The Map of Physics
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At the beginning of the Seventeenth Century Rene Descartes attempted to revive the philosophical doctrine of Rationalism, the doctrine based on the old Greek notion that Reason alone is sufficient to apprehend Reality. Because deductive logic generates undoubtably true statements (theorems) from self-evident axioms, he hoped to create an axiomatic-deductive system that would generate an accurate description of all Reality (which comprises things that are absolutely true). He wanted to start with at least one axiom about which he could have no doubt and which he could then use as the foundation of his Rationalist system. He understood that the act of doubt was the key to his axiom: in order to doubt he had to be thinking and in order to be thinking he had to exist, so the one thing that he absolutely could not doubt was his own existence (expressed in the famous statement ACogito; ergo, sum@ (I am thinking; therefore, I exist)).
Unfortunately, Descartes was unable to create a correct Rationalist physics. Indeed, his vortex theory of planetary motion was explicitly falsified by Isaac Newton in Newton= s APrincipia@ (first published in 1687), a work whose success in explaining the motions of the planets, their moons, and other celestial bodies gave the empirical-inductive system of Existentialist physics primacy over Rationalist notions. Now, though, we have a correct Rationalist physics, expressed in the Map of Physics, a flowchart that shows how the laws of physics can be deduced from three self-evident axioms. We want to see now whether we can find a relation between Descartes= ACogito@ and the Map of Physics.
When we require that our axioms be self-evident we are prescribing statements whose contraries we cannot reasonably hold to be possibilities. The ACogito@ is self-evident in this way in the sense that we cannot reasonably regard something like AI think; therefore, I do not exist@ as a real possibility: If I am thinking, I must necessarily, rather than possibly, exist. Do the axioms on which we have founded the Map of Physics have that same character?
But first a caution. Consider Euclid= s fifth common notion, Athe part is less than the whole.@ The self-evidence is clear: we need only imagine dividing some object into parts and notice that each part is less than the whole object itself. But then we come across an infinite set, such as the set of the natural numbers (the positive integers). We discover, among other things, that we can prove and verify the proposition that the set of the even positive integers has exactly the same size as does the entire set of the natural numbers. In that case the part is equal to the whole. That discovery obliges us to restrict the axiom by modifying it to say, Athe part is less than the finite whole.@ In that way we can refine our axiomatic-deductive system of logic.
AValid statements describing Reality must be true or false, but not both.@
This is the law of non-contradiction from classical Aristotelian logic. Actually I=ve introduced a minor error by leaving the realm of discourse off the truth values. Properly we should say that the statements are either true to Reality or false to Reality. In specifying the realm of discourse we ensure that our assignment of truth values to a statement is meaningful. For example, it is meaningful to say that a proposition stands true to mathematics because there are propositions that stand false to mathematics (e.g. 2+2=5). But it= s meaningless to say that an idea stands true to imagination because no idea can stand false to imagination: I must be able to imagine the idea in order to declare it false.
The contrary of non-contradiction, for comparison, is a statement that descriptions of Reality may be true, false, or both. But what does it mean for a description of Reality to be both true and false? We might be tempted to say that the statement is partly true and partly false and identify the concept of partial truth with that of probability.
The archetype of probability for us is the rolling of dice. Because the forces and torques imposed on a die when it is cast are unmeasurable and thus not available for a calculation in Newtonian dynamics, we profess complete ignorance of how the die will be oriented when it stops rolling and comes to rest. We can=t have perfect knowledge of the die=s reading in advance of our tossing the die, but we can have imperfect knowledge: we can calculate an expectation value. Because the die is a uniform, perfectly balanced cube, the possibility of appearing as the top face when the die stops moving is equal for all six faces. Thus we divide the 100% probability that at least one of the faces will come up into six equal parts and assign each of the resulting 16.666...% probabilities to each of the faces. That means that if we throw a large number of dice or throw a single die many times, we expect one sixth of the outcomes to show, for example, two spots on the upper face and five sixths to show Anot-two@ . So every time we throw a single die we have a one-sixth expectation that it will show Atwo@ and a five-sixths expectation that it will show Anot-two@ .
