The Linguistics Space (III) --- the New Mathematics

In the "Linguistics Manifesto", I stated that linguistics is roaming in a space which is composed of FGL (Formal - Godel - Life) system. That is, the Life system is the extension of the Formal - Godel systems which are both mathematics. Yet, there is no such a mathematics the same as the Life system thus far. The rule of game for both Formal and Godel systems is consistency while the rules for the Life System are,
2. reigning in infinities (renormalization).

I. What is renormalization?

It is a physics term, invented for quantum electrodynamics (QED). There are, at least, two types of renormalization.

a. Scale Renormalization

It was developed, at least, in two stages.
1. To avoid the divergence for a physics equation which has inverse-square. This kind of equation diverges when the distance between two interacting particles approaching to zero. Although infinities are understood mathematically, it cannot be handled or understood in physics. Any diverged physics equation has no meaning to physicists. Thus, if a physics equation is tested out valid before it diverges, our choice is to keep the equation by cutoff the diverging part. Of course, it takes a bit intelligence to know that where is the cutoff point. The game of finding the cutoff point is called "renormalization".

2. As an indicator to discover new physics. If a physics equation diverges at a point, it must become invalid after that point. That is, beyond that point, there must be a new physics, and we call that point, the phase transition point. For example, the equation of measuring the distance between two points (A, B), D = V (speed/hour) x T (time needed to get from A to B). Now, we have the following theories.
1. Air plane theory -- D = 500 miles / hour x the flight time. This theory is valid between all air ports. Yet, it cannot measure the distance between three blocks in a city.
2. Car theory -- D = 35 miles / hours x traveling time. This theory can measure all distances among city streets. It can even replace the air plane theory until it reaches Oceans.
3. The foot theory -- D = foot x number of steps. Both air plane theory and car theory fail for measuring the distance in a house.
This example shows that the original game of cutout divergences becomes a tool for discovering new physics.

b. Self Interaction Renormalization

Besides the scale divergence, there is another process, the self-interaction, which can create infinities. While the self-reference in the formal system is, often, creating paradox, the self-interaction is creating divergence. While there is no genuine way to eliminate the self-reference in formal system (based on Godel's theorem), the self-interaction is the essence in physics (all physical processes). For example, I am writing this paper "now" with the knowledge which I learned 30 years ago while my plan is to make this paper as a part of human heritage forever. That is,
1. I am recalling a person who existed 30 years ago, and
2. I am projecting a person who disappears for thousands years in the future.

Perhaps, many readers will discount the above as philosophical nonsense. Yet, this kind of self-interaction is the essence (the vital part) of every physical process. Let us look at an electron-electron scattering (colliding) process.
1. Electron is represented as a simplified wavefunction, with WF (wave front), WB (wave body) and WT (wave tail).
2. the location of the electron is identified by time coordinate, e(t1) = {WF(t1), WB(t1), WT(t1)}
3. the motion of the electron is defined with the following equations.
• WF(t2) = WF(t2), a new front
• WB(t2) = WF(t1), new body enters into the space of old front
• WT(t2) = WB(t1), new tail enters into the space of old body.
4. the duration of the collision is 10 time units (from t1 to t10).
5. the interaction radius is also 10 time units.

Now, an incoming electron (ie) sees the above electron e(t1),
1. at 10 time units away at t1 as {WT(t1), WB(t1), WF(t1)}
2. at t2, ie sees {WT(t1), [WB(t1), WT(t2)], [WF(t1), WB(t2), WF(t2)}, the e(t1) is still in ie memory while it sees also the e(t2).
3. at t3, .... t10.
That is, ie sees not a single electron but many, and they can be described as many virtue particles which is the result of "self-interaction". The summing over those virtue particles (self-interaction) can often lead to divergence. In fact, the self-interaction (the changes from e(t1) to e(t10)) is more complicate than the above description and can involve the following processes,
• vacuum polarization -- a photon creates a virtual electron-positron pair which then annihilate,
• self-energy -- emits and reabsorbs a virtual photon,
• vertex renormalization -- emits a photon, a second photon and reabsorbs the first.
• etc.
The above self-interaction processes can all create divergences, and they must be renormalized for any process to have a physical meaning.

II. Renormalization in Mathematics

Renormalization was "invented" as a trick to cutout the divergences in physics. Even its inventors were uncomfortable about it.

• Dirac wrote, "Most physicists are very satisfied with the situation. They say: 'Quantum electrodynamics is a good theory and we do not have to worry about it any more.' I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small -- not neglecting it just because it is infinitely great and you do not want it!"
• Feynman wrote, "The shell game [cut space into shells] that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate."
That is, the infinities are not just concepts in mathematics but are realities in physics. Yet, they cannot be deal in physics besides the tricks of cut them off with a game called renormalization. For the scaling divergences, they might not be caused by infinities but are caused by some phase transitions. However, the summing over self-interaction do involve infinities.

While Dirac said, "... not neglecting it just because it is infinitely great and you do not want it!" nonetheless, Dirac must accept the defeat. Mathematically, there "is" no way to transform an infinity into finite (which is the bottom line in physics as physics can never deal with infinities). In order for physics to handle infinity, all infinities must be able to transform into finite mathematically. So, the cutoff games are needed no more. Thus, there are two different types of renormalization now.
1. In physics, the cutoff games.
2. In mathematics, the transformation of infinities to finite. Yet, no such math is known thus far.

If a thing "TA" is the product of an infinity to finite transformation, then TA is the concrete representation of that infinity. Can we find such a TA?

