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Mathematics, physics and structure

This section examines how and why mathematics and physics have become purely structural. There are no fundamental entities with an intrinsic nature in these disciplines. The digitization of media is an example of how one can divorce structure from essence. The numbers on an audio CD describe the structure of a musical experience but have nothing of its essence.

Structure describes how a complex object is made out of simpler ones. For example, the structure of a house includes a foundation, walls and a roof. It would seem that when we examine the structure of an object we must ultimately come to some irreducible components that have an essence or intrinsic nature. For example we might consider the lumber, nails, concrete foundation and roofing shingles as being among the fundamental components of a house. Newtonian physics was constructed like this. All matter was ultimately composed of billiard ball like particles that had an intrinsic nature from which many of the laws of physics could be derived.

Contemporary physics has no such primitive entities. It is entirely mathematical as explained later in this section. Contemporary mathematics has deliberately and systematically purged itself of any objects with an intrinsic nature. Mathematicians did this because starting with objects that had an intrinsic nature, like lines, led them to make false assumptions. That story shows the value of separating structure form essence in mathematics.

For centuries, the parallel postulate of Euclid was considered to be a self evident truth. Two lines are parallel if they are both perpendicular to a third line. For example the legs of a well made table are parallel because they are perpendicular to the table top. No matter how far one extends the legs they will never meet. This is the parallel postulate. It seems self evident.

Now consider the laws of geometry on the surface of the earth. Sailors determine their location in the ocean by latitude and longitude. These are imaginary lines this circle the earth. Lines of latitude are parallel to the equator. Lines of longitude are perpendicular to the lines of latitude. Thus all lines of longitude are parallel with each other. However, if you look at a globe with the major lines of latitude and longitude marked, you will see that all the lines of longitude intersect at the north and south poles.

The surface of a sphere does not conform to our intuitive notions about parallel lines. We call geometries that obey the parallel postulate Euclidean. Many important geometries are not Euclidean including the surface of our planet. General relativity defines the geometry of our universe in contemporary physics. It too is not Euclidean.

Mathematicians wanted to avoid making assumptions that are not universally true like the parallel postulate. To that end they removed any fundamental entities like lines, planes or points from the formulation of mathematics. They invented set theory. In set theory there is a single primitive entity, the empty set, and a single primitive relationship, set membership. The only objects are the empty set and things constructed from the empty set. For example the number one is the set containing the empty set. The number two is the set containing the number one and the empty set. Thus the number two has two members. In general the number N contains the empty set and all numbers less than N. Of course the number 0 is the empty set.

This is an awkward way to do mathematics. Defining a line starting with the empty set is complicated. You need to define a topology of points. You do this with numbers but you need real numbers like 32.25, $5 \over 3$ and even $\pi$. Mathematicians do not necessarily think or work in terms of sets. But they know how to formulate the work they do in those terms. All existing branches of mathematics can be formulated in the language of set theory and all widely accepted mathematical theorems can be derived from the axioms known as Zermello Frankel Set Theory plus the Axiom of Choice or ZFC. These axioms can be easily written on a single page.

Set theory is a pure study of structure. When you complete the analysis of an object in set theory you wind up with the single irreducible entity of the empty set. This is not an object with an intrinsic nature. It is nothing at all. It is as clear a symbol as one could imagine that there is no essence in mathematics.

Just as mathematics has become completely abstract physics has become purely mathematical. The only connection between the mathematical formulas of physics and human experience is experimental technique. There still exist fundamental particles as in Newtonian physics but these are defined with mathematical formulas. They behave strangely and do not always have an individual identity. For example light consists of particles called photons. These seem to precipitate out of an intense beam of light like rain drops precipitating out of a cloud. The raindrops did not exist as individual entities when they were part of the cloud. Photons do not exist as individual entities in an intense beam.

A discussion like the above is at best a metaphor. Contemporary physics provides no model of what is ``really happening''. It only describes how probabilities evolve between observations. If we observe a particle at a particular location and time then quantum mechanics allows us to predict how likely we are to observe it at a different location and later time. Quantum mechanics says nothing about a particle traveling from one location to the next. Most attempts to fill in the blanks between observations have led to theories that make wrong predictions. Those that do not, like Bohm's[4], must be inconsistent with special relativity or quantum mechanics[2] and those discrepancies cannot be experimentally detectable with existing technology.

No other theory has come remotely close to the accuracy that quantum mechanics is capable of. Quantum mechanics allows us to build postage stamp size computer chips that do billions of calculations in a second. The theory accomplishes all this, yet it is more abstract than set theory. It does not model the structure of the physical world. It models only the evolution of probabilities.

The intuitive idea that complex structures have fundamental components with an intrinsic nature is not true in contemporary mathematics and physics. Mathematics builds everything from the empty set or nothing at all. Physics is pure mathematics and does not even model the structure of physical reality but only the evolution of probabilities.

The divorce between structure and essence in our scientific understanding is clearly illustrated in the digitization of media. Everything we see and hear can be encoded as a sequence of numbers on a CD of DVD. These sequences preserve the structure of the sound or image. In the case of a CD the numbers represent sound pressure level at a given instant. By recreating the sequence of sound pressure level we recreate the original sound. That is what a audio system does with the numbers on a CD. Similarly for images the numbers represent the intensity of the three primary colors, at a given point in an image and instant in time. Recreate the intensity levels of these colors at the correct location and time and you recreate the image. That is what a DVD player connected to a television does.

Digitization has its roots in information theory. Shannon defined information as that which allows us to reduce the number of states a system may be in. For example suppose we know that a flag must be red, blue, green or yellow. Then it can be in any of four states where each state corresponds to a different color. If we are now told the flag is green we have reduced the four possible states to a single state. The amount of information transferred is that needed to reduce four states to one state. This requires a number between one and four.

This measure of information is universal. It applies equally to color, sound, a page of text etc. Shannon's definition applies to everything we can communicate. We can always measure the information communicated in terms of how much we have reduced the number of possible states. We can always communicate that information as a number that selects possible states as long as both the sender and receiver have the same map between numbers and states.

We ordinarily communicate through sight and sound. The map that translates pressure waves in our ears or light waves in our eyes to sounds and images is in our nervous system and brain. This translation happens automatically. We have no sense of creating sounds or images in our heads like we do of projecting them on a movie screen. But we know a great deal about this mapping process and that knowledge allows us to explain the many ways in which our vision and hearing can be misled.

Recognizing that all information can be represented by numbers that select states was an important step in the removal of fundamental entities with an essential nature from science and mathematics. Shannon's concept of information and the set theory based notion of structure in this paper are close. One can assign a unique integer to every finite set. Thus gives a measure of the information needed to describe any finite structure. Every structure contains information and every structure can be fully described with information.

The separation of essence from structure in science and mathematics was an extraordinarily valuable achievement. It removed any intuitive notions from the fundamental elements in mathematics and physics. Of course intuition is essential in developing scientific theories and mathematical understanding. But at the end of the day one does not want to have ill defined intuitive ideas at the root of science and mathematics. The power of these disciplines comes in part from their precision. By leaving nothing undefined beyond the empty set, mathematics achieves the precision it needs. By becoming purely mathematical physics achieves the same result. By reducing information to the selection of states, the foundation of the Internet and the digital revolution was laid.

These achievements create a philosophical problem. What does it mean for a physical object to exist if it is not constructed out of components with an intrinsic character? What is the intrinsic nature (the greenness of green, the smell of a rose) that is so evident and pervasive in our stream of consciousness? The next section completes the groundwork for addressing that question by exploring the nature of immediate experience.


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