But those expectations are rendered invalid once the die stops moving. At that time it certainly shows either a Atwo@ or a Anot-two@ . The statement AThe die shows two@ is either true or it=s false; there=s no alternative. The probabilities that we assigned to the outcomes prior to the die=s coming to rest simply expressed our ignorance of the factors that determine the outcome. If we could know all of the forces and torques acting on the die as it gets tossed and as it hits the table, then we could calculate, with no uncertainty, which face would come out on top. So the probabilities that we assigned to the outcomes of the toss do not represent states of Apartly true and partly false@ but rather represent the state of Awe don= t know@ .
The notion of partial truth, even though we believe we know what it is (because we use it in social situations), is not valid. The liar=s tale is simply Ain part true@ and not Apartly true@ and once we make that distinction we can see that our axiom is indeed self-evident. We can=t even conceive a legitimate notion of Apartly true@, so we certainly can=t hold it in our minds as a real possibility alternative to Atrue or false, but not both@ .
But then we come to the quantum theory. In particular we come to the paradox of Schrödinger=s cat. In that imaginary experiment, before we open the box, we say that the cat is both alive and dead: surely that statement is equivalent to saying that the statement Athe cat is alive@ is both true and false. Actually we have misstated the case: properly we should say that the cat is neither alive nor dead; it= s in an indeterminate state. The statement Athe cat is alive@ thus has no truth value; it=s neither true nor false to Reality, because, according to the hypothesis of the experiment, the cat=s being alive or dead is not yet a part of Reality.
Again consider the roll of a die. As the die bounces and tumbles across the table we cannot assign a truth value to the statement Athe die shows two@ ; we can only assign a truth value to that statement after the die has come to rest. That latter fact means that Athe die shows two@ is not a valid statement describing Reality until the die comes to rest. Likewise, the statement Athe cat is alive@ is not a valid statement describing Reality as long as the cat exists in an indeterminate state.
Fundamentally Reality consists of a flux of events. Thus, most valid statements describing Reality and their truth values have only a temporary existence. But some valid statements describing Reality and their truth values have a permanent existence: those statements comprise the set we call the laws of physics, the object of our study.
AAll numbers assigned by measurement obey the rules of arithmetic.@
We name the counting numbers and define the basic operations (addition, subtraction, etc.), thereby defining the realm of mathematics. Then we can say that any proposition is true to mathematics (e.g. the sum of the first N odd numbers equals the square of N) or false to mathematics (e.g. 2+2=5). The proposition is/isn=t an element of the set Mathematics.
We establish the rules of arithmetic by defining the act of counting; that is, by establishing the cardinal numbers in a definite, unvarying sequence. Our system of counting (zero, one, two, three, four, five, six, and so on) is the simplest and most convenient to use, but others could be used. The chief advantage of our system is that we can array the numbers along a straight line so that immediate neighbors on the line differ from each other by a count of one. We can thus describe the addition of two numbers A and B very simply by saying that the sum (A+B) can be obtained by counting from zero to A and then counting B more steps away from zero. Likewise, we can describe the subtraction of B from A by saying that the difference between them (A-B) can be obtained by counting from zero to A and then counting B steps back toward zero.
All of the rest of mathematics needed to describe Reality can be built up from those processes. Multiplication is simply repeated addition and division can be carried out through repeated subtractions. Likewise, the fundamental operations of the calculus can be defined through simpler operations, integration being a combination of multiplication and addition and differentiation being a subtraction followed by a division. Even algebra, which we think of as abstract mathematics far removed from our grammar-school arithmetic, is nothing more than a manipulation of recipes for doing arithmetic, which recipes are written with letters representing numbers to be inserted later.
