How to think about a theory of knowledge?

I believe that what you now have in your hands is a theory of knowledge. It starts with the most fundamental idea; that knowledge is constructed of artifacts. People construct and choose artifacts based on uniqueness. We pay special attention to the important artifacts, assuring that they are generally unique; for we believe that unique artifacts, both physical and conceptual, are precious. Uniqueness gives us a flexible way of differentiating, handling, and organizing the vast realms of experience that we face; a way to choose what we must focus on and to construct artifacts and patterns of artifacts for holding our experience.

There are only three different ways that an artifact can be unique; by being different and thus distinct from other artifacts, by being the same across other artifacts, or by matching and mating other artifacts. These three forms of uniqueness define the three fundamental tools and, therefore, the three elements of thought and knowledge. We produce entities our names and definitions - by differentiating. We fashion sites - our categories and classes, our concepts and generalizations - by collecting entities based on their sameness. And finally, we make fasteners - our theories and explanations - to link different sites by matching.

We constructed this theory and then connected these unique artifacts to the Pattern of Knowledge to see whether these abstractions, which we derived from uniqueness, fit our legacy of intellectual history by explaining its periods, phases, and sequence. The fit seems exact and strongly suggests that all of the knowledge we create is explained by this theory. This is what a theory of knowledge looks like: a simple, purely logical argument which provides a template for constructing a pattern that then holds and connects experience in epistemology, intellectual history, natural language, and mathematics.

A Theory in Physics: Electricity and Magnetism

A short stroll into a very special part of physics

Since a theory of knowledge is unfamiliar, it may be instructive to look at what a great theory in the discipline of physics is like; and Maxwell's theory of the Electromagnetic Field is the most beautiful example I know. Forgive me for again choosing an example in physics, but this discipline does give us our purest and most well thought out theories. James Clerk Maxwell, by 1860, had already established his reputation as a first-rate physicist working on a wide variety of problems when he turned his attention to the work of Faraday and the difficulties in electricity and magnetism. It was this work that marked him as the greatest physicist of the 19th century, and while he is less well known, he is on a par with Newton and Einstein. Unlike mechanics - the study of the motions of ordinary objects - which was well understood and elegantly theorized by Newton and his followers, electromagnetism was then in a chaotic state.

First a little background

There were a variety of different and often complicated laws dealing with electricity and magnetism, which generally melted down to three key ones. The electric force between two static (not moving) charges was the easy one - Auguste Coulomb defined it in 1775, exactly mimicking Newton's Law of Universal Gravitation.

Coulomb replaced the masses (m) by the charges (q) and the constant of Gravitation (G) by a constant for electricity (k). The force of electricity, like the force of gravity, varied as the square of the distance, indicating that it spread out in straight lines. Coulomb carefully constructed an electric balance to assure that this law fit the experimental data. Other than being either attractive or repulsive, this force was very Newtonian. 

Magnetic forces proved to be much more complicated. Magnets always come with two poles locked together. The force between magnets does not spread out in straight lines. And to make matters worse, by Maxwell's time, the magnetic force was known to be produced by electricity. What was clear was that a new force, the magnetic force, was generated when an electric object moved - a very un-Newtonian notion. Andre Marie Ampere, in 1818, formulated a law describing the force between two parallel wires carrying electricity. 

It was a mutation of Newtonian laws; the force varied as the distance (d) between the wires and not the square of the distance. And while the force was proportional to the currents (I) in both wires, it was also a function of the length of the wire (l). Things were getting messy. 

Michael Faraday, a wonderful teacher, perhaps the greatest experimentalist of the 19th century, and the inventor of the dynamo, which produces nearly all of our electricity, found that moving a magnet produced an electric current. His Law of Induction was still generally Newtonian because it was still based on forces between objects, but these forces no longer followed lines that were straight (witness iron filings over a magnet) and Newtonian forces were straight lines. He called these curves "lines of force," and their whole, the "magnetic flux." If it flowed through a loop of wire and it produced a current in that loop. Faraday's Law of Induction related the electric force (EMF) that produced a current in a wire to the rate of change of this flux. 

More than 75 years of attempts by world class physicists had produced these and other laws of electricity and magnetism under the Newtonian umbrella. Each was descriptive, based on a familiar pattern and not on a fundamental element. Each was very different and generally unrelated to the others in its form, in the way it looked, and in the way it worked. The result was complex and ugly.

Maxwell opened his great paper "A Dynamical Theory of the Electromagnetic Field published in the fall of 1864:

The mechanical difficulties, however, which are involved in the assumption of particles acting at a distance with forces which depend on their velocities are such as to prevent me from considering this theory as an ultimate one...

I have therefore preferred to seek an explanation of the fact in another direction, by supposing them to be produced by actions which go on in the surrounding medium as well as in the excited bodies, and endeavouring to explain the action between distant bodies without assuming the existence of forces capable of acting directly at sensible distances.

Even without a background in mathematics or physics you will be able to appreciate the beauty and the simple of this work.

