Mathematical Metaphors

Theories in all areas of science tell us something about the world. They are images, or models, or representations of reality. Theories tell stories about the world and are often associated with stories about their discovery. Like the story (probably apocryphal) that Newton invented the theory of gravity after an apple fell on his head. Or the story (probably true) that Kekule discovered the cyclical structure of benzene after day-dreaming of a snake seizing its tail. Theories are metaphors that explain reality.

A theory is scientific if it is precise, quantitative, and amenable to being tested. A scientific theory is mathematical. Scientific theories are mathematical metaphors.

A metaphor uses a word or phrase to define or extend or focus the meaning of another word or phrase. For example, "The river of time" is a metaphor. We all know that rivers flow inevitably from high to low ground. The metaphor focuses the concept of time on its inevitable uni-directionality. Metaphors make sense because we understand what they mean. We all know that rivers are wet, but we understand that the metaphor does not mean to imply that time drips, because we understand the words and their context. But on the other hand, a metaphor - in the hands of a creative and imaginative person - might mean something unexpected, and we need to think carefully about what the metaphor does, or might, mean. Mathematical metaphors - scientific models - also focus attention in one direction rather than another, which gives them explanatory and predictive power. Mathematical metaphors can also be interpreted in different and surprising ways.

Some mathematical models are very accurate metaphors. For instance, when Galileo dropped a heavy object from the leaning tower of Pisa, the distance it fell increased in proportion to the square of the elapsed time. Mathematical equations sometimes represent reality quite accurately, but we understand the representation only when the meanings of the mathematical terms are given in words. The meaning of the equation tells us what aspect of reality the model focuses on. Many things happened when Galileo released the object - it rotated, air swirled, friction developed - while the equation focuses on one particular aspect: distance versus time. Likewise, the quadratic equation that relates distance to time can also be used to relate energy to the speed of light, or to relate population growth rate to population size. In Galileo's case the metaphor relates to freely falling objects.

Other models are only approximations. For example, a particular theory describes the build up of mechanical stress around a crack, causing damage in the material. While cracks often have rough or ragged shapes, this important and useful theory assumes the crack is smooth and elliptical. This mathematical metaphor is useful because it focuses the analysis on the radius of curvature of the crack that is critical in determining the concentration of stress.

Not all scientific models are approximations. Some models measure something. For example, in statistical mechanics, the temperature of a material is proportional to the average kinetic energy of the molecules in the material. The temperature, in degrees centigrade, is a global measure of random molecular motion. In economics, the gross domestic product is a measure of the degree of economic activity in the country.

Other models are not approximations or measures of anything, but rather graphical portrayals of a relationship. Consider, for example, the competition among three restaurants: Joe's Easy Diner, McDonald's, and Maxim's de Paris. All three restaurants compete with each other: if you're hungry, you've got to choose. Joe's and McDonald's are close competitors because they both specialize in hamburgers but also have other dishes. They both compete with Maxim's, a really swank and expensive boutique restaurant, but the competition is more remote. To model the competition we might draw a line representing "competition", with each restaurant as a dot on the line. Joe's and McDonald's are close together and far from Maxim's. This line is a mathematical metaphor, representing the proximity (and hence strength) of competition between the three restaurants. The distances between the dots are precise, but what the metaphor means, in terms of the real-world competition between Joe, McDonald, and Maxim, is not so clear. Why a line rather than a plane to refine the "axes" of competition (price and location for instance)? Or maybe a hill to reflect difficulty of access (Joe's is at one location in South Africa, Maxim's has restaurants in Paris, Peking, Tokyo and Shanghai, and McDonald's is just about everywhere). A metaphor emphasizes some aspects while ignoring others. Different mathematical metaphors of the same phenomenon can support very different interpretations or insights.

The scientist who constructs a mathematical metaphor - a model or theory - chooses to focus on some aspects of the phenomenon rather than others, and chooses to represent those aspects with one image rather than another. Scientific theories are fascinating and extraordinarily useful, but they are, after all, only metaphors.

