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 chose 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.

The Electromagnetic Field

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. It can be measured by placing an electric or magnetic body at a point and plotting its direction and its magnitude. We can 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.

This is a Theory

It is a logical construction by which a fastening artifact defines and connects sites, to which the sites and entities of experience can be linked.

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

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, which liberates 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 which is 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 in this short work, I have deleted. 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 can not 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

Theories Take Us Further

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.