These basics are not presented in as rigorous a manner as the axioms, but they serve to clarify and build upon the axioms.

#### **(1) A unit is anything that can be grasped as an entity.**

**(1) A unit is anything that can be grasped as an entity.**

In mathematics, the most fundamental idea is that of a unit. A unit is commonly represented by the numeral “1” (one).

**(2) A unit has the property that it can be counted one at a time. **

Counting provides natural numbers that are commonly represented by the numerals, 1, 2, 3, 4, 5, and so on.

**(3) If a number is n, then the next number is “n+1”.**

The next number is obtained by counting one more. This gives us the basic function of adding. Addition is represented by the sign “+”. Thus, the next number after 1 is “1+1” written as 2; the next number is “2+1” written as 3; the next number is “3+1” written as 4, and so on**. **One may keep on counting forward without limit.

**(4) All numbers are referenced from 0 (zero).**

Any quantity present is relative to no quantity. Therefore, all numbers are referenced from the idea of “no units.” This concept is commonly represented by the numeral “0” (zero).

**(5) Therefore, the number n is fully defined by “0+n”**.

Therefore, the number 1 is fully defined by “0+1”; the number 2 is fully defined by “0+2”; the number 3 is fully defined by “0+3”, and so on.

*(6) The number “ 0+n” is abbreviated as the positive integer +n.*

The number “0+1” is abbreviated as +1**; **the number “0+2” is abbreviated as +2**; **the number “0+3” is abbreviated as +3, and so on**. **The numbers +1, +2, +3, +4, +5, etc. are called positive integers.

* (7) If a number is n, then the previous number is “n-1”. *

The previous number is obtained by counting one less. This function of taking away (subtracting) is the opposite of adding. Subtraction is represented by the sign “-”. Thus, the number previous to 3 is “3-1” or 2; the number previous to 2 is “2-1” or 1**;** the number previous to 1 is “1-1” or 0.

**(8) The counts previous to 0 (zero) account for units that are absent.**

As mentioned in (4) above, 0 (zero) represents the reference point of “no units”. The number previous to 0 would be, 0-1; the number previous to 0-1 would be “0-1-1” or 0-2; the number previous to 0-2 would be “0-2-1” or 0-3, and so on**. **These counts define units that are absent (missing or lacking). One may thus keep on counting backward without limit.

**(9) For a quantity absent, the number is defined by “0-n”**.

“0-1”, “0-2”, “0-3”, etc., provide a count of units that are not there. An example would be a count of the money that one owes.

*(10) The number “ 0-n” is abbreviated as the negative integer -n.*

The number “0-1” is abbreviated as -1**; **the number “0-2” is abbreviated as -2**; **the number “0-3” is abbreviated as -3, and so on**.** The numbers **“**-1, -2, -3, -4, -5, etc.” are called negative integers.

*(11) Zero (0) is neither positive nor negative.*

Zero (0) is at the intersection of quantity that is present and the quantity that is absent. *The undefined concept in mathematics boils down to 0 (zero).*

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Mathematics consists of the finest of logic. The greatest concepts are that of 1 and 0.

Also see Space, Time & Knowledge

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Mathematics is a means of getting to the bottom, as Einstein proved.

But Love, as attention, can get you there, or even straight attention, (as Idenics) can do it. Or, likely a few thousand ways we are not familar with. It all adds up to love, attention, understanding equals God. At some point, my opinion.

Knowledge in all forms is useful. It is not one bit of knowledge against another bit of knowledge. It is how the whole of knowledge works together.

Knowledge should be laid out fully in the simplest form for everyone to see. That is my opinion.

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(1) The basics of mathematics boil down to dealing with something and nothing.

(2) Nothing (0) functions as the reference point from which comes the awareness of something.

(3) Something (1) functions as the basis of computation in the form of replication of itself. See Fractals.

(4) The first level of computation is counting where 1 is used as the basis of additional something.

(5) From counting comes the ideas of numbers and addition.

(6) Simultaneous with addition is the opposite idea of subtraction.

(7) Addition and subtraction give us positive and negative numbers with reference to 0.

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Vinaire,

What is it that you are trying to accomplish with these clarifications? Are you saying that based upon some axioms, these are conclusions that you draw? And if so, what do you believe that are you contributing to the understanding of mathematics that has not been put forth before?

Those are excellent questions, Ho Tai. These questions get me to examine my interests and intentions.

My interests seem to be as follows:(1) To explore the fundamentals of knowledge for inconsistencies.

(2) To present the consistencies among these fundamentals in the simplest of manner to forward education.

I see mathematics as a subset of logic. I find logic to be fundamentally circular. By that I mean that we essentially prove what we assume. In this process we build a structure that can be quite complex, but the basic criterion is that every element of that structure must be consistent with every other element of the structure. This may bring us to a certain precise set of axioms, which are used to ensure that consistency.

But where do these axioms come from? These axioms are probably the simplest of assumptions that are consistent with what we observe. We then build up on the axioms to explain more of our observations. Thus we build up a mathematical structure based on certain rules of consistency that do not fall outside the scope of the axioms. This leads us into the exploration of thought itself. The ‘fundamental particles’ of thought in any subject are its axioms.

If we can lay out the structure of a subject very clearly in terms of its ‘fundamentals’ so it is easy to communicate and understand, then we can make great strides in the field of education. It is that ease of communication and understanding that I am exploring, especially in mathematics.

Each person is different in their capacity to understand. So there may be the possibility of formulating the relations among these fundamental in different ways to get a person started on a path to understand the structure of a subject.

If one focuses on such formulation being absolutely right in the most impeccable sense in some standard way then it might lose its utility in education. The attention in education should be on bringing about an understanding that is relatively consistent at the fundamental level. Beyond that the education should focus on bringing about the ability in the student to spot inconsistencies.Actually, that is what we are doing at any stage of our learning. It is during the process of exploring some inconsistency that new discoveries are made.

What is important in education is not just imparting data, but educating the person to spot inconsistencies among data. If that is accomplished at the beginning of one’s education then the person would be able to establish better correctness of data for oneself. Thus, it becomes very important in education to communicate the fundamentals of a subject in the simplest of ways that they can be understood in a consistent manner.

I think that is where I am trying to go, even though my attempts may appear quite chaotic.

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Beyond visual perception lies the perception that is derived from mathematical modeling of experimental data. This is the perception we employ when looking at phenomena at atomic levels and cosmic scales. Quantum mechanics is based on mathematical modeling, and so are cosmic theories of Big Bang and Black Holes.

Mathematics seems to have started with Geometry, which has to do with visualization. Geometry creates space and forms, in a manner that is systematic and consistent.

Earliest use of mathematics has been for weights and measures (trade), surveying (architecture), calendar, (astronomy), and geometrical patterns (art). Numbers were needed for weights and measures.

A positive integer is said to be the primary undefined concept in mathematics. But I believe zero (0) to be primary undefined concept. Please see points (4), (5) and (11) of the OP above.

Also see, THE FUNDAMENTAL INCONSISTENCY

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To not drive ourselves insane, the zero and the irrational numbers, etc., have to be thought of contextually. Either that or let loose your anchor points because “kansas is goin’ bye-bye!”

I had quite a good success this week applying my new understanding of the “orders of operation” when coaching two of my children in math. This was for me a sticking point and also for my children. All three of us have new confidence when performing mixed operations.

That is wonderful!

Mixed operations have the following logic,

Shorthands for Counting.