preparation and service falls into the same category. Even many workers in health care receive vocational training rather than scientific or artistic education. One does not require knowledge of algebra or drama or history to be effective in any of these jobs. Debate continues, though, about how much exposure to science and the arts helps people to be fully human and to be happy in their vocations while living among their common human beings. J.

Philosophy

One might say that every person is a philosopher, just as every person is a scientist, and every person is an artist. We all seek to understand the world around us and what it contains; with a combination of observation and experimentation, we all try to gain information and clarity about our world. We all express ourselves, from time to time, by humming a tune or doodling a design or telling a story. We all ask the Big Questions, at least once in a while, such as, “Who am I?” and, “Why am I here?”

But only a few become professional scientists, studying a science and earning a degree and taking a paid job to work with science. Only a few become professional artists, making a living in music or painting or storytelling or another of the arts. Only a very few become professional philosophers, and most of them end up teaching about philosophy in universities. In general, people understand why science and the arts should be funded. They don’t always know why philosophers should be paid for what they do.

Probably the earliest humans were scientists and artists and philosophers. The Neolithic Revolution—that time when people began to settle in communities and raise food rather than hunting and gathering food—allowed more specialization in such matters. Stone Age people and Bronze Age people had thoughts and ideas about their place in the world. These thoughts and ideas tended to emphasize relationships. People knew who they were by seeing their place in their families and their communities and by understanding where they stood in relation to their environment and their gods.

About twenty-five centuries ago, after the establishment of the Iron Age, a new wave of thinking arose in the world’s most established gatherings of people. Many of these new approaches are studied today among the world’s religions: from China, Confucianism and Daoism; from India, Buddhism and Upanishad Hinduism; and from Persia, Zoroastrianism. Each of these focused more than earlier teachings upon the human individual as an individual. Each taught followers to look within themselves for virtue and for truth. These new approaches have been described as the Axial Age in religion and philosophy. But the Greek experience of the Axial Age differs in some ways from what was happening at the same time among other cultures.

Greek Axial thinkers approached the world with questions about its nature, and they tried to answer these questions without resorting to religious formulas. The earliest asked about the structure of the world—from what are all things made? One thinker suggested water, another suggested fire, and still another suggested numbers. Thinkers debated whether the true world is always in motion or always at rest. After a time, such discussions degenerated into sophistry, as teachers offered to train their students in rhetoric, promising that they could win any argument, no matter which side they chose to defend. But another series of thinkers broke the pattern of sophistry by asking about what is good. How do we define goodness? How do we recognize goodness? How do we make ourselves good? Socrates and Plato and Aristotle are associated with this approach, and the Greek and Roman worlds were shaped by their thinking and by the generations of thinkers who followed their approach.

The Greek word for this kind of thought is “philosophy.” Literally, that word translates into English as “love of wisdom.” But the translation only requires us to define two words instead of one. “Philo” denotes love as in friendship or loyalty; it is neither the erotic love of romance and marriage nor the “agape” love for God and for one’s neighbors. “Sophia” as wisdom is not the ability to learn and retain a list of facts, nor is it the mechanical ability to use those facts to shape and change the world. Wisdom is not even the steady and helpful thought process that has been mislabeled “common sense.” Wisdom is a broader understanding—but an understanding of what? The most helpful description of wisdom is to view it as pursuit of the Good, the True, and the Beautiful.

For Greek philosophers, wisdom was distinct from religious knowledge and comprehension. Earlier thinkers in Israel had said the opposite. “The fear of the Lord is the beginning of wisdom,” the Bible says in the books of Psalms and Proverbs. Psalm 14 adds, “The fool says in his heart, ‘There is no God.’” Wisdom, as described in the book of Proverbs, resembles what the apostle Paul would call “faith.” For this reason, James wrote, “If any of you lacks wisdom, let him ask God, who gives generously to all without reproach” (James 1:5). Thus, we encounter two competing paths to wisdom—one which has the thinker look within, seeking the Good and the True and the Beautiful; and the other looking to God, hoping to find in God the Good and the True and the Beautiful. Which is fascinating, since one of the major questions asked by those on the first path is, “Does God exist?” J.

The Late Middle Ages

Efforts to distinguish the High Middle Ages of Europe, the Late Middle Ages, and the Renaissance are as arbitrary and capricious as are efforts to distinguish the several generations of recent American history. A steady process of development and growth marks European culture throughout this time span. Historians traditionally try to place any good developments of medieval times into the High Middle Ages, treating the Late Middle Ages as an era of trouble and collapse, thus introducing a splendid and sparkling Renaissance or rebirth in Europe. But the good and the bad are intertwined, as they always are in human history, and the Renaissance is more a continuation of medieval progress than it is any rediscovery or rebirth of ancient culture and virtue.

