Unification of physics and mathematics

Copyright (C) May, 1996 by Dr. Tienzen (Jeh-Tween) Gong

I: Three mathematics schools

What are laws of mathematics? Are they laws of God (absoluteness)? Or, they are just invented by men (arbitrariness)!
Greeks, Descartes, Newton, Euler, and many others believed that mathematics to be the accurate description of real phenomena and that they regarded their work as the uncovering of the mathematical design of the universe. In other words, mathematics is absolute. But, gradually and unwittingly mathematicians began to introduce concepts that had little or no direct physical meaning. After the introduction of quaternions, non-Euclidean geometry, complex elements in geometry, n-dimensional geometry, bizarre functions, and transfinite number, almost all mathematicians today accept and recognize the artificiality of mathematics. That is, mathematics is no longer absolute but arise arbitrarily. But if mathematics is only the invention of men, then why is it so consistently successful as a description of physical world?

Because of this invented-discovered (arbitrariness-absoluteness) paradox of mathematics, there are at least three schools of mathematics -- formalism, intuitionism and logicism. Logistic school was led by Russell and Whitehead. They started with the development of logic itself, from which mathematics follows without any axioms of mathematics proper. Nonetheless, the development of logic consists in stating some axioms of logic, from which theorems are deduced that may be used in subsequent reasoning.

The formalist school was led by Hilbert. He tried to provide a basis for the number system without using the theory of sets and to establish the consistency of arithmetic. He developed a proof theory, a method of establishing the consistency of any formal system. Controversial principles such as proof of existence by contradiction, transfinite induction, and the axiom of choice are not used. The formalists also maintain that logic must be treated simultaneously with mathematics. In short, to the formalist, mathematics proper is a collection of formal systems. Mathematics becomes not a subject about something, but a collection of formal systems, in each of which formal expressions are obtained from others by formal transformations. Since a formal system can be unending, metamathematics must entertain concepts and questions involving at least potentially infinite system. However, only finitary method of proof should be used.

The intuitionist school was led by Kronecker. He viewed Cantor's work on transfinite numbers and set theory was not mathematics but mysticism. He was willing to accept the whole numbers because these are clear to the intuition. These were the work of God, and all else was the work of man and suspect. To the intuitionist, mathematical ideas are imbedded in the human mind prior to language, logic, and experience. The intuition, not experience or logic, determines the intuition is opposed to the world of causal perceptions. Language serves to evoke copies of ideas in man's minds by symbols and sounds. But thoughts, especially mathematical thoughts, can never be completely symbolized. Mathematical ideas are independent of language and in fact far richer.

On the other hand, logic belongs to language. Logic is not a reliable instrument to uncover truths and can deduce no truths that are not obtainable just as well in some other way. Logical principles are the regularity observed a posteriori in the language. They are a device for manipulating language, or they are the theory of representation of language. The most important advances in mathematics are not obtained by perfecting the logical form but by intuitively perceiving the mathematical truth itself. Thus, logic rests on mathematics, not mathematics on logic.

The intuitionists therefore proceed to analyze which logical principles are allowable in order that the usual logic conform to and properly express the correct intuitions. Consequently many existence proofs are not accepted by the intuitionists. The law of excluded middle can be used in cases where the conclusion can be reached in a finite number of steps. In other cases the intuitionists deny the possibility of a decision. This gives rise to a new possibility, undecidable propositions. The intuitionists maintain, with respect to infinite sets, that there is a third state of affairs, namely, there may be propositions which are neither provable nor unprovable.

The battle among these schools on the issues about what the essence of mathematics is or what the correct methodology for mathematics is has a great significance which goes way beyond mathematics itself but enters into the domains of philosophy and theology. At least three issues in mathematics are beyond the reach by mathematics.

In formalist school, Hilbert reduced the consistency of geometry to the consistency of arithmetic, but the consistency of the latter remains to be an open question. The logistic school reduced arithmetic to logic. But, logicism does have many undefined ideas, as any axiomatic system must have since it is not possible to define all the terms without involving an infinite regress of definitions. The consistency of logic axioms remains to be an open question. Seemingly, mathematics is built on sinking sand, and it can never find a solid bed rock to rest upon it. However, if physics laws must be the direct consequences of mathematics principles, then the arbitrariness of mathematics can, then, find a solid bed rock.