To assign numbers to physical entities, we count those entities. Things like apples in a basket can be counted directly because those things come to us as discrete and therefore countable units. Things like distances are not discrete and countable, so numbers must be assigned to them by measurement, a process of comparing the entity whose count is desired with some arbitrarily defined, commonly accepted standard unit of that entity. We take the length of the king of England=s foot, for example, and cut straight wooden slats to the same length. Confronted with an arbitrary distance, then, we can lay our slats end to end along that distance and then count the number of slats we=ve used, expressing the distance as Aso many feet@ .
Of course, not all of the numbers that we assign to various physical entities are measurements. Velocity, for example, is not a measurable quantity, but is rather the ratio of two measurable quantities, distance crossed and time elapsed (the standard of comparison by which time is measured being some repeating phenomenon, such as the sway of a pendulum). When we claim to measure velocity we are actually measuring the displacement of a pointer of some kind, the displacement being caused by some phenomenon related to the velocity we wish to know. Velocity is an inferred quantity, rather than a measured one, so we don=t expect compound velocities to obey the simple rules of addition and subtraction (and velocities near the speed of light don=t).
Our axiom claims that if we divide a measurable entity into two parts, measure the two parts separately, and measure the entity as a whole, the measure of the whole entity simply equals the sum of the measures of the parts and can be no other number. Euclid expressed this in his Acommon notion@ that the whole is equal to the sum of its parts. The contrary to this axiom thus claims that the measure of the whole differs from the sum of the measures of its parts. Can we accept that contrary as a reasonable possibility alternative to our axiom?
If two people, starting near the middle of a path and walking in opposite directions, lay down a series of foot-long rulers end to end along a well-defined line, counting as they go, then when they reach the ends of the path they can calculate the total number of rulers laid down by adding together the numbers to which they counted. If I then walk along the path from one end to the other, picking up the rulers and counting them as I go, I expect that the number I have at the end of my traverse will be identical to the one that my assistants calculated from their counts. If the two numbers were to differ, we would assume that some rulers were lost or gained and not that the process of counting is inapplicable to them. The use of discrete objects to measure other objects by direct comparison and counting leaves no room for the numbers obtained to differ when the same objects are counted in different sequences. And so long as the objects used to measure have a nature similar to that of the objects being measured in what is being measured (as they must necessarily do since we could not, for example, measure length with something that has no extension in space), then the process is certainly valid. The idea that we could count a collection of objects in different ways and come up with different totals is absurd, so we cannot reasonably hold the contrary of our axiom to be true.
AThe Universe exists with its boundary touching Absolute Nothingness.@
The self-evidence of this axiom seems clear enough. We define the Universe to comprise everything that exists, so anything outside the Universe, which makes up the context in which the Universe exists, must necessarily comprise what does not exist, what we might describe mathematically as the empty set.
But while our statement is truly an axiom, it incorporates a statement that is not axiomatic at all B AThe Universe exists@. Though we certainly have reason to believe that it=s true, the statement doesn=t have the self-evident character of an axiom. After all, the world that seems present to our senses could actually be an illusion, a kind of hallucination that we create ourselves to provide a context in which to interpret our existence. It can=t be an axiom, so the best we can hope to find is that the statement turns out to be a theorem, a statement deduced from an axiom or axioms.
This axiom has the consequences that we seek in laying out the Map of Physics. We know, to begin, that the boundary has zero thickness: anything that gave it a non-zero thickness would merely be a non-boundary item adjacent to the boundary. Thus whatever property the boundary has on one side, it must have on the other. But it can have no properties on the side facing Absolute Nothingness, so:
1. It has zero measure (length, area, volume, etc.); therefore it exists as a single point.
2. Time does not elapse on it.
3. The Universe and its contents must have zero linear motion at all times; that is, all of the linear motions in the Universe must always add up to a net zero. Therefrom we deduce Newton=s laws of motion.
4. The Universe and its contents must have zero rotary motion at all times; that is, all of the rotary motions in the Universe must always add up to a net zero. Therefrom we deduce the rotary equivalents of Newton=s laws of motion.
