From the cognition point of view:

As a kind of mathematical language, the roles it plays are decided both mathematically and linguistically.

As a basic numerical (number) element, it locates at the right position in the Table of the Numerical Elements (Number Spectrum) such as Table of the Chemical Elements in chemistry and Light Spectrum in physics.

Following three questions are important for us to understand what 0 (zero) is:

1, How does 0 (zero) exist as mathematical language with or without numerical value meaning itself?

2, What position dose 0 (zero) locate in the categorical spectrums of numbers?

3, Do we need a negation of 0?

Zero means different things, depending of the structure we consider around it. Take the classical tower of structures N \subset Z \subset Q \subset R \subset C. In N, zero is the first element, and it generates N using only the successor function. In Z, zero is divisible with all integers, because it belongs to all ideals of this ring. It is not just the hasard that in all languages one says "zero cats" for no cat, and not "zero cat". In particular zero is an even number. In Q, R, and C zero is the only non-invertible number, and one found the infinity to express 1 / 0. In R and C one can start with the affine line or the affine plane, and put the zero where ever one whished. If zero chosen, one direction and one unit, then we get back R or C. So here zero becomes again a kind of generator. There are just some roles of the object called "zero" from a potential infinity.

Dear Geng Ouyang,

In the axiomatic bulding up of number theory (set theory) 0=empty set, 1={empy set}, 2={empty set,{empty set}},....  This means that zero in mathematical point of view can be identified  with a set the so-called empty set which is a non-existent set in empirical point of view. My opinion is that the mentioned two concepts hold the same information, so they are agree.

This is an important concept.  For instance if we have a rotating dipole it creates a field around the dipole that oscillates.  However if there is nothing around it or it is in empty space there is no way to tell if it is oscillating and therefor the field is static or may be considered to not exist.  My point here is that the definition may change given the way we look at the particle.  From an observers standpoint there is no way to tell if we are moving with respect to the object or the object is static in space as we are.  Both ways to look at the object give us different conclusions as to if there is a moving field or no field moving in space.

This is one of the reasons that it is so hard for us to make headway in unifying a field theory.   

There is no absolute in the area of Zero, motion or rotation.

I like this question.

Dear friends, thank you.

"Zero” is there in our science and in our life; we can develop our Zero relating knowledge in several directions.

Sincerely yours,

Geng

Zero has to have a relativistic meaning.  In other words it has to be relative to the observer and their location relative to the thing they are measuring.

You are right, dear George.

Our above researchers have presented many different observation angles (directions) to Zero, if we can systematically seek them out and develop; each will brings us fruitful result.

Geng

Dear researcher friends,

The topic we discuss here is not only mathematical but very philosophic too, hard and challenging.

Sincerely yours,

Geng

I observed that many posts here are more or less of physical nature, so I come back with another illustration of the relativity of zero. It is about temperatures and termodynamic. 0 °C is roughly speaking the temperature needed for water to go into the solid state ice. 0 °F is the temperature needed to cool brine. 0  °K and 0 °R have an absolute nature - the temperature where there is no atomar vibration at all, in any material body. This definition is hard to be harmonized with new research in quantic physics and already contested. It is not the point now to discuss what could mean -1 °K - but just to recall the principle. Take an affine line, choose one zero point, one direction and a unit (a second point in this direction) and you get the field of real numbers. Of course, every zero defined above (and also the units) have a lot of significations and it have been chosen accordingly. They are very useful for the corresponding domain of interest.

It is much more difficult to speak about velocity zero in an universe where all moves (and all rotates...) Here again all is just about choosing a system of reference. its origin is called "zero"!

Dear Mihai,

It is true just like what you say in your last post, that in many places zero serves as a kind of reference (generator, starter, beginner, origin, …) in a scientific system (a chosen system). Its “location” may be in the middle of left-right or up-low ------it is us human that give “something” in universe a name of zero in a mathematical language system（we could even just give it another name like ” generator” or “middle” or… ）.

As a kind of reference, this “zero” serves as a kind of number and locates in the middle of somewhere both in universe and our mind------in number spectrum of our mathematics.

Sincerely yours,

Geng

The above discussions remind us an important relationship between cognitions and languages.

Different “languages (such as mathematical language, physical language, literature language, … )” are needed to let us human know what human sensory cognitions (scientific or non-scientific) have been got. People have been trying all ways to build the bridge between cognitions and languages.

Mihai's examples of temperature measurement and (more generally) of measurement by marking an initial point as "0" and by marking a second point as the unit "1" of measure, clearly illustrate the respective role of these numbers. This is highly compatible with a mathematical viewpoint, where "0" and "1" are simply the names given to numbers with are neutral for addition, respectively, multiplication. This terminology extends to all algebraic structures which can be considered as a "number system".

