What is a number? ( By Owen Prince)

A number is a name given to a symbol, that represents a group(class), that represents an address in a three dimensional space, that has a sequential list of order incremented by one plus one (counting), that locates an object (thing or unit) in that space. 

The question:  What is a number has been asked by many people over many years. In the past many famous mathematicians have tried to define what a number is and according to Bertrand Russell it can not be done.  I do not believe they were able to define it in a way that children can understand it.  In fact many adults do not know what a number is.   I will quote some of the famous  mathematicians from the Mathematics Encyclopedia Britannica that will show that they struggled with this concept.

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Bertrand Russell (Principia Mathematica 1910-1913)

"The number 2,...is a metaphysical entity about which we can never feel sure that it exits or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematical number 2 which must always remain elusive." Accordingly the following definition:

In this statement he claims that the number 2 is elusive and will always remain that way. He then makes an attempt to define number 2 after he said it cannot be done. If you were to define an apple, the way he defines a number, it would look like this.

The item of an apple is the apple of all those apples that are similar to it.

This does not define an apple and his definition does not define a number. The worlds’ best math minds struggle with this question. "What is a number?". Yet we expect children to solve this complex metaphysical problem when the experts have trouble with it.   To say an apple is an apple does not define an apple. You cannot use the same word in a definition to define the word. These masters of math were not able to define what a number is. They then tried to prove that 1+1=2 and that 2+2=4.

Alfred North Whitehead (1861-1947) (Principia Mathematica)

Principia Mathematica has been called one of the greatest contributions to logic. It set the pattern of symbolic language for mathematics. This took ten years and created three volumes to find what they call the proof of 1+1=2. The problem with this is  if you cannot define what a number is you cannot prove what you do with a number is correct.

Lacelot Hogben (1895)

"You will often hear people say that nothing is more certain than two and two make four. The statement that two and two make four is not a mathematical statement. The mathematical statement to which people refer...is as follows: 2+2=4 This can be translated to two add two to get four. This is not necessarily a statement of something which happens in the real world."

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Pawel Gora (10/27/95)

"This is an original question I have been asked by a student": Why is 2+2=4?

Pawel asked for help on the internet from the math experts. They gave some information about Alfred and Bertrand and then said it was out of their scope. One response was "when kids are told by their math teacher that proving why 1+1=2 is hard, it usually means two things: (1) the familiarity with concepts required to construct the integers is not attainable by elementary schoolchildren, and (2) aside from this, the teacher ... doesn’t know ...."

It is not an easy thing to prove 2+2=4 Before I can prove 2+2=4, I must prove what a number 2 is. First we have to find out what we agree on as truth. This will save time and give us a place to start debating the topics we do not agree on. We start from there and apply different tests to find truth.

Truth is: When all your senses send you the same message, and they agree with a law of nature (like gravity). The inputs are visual, logic and observation. To prove a statement true I must show it is true with one of the following:  logical geometry   or  algebra  or  counting and others must be able to reproduce the same results.

Next I will show the definition of a number and how to prove it is true.(page 205 )

What is a number?

A number is one group of one unit. ........ word sentence

1N = 1 group times 1 unit ......... Number sentence

1 N = 1 g x 1 u ............ Algebraic expression

N = g x d ............. Dots are the units

There are two parts to a number. Groups and counters. If you do not have both parts, you do not have a number. You would have a single line symbol that means zero. This is the basic primary fundamental formula in algebra therefore

a number is an algebraic expression. I put dots for the units to make the formula:

Number = group x dot

     N     =  g    x  d

I translated the word sentence into a number sentence and then into an algebraic expression. This is not as earth shaking as E=mc^2 but it is a major breakthrough. . I have constructed a truth table chart that shows how to prove 2+2=4 with logical geometrical construction and with the missing link dot symbols (counters) and with algebra. The number one is an algebraic expression. (1n= 1gx1d)

The dot is the number and we represent it with the single line symbol "(1)". If there are no dots, we represent it with the single line symbol "(0)" (p186). If we have zero dots (no units) then we must use zero in our formulas because we do not have a number. Zero is a place holder and not a number. We cannot multiply or divide by zero because it is not a number. It is a place holder that helps us define a single line symbol by its location. The single line symbol 2 can change its meaning by changing its location. It can be 2, 20, 200 etc. It is the zero as a place holder that can change the single line symbol 2 by changing its location. The same is true of groups and counters. The single line symbol 2 can be a group or counter.

The single line symbol 2 is a name we give to a group or to the total number of objects in a group. Think of a group as the space inside a circle or ball (excluding the circle or ball). The counters are all the things inside the space of a circle or ball. We can put anything of any size inside the circle or ball.

Names for counters are:

{an object} = {l} = {1d} = {1} = {1/1} = {undivided} = {one}

All these symbols are names and they mean the same thing. We use them to communicate an idea to other people. Other people must know and agree on what the symbols mean or we cannot communicate. The purpose of language and math is to communicate to each other our ideas and to understand the world and the universe.

There are two parts to a number. (P205) N = g x d

1) The first part is the group symbol. We use a small g for groups

2) The second part is the unit symbol. We use a small u or d for units.

Note: units = counters = dots = elements = things = objects

1. Groups: (Collection)       (p208 )

          Single line symbols for groups = 1, 2, 3, 4, 5, 6, 7, 8, 9

          (We can add or multiply groups to groups). (p70)

2. Counters: (Objects) ( formerly numerals)

          Single line symbols for counters: 1, 2, 3, 4, 5, 6, 7, 8, 9

          (we add or multiply counters to counters) (p68)

Note: We cannot add groups to counters we only multiply them. (p213)

The most important thing to understand at this point is:

Single line symbols for groups and counters are twins!!!

