When one considers whether there is any meaning or purpose or destiny in reality, one must ask if there are any overriding principles that governs the universe. Questions of whether there is any justice or righteousness in creation require that there be some underlying correctness and consistency to answer. And if there are any answers that can be definitely asserted yes or no, then whatever principles govern the universe must start with a logic of true and false.
Some may think that logic is just a mental exercise, or an engineering tool to approximate nature. Or they may think that logic is just a psychological convention that we have invented to placate an emotional need for consistency. But the methods of logic are used in every area of study to advance our knowledge of the universe and our place in it. Until now, no one even dares to suggest that nature is truly illogical.
So this begs the question as to how relevant logic really is to events we observe. Does it apply to all or just some parts of reality? Some believe that logic is only useful as a tool when we plug some starting facts into the equations of logic in order to derive conclusions. Since they believe that logic is only in our minds and is not an integral part of nature, they insist that logic alone cannot produce the laws of physics. They are content to measure data and discern the mathematical relationships between events. But this will never answer why things are as they are. It can only answer how they mathematically relate to each other. Yet the only alternative to a physics derived from logic is to suggest the existences of some event at some level which has no logical basis for its existence; it just is. But this would contradict any answer to questions of how it came to be.
Besides, we may have no choice but to rely on logical consistency to test competing theories of creation. For it would take all the energy in the universe in order to recreate the energy levels of creation so that we could confirm our theories by direct experimental measurements. And if we are trying to discern how the very first particles evolved, then it is not possible to suppose that they are the consequence of other prior existing particles used as the premises for deriving them as a conclusion. We are only left with deriving the first events from nothing more than principle itself.
So I cannot help but think that the structure of the universe, the nature of space and time, mass and energy, and all the events that follow can indeed be derived from the principles of logic alone. The quest is to find some method of expressing the principles of logic that is also capable of expressing the mathematics of physics. This common means of expression must be capable of expressing the true and false of propositional logic, the quantifier concepts of predicate logic, the probabilities of quantum mechanics and the tensors used in general relativity. And once we find a way of expressing logical principles in mathematical form, it may be possible to derive the mathematical laws of physics from principle alone.
As can be shown, Venn diagrams can be used to demonstrate the principles of Propositional and Predicate logic, Set theory, and the concepts of Probabilities. So it occurs to me that if we impose an arbitrary coordinate system on these diagrams, we have a way of expressing logic in a mathematical way. And since logic is used to describe the relationship between events in general, the math we impose on this logic should give the mathematical rules that govern events in general, or in other words, the mathematical laws of physics.
The article entitled, A Graphical Representation of Logic, explains how Propositional Logic, Predicate Logic, Set Theory, and Probabilities can all be represented graphically as objects within regions of space. Each of these disciplines of Logic describes a different aspect of the same graphical situation. Propositional Logic is only concerned with the regions where samples may exist for which propositions may be true. Predicate Logic is also concerned with whether samples actually exist in these regions. Set Theory explicitly lists the samples that exist in a particular region. And Probabilities compare the number of samples in various regions.
And the article entitled, Probability Densities, describes how a mathematical function can be defined that gives information as to where samples are located in space once a coordinate system is imposed on the sample space. Regions in sample space are called events, and the probability of an event is derived by integrating the probability density function throughout the region of interest. The various disciplines of logic use the information contained in the probability density function to varying degrees. The study of probabilities uses all the information contained in the probability density function. Predicate logic is only concerned with whether the function is zero or non-zero in a given region. And Propositional logic is only interested in the regions where the function may be integrated.
So why should we think that samples in regions have anything to do with the physical world? It seems every physical situation is described by objects contained within regions with some particular set of properties: The universe contains galaxies, galaxies contain stars, nations contain cities, cities contain homes, rooms contain furniture, matter contains molecules, molecules contain atoms, atoms contain protons, protons contain quarks, etc. And when we ask if some particular thing exists, we are asking whether there is a sample with a certain set of properties.
And why should we think that physical events have anything to do with properties and probabilities? Since there is no other alternative that could possibly exist other than existence itself, the existence of the universe as a whole is an absolute certainty. Like dice with the same number on each face, since there is no other alternative, the outcome is 100% certain. That the universe exists is certain, 100% certain; that much we know beyond any doubt. And the only way to describe reality beyond the universal property of existence with 100% certainty is to describe portions of it as having other properties with less than 100% certainty. For 100% of an entire sample space is equal to the addition of all the portions of it that have less certainty.