MOOCs and the Unknown

MOOCs - Massive Open Online Courses - have fed hundreds of thousands of knowledge-hungry people around the globe. Stanford University's MOOCs program has taught open online courses to tens of thousands students per course, and has 2.5 million enrollees from nearly every country in the world. The students hear a lecturer, and also interact with each other in digital social networks that facilitate their mastery of the material and their integration into global communities of the knowledgable. The internet, and its MOOC realizations, extend the democratization of knowledge to a scale unimagined by early pioneers of workers' study groups or public universities. MOOCs open the market of ideas and knowledge to everyone, from the preacher of esoteric spirituality to the teacher of esoteric computer languages. It's all there, all you need is a browser.

The internet is a facilitating technology, like the invention of writing or the printing press, and its impacts may be as revolutionary. MOOCs are here to stay, like the sun to govern by day and the moon by night, and we can see that it is good. But it also has limitations, and these we must begin to understand.

Education depends on the creation and transfer of knowledge. Insight, invention, and discovery underlay the creation of knowledge, and they must precede the transfer of knowledge. MOOCs enable learners to sit at the feet of the world's greatest creators of knowledge.

But the distinction between creation and transfer of knowledge is necessarily blurred in the process of education itself. Deep and meaningful education is the creation of knowledge in the mind of the learner. Education is not the transfer of digital bits between electronic storage devices. Education is the creation or discovery by the learner of thoughts that previously did not exist in his mind. One can transfer facts per se, but if this is done without creative insight by the learner it is no more than Huck Finn's learning "the multiplication table up to six times seven is thirty-five".

Invention, discovery and creation occur in the realm of the unknown; we cannot know what will be created until it appears. Two central unknowns dominate the process of education, one in the teacher's mind and one in the student's.

The teacher cannot know what questions the student will ask. Past experience is a guide, but the universe of possible questions is unbounded, and the better the student, the more unpredictable the questions. The teacher should respond to these questions because they are the fruitful meristem of the student's growing understanding. The student's questions are the teacher's guide into the student's mind. Without them the teacher can only guess how to reach the learner. The most effective teacher will personalize his interaction with the learner by responding to the student's questions.

The student cannot know the substance of what the teacher will teach; that's precisely why the student has come to the teacher. In extreme cases - of really deep and mind-altering learning - the student will not even understand the teacher's words until they are repeated again and again in new and different ways. The meanings of words come from context. A word means one thing and not another because we use that word in this way and not that. The student gropes to find out how the teacher uses words, concepts and tools of thought. The most effective learning occurs when the student can connect the new meanings to his existing mental contexts. The student cannot always know what contexts will be evoked by his learning.

As an interim summary, learning can take place only if there is a gap of knowledge between teacher and student. This knowledge gap induces uncertainties on both sides. Effective teaching and learning occur by personalized interaction to dispel these uncertainties, to fill the gap, and to complete the transfer of knowledge.

We can now appreciate the most serious pedagogic limitation of MOOCs as a tool for education. Mass education is democratic, and MOOCs are far more democratic than any previous mode. This democracy creates a basic tension. The more democratic a mode of communication, the less personalized it is because of its massiveness. The less personalized a communication, the less effective it is pedagogically. The gap of the unknown that separates teacher and learner is greatest in massively democratic education.

Socrates inveighed against the writing of books. They are too impersonal and immutable. They offer too little room for Socratic mid-wifery of wisdom, in which knowledge comes from dialog. Socrates wanted to touch his students' souls, and because each soul is unique, no book can bridge the gap. Books can at best jog the memory of learners who have already been enlightened. Socrates would probably not have liked MOOCs either, and for similar reasons.

Nonetheless, Socrates might have preferred MOOCs over books because the mode of communication is different. Books approach the learner through writing, and induce him to write in response. In contrast, MOOCs approach the learner through speech, and induce him to speak in response. Speech, for Socrates, is personal and interactive; speech is the road to the soul. Spoken bilateral interaction cannot occur between a teacher and 20 thousand online learners spread over time and space. That format is the ultimate insult to Socratic learning. On the other hand, the networking that can accompany a MOOC may possibly facilitate the internalization of the teacher's message even more effectively than a one-on-one tutorial. Fast and multi-personal, online chats and other networking can help the learners to rapidly find their own mental contexts for assimilating and modifying the teacher's message.