His theory, today written in the simplified notation of vector Calculus with just four equations, is so elegant and esteemed that we decal it on sweatshirts for college students and babies. While the symbols used may appear foreign, the fundamental ideas can be appreciated by all of us, just as we can feel the wonder of Beethoven's 9th Symphony even if we can't read a note or play an instrument.

The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric or magnetic bodies, and it may be called a Dynamical Theory, because it assumes that in that space there is matter in motion, by which the observed electromagnetic phenomena are produced.

A field is an environment. We can describe it with vectors.

The electromagnetic field is an environment - a continuum - spreading out in space. It is a "substance." Maxwell believed that this "ethereal medium" was real. While there are a variety of ways to imagine such an invisible medium that acts on only electric and magnetic objects, if you think of this environment as an atmosphere with winds which blow only electric and magnetic particles, then you will have a good metaphor. This wind, this field, has a strength and a direction at every place in this continuum. The field can be measured by placing an electric or magnetic body at a point and plotting its direction and its magnitude. Imagine filling the field with electric wind-vanes and magnetic compass needles, each pointing in the direction of the field and varying in length to show the field's strength. Such arrows are vectors, mathematical representations for both the magnitude and direction of the fields. 

In a simple case, a single charged particle produces an electric field radiating spherically out from it, the electric field vectors all pointing straight from the center like a pin cushion. This field produces a force on a charged particle along that radiating vector. Even in this static case, the field proved a powerful idea allowing Maxwell to no longer think of forces "acting at a distance" between the charges, but instead, of forces as the result of a field generated by a charge acting on another charge. Forces no longer needed to reach across space at infinite speed. But it was in the dynamical situations with forces being produced by motions that the power of this idea became apparent. The motions of electric charges produce a magnetic field and the motions of magnets produce electric fields.

The Calculus is the mathematics of change, describing how fast something changes or how much change has occurred.

The search for explanation and not just for description drives physics. The explanation of change is the heart of the matter. Whether it be motion or fields, the uniform is the starting point, the natural condition. The changing field, like the changing motion is where the action is. We explain it by connecting a cause to the rate at which the change occurs. For Newton force was connected to accelerations, for Maxwell, the cause was connected to a changing field. The rate of change is the realm of the Calculus, invented by Newton and Leibniz, and initially developed in one dimension along the path of the motion. But the field is three-dimensional and fortunately by the 19th century the Calculus was extended to rates of change in three dimensions using double and triple integrals and partial differentials. You may have seen the "curly d" , perhaps on one of those Maxwell sweatshirts, that represents the "partial" derivative, which is the rate of change of a function with multiple dimensions in just one of those dimensions. Thus a vector with three dimensions has three partial derivatives, one in each direction.

Now the last piece of the logical pattern - there are two different forms of change of the vector field - Divergence and Curl.

Vector Calculus, developed soon after Maxwell's great work, greatly simplified the model and the notation, enabling us to combine his original 20 equations into just four. A single symbol  pronounced "del" represents the sum of all of these partial derivatives in all three spatial dimensions. Finding the rate of change of a field is just a matter of multiplying  by the field. But vector multiplication unlike normal multiplication, generates two different kinds of products not one - the "dot" product () and the "cross product" (). The dot product of  and a vector field is a scalar, a number. It is so important in physics that it has its own name, the "divergence," because it represents the change in a field generated by a point source. Like the pincushion, this field diverges radially and never changing in direction only in magnitude (weakening with distance as the sphere through which it passes gets larger).

The cross product is a vector that we call the "curl." Like its name implies, this field is always changing direction, turning or curling around its source. Returning to our wind metaphor, if we blow from our mouths, we create a source and that wind goes straight out, diverging and weakening as it gets further from our face. Wind in nature, however, is always curling, rotating clockwise around highs and counterclockwise around lows. We see it in dust devils, tornadoes and on a larger scale in the satellite images of cloud formations and of hurricanes.

And 2 kinds of fields:
Electric
Magnetic

There are two kinds of fields in electromagnetism, the Electric (E), and the Magnetic (B), (M is an already overused symbol in physics). Here, then, are the logically defined left sides of Maxwell's equations; the divergence and curl of the electric and magnetic fields. There are four, and only four, possible forms. If the field is fundamental then we should be able to connect this complete structure to the experience of electricity and magnetism - to the descriptive laws of Coulomb, Ampere, and Faraday, rewritten in terms of fields rather than forces. That is what Maxwell did!

1. The first equation connects a diverging electric field radiating in space with an electric charge. It restates Coulomb's Law.

2. The second defines a diverging magnetic field, which would be the result of a magnetic "charge." But none has ever been found and thus the magnetic divergence is set equal to zero.

3. The third links a circulating magnetic field, curling through space, with an electric field changing with time (an electric current). It expands Ampere's Law.

4. And the fourth ties a circulating electric field which would produce a current in a loop of wire, with a magnetic field changing with time (a moving magnet). It is Faraday's Law.

The aim of science is, on the one hand, a comprehension as complete as possible, of the connection between sense experiences in their totality, and, on the other hand, the accomplishment of this aim by the use of a minimum of primary concepts and relations.
Albert Einstein

A logical construction -- a fastening artifact defines and connects sites -- a framework on which sites and entities of experience are linked.

