Integer

An integer is a difference between two multitudes, considered together as an ordered pair. The concept of integer is sometimes denoted by a symbol, $\mathbb {Z}$.

Examples of integers

In this section, I will give examples of concretes subsumed under specific integers, like -2, 0, 3, etc.

Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."

• If pile #1 contains 3 apples, and pile #2 contains 5 apples, then together they are a unit of the concept "-2." One may say that the signed difference between them is -2 apples.
• If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0." One may say that the signed difference between them is 0 apples.
• If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3." One may say that the signed difference between them is 3 apples.

Consider two sets of gold coins. "Set #2" is the set of gold coins which Bob has withdrawn from his bank account. "Set #1" is the set of gold coins which Bob has deposited in his bank account.

• If set #1 contains 6 gold coins, and set #2 contains 5 gold coins, then together they are a unit of the concept "1." His balance is said to be 1 gold coin.
• If set #1 contains 100,000 gold coins, and set #2 contains 200,000 gold coins, then together they are a unit of the concept "-100,000." His balance is said to be -100,000 gold coins.

Consider two points A and B on a ray. Let $L_{A}$ denote the distance from A to the origin, and let $L_{B}$ denote the distance from B to the origin. Choose the ordering of the set $\{L_{A},L_{B}\}$ according to which $L_{A}$ is first, and $L_{B}$ is second. Then

• If $L_{A}$ is found to be 17 inches, and $L_{B}$ is found to be 31 inches, then together they are a unit of the concept "14." One may say that the signed distance between A and B is 14 inches.
• If B is the origin of the ray, and $L_{A}$ is found to be 10 inches, then together they are a unit of the concept "-10." One may say that the signed distance between A and B is -10 inches.
• If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a rational number.

Consider a point P on a line with an origin. Let $L$ denote the number of centimeters between P and the origin which are to the left of the origin, and let $R$ denote the number of centimeters between P and the origin which are to the right of the origin. Necessarily, either $L=0$ or $R=0$. The tuple $(R,L)$ is an integer (if L and R are integral), and it is the linear coordinate of P.

Difference

The difference between two quantities A and B, is the quantity which would have to be added to the lesser quantity in order to make it equal to the greater quantity. In the case where A and B are the same quantity, we say that the difference between them is zero, or 0.

Examples

The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

The difference between a pile consisting of 5 apples, and a pile consisting of 3 apples, is 2 apples.

The difference between 3 and 5 is 2.

Non-examples

"A difference between me and my friend is that I like chocolate ice cream, but he doesn't." This is a perfectly valid use of the concept of difference, but it's not a difference of quantities.

Signed differences

[TODO this section needs a lot of work]

A difference is something that is identified with respect to two quantities. For a difference, the order of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.

A signed difference, which Objective Mathematics sometimes calls a sifference, is a concept much like a difference, except that it keeps track of the order the two quantities under consideration. Let A and B denote two quantities, where A is greater than or equal to B. The sifference between A and B is the difference between A and B; the sifference between B and A is the difference between A and B, but with a slight asterisk to remind us about the order.

Describing things like I have, in the English language, may give the reader a slightly incorrect idea, because "and" is often considered to be symmetrical. Indeed, "Bob and Jane" usually means the same thing as "Jane and Bob." In our context, however, it is very important to distinguish between the two noun phrases. This emphasis on the order of "and" is not completely foreign to English, however: authors know that at times, the subtle change in emphasis between "Bob and Jane" and "Jane and Bob" matters.

Symbolically, we write the sifference between $X$ and $Y$ as $X-Y$, and we write the sifference between $Y$ and $X$ as $Y-X$.

Examples

The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

in fact there is a symmetry between them.

Addition of integers

To add two integers $n$ and $m$ is to regard them as a single ordered pair $p$, such that the first element of $p$ is the sum of the multitudes in the first element of $n$ and the first element of $m$, and such that the second element of $p$ is the sum of the multitudes of the second element of $n$ and the second element of $m$. The resulting integer is denoted by $n+m$.

In symbolic notation, $(n,m)+(a,b):=(n+a,m+b)$.

Examples

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Negation of integers

To negate an integer $n$ is to reverse the ordering of the pair to which it refers. The negation of $n$ is denoted by $-n$.

The ordering is something that your consciousness added to the pair. So in some sense every integer (meaning every object referred to by an integer) carries a $\mathbb {Z} _{2}$ symmetry. It's like a gauge symmetry (change in description) rather than a true symmetry.

Examples

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Multiplication of integers

To multiply an integer $n$ by a natural number $k$, is to add $n$ to itself $k$ times. The number measuring the resulting difference may be denoted as $kn$ or $k\times n$.

We can extend this to multiplication of integers by integers, as follows. To multiply $n$ by 0 is 0. To multiply $n$ by a negative integer, $-k$, is to add the negation of $n$ to itself $k$ times.

In symbolic notation, we have described an operation$(n,m)\times (p,q):=(np+mq,nq+mp)$.

Examples

[TODO put tons and tons of examples here]

The traditional concept

[TODO needs a lot of work]

The integers are sometimes taken as an irreducible primary in mathematics. The mathematician Leopold Kronecker said

God made the integers, all the rest is the work of man.

and I think this quote is famous because some mathematicians sympathize with it.

Integers are literally taken as an irreducible primary by ZFC set theory, which has the axiom of infinity (TODO write something else. Calling this the integers, and not---say, the natural numbers---is an oversimplification.)

The Grothendieck construction

There's a construction I have seen called the Grothendieck construction, which is in some ways similar to Objective Mathematics' way of viewing the integers. The idea is to view the integers as the set $\mathbb {N} \times \mathbb {N} /\sim$, where $(n,m)\sim (p,q)$ if there exists some $k\in \mathbb {N}$ such that $(p,q)=(n+k,m+k)$.

I am sure that versions of this construction long predate Grothendieck, but it was made famous by Grothendieck because it's the starting point of K theory.