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How to pull zero and negative intgegers out of a hat. Pt 1

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This is part 1 of a way to develop negative integers and zero from the natural integers 1,2,....

The point of this presentation is to show how mathematical concepts can be extended by abstraction and analogy, as opposed to connection with particular instances. Much of mathematics is produced this way. This presentation is very similar to the way Eudoxus (the greatest Greek mathematician after Archimedes) developed the theory of ratios for non-rational numbers. His development is the contents of Book VI of Euclids Elements). So button up, put on you thinking caps and join me in adventure of abstraction and analogizing.

Let N be the set of natural integers 1,2, ...

Let P = N x N (the cartesian product of N with itself, the set of pairs (n,m) where n, m in N).

We refer to the elements of P as Pairs

First we state the postulates we use for natural integers (they are derivable from the Peano Axioms, but we shall simply state them as postulates). + is ordinary addition of the natural integers, X is ordinary multiplication of the natural integers. = is ordinary numerical equality.

1. For n, m in N n + m = m + n (ordinary integer addition is commutative).

2. For k, m, n in N k + (m + n) = (k + m) + n (ordinary integer addition is associative).

3. For k, m, n in N if m + k = m + n then k = n

4. For n, m in N n X m = m X n (ordinary multiplication for the natural integers is commutative.

5 . For k, n, m in N k X (m X n) = (k X m) X n (associativity of X

6. For n in N, 1 X n = n X 1 = n (1 is the multiplicative identity)

7. For k, n, m in N k X (m + n) = (k X m) + (k X n) (distributive law for natural integers)

For the set of Pairs, P we shall define operations +' (an analog to integer addition), X' (an analog to integer multiplication) and =' (an analog to equality for natural integers).

First we define how =' works for Pairs. Let (m,n), (t, u) be in P. The we define

(m,n) =' (t,u) if and only if m + u = n + t. =' is not an identity operation like =, but it is an equivalence operation and we can define equivalence classes of Pairs. We can show

a) (m, n) =' (t, u) if and only if (t, u) =' (m, n) (symmetry of =')

B) (m, n) =' (m,n) (reflexivity of =')

c) (m, n) =' (r, s) and (r, s) =' (t, u) imply (m, n) =' (r, s) (transitivity of =')

We can then define equivalence classes of Pairs using these results.

Next we define an analog to +, written as +'. The operation +' operates are pairs of elements of P (pairs of Pairs, if you will). It is defined as follows:

(n, m) +' (t, u) is by definition (n + u, m + t). Note that it is kosher to use + inside

the parens since + is well defined for the natural integers and n, m, t, u are all natural integers. (see definition of the set P). Ditto for using X inside the parens.

(n, m) X' (t, u) is by definition ( {n X t} + {m X u}, {m X t} + {n X u} )

Thus ends part 1. In future parts we shall show how an arithmetic zero is defined from operations on Pairs and how arithmetic negatives are defined for Pairs. We will have (in effect) pulled zero and the negatives out of a hat --- sort of.

Further to follow....


Bob Kolker

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  • 9 months later...

I admire what you're trying to do. That said, I think it is better to talk about why adding the parenthesis doesn't mess up your addition problems, as opposed to saying that ordinary integer addition is associative. Be real with the people. That's just my opinion. To me, it seems like what is really going on is putting the parenthesis around a math problem like this -> {5+4} tells you that you'd better be careful to resolve what is inside the parenthesis before you do something else.

So, (5+4}+3 is different from 5+(4+3) because someone is telling you about the order in which you add things up. Now, what does it matter since, you're only adding things up. They all get summed together eventually right? That seems like a clearer way to say what is going on than to say addition is associative, but that's just me. I'm more than willing to have somebody tell me I'm wrong, and I should be eager to use the bigger words to describe it.

Maybe "more than willing" is an overstatement. I just don't have much use for long strings of letters and fancy jargon.

Another example is the distribute law for natural integers. The "distribute law" is a "law" because multiplying 10 by 2 is the same thing as multiplying 10 by 1 twice. Or adding up 10 twice. Or adding up 2 ten times. Six times Seven is the same thing as Seven six times. Or Seven 3 times and Seven 2 times and Seven 1 time all summed together. Because multiplication is shorthand for lots of adding. And exponents are shorthand for lots of multiplication.

Edited by Brian9
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The concept "number" as used in phrases like "zero", "negative number", "rational number", "real number", "complex number", is a misnomer. The proper mathematical term for these cases is "scalar", i.e., an element of a mathematical field.

Let me separate the conflated concerns a bit, for the sake of clarity.

First, you get counting numbers by simply counting, this is the simplest notion of measure one can discover -- how many crows flew off when the man entered the field? Oh, and really you only need to count to two, and have a means for holding place, to express any other counting number as a binary string.