This is what a theory looks like. It is a logical system (the left side of Maxwell's Equations) - defined in this case by the unique mathematics of a new element, the environment; which creates and connects a set of logical sites. When these fastening artifact sites are linked to the empirical sites - the patterns of experience, then the theory encompasses and connects our experience. When that happens, the empirical sites become connected, fastened into a new unity, which now brings uniqueness to our knowledge liberating us from the arbitrariness of the experiential names for the sites and the pretense of their links. It is a very powerful thing. We suddenly have a logical understanding, a model for our experience drawn together by simplicity and the power of human thought. It is a fastening artifact, the electromagnetic field, which has four dynamic forms that now join together the experiential patterns. Each form of the fastening artifact is connected to an empirical site (a well-defined collection of experience as described in each law). And the fastening artifact, in its full glory, now links all of these descriptive sites together into a unified picture.

The theory of knowledge has the same form.

The theory of knowledge fits this formula. It is constructed with a new element - the artifact - and builds a logical structure for sites based on the forms of uniqueness and thus the uniqueness of artifacts. We then linked those basic forms to the descriptive Pattern of Knowledge - the empirical sites. And by that action, the unique artifact brings a fundamental and beautiful unity to this pattern. No longer are the sites or their names arbitrary. No longer do we wonder if this pattern is unique. No longer do we wonder why it exists. For we have a theory, a theory that connects the sites we pulled from experience by linking them to a unique purely logical form.

There are three and only three forms of uniqueness: difference, sameness, matching. With these three forms we construct the three elements of knowledge and only three: entities based on difference, sites based on sameness, and fasteners based on matching. We can further differentiate these elements based on uniqueness: the entities into singular and plural (difference and sameness), the sites into parts and wholes (difference and sameness), and the fasteners into connections, relations, and transformations (difference, matching, sameness. We build knowledge by starting with a unique entity, a template and tool fundamentally different from any other: a symbol, a universal, an object, an environment, and now an artifact. We fashion with it our individual entities, from which we choose a few unique ones to become sites that group and collect the others, and finally, from a unique site we build a fastener that connects the other sites together. This is how we construct knowledge. And when we lay out all of the possible forms, they build the structure of the Pattern of Knowledge

I hope that you now find that all of those mysterious symbols in Maxwell's Equations were worth following. For without seeing them, it is difficult to really understand how simple and yet powerful a great theory can be, and how we build them. I apologize to those physicists who may complain that I have left out some of the fine detail. There is a bit more pattern that I have deleted in this short work. But the essence is here and I hope that you can see why Einstein loved it so; and why, though a full understanding and more importantly a useful application of a theory may be complicated, its basic elements are so very simple.

Without a theory the way we name and unify a pattern of knowledge is quite arbitrary. I had all of the elements of the Pattern of Knowledge by the mid-1970's, but I did not publish. The names I used for each of the forms were arbitrary; they came from the best description of the empirical content and did not represent any theoretical-logical meaning. I did not publish because I did not want the wrong names, the mislabeling of these ideas, and so I struggled with the theory to get the names right. When we create sites outside of theory, we get interesting names like those which label Quarks - color, flavor, up, down. Without theory we make up names and hope that they illuminate. Without theory we cannot change our vision of experience. Without theory we do not extend our ideas.

This started it all for me.

Before we start to follow our theory of knowledge into new and uncharted territory, let us linger for just a moment longer at the wonder of Maxwell. In my favorite passage in all of the literature of physics and the one that more than any other single thing enabled me to understand the great leap of Maxwell and the others in the 1860's, Einstein describes the genius of Maxwell's contribution.

Neglecting the important individual result which Maxwell's lifework produced in important departments of physics, and concentrating on the changes wrought by him in our conception of the nature of physical reality, we may say this: before Maxwell people conceived of physical reality -- in so far as it is supposed to represent events in nature -- as material points, whose changes consist exclusively of motions, which are subject to total differential equations. After Maxwell they conceived of physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. This change in the conception of physical reality is the most profound and fruitful one that has come to physics since Newton....
Einstein & Infeld, The Evolution of Physics, 1938, Norton

They connect aspects of experience that are totally unexpected.

Beyond the sense of completeness and clarity that Maxwell's unification brought to laws of electricity and magnetism now directly tied together, his theory also produced some exciting surprises and totally unexpected results. Not only did his theory join together Coulomb's, Ampere's, and Faraday's laws, but it made sense of and integrated many other electrical phenomena including capacitance. And in one great and totally unexpected result, a direct extrapolation of these equations - Maxwell joined light to electricity and magnetism and paved the way not only for an understanding of light, but also for the development of radio, all of our electronics, and our electromagnetic communications. Our theory of knowledge, if it is powerful, should extend to other parts of knowledge that were not included in the development of the theory and provide us with some wonderful surprises.

Unique Artifacts apply to children as well as to adults and explains the stages of Piaget.