One might say that "0" and "1" are special numbers because they calibrate the system. In case of measurements of any kind, 0 and 1 each represent the outcome of some agreed default situation. As to "0", this default is often some kind of failure (zero distance, zero dollars on your US bank account). But "zero degrees Celcius" and  "zero-th floor" (downstairs, at least in British English) are of a different nature.

It seems that "0" is only occasionallly related with "non-existence" and even then, it is usually some kind of agreement.

Can we put 0 (zero) into three categories:

(1) A kind of reference ------- generator, middle, neutral, beginner, origin, placeholder, marker,…,

(2) Absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness,

(3) Relatively non-existent ------- with numerical value meaning, subjectively nothingness, the approximate nothingness, the result of infinitesimal limit , ….

Dear Wes, thank you for your constructive opinion.

I think as a mathematical language, zero is illustrated by following “zero tree” through out our mathematics history.

(1) a kind of reference ------- generator, middle, neutral, beginner, origin, placeholder, marker,…,

(2) absolutely non-existent ------- without numerical value meaning, the negation of being, objectively nothingness,

(3) relatively non-existent ------- with numerical value meaning, subjectively nothingness, the approximate nothingness, the result of infinitesimal limit , ….

The above “zero tree” give us three different usages (meanings) of zero. Absolutely nothingness plus absolutely nothingness or their repetitions is absolutely nothingness, and negative numbers are in category (1).

May be people put category (2) and category (3) in one category because it is believed that one is included in another.

Regards, Geng

Dear Wes，

Zero (nothingness, NON-EXISTENCE) doesn’t exist on its own but exists as a special form of number------mathematical language created for us human to understand and describe things in universe.

Zero is NON-EXISTENCE as a special form of number in the categorical spectrums of numbers (number tree). But in present infinite related number spectrum (number tree), several forms of number are absent (including SOMETHING which is the newly discovered “complete set for ABSOLUTE EXISTENCE” opposite to “zero set for ABSOLUTE NON-EXISTENCE”) and negative numbers are in category (1).

My regards, Geng

One thing has been troubling many people for long: at the time when 0 was born as a kind of number in our science, must it be proved  mathematically or just came out as needed and staying there without proof?

There is an excellent article in Wikipedia on the history of "zero"

http://en.wikipedia.org/wiki/0_%28number%29

It carefully distinghuishes the two uses of "zero": as a placeholder in a positional numeral notation, and as a value. The first has a complex and ancient history (obviously without proofs) whereas the second seems to date from medieval times in Europe.

Anyhow, as zero is a concept, it goes with a definition or agreement. It needs no mathematical proof for justification. It did trigger some discussions of a philosophical nature ("naming emptyness"). The mathematical "universe of discourse" is independent of reality.

Dear Marcel，thank you for your helpful offer.

But 0 is a kind of number, too. We have many different kinds of number forms: 0 (zero), natural number, fuzzy number, …, but at the time when they were born in our science, must they be proved mathematically or just came out as needed and staying there without proof?

My regards, Geng

Dear Wes,

I think your above two posts are very interesting with new ideas in analyzing one of the mathematical meanings of 0; you start a new way (linguistics ways, parse tree?) in the fundamental research on some mathematical concepts. For sure that any concepts（including mathematical concepts）have their primitive linguistics roots, it is wonderful you are on the way to build a bridge between linguistics and mathematics theoretically and practically.

Yours, Geng

A message board may hold the message "no messages today". Some consider this a joke or a contradiction, but it is meaningful communication. "Empty set" and "zero" play similar roles in mathematics.

Abstraction is really a very important characteristic of mathematics.

I would like to add in this discussion the logarithmic relationship of Zero:

Logarithm [ to any base] of 1 is defined as zero.

This zero point is met in the Archimedes Spiral [or Logarithmic Spiral] , where spiral starts.

http://www.stefanides.gr

Dear Mr. Panagiotis Stefanides,

Do you think this kind of 0 be called “placeholder”?

Yours, Geng

>....Do you think this kind of 0 be called “placeholder”?...,

Dear Geng, thank you.

 According to :http://www.stefanides.gr/Html/why_logarithm.htm,

it is just the zero crossing point of an  X-Y  polar  axes  plane  of reference

of vectors [ on the curves of Spirals to various logarithmic bases- not crossing zero point]

related to corresponding logarithms [ vectors on the Logarithmic  Spiral- crossing zero point].

Regards

Panagiotis Stefanides

Dear Panagiotis,

I visited the website you offer, thank you for a new idea with a very demonstrative figure.

Yours, Geng

Dear Geng, thank you.

Panagiotis

With numerical value meaning, without numerical value meaning, as a placeholder, …, in its own way, 0 carries information in our science and in our life.

Yes we have a -ve of zero. New interpretation for zero. new philosophical view merging infinity, imaginary number and zero

Article New Philosophical View Merging Infinity, Imaginary number an...

Dear Dr. Salah A. Mabkhout, thank you.