Many people have complained that the internet undermines the permanence of the written word. No document is final if it's on the web. Socrates might have approved, and this might be the greatest strength of the MOOC: no course ever ends and no lecture is really final. If MOOCs really are democratic then they cannot be controlled. The discovery of knowledge, like the stars in their orbits, is forever on-going, with occasional supernovas that brighten the heavens. The creation of knowledge will never end because the unknown is limitless. If MOOCs facilitate this creation, then they are good. 

Habit: A Response to the Unknown

David Hume explained that we believe by habit that logs will burn, stones will fall, and endless other past patterns will recur. No experiment can prove the future recurrence of past events. An experiment belongs to the future only until it is implemented; once completed, it becomes part of the past. In order for past experiments to prove something about the future, we must assume that the past will recur in the future. That's as circular as it gets.

But without the habit of believing that past patterns will recur, we would be incapacitated and ineffectual (and probably reduced to moping and sobbing). Who would dare climb stairs or fly planes or eat bread and drink wine, without the belief that, like in the past, the stairs will bear our weight, the wings will carry us aloft, and the bread and wine will nourish our body and soul. Without such habits we would become a jittering jelly of indecision in the face of the unknown.

But you can't just pull a habit out of a hat. We spend great effort instilling good habits in our children: to brush their teeth, tell the truth, and not pick on their little sister even if she deserves it.

As we get older, and I mean really older, we begin to worry that our habits become frozen, stodgy, closed-minded and constraining. Younger folks smile at our rigid ways, and try to loosen us up to the new wonders of the world: technological, culinary or musical. Changing your habits, or staying young when you aren't, isn't always easy. Without habits we're lost in an unknowable world.

And yet, openness to new ideas, tastes, sounds and other experiences of many sorts can itself be a habit, and perhaps a good one. It is the habit of testing the unknown, of acknowledging the great gap between what we do know and what we can know. That gap is an invitation to growth and awe, as well as to fear and danger.

The habit of openness to change is not a contradiction. It is simply a recognition that habits are a response to the unknown. Not everything changes all the time (or so we're in the habit of thinking), and some things are new under the sun (as newspapers and Nobel prize committees periodically remind us).

Habits, including the habit of open-mindedness, are a good thing precisely because we can never know for sure how good or bad they really are.

Why We Need Libraries, Or, Memory and Knowledge

"Writing is thinking in slow motion. We see what at normal speeds escapes us, can rerun the reel at will to look for errors, erase, interpolate, and rethink. Most thoughts are a light rain, fall upon the ground, and dry up. Occasionally they become a stream that runs a short distance before it disappears. Writing stands an incomparably better chance of getting somewhere.

"... What is written can be given endlessly and yet retained, read by thousands even while it is being rewritten, kept as it was and revised at the same time. Writing is magic." 

Walter Kaufmann

We are able to know things because they happen again and again. We know about the sun because it glares down on us day after day. Scientists learn the laws of nature, and build confidence in their knowledge, by testing their theories over and over and getting the same results each time. We would be unable to learn the patterns and ways of our world if nothing were repeatable.

But without memory, we could learn nothing even if the world were tediously repetitive. Even though the sun rises daily in the east, we could not know this if we couldn't remember it.

The world has stable patterns, and we are able to discover these patterns because we remember. Knowledge requires more than memory, but memory is an essential element.

The invention of writing was a great boon to knowledge because writing is collective memory. For instance, the Peloponnesian wars are known to us through Thucydides' writings. People understand themselves and their societies in part through knowing their history. History, as distinct from pre-history, depends on the written word. For example, each year at the Passover holiday, Jewish families through the ages have read the story of the Israelite exodus from Egypt. We are enjoined to see ourselves as though we were there, fleeing Egypt and trudging through the desert. Memory, recorded for all time, creates individual and collective awareness, and motivates aspirations and actions.