Thanks to Jean Piaget, we know a lot about how children construct knowledge. His stages of development of knowledge are well documented in children across a wide variety of societies and cultures. The sensori-motor stage runs from birth to 2, at which point most children begin to speak in sentences and become pre-operational. At about 6 years of age, children become concrete operational and are able to apply and use standard operations on symbols and conventional classification. Lastly, between 12 and 14, children become formal operational and start to use logic, abstract metaphor, and formal reasoning. If we look at the new things children do at the onset of each of these stages with the inventions at the start of the periods of knowledge, the connection between the pattern of knowledge in children and in human history is plain.

Concrete operations uses plural symbols - empire knowledge meets 6 year-olds

At the onset of concrete operations, children's learning explodes suddenly. They "get" reading, going from words to sentences nearly overnight; they learn do mathematics, count to any numbers, add and subtract, tell time, follow calendars, understand sophisticated classification, play their own and adult games with complex rules, and work together in large organized groups on major projects. They even draw with standard methods, often that they invent, and they draw full-featured human faces and figures in profile or frontal poses. These knowledges match those of the empires. It is uncanny how close the resemblance is. The inventions are the same!

The inventions are the same because the tools are the same - both concrete operational children and "empire" adults use plural symbols to construct knowledge, symbols based on sameness. We can teach concrete operational children to read and write because they can use a few symbols, common elements, to represent all words. They invent well-defined representations in their drawings because these are visual underlying symbols that remain the same throughout a wide variety of pictures. They can do mathematics because they can use number and operation, categorical symbols, and see these same elements as common to anything that is countable. And they can play rule-based games because games are built on symbols, fastened by rules which are entirely independent of the players. A quarterback is a position and a type of player, and an touchdown is a well defined rule for the interaction of these players. We could, in fact, call concrete operations the game phase. Concrete Operations is the use of rules on symbols. To have rules, symbols must be constructed on sameness; they must represent a class or category. Thus the knowledge constructed or learned by 6 to 12-year-old children and by the empires appears, and is, fundamentally the same.

Formal operations is singular universals - Greek knowledge and 13 year-olds

Formal operational children search for truths; construct proofs, attempt to build logical systems, use variables, and start to argue formally and universally. Their entry into formal operations is accompanied by explosive growth. They change before our eyes both physically and mentally; suddenly making arguments and explanations that are adult-like abstractions; they enter into conversations about religion, society, evolution, politics, and all manner of philosophical subjects. They can make and follow a long, logical argument. They can learn to solve logical puzzles, and they can use variables. We can teach them to prove geometric theorems, to be critics of essays, and to understand abstract metaphors. Their inventions, their interests, and their reactions are very much like those of the early Polis Greeks. They are formal operational, they use universals instead of symbols, seeking truths and logical meaning.

The universal enables formalisms, proofs, logic, abstractions, theories, and, of course, true metaphor where an entire idea is given broader meaning, universality, by being associated with a new word. Like the Polis Greeks, the world of logic for formal operational children starts with singular universals, absolute truths, and individual visions of themselves and their world. They become very independent, creating their own rules of behavior which they consider logical. They form identities separate from the family, around local universals, local "guilds," joining gangs, and forming small but very strongly defined groups. And they love making new rules for society- formed to be just, ideal, and equitable. This is the singular universal as the unique artifact for constructing knowledge.

Pre-operations is singular symbols - tribal knowledge and 2 year-olds.

We do not find it at all odd that developmental psychologists call the pre-operational period "the magic years." Most children burst into language between two and two-and-a-half-years; jumping, nearly overnight, from using just a handful of individual words to speaking in sentences with rapidly expanding vocabularies. They tell stories, develop rituals, believe in magic, make up names, fashion fantasies, recognize traffic signs, become fascinated with familial relationship and kinship systems, and create elaborate magical formulas and mythical explanation for all sorts of things. They construct stories and explanations that are "magic" and ignore the constraints of the real world. They personify objects and natural forces, telling us "the thunder is angry." They confound fantasy and reality, cause and effect (the clouds make the wind). And they invent names out actions. Their magic orientation, their new capabilities, and their inventions- including their art - look just like those of the tribal societies. They are using symbols for the first time, and whether their symbols are borrowed or invented, they are recreating the world. While their language certainly has a mimicry component, must of their syntax is of their own invention. Their creations and actions and those of tribal peoples are very similar in form and in kind. They even like to dress-up, decorate, engage in socio-dramatic play, and tell action stories.

Knowledge building in children and adults follow the same pattern because both are based on the same tools - that is - the same theory applies to each.

This connection between the development of knowledge in children and the historical development of knowledge is not, definitely not, part of that weary ontogeny-phylogeny debate. The issue isn't one of recapitulation, but rather that the pattern to the development of knowledge in both children and in intellectual history is constructed with the same tools. Children and adults construct knowledges that are fundamentally the same, because the sequence of unique entities and entities, sites and fasteners that both use is based on uniqueness and cannot diverge. Thus, the patterns we find in their respective development of knowledge must be the same. The Unique Artifacts theory applies to the construction of knowledge by children just as it applies to the construction of knowledge by adults.