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II: The Corresponding Principle

The traditional physics was and still is built up from the bottom up, that is, from observations to form hypothesis, then verify it with experimentations and more observations, then on and on. This bottom up method can discover those constants of nature but will not be able to understand why they arise. On the other hand, the Final TOE must be able to construct physics from top down, that is, from a principle -- the corresponding principle -- to deduce a complete physics system.

The Corresponding Principle states --- the complete physics system (the current mainstream physics is not yet complete) is isomorphic to the complete mathematics system (the current mathematics is not yet complete).

Seemingly, this principle is too vague to be useful and the term complete is not yet defined at this point. Nonetheless, the following theorem can be very useful.

Theorem 1: Every mathematical issue (especially those unsolvable mathematics problems) has a corresponding issue in physics or in physical world.

Corollary 1: If an unsolvable mathematics problem has no obvious corresponding issue in physics or in physical world, then there is a process which has smoothed out that problem in the physical world.

With this corresponding principle, two things have changed completely.

Because we now do have plenty experimentally verified physics on hand already, they can be the landmarks to verify the validity of this new physics. But, from the zillion mathematical ideas (many of them not yet discovered), which ones could or should be chosen as the starting points. I am choosing three mathematics issues --- computability, countability, and uncountability --- as the foundations of a new and complete physics.

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III: The computable and the uncomputable

The definition of recursive function in mathematics is quite technical; I will not repeat it. The functions in the sequence sum, product, ... belong to the class of primitive recursive function. Today, the following three theorems are proven in mathematics.

But, are all computable functions Turing computable? No one knows, but Alonzo Church's thesis (which does not admit of mathematical proof) says yes.

Turing computer differs from any real computer at least in two ways.

Church's thesis is, however, not based on these two marvelous abilities of the Turing machine but is based on its inability of knowing that enough is enough, not knowing when to halt the calculation. This inability of Turing machine is called busy beaver problem or the halting problem.

With these two marvelous abilities, Turing machine can calculate a function forever and ever. Thus, both Turing computer and we will never know whether that function is computable or not. If we or Turing machine itself can find a procedure to tell it to halt at a certain point and to print out one of the two answers:

  1. the value of this function is ****, or
  2. this function is uncomputable,
then this busy beaver problem is solvable.

It is now mathematically proven that if Church's thesis is true, then busy beaver problem is unsolvable. This proof is in no means implying that Church's thesis is true.

Church's thesis itself (that all computable functions are Turing computable) is not as important as the fact that it itself is unprovable. The fact that Church's thesis defies the mathematical proof clearly points out that there is a hole existing in mathematics. This hole will become the rock solid foundation of a new physics. That is, there must be a corresponding difficulty in physics, or there is a process that has smoothed out that difficulty in the physical world.

By accepting Church's thesis and combining it with the busy beaver problem, we can easily prove an existence theorem for uncomputability.

Theorem 2: There are functions which are not computable (by Turing machine or by anything else).

Corollary 2: There is at least one uncountable infinity. (Georg Cantor proved this long ago).

Definition 1: The universe is the union of two sets A and B. Set A is the set of all computable functions; B is the set of all uncomputable.

Theorem 3: Every computable space can be represented with a two codes space, such as: (0, 1), (Yin, Yang) or (Vacutron, Angultron). Note: Many college math books have proof on this theorem.

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IV: The ghost world of the mathematics universe

The concept of a ghost world in the physical universe is derived from a fact of physics --- the intrinsic spin. According to the corresponding principle, there must be a corresponding ghost world in mathematics.

In order to finish this discussion in a few pages, I will discuss this topic in a conceptual level and with examples. I will not provide leak proof definitions and theorem proofs.

Example 1: A = {0, ', +, *, =}. From set A, we can make some arbitrary definitions, such as: 0 = 0, 0' = 1, 0'' = 2, 0''' = 3, ... . F1 is a function and F1(0'''') = 0'', F1(0''''''''')=0''', etc.. F1, F2,... are functions which contain no symbols in addition to those in set A, and they are elements of set FA = {F1, F2,...}. The values from functions of set FA form set R = {0, 0', 0'',...}.