So at the very first instant of creation, when the very first thing that could be distinguished from non-existence came into being, the first particle, if you will, this first particle represented absolute truth and was completely distinguished from non-existence which represents falsehood. It demonstrated the fundamental principle of logic called material implication wherein truth can still be derived from a false premise, but falsehood cannot be derived from truth. And the only thing possible for this one, absolutely certain particle is to distribute into other particles whose properties are not absolutely certain. This sounds a lot like the postulates of quantum mechanics.
If reality is logical at every level, then we are left with trying to derive the laws of physics from the mathematics imposed on the sample space used to illustrate the principles of logic. For this logic is used to derive events as a consequence. And there seems to be some intuitive sense that this approach will succeed. The questions is how to derive the dimension of time and the 3 dimensions of space along with the particles that travel through these dimensions and the forces exerted between them all from a probability density function that exist in regions of sample space. For if we can derive even a few of the familiar relationships of physics, then we will know that we have succeeded in discerning physics from logic. For we presume that logic and physics are each a consistent system, so if they agree at any point, then they must be equivalent.
Now it has been demonstrated in the articles above that an event is represented as a region in sample space in which a probability density function exists. Even though this event may be described by a complicated set of properties in conjunction, still we are talking about one region of sample space per event. And we are tasked with discovering how one event mathematically relates to another and how one event can change into another. It is hoped that these mathematical relationships in sample space will some how translate into the familiar math of physics in real space. Perhaps this requires a mathematical transform from sample space to real space. Or perhaps it requires an interpretation of the math of sample space in order recognize them as physical phenomena.
Regions in sample space can be defined by a mathematical function once a coordinate system is imposed. For instance, a regions is defined as a line enclosing an area in a two dimensional system, or a regions is defined as a surface enclosing a volume in a three dimensional system. In the figures that were used in the articles above, we chose to put the origin of the coordinate system in a place that made calculations easier for us. But there is no logical or mathematical reason that necessitates putting the origin of the coordinate system in any particular place. We could have just as easily drawn the origin someplace else. The calculations might have become more complicated, but the equations would have been no less valid, and the results would have been the same. So the origin and the scale and the orientation of any coordinate system we may use is completely arbitrary. And the mathematical principles we seek must be the same no matter what coordinate system we chose. In the language of tensor analysis, we are looking for mathematical principles that are invariant with a change of coordinate systems.
It was shown that the logical operations of AND and OR and the concepts of Union and Intersection could be represented by regions of any dimension. And we have no mathematical reason at this point to prefer one number of dimensions over another. So we must start with equations that have the same form for any number of dimensions. Tensor analysis is especially suited for this purpose.
Samples of events can be described by a list of measurements. For instance, a sample may consist of a measurement of the size, weight, color, temperature, location, and time of occurrence. These would be listed in vector form as (s,w,c,T,l,t) and would describe a six dimensional sample space. Each of the variables, s, w, c, T, l, and t are called random variables, and the vector, (s,w,c,T,l,t), is called a random vector, because we are not certain what the measurements will be before we do the measurements. Of course, no instrument is capable of measuring with absolute precision. There is always a plus or minus tolerance associated with the measurement of each of the variables in the vector. This tolerance spread in each of the variables constitutes a region in sample space. Samples are simply events specified with greater accuracy inside larger events specified with lesser accuracy. And there is always a greater probability of measuring events with less accuracy.
The principle in nature that we can discern the existence of one subset of reality based on other subsets of reality seems to be applicable at every level of reality from the most minutest subsets (subatomic particles, parts of strings, etc) to the universe as a whole. This suggests that the most fundamental constituent of reality is a subset in essence, or a region of sample space. And the characteristics of space, time, matter, and energy can be describe by the characteristics of these regions of sample space within a probability density function.
Events are described as regions of sample space, and regions can only be specified as being bounded by some surface or another. These surfaces can be described with mathematical equations. And everything knowable about an event must be obtainable from information on the surface. For as soon as you consider information inside or outside the surface, that information is not specific to the event in question. It is information common to an infinite number of other possible smaller or larger events. So we are left with integrating and differentiating information contained in the probability density function along the surface of event. We can only scratch the surface of things, as it were.
All this allows us to transform the description of a proposition or event into a problem in geometry using vector and tensor analysis. Since the description of physical phenomena form the propositions we use in logic, and since we have a mathematical description of propositions, then we have a mathematical description of physical phenomena - or, in other words, a mathematical description of physics.
It should not surprise you, then, to consider that this approach leads to mathematics that looks very much like Superstring theory which is the most promising theoretical model of physics to date. Click here to learn about the similarities between String Theory and this sample space approach.