What is different, of course, is the nature of the knowledge that is constructed. Children replicate the adult construction of physical artifacts; they make pictures, build block buildings, dress up, cook pretend food, dig ditches in the sand, carve shapes in clay, and invent all kinds of things that mimic our adult creations. But they do not create the kinds of wonderful, complex, and profoundly beautiful artifacts that adults do. They may build knowledge artifacts out of the same kinds of tools, but they do not build the same qualities of knowledge. They do not have the patience, maturity, or skills to do so. It is no different for conceptual artifacts than it is for physical artifacts, the tools and the types are the same, but the results are different. What is clear is that forms of these artifacts, no matter how crude or fine, are totally dependent upon the tools that are being used, and these tools follow the same, the exact same, patterns of uniqueness. The unique artifacts from which humans construct knowledge are the elements of all knowledge.

Children seem to have natural mental maturations at Piaget's key ages, and if their society has enabling entities, they naturally jump to the next step. Indeed, if we were to analyze the development of knowledge in children more carefully, we would actually find the same pattern of phases we found in historical knowledge. These are not fully delineated in Piaget's work, but they are not difficult to articulate when we look at how and what we teach children at each grade level or at how they behave before formal schooling.

The Sensori-motor stage is pre-symbolic and does not have a comparable phase in the Pattern of Knowledge

I did leave out the definition of a phase that exists in children but does not have a counterpart in the Pattern of Knowledge. Sensori-motor is the stage from just after birth to about 2 years of age and the one Piaget initially studied. It is the stage before formal language, where words are things and drawings are scribbles. It must represent a different entity, one that I call signals. A symbol is a name for an experience, a signal is that experience: hunger, thirst, fear, joy. It may come from the external world to be dealt with or it may emanate from inside the organism as an expression. For most children up to 2 years-of-age, words are such signals, as is crying or laughing or making signs with their hands. Their art works are signals, like so many of their physical actions. This entity was undoubtedly used by our pre-symbolic ancestors, and likely underlies learned animal behavior in chimps, dogs, horses, and other species. While signals, particularly shared signals can be complex; they are limited in number because each represents a single action or sequence of actions. Trained primates seem to be limited to about 250 signals. Though a new entity, it follows the same sequence of phases as the other entities - phases that can, with minor variations, be connected to Piaget's stages of sensori-motor development.

Unique Artifacts applies to thought as well as to epistemology.

Psychology, established a little over 100 years ago as a separate discipline, rapidly grew beyond its initial study of the way individuals behaved in different physical environments. It quickly accrued cognitive development, abnormal behavior, and learning theory, which had been elements of either philosophy or medicine during its first two decades, and it has continued to add interesting parts of other disciplines, like linguistics from the humanities, or build new disciplines like cybernetics. Psychology has been the vibrant discipline of the 20th century. Today we learn about thinking in courses mixing behavioral psychology, the learning process, human development, the nature of language, and advanced programming and the visual display of information. We do not learn about thought in connection with knowledge. And while Piaget and his followers may describe themselves as epistemologists, they do not connect the study of thinking and intellectual history. Detached from philosophy, cognition severed its ties to knowledge.

And there it remains today, thought and knowledge in separate realms, considered to be completely different, studied in different disciplines, and virtually unrelated. Obviously this cannot be the case. Perhaps it is time to extract knowledge from philosophy and thought from psychology; to bring them together into a new discipline. While this short work is not the place to indulge in a thorough analysis of thought and its connection to knowledge; we should, before we conclude, look at a few familiar elements of cognition and their obvious connection with unique artifacts and thus with knowledge as a means to illuminate both. I am convinced that there should be no fundamental separation between thought and knowledge - that the way we construct artifacts and the artifacts we construct are fundamentally connected.

I know of no comparable pattern of thought in adults to Piaget's stages. The closest thing to even a list is Benjamin Bloom's Taxonomy of Cognitive Objectives developed by a committee of educators and scholars headed by Bloom and published in 1956. Bloom listed six stages of thought- knowledge, comprehension, application, analysis, synthesis, evaluation- a comprehensive hierarchy, as we would expect from a pattern developed in a wholistic relations phase. The last three of his objectives, Analysis, Synthesis, Evaluation, were drummed into our teacher heads as the most important aspect of what we were to do with kids, the higher order thinking skills! Much of the focus of today's educational curricular reform can be traced to this Taxonomy. And while, it is not at all clear that the Bloom Taxonomy is a complete description of the tools we use to think, it provide us with a much needed and well-established pattern that we can work with. We will start with the higher order skills, and return to look at the complete pattern which has, I find, an interesting structure.

Analysis

Analysis is the act of differentiating. Bloom describes it using verbs like - compare, contrast, differentiate, discriminate, examine, experiment. Analysis is a fundamental cognitive process because it is the search for difference. Whenever we take apart something, be it ideas or physical objects we are either creating or looking for difference. This is the act of cutting up experience. It is not difficult to see that our tool - analysis - is based on the uniqueness form - difference.

Twilight is a good time for us to see how the world looks to babies and to see how we come to analyze a section of experience for the first time. If there is just that right level of darkness after it has gotten just dark enough for our color cones to no longer work well and while there is just enough light to enable us to still discern shapes if we work at it. And if we are in a new place, perhaps driving down a highway or waking in a strange room in the middle of the night. And if we are suddenly brought to attend and look out at that world, we catch a glimpse of unrecognizable experience. We suddenly do not know what we are looking at. Everything seems strange.