Definition 2: All mathematical functions are called function.

Definition 3: Set A has finite number of elements. Its elements are symbols of functions. Set A is called a base set.

Definition 4: All elements of set F are functions. F is a function set.

Definition 5: All functions of set FA contains no symbols in addition to those in base set A, FA is a base A function set.

Definition 6: The elements of set RA = {0, 0', 0'',...} are values of functions of FA. RA is a range of FA.

Definition 7: The elements in set LA are logical operators (L1, L2, ...) defined by any logic. Set LA is called logic set of base A.

Definition 8: F belongs to FA. L belongs to LA. P = LF is called a statement. P belongs to set PA

Definition 9: If set SA is the union of set A (base), set FA (functions), set RFA (range), set LA (logic), and set PA (statement), then SA is a system of base A.

Definition 10: P1, P2, ..., Pn are statements. If P = Pn with a finite number n, P is True in SA, otherwise P is False.

Definition 11: F is an element of FA. F is a legitimate function in SA if F has at least one value which belongs to SA.

For example F is the function of square root. F(2) is not in SA, but F(4) is. Thus, F is a legitimate function in SA.

Definition 12: H is a hole of system SA if H = F(x) not belongs to SA and F is a legitimate function of SA, or if H = P (a statement) and P is False.

So far, I did not invent a new mathematics but construct a new language in order to reinterpret an old theorem.

Kurt Godel's incompleteness theorem --- it states, "If any formal theory T adequate to embrace number theory is consistent and if the axioms of the formal system of arithmetic are axioms or theorems of T, then T is incomplete." That is, there is a statement S of number theory such that neither S nor not-S is a theorem of the theory. This incompleteness cannot be remedied by adjoining S or not-S as an axiom because there will be other statements unprovable in this enlarged system. This result applies to all mathematics schools --- logicism, formalism and intuitionism. Thus, no system of axioms is adequate to encompass, not only all of mathematics, but even any one significant branch of mathematics. There always exist statement whose concept belongs to the system, which cannot be proved within the system but can nevertheless be shown to be true by nonformal arguments.

This incompleteness theorem not only dealt a death blow to axiomatization of mathematics and placed a limitation on mathematical reasoning, but it is in fact the intrinsic property of reality. The prevailing interpretation among physicists and mathematicians about this incompleteness theorem is the conviction that this theorem places a limitation on any attempt to understand the ultimate nature of universe.

On the contrary, I see this incompleteness theorem is the only gateway to Ultimate Reality.

The above discussion points out three new concepts.

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V: Mathematics gives rise to physics

On the time sequence, it is the laws of physics which govern the evolution of the physical universe. Humans arose 15 billion years after the creation of those physics laws, and mathematics were created a few million years after the rise of humans. However, in order truly to unify physics and mathematics, mathematics must also be able to give rise to physics laws, that is, the laws of physics must be direct consequences of laws of mathematics.

According to the previous sections, the real mathematics universe is permanently entangled with the ghost mathematics world. For the sake of convenience, I will approximate the entire mathematics universe (union of the real and the ghost) by dividing it into three parts --- computable-, countable infinite- and uncountable infinite-universes.

Per theorem 3, the computable universe can be fully represented by a two code space, such as (Yin, Yang) or (Vacutron, Angultron). Thus, quarks cannot be the rock bottom building blocks of the physical universe. Prequark model (Vacutron, Angultron) is indispensable.

Someone might argue that there must have more layers of inner structure inside of the prequarks. Again, we can prove that that idea (more inner structure of prequarks) is wrong simply from the argument of mathematics.

If there has only one subparticle, named D, underneath prequarks, then we can consider two cases.

If there has two subparticles, named (D, E), underneath prequarks, it is definitely a tautology.

If there has three or more subparticles, named (D, E, F), underneath prequarks, then is there not only no reduction but an increase to complexity.

Thus, there is no reduction by adding one more layer of structure underneath any two code system.

Again, because there is an uncomputable universe in mathematics, the physical universe can never be represented with just two codes only. More dimensions are needed in the physical world to represent that uncomputable universe.

a: The finitude of physical universe

In 1823, Heinrich Olbers showed with a simple mathematical model that the night sky ought to be as bright as day if the universe is infinite in size and if the Copernican Cosmological Principle (CCP) is true. The fact that the night sky is dark means that those assumptions are wrong or there are some unknown factors at work, such as the universe is expanding or it had a finite beginning (not infinite in size).

