- I: Three mathematics schools
- II: The Corresponding Principle
- III: The computable and the uncomputable
- IV: The ghost world of mathematics universe
- V: Mathematics gives rise to physics
- VI: Dimensionality of the universe
- VII: The Essence of Mathematics --- ball-donut transformation

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.

- First, there are many undefinable concepts in mathematics.
- Second, all axioms seemingly arise arbitrarily.
- Third, there is a consistency issue in every school.

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.

- One, mathematics is no longer only as a tool or as a language for physicists to describe physics.
- Two, we can construct a complete physics by using mathematics as the foundation and the starting points.

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.

- Theorem A: All recursive functions are abacus computable.
- Theorem B: All abacus computable is Turing computable.
- Theorem C: All Turing computable functions are recursive.

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

- One, it has unlimited storage space.
- Two, it is an ideal machine and does not require electricity to run it. So, it can still do the calculation even if the universe would have collapsed.

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:

- the value of this function is ****, or
- this function is uncomputable,

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.

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.

- First, I will restate Godel's theorem with the new language I constructed above.
**Theorem 4**: If system Ar contains system Ap and system Ap obeys the incompleteness theorem, then system Ar has at least one**hole**.

For example, in a system which discusses the computability, the Church's thesis and the halting problem form a hole. At least one of them is unprovable.**Theorem 5**: If the system Af has an uncountable infinite number of elements, then Af has at least two holes.

**Definition 13**: Set T is the union of Af (which has an uncountable infinite number of elements) and its**holes**, T is called**totality**.

- Second, we can use Godel's theorem to build a process. We can start from system A1, and it has one unprovable statement S1. Then we can construct system A2 from A1 by adjoining S1 as an axiom. Per Godel's theorem, there will be a S2 which is unprovable in A2. And this process can go on with countable infinite number of times. See graph below.

Definition 14: Ar is a system. H = {h1, h2, h3, ...} and h1, h2, h3,... are holes of Ar. Ar is called a**real mathematical universe**(RMU in short). H is called the**ghost mathematical world**(GMW) of Ar.

Definition 15: Ar, Ap are RMUs. Ap contains Ar. Sr is an unprovable statement in Ar but provable in Ap. Sr is called a**mathematical quanta**.

Definition 16: Sr is an unprovable statement of Ar. Ap is constructed as the union of Ar and Sr. The process of constructing Ap is called**System Construction Process**(SCP in short).

Definition 17: Ar is a system which obeys Godel's theorem. AF is a system constructed after repeating SCP a countable infinite number of times, AF is called**Final System**.

**Conjecture 1**: T (totality) contains AF (Final System).

The above discussion points out three new concepts.

- First, there are
**holes**in mathematics. Those holes form a mathematical ghost world which is a shadow of its associated mathematical system. In fact, any mathematical system is permanently entangled with its ghost world. - Second, there are
**mathematical quanta**. With mathematical quanta, the big bang of mathematical universe arises. - Finally,
**totality**and**final system**are defined. The final system is the largest mathematics system can ever be constructed by any intelligent being, but the totality is still larger than the final system.

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.

- One, if V = D and A = D, then V = A. It is a contradiction to the Prequark Model.
- Two, if V = D, A = (D, D), ..., then A does not carry electric charge. Electric charge must be represented in another way. Then, it turn out to be a two code system again. It is a tautology.

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.

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.

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?

- One, it takes countable infinite number of steps to trisect an angle. That is, the countable infinity is transformed into finite as
**Angultron**via a**trisecting process**. This process also gives rise to the followings:- Space is trisected into three seats.
- Time is trisected into three generations.
- Charge is trisected into 1/3 e.
- Intrinsic spin arise. That is, this trisecting process also gives rise to Planck constant (h).

**measurable and causal universe**, the Prequark Model. - Two, the uncountable infinity can be transformed into finite via a soft compass (Archimedes spiral). This process creates time. The projection of this Archimedes spiral is Einstein's time cone.

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.

- One, the blue number system is not complete, that is, the value of some operations will fall out of its boundary, In short, the blue number system has
**holes.** - Two, the number system is
**indivisible**. That is,it cannot truly be divided into red or blue numbers. The red and the blue numbers are permanently confined to or entangled with one another.

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

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.

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.

- For biological life, a single fertilized egg turns into a tube-like system.
- On the scale of Cosmos, there are two holes --- the eternal past (before the Big Bang) and the eternal future.
- On the micro-level, there are seven color dimensions for elementary particle, and these seven color dimensions are result of the ball-donut transformation.