5. Nothing can touch it (space and time must be so shaped that no object can reach the boundary.); Space and time represent determinate sequences of objects and events; therefore, they must have finite extent. Thus the boundary must move away from all objects at a finite speed no object can reach. Therefrom we deduce Relativity.
The A Cogito@
The question of whether the Universe truly exists came to Descartes when he was seeking to establish the foundations of his Rationalist worldview and for reasons similar to that given above he rejected the existence of the Universe as an axiom. The prime axiom, the archetypal undoubtable statement, upon which Descartes hoped to build his doctrine, was his ACogito; ergo, sum@. That seems a rather sparse statement from which to deduce a full description of Reality and, in fact, Descartes was unable to deduce a description that was true to the discoveries of modern physics. But we have more information than Descartes had, so we can approach the problem anew with some hope of success.
Let=s define Reality as consisting of the objects of perception (things that exist independent of the action of my mind) and their relationships. This differs from our definition of the Universe and we want to develop a proof that they are identical. We have defined me and all of the things that exist independent of my existence to comprise the Universe. Their property of necessary existence is contrasted with necessary non-existence, so we assert that the Universe exists in a context of Absolute Nothingness. On a Venn diagram AI@ am represented as a subset of Reality. Reality is impervious to direct action of my mind; it is independent of and outside me. I assume that AI@ am created and maintained by direct action of my mind; dependent on and within me. The boundary, the perceptual/conceptual interface, between AI@ and AReality@ is the sensorium.
We take as our fundamental axiom, as Descartes did, the statement that AI think; therefore, I exist@ but not AI exist; therefore, I think@. I can exist without thinking but not vice versa. Thinking is a subset of existing. Thus I can restate the ACogito@ as AThat which creates thought in me necessarily exists@.
AI think; therefore, I exist@ means that I am my thoughts (neither more nor less). My thoughts consist of percepts (created in my mind by action of Reality) and concepts (created in my mind by action of mind). If I assume that all percepts are concepts, then my mind is all of Reality. But I don=t believe that statement.
When I say, AI=m thinking; therefore, I exist@, I am tacitly claiming to know something about thinking. What is it that I know?
1) Thinking is the process of showing myself images one after another. That sequentiality necessitates that my thinking and, therefore, my existence be inherently associated with the concept of time.
2) The images that comprise my thinking are all different one from another; that is, the images change from moment to moment. Thus thinking is a creative act, one that continuously creates new thoughts or renews old ones. The essence of my existence, then, is that creative act.
3) Some of my thoughts are true and some are false (but none are both). If that statement is false, then my thoughts must either all be true or all be false. But that statement itself automatically eliminates the possibility that all of my thoughts are true. I am compelled to assert that all of my thoughts are false. But if that statement is true, then I must affirm my original statement: some of my thoughts are true and some are false. It was this statement, less rigorously worked out, that Descartes confronted in working out his ACogito@ . We seek a way to determine which of our thoughts are true.
In considering the axioms I can think of things that are true and things that are false. That thoughts can be true or false is an axiom because if I can=t think of things false, my axiom is false, but I thought of it nonetheless. And if I can=t think of anything true? Then my necessary conclusion that I can think only of falsities is true. This statement connects to the First Axiom.
4) Ex nihilo nihil fit. My thoughts could not have risen spontaneously out of nothingness, so I must have in my mind thoughts that I did not create. My thinking is based on those thoughts that I did not create.
5) Things that I did not create are things independent of me and originate in things independent of me. Therefore, something independent of me created some of my thoughts and put them into me. Therefore, something necessarily exists independent of my existence.
6) Some of my thoughts are true and some are false. Thoughts of objects in space that are true refer to things whose existence is independent of my thinking (i.e. I can= t change them merely by thinking of a change). Therefore, something that consists of objects in space exists outside my will; that is, outside me as defined by the ACogito@ . This connects to the Third Axiom.
Thus we relate the ACogito@ to the axioms on which we base the Map of Physics.
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