We start to stare at things, to try to make out shapes. We look for edges; we search for differences to start building shapes. We cut out a form here, and then another there, looking for something we recognize. Once we have constructed a few of these entities by difference, we usually know where we are and the rest of the vaporous shapes fall into a pattern. But it is in that first few seconds that we can see our minds at work. We can see how the infant begins to shape their world. And we can see what we do when we analyze anything. We look for edges, for discontinuities, for differences, and we begin to construct artifacts from that experience by cutting it up. Thus analysis is the thinking process we use to create entities, the basic artifacts fashioned by difference.

Synthesis

Bloom describes synthesis with words like - arrange, assemble, categorize, organize, plan - the act of putting things together. To synthesize is to find sameness, to collect common artifacts, to build patterns, even taxonomies. It is the opposite of analysis just as sameness is the opposite of difference. Here are the two forms of uniqueness - difference and sameness- playing such a fundamental role in cognition.

When we synthesize knowledge, we are literally finding sameness. While in everyday language we may talk of synthesis as both the collection of things and the making of theories, we will, as Bloom did, connect it to the former. A synthesis is the making of a collection, the finding of a pattern. In our language it is the construction of sites. It is thus fashioning using sameness. When we synthesize, we make a new container for our experience, grouping disparate experiences into a single unified artifact. We do so by looking for what is the same in that collection.

Evaluation

At first glance, evaluation does not seem to fit our pattern. And perhaps there is a better word for what it is we do at this level of thought. But when we look again at the descriptive verbs that Bloom uses - attach, judge, evaluate, argue - words that we connect to fastening, to the linking of artifacts into a theory. When we create theory, we explain, we give reason, we join, we integrate. It is theory that allows us to evaluate. For in evaluation we are making a match and matching is the third form of uniqueness. It is the fastening artifact that connects our world together that enables us to join disparate experiences and the act of doing that is in essence an act of evaluation. That is why Bloom and his group have chosen this odd word, and that is why it is based on matching.

The higher order skills of Bloom's Taxonomy, the cognitive skills that most of us would agree our young must learn and master, are based on exactly the same forms of uniqueness, as are the fundamental artifacts of knowledge, entities, sites, and fasteners. Analysis, synthesis, and evaluation are based on difference, sameness, and matching. When we analyze we seek differences, when we synthesize we seek sameness, and when we evaluation we seek matches.

The whole pattern

It is not surprising that uniqueness plays this essential role in cognition. We would hope not to find a fundamental separation between knowledge and cognition, but we can now see that they are constructed on the same basic forms. As I said earlier, the full pattern of Bloom's Taxonomy, which includes knowledge, comprehension, and application, is fascinating. If we now go back and look at the first three stages - knowledge, comprehension, application - we notice that they look surprisingly similar to what we have already done. When we create knowledge (in Bloom's use of the term, actually factual knowledge), we define, label, name, recognize; in other words we differentiate. And when we comprehend, we classify, select, describe; we are collecting and finding sameness. And finally, when we apply, link, connect, use; we are matching disparate ideas. Knowledge, comprehension, and application, like analysis, synthesis, and evaluation are our use of difference, sameness, and matching.

The first group applies to singular experiences and the second to plural experiences. Thus the second group of stages appear to us to be much higher order thinking skills because they apply to much more complex ideas. With good reason we reserve teaching them to our middle school students and above. And with good reason the forward pedagogical thinkers emphasize them in a curriculum that has too often, simplistically, focused on the most minuscule parts in an attempt to make evaluation easy. But when we look at both sets from the standpoint of Unique Artifacts, we see them as the same, the tools to construct artifacts based on difference, sameness, and matching. These are clearly the essential tools of cognition. They are intimately connected to entities, sites, and fasteners. Knowledge and analysis construct entities. Comprehension and synthesis construct sites. Application and evaluation construct fasteners. They are the same because thought like knowledge is the fashioning of unique artifacts.

Simplicity

'Tis the gift to be simple, 'Tis the gift to be free;
'Tis the gift to come down where we ought to be;
And when we find ourselves in the place just right,
'Twill be in the valley of love and delight.
When true simplicity is gain'd,
To bow and to bend we shan't be asham'd
To turn, turn will be our delight,
'Til by turning, turning, we come round right.
Elder Joseph Brackett (1797-1882) Simple Gifts c.1875

The odd dichotomy

This wonderful old Shaker song seems so at odds with the real world, much as the Shaker's themselves were with the rest of 19thcentury America. The world is complex, broad, and multifaceted. We expect theories to be complicated and large. Our disciplines are driven by great organizations - numerous practitioners jointly sign new works, large scale funding is required to advance the state of the art, and we have the belief that significant new knowledge will come out of massive collaborations, “Manhattan Projects." And while we may yearn for simpler times, as did the Shakers, we are generally unshaken in our belief in the complexity of our world.