Without writing, much collective memory would be lost, just as books themselves are sometimes lost. We know, for instance, that Euclid wrote a book called Porisms, but the book is lost and we know next to nothing about its message. Memory, and knowledge, have been lost.

Memory can be uncertain. We've all experienced that on the personal level. Collective memory can also be uncertain. We're sometimes uncertain of the meaning of rare ancient words, such as lilit in Isaiah (34:14) or gvina in Job (10:10). Written traditions, while containing an element of truth, may be of uncertain meaning or veracity. For instance, we know a good deal, both from the Bible and from archeological findings, about Hezekiah who ruled the kingdom of Judea in the late 8th century BCE. About David, three centuries earlier, we can be much less certain. Biblical stories are told in great detail but corroboration is hard to obtain.

Memory can be deliberately corrupted. Records of history can be embellished or prettified, as when a king commissions the chronicling of his achievements. Ancient monuments glorifying imperial conquests are invaluable sources of knowledge of past ages, but they are unreliable and must be interpreted cautiously. Records of purported events that never occurred can be maliciously fabricated. For instance, The Protocols of the Elders of Zion is pure invention, though that book has been re-published voluminously throughout the world and continues to be taken seriously by many people. Memory is alive and very real, even if it is memory of things that never happened.

Libraries are the physical medium of human collective memory, and an essential element in maintaining and enlarging our knowledge. There are many types of libraries. The family library may have a few hundred books, while the library of Congress has 1,349 km of bookshelves and holds about 147 million items. Libraries can hold paper books or digital electronic documents. Paper can perish in fire as happened to the Alexandrian library, while digital media can be erased, or become damaged and unreadable. Libraries, like memory itself, are fragile and need care.

Why do we need libraries? Being human means, among other things, the capacity for knowledge, and the ability to appreciate and benefit from it. The written record is a public good, like the fresh air. I can read Confucius or Isaiah centuries after they lived, and my reading does not consume them. Our collective memory is part of each individual, and preserving that memory preserves a part of each of us. Without memory, we are without knowledge. Without knowledge, we are only another animal.

Alone

[S]ince there is an infinity of possible worlds, there is also an infinity of possible laws, some proper to one world, others proper to another, and each possible individual of a world includes the laws of its world in its notion. Gottfried Wilhelm Leibniz

On simple matters we can agree. Water freezes and wood burns. People can agree on social or political issues, though often more from self interest than from reasoned argument.

Agreement is rare or flitting on what is good or bad, worthy or worthless, humane or heartless. Are we simply not wise or intelligent or patient or convincing enough to find consensus?

Agreement is rare because the realm of possibilities is boundless. Every thought or vision carries a cosmos of variations and extensions. A good idea is one that spawns new good ideas, on and on. We are told that God the creator created man and woman in his image: as creators, to be fruitful and to multiply children, and ideas, and worlds.

At first we think that we are the entire world. Then we discover other worlds - things and people - and we think that they are the same as us. Then we discover that they have minds that, like ours, create their own worlds. We learn to communicate with those minds out there. We think that our meanings are their meanings, and this is true for many things, and even for many thoughts. But not for all of them. Then we discover that our deepest feelings are ours alone, and that we have created a continent whose shores are only lapped by waves from distant lands. 

I am a Believer

There are many things that I don't know. About the past: how my great-great-grandfather supported his family, how Charlemagne consolidated his imperial power, or how Rabbi Akiva became a scholar. About the future: whether I'll get that contract, how much the climate will change in the next 100 years, or when the next war will erupt. About why things are as they are: why stones fall and water freezes, or why people love or hate or don't give a damn, or why we are, period.

We reflect about questions like these, trying to answer them and to learn from them. For instance, we are interested in the relations between Charlemagne and his co-ruling brother Carloman. This can tell us about brothers, about emperors, and about power. We are interested in Akiva because he purportedly started studying at the age of 40, which tells us something about the indomitable human spirit.