Can the Zero Theory you introduce in your paper above solve the newly discovered Harmonic Series Paradox which has much to do with “zero related infinitesimal”?

Following divergent proof of Harmonic Series can still be found in many current higher mathematical books written in all kinds of languages:

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                     （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．       （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．                （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                               （4）

Each operation in this proof is really unassailable within present classical “actual infinite--potential infinite” related science theory system. But, the unavoidable practical trouble is how many items in （1） can be added up by “brackets-placing rule" to produce the infinite items in （3） each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or…?

How to deal with those items of Un--->0 in Harmonic Series decides whether or not we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

If those items of Un--->0 are treated as “with-quantitative-value 0”, we can produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule"; if those items of Un--->0 are treated as “without-quantitative-value 0”, we can not produce infinite items each bigger than 1/2 or 1 or 100 or 100000 or 1000000000000000 or… from Un--->0 infinite Harmonic Series by “brackets-placing rule".

Your idea on the above “with-quantitative-value 0” and “without-quantitative-value 0” would be very much appreciated. Thank you again.

Best regards, 

Geng

In fact, Zeno’s Achilles--Turtle Race Paradox has became the greatest paradox in human science and mathematics:

1, it has never been solved and suspended for more than 2500 years.

2, it is a huge paradox family with so many family members.

3, its family member (Harmonic Series Paradox) has been proven by the strictest mathematical way:

In present mathematics, it is “scientific and reasonable” to use the well known Oresme's “bracket computing operation” of creating infinite items each greater than 1/2 or 1 or 100 or 1000000000000000 or 1000000000000000000000000000000 or ... from the Harmonic Series to change the Un--->0 Infinite Harmonic Series into a “Vn ＞any positive constant” infinite series (such as the positive constant of 1000000000000000000000000000000):

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1 + 1+ 1 + 1 + 1 + 1 + 1 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 100 + 100 + 100 + 100 + 100 + …--->∞

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ... ＞ 1000000000000000 + 1000000000000000 +1000000000000000 + 1000000000000000 + …--->∞

…

Thank you for your idea dear Marcel Van de Vel

1， The idea of Oresme's proof is modern limit theory.

2， The result of Oresme's proof can be “now transformed in a completely finitely argument”.

3， My work care for “the idea of Oresme's proof-------the foundation of modern limit theory”.

4, Whether the Harmonic Series is convergent or divergent or something else is another issue and will be discussed in the future.

Sincerely yours,

Geng OUYANG

Nine of my published papers have been up loaded onto RG to improve the work:

1，On the Quantitative Cognitions to “Infinite Things” (I)

https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I

2，On the Quantitative Cognitions to “Infinite Things” (II) --------The Mistakes and Defects in the Classical Definition of “Infinite” and the 2500--year Obstacles in Quantitative Cognitions to “Infinite Things”

https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II

3，On the Quantitative Cognitions to “Infinite Things” (III) ---------Fourteen Challenging Conundrums of Infinite Things’ Quantitative Cognitions Produced by the Absence of the “Infinite Carriers’ Theory” and the Confusion of “Potential Infinite” and “Actual Infinite”

https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III

4，On the Quantitative Cognitions to “Infinite Things” (IV) ---------“Infinite”, “the Infinite Mathematical Carriers” and “the Infinite Number Forms” in Mathematical Analysis

https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV

5， On the Quantitative Cognitions to “Infinite Things” (V) ---------Solving eight challenging “quantitative cognition” conundrums in present infinite set theory

https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V

6， On the Quantitative Cognitions to “Infinite Things” (VI) ---------A New Way Solving Fatal Defects in the Foundations of Present Classical Set Theory and Its Quantitative Cognizing Tool of “One to One Correspondence”

https://www.researchgate.net/publication/331428737_On_the_Quantitative_Cognitions_VI

7， On the Quantitative Cognitions to “Infinite Things” (VII) ---------The Underlying “Self-refutation Mechanism” Origin of “Infinite Paradoxes Syndrome” and an Exactly Same Paradoxical Phenomenon in Present Set Theory and Mathematical Analysis

https://www.researchgate.net/publication/335110516_On_the_Quantitative_Cognitions_to_Infinite_Things_VII

8， On the Quantitative Cognitions to “Infinite Things” (VIII) ------------ Three Exactly Same Fundamental Characteristics in the Old and New Generations of Infinite Set Theory and Mathematical Analysis

https://www.researchgate.net/publication/341754494_On_the_Quantitative_Cognitions_to_Infinite_Things_VIII

9， On the Quantitative Cognitions to “Infinite Things” (IX) --------- "The Infinite Carrier Gene”, "The Infinite Carrier Measure" And "Quantum Mathematics”

https://www.researchgate.net/publication/344722827_On_the_Quantitative_Cognitions_to_Infinite_Things_IX_---------_The_Infinite_Carrier_Gene_The_Infinite_Carrier_Measure_And_Quantum_Mathematics