Next, you get positive rational numbers, or fractional quantities, by using a unit that does not divide equally into the measured value. This is always the case when looking at statistically large agglomerations where the individual parts (e.g., atoms) cannot be resolved and one must resort to an approximation. In other words, one can form ratios of counting numbers. And the nature of these ratios is not the same as the nature of the counting numbers from which they are formed -- the conceptual "units" are different, the context of application is different: the fact that 4/2 corresponds to 2 does not imply any correspondence between the whole ticks ... just because we can put 2 and 2/3 on the same line does not mean they are commensurable -- all one can claim is the correspondence, one cannot count with rational numbers, nor approximate with counting numbers, per se.

Then comes so-called irrational (but not transcendental) numbers (?!). But these, again, have different context than do counting or rational numbers ... they are solutions of algebraic equations, and are only nominally related to the other classes of numeric concepts, via the crude, conflationary context of the so-called "real" number line.

Finally comes the transcendental numbers, which can always be approximated to arbitrary precision as finite sequences of rational numbers (so can irrationals, for that matter).

The most general context is not some imagined "real" line, but rather the fact that each and every number can be approximated with a finitely generated sequence of rationals. Rationals and counters happen to be wholly finite sequences, whilst irrationals and transcendentals cannot be precisely determined (what a surprise ...). The other perspective is solutions of algebraic equations, but does not capture the transcendentals.

Note that human sense perception reduces to counters and/or rationals ... nobody can observe an irrational number, nor a transcendental one, because measurements are by nature ratios to a unit.

As for negative numbers, they aren't scalars, they are vectors! A vector is a directed line segment; -1 means one step to the left, +1 means one step to the right (dropping the '+' in '+1' is facetious and confusing practice that allows the inveigling of young minds with the false notion of negative "numbers" as scalars). Note that mathematicians have managed to cast negatives as scalars only with a particularly irrational assumption that adding two scalar quantities together can lead to zero. There is no such thing as a negative quantity: quantity implies something, as opposed to less than nothing (!?!?). But of course it is easy to do the math with vectors -- go left one step, then go right a step, and you are back where you started, your net displacement is indeed vanishing (albeit, time has passed -- that is another thread). So negatives aren't really numbers.

The concept that mathematicians traditionally call "negative integers" is, properly analyzed, (isomorphic to) the set of equivalence classes of pairs (s,t) where s,t are counting numbers and the equivalence relation is defined, naturally, as:

(s,t)==(u,v) iff s-u==t-v

(in English, two pair are in the same equivalence class if and only if the absolute values of the differences between their respective element are equal).

Best way to visualize this: Consider a Cartesian grid, and draw the "diagonals", i.e., all lines parallel to x=y and crossing grid points. All the points on any one of these lines are equivalent, and the set of lines represent the integers. The mapping is pretty easy: x=y corresponds to the integer '0', and x=y+N corresponds to the integer +N, and x=y-N to the integer -N.

Hope this helps, but expect from experience it may light a fuse or two ...


- ico

Edited by icosahedron
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I've rebutted your assertion that -1 belongs on a different dimension of a vector from +1 on another thread. Basically, they are not orthogonal to each other. How someone who is (properly) willing to accept "imaginary" numbers can insist that -1 is fundamentally bogus is beyond me.

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A negative number permits the integration of via an act of cancellation.

For instance, in monetary matters, it allows you to count the multiplicity of your assets, count the multiplicity of your debts, and then integrate them together by comparing the results, one to another.

In physics, it permits forces working in different directions to be integrated into a single calculation, where motion in one direction is canceled out by motion in another without having to necessarily consider them separately.

Is the posulation method outlined in the OP really necessary to validate number?

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I've rebutted your assertion that -1 belongs on a different dimension of a vector from +1 on another thread. Basically, they are not orthogonal to each other. How someone who is (properly) willing to accept "imaginary" numbers can insist that -1 is fundamentally bogus is beyond me.

Hi Steve, thanks for noticing in both threads!

Check the other thread for more detailed comments, but suffice to say here that your rebuttal was based on a premise I question, and ask that you re-examine your conclusions in light of my thoughts.

For the sake of this thread, my argument in a nutshell: Any two distinct rays emanating from the same point are linearly independent in the simplest sense: linear displacements along one ray can never get you to a point on the other. The premise I question is that opposing rays somehow coalesce into linear dependence, whilst rays at a tiny angular displacement from opposition are linearly independent.

Here's the link to the related thread: http://forum.ObjectivismOnline.com/index.php?showtopic=18245&st=60

My detailed response to your rebuttal, Steve, is #77 in that thread.


- David

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