Yet there are a few bits and pieces which should cause us to pause. When we listen to those who worked with the great thinkers, Einstein, Bohr, Fermi, we are told of their most special qualities, the ability to ask simple questions. We love the child-like nature of Picasso, of Richard Feynman, of Linus Pauling. And when we describe people as child-like we are most often describing their essential simplicity. It does seem odd that often the deepest quality of great inventors should be described in these ways. Just maybe, that old Shaker prayer really does express something truly profound.

Simplicity - An auto example

We can get a better sense of this by looking at our most prominent physical artifacts, automobiles. The Model A Ford, introduced in 1928, was wonderfully simple, indeed it was much simpler and easier to use than its predecessor the Model T. Perhaps the first truly modern car - it was manufacturable, had all of the same components as today's cars, and was designed to be cheap and easy to assemble. When we try to restore one, we see all of the parts that are still fundamental to today's cars. Compared to our automobiles, its parts were much simpler, but it was actually more complex to construct. Despite the great strides we have made in the sophistication of our automobiles, we have actually made them simpler to put together. The roof of the Model A was fabric, covering metal cross bars, screwed, clipped, nailed, and hooked into the body. Compared to a modern car with a single stamped welded roof, it was very complex with many parts.

Starting my son's Model A requires turning on the gas valve, setting the spark advance, the choke, and the idle speed, pumping the gas petal, turning on the key, holding in the clutch, and pressing the starter button with your foot. Starting my Saab requires turning the key.

We no longer make a separate body and frame (chassis) of a car, but instead mold them together in giant presses of the same steel. We no longer make the floorboards of the car from multiple pieces of plywood carefully cut to shape, but of a single sheet of molded steel. The parts of a modern car are certainly much more sophisticated than those of the Model A, they are no longer machinable or fixable by a home mechanic, but the overall construction is actually simpler. The automobile companies make it easier and cheaper to put together.

The history of the automobile is a mirror of the history of knowledge. The first horseless carriages were very simple affairs, equivalent to a golf cart. As they became automobiles, they grew more and more complex with new parts added year after year, lights, brakes, doors, locks, transmissions, reverse, electric starters, fuel pumps, heaters, automatic transmissions, and on and on. At first, each of these new parts was just added on, effectively bolted onto the machine. Thus the trunk was actually a wooden "truck" that could be found in any home, strapped to the rear of the car. Then came an integration, when separate parts were collected into a new part or a new whole. The car looked different, worked differently, and was manufactured in a new way. The Model A was such a car. The pattern follows an increasing complex collection being replaced by more complex and sophisticated elements and more highly integrated wholes. The automobile industry has just been through another such cycle, today producing a car that is much better built and much more satisfying to drive then those of the 1970's and '80's.

Simplicity and Knowledge

It is the same with conceptual artifacts and the Pattern of Knowledge. We start simply; build complexity taking in more and more parts and more and more experience. The sites and fasteners become complicated. We then invent new fundamental elements greatly simplifying the sites and fasteners.

Simplicity is the answer to an interesting question about the elements of knowledge.

This is exactly what happens when a new element of knowledge is invented. It can hold much more, and it can contain a much wider variety of experience. It enables us to greatly simplify our constructions, the building of other artifacts. Simplicity is the answer to an important question. Why do so many, but not all, of the greatest ideas appear at the biggest changes in the "Pattern of Knowledge?" Why did the revolutions of the 7th century, of the 1500's and the 1860's produce so many new and fundamental works? It is because such new fundamental elements enable great simplifications in our knowledges.

The new element is more powerful, more sophisticated, capable of holding much more. And the fasteners are so much simpler. It makes our world look simple and understandable. Once such an element is fashioned, it creates the potential for such vast simplification that it opens the floodgates to invention and with lightning speed passes from discipline to discipline.

While at first blush, we may think of a new element as complicated, as difficult, as sophisticated, as hard to create; as we come to understand it, we see it as profoundly simple. We stop seeing what it took to construct this element and begin to see it as our essential building block. It is more abstract, and harder to get our arms around initially, but as we do learn to understand it, we see its essential simplicity and how it simplifies the world around us.

We thus do treasure simplicity as the Shakers did.

Parsimony

One comes nearer to the most superior scientific goal, to embrace a maximum of experimental content through logical deduction from a minimum of hypotheses.
Albert Einstein

Uniqueness explains the shifts between the Elements - symbols, universals, objects, environments, artifacts.

Abstraction has proven to be a very powerful and yet very elusive idea. While we use it all of the time to describe thought, we never seem to get a full grasp of it. Uniqueness provides a measure of clarity. The abstractness and uniqueness of artifacts are clearly connected. The more abstract an artifact is, the broader its reach and the more likely it is to be unique. When we think, we try to construct the largest idea that we can to hold our experiences, for that idea will necessarily be rarer, more unique. It will also be more abstract. We are always trying to replace many entities with fewer entities, many sites with fewer sites, many fasteners with fewer fasteners. We try to make larger artifacts that will replace a multiplicity of smaller ones. Such larger artifacts, further from individual experiences because they encompass greater quantities, are more abstract. Our drive to uniqueness is necessarily a drive toward abstraction.