We must be able to tell the twins apart. The best way to do this is to assign them a location. We define a single line symbol as a group or counter by its location.

1.) Group symbol: If no units are given the symbol is a group symbol.

2.) Counter symbol: If the units are given the symbol is a counter symbol

Note: You can have groups with zero counters (empty space) but you cannot have counters without a group. We always put the counters into groups, or things into containers.

(Q). Why do we use single line symbols?

(A). It is faster to make a single mark than to write out words that are very long. It would take too long and too much time to calculate with words so we use a short hand that represents the words or things. If we do not know what the symbol means we cannot do the math.

(Q). Where did the number symbols come from? For example why do we use the symbol 4 to represent four things instead of a different symbol?

(A). I believe the original four was made with a stick in the sand. The symbol 2 was put on top of a second symbol 2 so they overlapped to make a four(see truth table). I have 300 fonts on my computer and they all have a different version of what a four should look like. The 300 types of fours I have on my computer were created by artists. They used artistic licence to change the math symbols to what they though they should look like. There is a real danger in the misuse of artistic licence. The following is a true story to explain the danger in the misuse of artistic licence.

Engineers and designers made a fleet of unsinkable ships. On the first trip to sea all the ships sank in calm water. They sank so fast that no call for help could be made. It took a long time to find what went wrong. The painter had moved a line on the side of the ships because he did not like how it looked. He felt that artistic licence gave him the right to move the line. He felt the designers had no appreciation for art so he moved the line to where he thought a racing stripe should be. The painter sank a fleet of unsinkable ships with a paint brush and killed every man on every ship. The line he changed was not a racing stripe - it was the overload line. The ships were overloaded before they left the harbor so it was a miracle they did not sink in the harbour.

The painter’s contempt for the people that work in the applied math field is wide spread. Many people in society call them nerds and other disrespectful names.

Math in the elementary grade is applied math used by the workers of the world.   Architects, engineers, draftsman and accountants are the workers of the world that make the world work. When they speak, everyone should listen.   They have said you must have some thing to add or you cannot add.  (You must have units or the  single line symbols (SLS) have a value of zero)

Conclusion:

The masters of math, the experts on the internet and teachers say they couldn’t define the number 2 or prove that 2+2=4. I was able to prove 2+2=4 three different ways. (1) geometry   (2) counting   (3)algebra.

(1) geometry: The symbol 2 placed over top of another symbol 2 make a symbol four.

(2) counting: The dots placed around the number symbol to define the value of the symbol.

(3)algebra: The formula that I designed to define what a number is. This formula conforms to the dictionary definition and to the examples in every math text.   The formula is:      N= g x d

A  Number is(=) a collection (group) of  things (dots)

I was able to define what a number is and how to identify its parts. The masters of math made three big books to explain their theory but very few could follow it and even less could duplicate the results. My work is on three pages and most people can reproduce the results, even children. 

Owen: (what is a number?)

A number is a name given to a symbol, that represents a group(class), that represents an address in a three dimensional space, that has a sequential list of order incremented by one plus one (counting), that locates an object (thing or unit) in that space.

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Webster’s New world dictionary defines a number as:

A number is a collection (group) of things (objects or units)

(A)means one  (is) means equal to (of) means times. 

A    number        is  one group        of  one thing (unit)

1   number         =    1    group        x    1    dot 

          N            =          g               x          d

Therefore 2dots = 1group of 2dots,     3dots= 1group of 3dots... etc.

Therefore 4dots = 2groups of 2dots, 6dots= 2groups of 3dots....etc

This means you must have both a group and some thing in that group to have a number.  If you subtract the units (thing) from the equation you would have N = g x zero therefore n = zero because any thing times zero is zero.  You must have some thing to add or you can not add.

zero means that there are no things or units. It is the starting point, beginning, or a point that separates. It is a place holder for single line symbols.

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The answer is DotMath for kids:

The "DotMath for kids" system was the result of my personal 40-year study.  I made my first version in 1966.  It explains how dots can help us understand math. The dots are the missing link between the objects we want to represent and the single line symbol that we make that we call numbers. Children believe that everything they are taught is true. This is why we must be able to prove our answers are correct. We can no longer just tell them to trust us on what we say is true. We must show them proof.

An example of an error in standard math presentation is the question: "solve the following 2+2=n". If I gave "four" as an answer in my engineering class, I would get it marked wrong. I would get a zero percent for giving the answer "4." If it is wrong in college and university, it is wrong in grade one. The true answer to the question 2+2 is zero because there are no units. You must always have some kind of unit or you cannot add or subtract. A number symbol without units is a single line symbol that means you have groups with nothing in them. So 2+2=0 not 4.

2groups (nothing) + 2groups (nothing) is 4groups of nothing or "zero (0)."

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I place dot patterns around the number symbols in a way that they do not touch the symbol. This is very important because you cannot add to or take away from a number symbol without destroying it. If you add anything to a number symbol, you would destroy it as a universal standard symbol. Numbers are recognized around the world by everyone as a universal standard.   You should not put anythng on top of the number symbols or you destroy the number symbols.   I made that mistake in my first version in 1966 and it took me many years to recover from that mistake.