In 1922, Alexander Friedmann combined CCP with General Relativity and predicted that the universe cannot be static. His prediction was ignored by the world (including Einstein) but was confirmed seven years later in 1929 by Edwin Hubble.

Since the universe is expanding, then it must have a finite beginning. In 1948, George Gamow put forward the idea of the Big Bang theory, and the Olber's paradox was put to rest. The expansion of the universe will dim the light (through red-shift) by a factor of about 2. The finite size of universe (that is, the universe is still young, having a finite age) gives darkness to the night sky.

b: Transforming infinities into finites

According to the Corresponding Principle, those mathematical infinities must be transformed into finites in the physical world by some processes because the physical universe is a finitude.

Then, how to transform the infinities into finites?

In mathematics, the uncountable infinity gives rise to the concept of indivisibility. Let us presume that the intersection between the set of nature number B and the set of irrational number IR is an empty set. Then, we can color nature number in blue and irrational in red. The Fermat's last theorem tells us that we can never find a blue number (z) while its nth power equals to the sum of nth power of a blue number (x) and the nth power of another blue number (y) when n is larger than 2. However, we do able to find a z, but it must be a red number. This theorem points out two very important points.

This indivisibility of mathematics must show up in the physical world. And, it does, as the reality of non-causality via the spooky action.

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VI: Dimensionality of the physical universe

Time is one dimensional in Equation zero which gives rise to 64 space dimensions (subspaces). Then, they are reduced to 11 dimensions with some internal relationships, such as prequarks, quark colors and generations. These 11 dimensions consist of 4 spacetime dimensions and 7 colors dimensions which are curled up into a topological structure, a torus. How can one dimensional time line transform into 64 or 11 dimensions? Is this transformation allowed by mathematics?

In 1870s, Georg Cantor was working on the number theory. He proved that the set of rational numbers, seemingly many more numbers than nature numbers, can be brought into one-to-one correspondence with the set of nature numbers. Then, he raised a nagging question: Can the real numbers be also brought into an one-to-one correspondence with the naturals? Soon, he found his theorem: The correspondence is impossible; there are, in this sense, more real numbers than there are naturals.

If there are so many points on the real line, how many might there be in a plane, in a three- or four- or n-dimensional space? He finally proved that every n-dimensional space can always be brought into an one-to-one correspondence with the one dimensional line, that is, one dimensional line can give rise to n-dimensional space.

This dimensionality issue can also be understood with fractal geometry. In fractal geometry, there are many space-filling curves, an infinite number of them to be exact. I will discuss one of them here, the Hilbert curve. The Hilbert curve is generated by starting with a square with one side being removed. Then that same figure is applied to each of its own three sides, and some erasing is done. After the iteration of the Hilbert curve is carries out indefinitely, the curve crosses every point on a plane without crossing itself. See the figure on the left.

With the concept of space-filling curves, Cantor's theorem of dimensionality in number theory is now generalized as a Shadow Theorem in fractal geometry. This Shadow Theorem illustrates how apparently random orbits may actually be the shadows of deterministic motions in higher dimensional spaces. This Shadow Theorem unifies three sets of opposites, thus produces three cornerstones for the real world. These three sets of opposites are random versus deterministic, high dimensions versus one dimension, and simplicity versus complexity. Again, the mathematics principle allows the rise of Equation zero.

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VII: The Essence of Mathematics --- the ball-donut transformation

Godel's incompleteness theorem constructs a process, and it shows that all mathematical systems which are invented by men who have arbitrarily chosen a few finite number of axioms and mathematical reasoning rules will always come alive and puke up at least one mathematical statement which is beyond the comprehension of its inventors. In short, the fire of God's spirit will always burn a hole on every mathematical system invented by men. This process is a ball-donut transformation.

In Topology, a topological ball can never be transformed into a topological donut by a continuous deformation process, but the growing process of mathematics above is not a continuous deformation process. This ball-donut transformation process of mathematics must manifest in the physical world. In fact, it does.

In physical world, the ball-donut transformation manifests on all levels.

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