We sometimes get to the bottom of things and understand the whys and ways of our world. We see patterns and discover laws of nature, or at least we tell stories of how things happen. Stones fall because it's their nature to seek the center of the world (Aristotle), or due to gravitational attraction (Newton), or because of mass-induced space warp (Einstein). Human history has its patterns, driven by the will to power of heroic leaders, or by the unfolding of truth and justice, or by God's hand in history.

We also think about thinking itself, as suggested by Rodin's Thinker. What is thinking (or what do we think it is)? Is thinking a physical process, like electrons whirling in our brain? Or does thinking involve something transcendental; maybe the soul whirling in the spheres? Each age has its answers.

We sometimes get stuck, and can't figure things out or get to the bottom of things. Sometimes we even realize that there is no "bottom", that each answer brings its own questions. As John Wheeler said, "We live on an island of knowledge surrounded by a sea of ignorance. As our island of knowledge grows, so does the shore of our ignorance."

Sometimes we get stuck in an even subtler way that is very puzzling, and even disturbing. Any rational chain of thought must have a starting point. Any rational justification of that starting point must have its own starting point. In other words, any attempt to rationally justify rational thought can never be completed. Rational thought cannot justify itself, which is almost the same as saying that rational thought is not justified. Any specific rational argument - Einstein's cosmology or Piaget's psychology - is justified based on its premises (and evidence, and many other things). But Rational Thought, as a method, as a way of life and a core of civilization, cannot ultimately and unequivocally justify itself.

I believe that experience reflects reality, and that thought organizes experience to reveal the patterns of reality. The truth of this belief is, I believe, self evident and unavoidable. Just look around you. Flowers bloom anew each year. Planets swoop around with great regularity. We have learned enough about the world to change it, to control it, to benefit from it, even to greatly endanger our small planetary corner of it. I believe that rational thought is justified, but that's a belief, not a rational argument.

Rational thought, in its many different forms, is not only justified; it is unavoidable. We can't resist it. Moses saw the flaming bush and was both frightened and curious because it was not consumed (Exodus 3:1-3). He was drawn towards it despite his fear. The Unknown draws us irresistibly on an endless search for order and understanding. The Unknown drives us to search for knowledge, and the search is not fruitless. This I believe. 

We're Just Getting Started: A Glimpse at the History of Uncertainty

We've had our cerebral cortex for several tens of thousands of years. We've lived in more or less sedentary settlements and produced excess food for 7 or 8 thousand years. We've written down our thoughts for roughly 5 thousand years. And Science? The ancient Greeks had some, but science and its systematic application are overwhelmingly a European invention of the past 500 years. We can be proud of our accomplishments (quantum theory, polio vaccine, powered machines), and we should worry about our destructive capabilities (atomic, biological and chemical weapons). But it is quite plausible, as Koestler suggests, that we've only just begun to discover our cerebral capabilities. It is more than just plausible that the mysteries of the universe are still largely hidden from us. As evidence, consider the fact that the main theories of physics - general relativity, quantum mechanics, statistical mechanics, thermodynamics - are still not unified. And it goes without say that the consilient unity of science is still far from us.

What holds for science in general, holds also for the study of uncertainty. The ancient Greeks invented the axiomatic method and used it in the study of mathematics. Some medieval thinkers explored the mathematics of uncertainty, but it wasn't until around 1600 that serious thought was directed to the systematic study of uncertainty, and statistics as a separate and mature discipline emerged only in the 19th century. The 20th century saw a florescence of uncertainty models. Lukaczewicz discovered 3-valued logic in 1917, and in 1965 Zadeh introduced his work on fuzzy logic. In between, Wald formulated a modern version of min-max in 1945. A plethora of other theories, including P-boxes, lower previsions, Dempster-Shafer theory, generalized information theory and info-gap theory all suggest that the study of uncertainty will continue to grow and diversify.

In short, we have learned many facts and begun to understand our world and its uncertainties, but the disputes and open questions are still rampant and the yet-unformulated questions are endless. This means that innovations, discoveries, inventions, surprises, errors, and misunderstandings are to be expected in the study or management of uncertainty. We are just getting started.