One key development in world history overlaps the High Middle Ages of Europe: the sudden appearance of the Mongol Empire in Asia. Genghis Khan (born Temujin) assembled in his lifetime the largest landmass under one government in all human history. (Wikipedia quibbles regarding this achievement, suggesting that some World War II developments achieved greater control over the Earth, but the Mongol Empire remains the largest by any reasonable definition of “empire.”) The land ruled by the Khan included China (formerly under the Chinese Song Empire), other central Asian states, Persia, western Asia almost to Egypt, and northeastern Europe covering most of modern Russia as well as parts of Poland and other east European lands. Under his successors, the Mongol Empire would divide into four cooperating governments; Kublai Khan, the grandson of Genghis, would attempt to add Japan to his Japanese holdings; his failure, largely due to adverse weather, is as important to Japanese history as the Persian invasion is to Greek history and the Spanish Armada to British history. The greatest impact upon medieval Europe from the Mongols was indirect; controlling much of the Silk Roads network, they facilitated the import of Asian products into Europe, enriching the economy and creating a greater demand for Asian products in Europe.

Commodities traveled along the Silk Roads. So did ideas. So did disease. Bubonic plague had been known in the Mediterranean world long before the time of the Mongol Empire, but a new virulent strain of the disease traveled along the Silk Roads west into Europe and east into coastal China, leading to outbreaks of sickness and death commonly called the Black Death. This plague killed at least a quarter and perhaps more than a third of the population of Europe in the fourteenth and fifteen centuries. Periodic outbreaks of the plague continued in later centuries. No one was immune—rich or poor, noble class or peasant, church worker or casual worshiper or secret unbeliever. Some members of the European communities turned to the Christian faith hoping for supernatural protection from the disease; others rejected religion and followed the motto, “Eat, drink, and be merry, for tomorrow we will die.” The population decline across Europe added value to the remaining lives, especially in the working class. Peasants demanded more from the noble and the wealthy in exchange for their work; when the noble and wealthy refused, wars of rebellion broke out on occasion. This class warfare set the stage for greater change in Europe during the coming centuries.

Another important event of the Late Middle Ages was the death of Charles IV of France, last of the Capetian line of kings. He had no sons or brothers to inherit the throne; his nearest male relative was Edward, the nephew of Charles, who was King of England. French officials refused to acknowledge Edward as King of France; instead they crowned a cousin of King Charles, beginning the line of France’s Valois kings. Edward did not take this insult sitting down. He brought the English army into France, seeking to claim the throne that he considered his. Instead, he began the Hundred Years War between England and France.

The Hundred Years War actually lasted 116 years, but those years included two lengthy peace treaties between the French and English governments. English fighters had superior training and weaponry with their longbows, but they were unable to defeat the French in any conclusive manner. Instead, in the last years of the war, the French forces were rallied by a teenage girl named Jeanne Darc (Joan of Ark in English), who heard voices that told her what the English were planning and how they could be defeated. Eventually Jeanne was captured in battle, tried for witchcraft, condemned, and executed. But the Hundred Years War ended with the French government taking control even of lands that had belonged to the English crown, while the English government disintegrated into a civil war known as the War of the Roses.

Other unpleasantness at the same time as all these events was the highly unexpected Spanish Inquisition. Several governments in Europe had inquisitions—judicial tribunals of the Church that identified heretics, traitors, and other undesirable members of the citizenry and handed them over to the civil government for punishment. The atrocities of the Inquisition have been exaggerated by many writers, but the work of the Inquisition was far from modern judicial systems that respect the rights of the accused and grant them a hearing before a jury of their peers.

In spite of the Black Death, the Hundred Years War, and the Inquisition, European culture continued its progress during these years. Philosophers such as Roger Bacon, Nicholas of Cusa, and William of Ockham (famous for his principle of Ockham’s Razor) helped to invent the scientific method of observation, prediction, and experimentation. Great literature was being written by Dante and Petrarch, by Chaucer, and by numerous poets who built the romantic legends of King Arthur and his knights. Meanwhile, a mystic tradition of Christian devotion was growing, a tradition that helped to prepare the Church for its Reformation and for its existence and growth in the Early Modern world. J.