If such a TA does exist, it must be produced with "infinite" steps. Then TA and infinity (TA) are two sides of the same coin. That is,
• From TA, we get infinity (TA)
• From infinity (TA), we get TA.
In mathematics, there are two kinds of infinity, countable and uncountable. There is a commonly accept belief that there is no way to trisect an angle in Euclidean geometry. Yet, a trisected angle is always a physical reality. In fact, if we can evenly divide an angle, we can always trisect that same angle with the following process.
1. Divide angle A evenly.
2. Divide the two angles evenly again.
3. The top and the bottom angles are now 1/4 of A. As they are symmetrical, we will look the top angle AT only.
4. The middle angles are divided evenly again. They become 1/8 of A.
5. Let AT plus 1/8 A, and it is larger than 1/3 A.
6. Divide the 1/8 A evenly again, 1/16 A.
7. Let AT (1/4 + 1/8) minus 1/16 A. "AT" is now smaller than 1/3 A.
8. the above process goes countable steps.
Then, AT = 1/2 - 1/4 + 1/8 - 1/16 + 1/32 - 1/64 + 1/128 - 1/256 + 1/512 - 1/1024 + 1/2048 -... +...
= .33349 - ... + ... = .3333333333333..... = 1/3

With countable steps, the above process trisects the angle A "exactly". That is,
• with countable infinity, we get AT,
• With AT, we get countable infinity.

That is, "AT" (the trisecting angle process) has "renormalized" the countable infinity. For the countable infinity, it can show up as a finite and concrete entity as a trisected angle. This fact is described in detail in the book "Super Unified Theory" (Library of Congress Catalog Card number 84-90325, with the US copyright number TX 1-323-231, issued on April 18, 1984). This book is available at,
Then, can the uncountable infinity be renormalized? That is, can we find a concrete object which encompasses the uncountable infinity? The answer is, of course, Yes. Please read the article "Metaphysics of Linguistics" at http://www.chinese-word-roots.org/cwr018.htm

However, we can address this uncountable issue with a different question. We can reach countable infinity by counting. How can we reach uncountable infinity, with what process?

With the above trisecting angle procedure, we know that 1/3 = 0.333333..., that is, 1/3 has countable number of digits. In mathematics, we do know that the number pi (=3.14159...) is a normal number, that is, it has uncountable number of digits in it. Then, how can we reach pi (precisely)? Well, we can reach it with the following equation.

Equation A:
The circumference of a circle with a radius of 1/8 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ... + ... (with "countable" infinity steps)
= pi / 4

That is, the number pi (which contains uncountable number of digits) is reached with a "countable" steps. So, the uncountable infinity can be renormalized with countable infinity, and countable infinity can be renormalized with the trisecting angle procedure.

Note 1: There is something interesting in equation A. The circumference of a circle with a radius of 1/8 (with an even number denominator) can be reached only with a sequence of odd number denominators.

Note 2: The area of this circle = pi / 64; then, the number "64" must be fundamental for this uncountable infinity renormalization process.

Now, "renormalization" is no longer a term of physics but is a term in mathematics. Both infinities can be renormalized, that is, be concretized. Now, any physics theory which does not encompass these two infinities can never be a valid theory at the end.

The key rule in mathematics is consistency, noncontradiction. Thus far, no mathematics is centered with contradictions. In my previous articles, I stated that one of the key feature of Life system is mutual immanence; the contradictions are the sources (roots) of one another. For many readers, it is a kind of philosophical arguments. However, they are, in fact, the result of a new math.

The major contradiction in math is the following statement.
If a, b both are real numbers, and if a - b = 0, then
a <<>> b (a not equal to b)
That is, a - b = 0, yet a <<>> b. Of course, this situation can arise in some special mathematical cases, not while a and b are real numbers. When this contradiction becomes the "general" consequence, a new mathematics arises. In this new math, the concept of continuity must be changed. For every "point" of real number line, it is no longer "one" number but has many numbers (colored numbers). The consequence of this new number system leads "directly" to the "renormalization" of the infinities. That is, the renormalization and the mutual immanence are mutually immanent, again, the two sides of the same coin. This new math (colored numbers) is described in detail in the article "The Philosophical Meanings of Fermat's Last Theorem" at http://www.prequark.org/Nf4.htm

Conclusion

Now, the essences of the Life System (
1. Mutual immanence -- encompassing all contradictions. In fact, contradictions are the sources of each other.
2. Renormalization -- all infinities are having concretized forms in mathematics
) are now mathematics.

In the current paradigm, there is no way of any kind to transform infinities to finite (numbers). In this new mathematics, infinities can be renormalized (concretized) into concrete objects.
1. countable infinity -- concretized as trisected angle which is a physical concrete object.
Note: the hinge of this renormalization is the number 3 or 1/3.
2. uncountable infinity --
• the first renormalization -- concretized as geometrical circle, a concrete object.
Note: the hinge of this renormalization is the numbers of pi, 64 or 1/64.
• the second renormalization -- concretized as a torus. See "Metaphysics of Linguistics".
Note: the hinge of this renormalization is the number 7 (the Heawood conjecture).

If the physical universe is the concretization from infinities and is returning to infinities during this expression (expansion), then it is hinged on four numbers (3, pi, 7, 64). That is, the final physics theory on universe must have those four numbers as its essence. If these four numbers do not show up in a physics theory (such as Higgs mechanism or the whatnots) as fundamental parts of it, then that theory can never be the final theory.

With this new mathematics, the "Linguistics Manifesto" is now complete. And, any physics theory which does not encompass the principles of this Life System can never be a valid theory at the end.