We often confuse and denigrate the abstract, when we are given a generalization as an explanation. It is common for people to say - "It is the environment. It is human nature." - and for us to feel that we have been told nothing useful. This is no more mysterious than the difference between a complete physical artifact and one that is just beginning to take shape. The unfinished project may appear wonderful to the artisan, because their finished vision is clear. But for the rest of us that vision must be fully constructed. So it is for the artifacts of our imaginations. They must be fully constructed for us to appreciate and accept them. Thus a broad generalization is not necessarily an abstract artifact, unless it is complete.

Abstraction explains the differences between the artifacts developed by children and adults, even when they are based on the same unique entities. The adult's artifacts, generally richer in experience, are more abstract than the child's. They are thus more unique and more powerful. While superficially the artifacts seem the same, upon closer examination they will show substantial differences in abstraction. As our minds grow stronger and our experience increases, we seek artifacts that are more abstract for they are more unique as vessels for larger quantities of experience.

In this same vein, abstraction helps us to better understand the differences between everyday artifacts and the great human artifacts. They differ in abstraction, in the quantity of experience they can hold. The great human artifacts are abstract; they are models for other artifacts. Of course, they must be finished, and they must be complete. The difference in abstraction is thus the difference in uniqueness. Abstraction is a measure of uniqueness.

Now, finally, we return to those artifacts that started our quest, the unique entities (symbols, universals, objects, environments, and now artifacts). These elements of knowledge are also the greatest abstractions that we have. And it is of great interest that our most powerful abstractions should only come in these few forms. At its most fundamental, abstraction must then not be continuous. It must have discrete levels. We build abstraction on the broadest scale in large and singular steps. Symbols, universals, objects, environments, and now artifacts are the most abstract and the most unique ideas that we have. Each is the next largest idea that encompasses the previous one. We can make environments bigger and bigger, more and more abstract; but if we are to construct an element that is different, that enables simplification and not just greater abstraction; then our invention jumps a significant level of abstraction. Each of these Unique Elements is fundamentally different from the other. Each is a new stage of abstraction.

Here we come to a crucial point. There are two great thought drivers, simplification and abstraction. We seek to construct ideas that are simpler and thus more unique. And we seek to construct ideas that are more abstract and thus more unique. Artifacts are unique when they are different and when they are the same. When we seek simplicity we are searching for difference. When we seek abstraction we are looking for sameness. The essential tools for the construction of knowledge, sameness and difference are also the essential tools that drive thought. To simplify is to group together, to fashion sameness in our artifacts. To abstract is to differentiate, to separate, to define a difference from other artifacts.

This combination of simplification and abstraction is the basis for the unique entities. Each is more abstract that the previous one. Each produces a great simplification. Without simplification, abstraction only leads to complexity. And without abstraction, simplification leads to triviality. These ideas are the heart of thought because they are the fundamental unique elements of thought in the construction of singular artifacts.

Incredible as this may seem, it provides us with an explanation for the unique entities. We commonly think of concrete to abstract as a continuum, but when artifacts are most fundamental, their abstractness comes in very discrete packages. Abstraction also helps us to understand the shifts in the unique entities from symbols, to universals, to objects, to environments, and now to artifacts in the Pattern of Knowledge. The sequence is growth in abstractness of our fundamental elements. Each is the next level of uniquely abstract entity. Each element allows a new level of abstraction, a new unique step in the capacity of our artifacts to hold experience. And each brings with it a new level of unity.

And here, finally, we return to the beginning of this work, where we found the fundamental elements that produced the great periods of knowledge. These elements - Symbols, Universals, Objects, Environments, and now Artifacts- are the basis for knowledge, the templates upon which we design the artifacts we use, the forms for the entities and thus the forms for all of the artifacts. They are the very essence of our knowledge, the most fundamental building blocks. We have already said that each is unique, fundamentally different from the others. And now we can say why.

Each new element is unique because it is different from the one that came before it. Each element is unique because it is a union of the one that came before it, collecting all of those that came before it as the same. Each is unique because it is different, fundamentally different from those that came before it.

Thus we can fashion ever increasingly complex environments, adding more and more, larger and larger environments together, but we fail to produce increased uniqueness or even increased abstraction. For to create fundamentally new uniqueness we have to construct a new kind of artifact, a new element. It must be different from those that came before and yet include them. Such are the unique elements.

Now, since we have made one, why can't we construct others? We have the formula! We have the formula - but unlike singular, plural, parts, and connections, relations - the unique entities do not form a regular, repetitive pattern that we can apply an algorithm to. The unique entities are free inventions, we cannot imagine the next one until we have used, fully explored the current one. We make a new one by fashioning a union of the old ones and then creating something fundamentally different. That is what makes them unique. This last is an invention, an act of creation, and we have been witness to a history of new invention that cannot leapfrog the pattern.

We can thus predict the pattern of knowledge in broad brush for the artifact period, but we cannot construct it out of sequence. And we shall have to wait until this period is complete before we will find what it is that will govern the knowledge building of the next. We need not at all fear that we can see the end of our construction of knowledge. Indeed it is just beginning. And it remains, as it has always been, an act of profound and wonderful invention.