Re: a mathematical question .. unanswered !!!



RP wrote:



Richard Herring wrote:

In message <yuednUpAOKregUzeRVn-vw@xxxxxxxxxxxxxx>, RP <no_mail_no_spam@xxxxxxxxx> writes



Richard Herring wrote:

In message <rlYzf.163306$vl2.18559@xxxxxxxxxxxxxxxxxxxxxxxxx>, Hexenmeister <vanquish@xxxxxxxxxxxx> writes


<mmeron@xxxxxxxxxxxxxxxxxx> wrote in message
news:VZRzf.23$25.2120@xxxxxxxxxxxxxxxxxxxx

In article <UnRzf.504321$zb5.485701@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Michael J. Strickland" <qualityser@xxxxxxxxxxxxxxxx> writes:

<mmeron@xxxxxxxxxxxxxxxxxx> wrote in message
news:HLbzf.15$25.1200@xxxxxxxxxxxxxxxxxxxx

In article <RW9zf.491417$zb5.158732@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Michael J. Strickland" <qualityser@xxxxxxxxxxxxxxxx> writes:

<mmeron@xxxxxxxxxxxxxxxxxx> wrote in message news:QUQyf.9

...

You start with natural numbers alone and define addition and
multiplication the usual way.  Then you define negative numbers
additive inverses of positive ones, i.e a + (-a) = 0.  This is a
definition.


So then:
       (-a) = 0 - a
       (-a) = 0 + (-a)


This is a circular reference.




Nope, nothing circular about it. 0 - a is just a shorthand for 0 (-a) 





We are attempting to define the additive inverse (unary) operation.
The original definition is:
(1)   a + (-a) = 0

I'm looking for a definition of the form:
 -a = (expression using previously defined objects and operations).





I've scant interest in what you're looking for.


The only way I see to get there from (1) above is:

(2)    -a = 0 - a

But this defines the (-a) in terms of the subtraction operation which has
not
been defined (at this state of the development of algebra). 




Indeed, it hasn't.


If you define subtraction (binary operation) as adding the additive
inverse
(unary operation),





Yes, tha't s what you do.

then you have:


(3)   -a = 0 + (-a)

which is a circular reference. 





No, it is not a reference at all, just tautology.  Since by the
definition of 0, 0 + x = x, then obviously

0 + (-a) = (-a)

The (-) here is a signe, not an "subtraction operation".




It is called a unary operator.


Nope. (-a) is a single symbol which happens to be made from four characters (or two if you drop the parentheses). It stands for the object "minusa", which is defined as the additive inverse of a.



Which is equivalent to (0 - a)



You're begging the question, because we haven't yet defined the subtraction operator. We're trying to *prove* that subtraction is equivalent to adding the inverse.

(-a) is just shorthand for the above. Though we can pretend that the (-) operator has disappeared from the math, it is only hidden from view, since it is contained in the full sense of (-a).



Not if you follow the logic in the right order. To begin with we just have natural numbers and addition. Then we postulate the existence of an additive inverse and show that the expanded set of numbers still behaves consistently under addition. Only then can we deduce the existence of a unary negation operator and a binary subtraction operator and define them in terms of addition and additive-inverse.


(0 - a) is the additive inverse of a, since

No, because you haven't yet defined subtraction.

(0 - a) + a = 0 as well. 



That indeed is what we would like to prove.


thus (-a) is nothing more than a variable that represents an expression rather than a number.



Of course. But we have to *prove* that it can be used consistently instead of a number.


(-a) is just a variable that is assigned the value of the expression (0 - a), and indeed as such it can be called anything at all. We could script it in any way whatsoever to our liking. We could call the inverse additive of a something like

h_a

and in this form the - symbol isn't even present. We would still have, however, that:

h_a = (0 - a).

It's just simple friggin algebra!



Of course it is, once you have constructed the algebra, but you seem to be confusing construction with use. It's a demonstration that the "subtraction" component of algebra can be defined in terms of the "addition" part. What you're overlooking is that before you can legitimately apply algebra to arbitrary sets of symbols, someone has to define each operation and the set of objects to which they can be applied, and prove that they are all consistent with one another.

There are some people, however, that choose for whatever reason to believe that by the simple act of describing (0 - a) as the *additive inverse* of a, and then ascribing an arbitrary symbol to represent this expression, that the expression, including its operands and operator, no longer exist. This is unfortunately why so many articles have been written about people lost in the abstract, losing touch with all reality. The famous quote "Much learning doth make thee mad" takes on real substance in instances such as this one. I wasn't addressing this remark toward you Mr. Herring, since you did nothing more than to define the additive inverse once again. 



And did so without invoking the notion of subtraction.

I have no idea whether you agree with Meron that there is no such operation as subtraction required. It is however directly implied by the definition of additive inverse, a point that he apparently steadfastly rejects.

I agree with what he's saying, which doesn't correspond to your paraphrase of it.



"[...] We define addition and multiplication of integers in the following way. Let x = a - b
and y = c - d be integers. Then
x + y = (a - b) + (c - d) = (a + c) - (b + d),
x · y = (a - b) · (c - d) = (b · d + a · c) - (a · d + b · c).
Note that each of the symbols + and · are used here to represent two different operations,
namely operations on integers as well as operations on natural numbers.
The operations of addition and multiplication of integers are well-defined (independent
from the representatives chosen), associative and commutative. Also multiplication is distributive
over addition.
The integer a - b is called positive if b < a and negative if a < b. The integer a - a is
called zero, i.e., a - a = 0. The integers a - a and (a + 1) - a are identities with respect
to addition and multiplication, respectively. Also every integer a - b has an inverse with
respect to addition (called an additive inverse) which is denoted by -(a - b) (pronounced
minus a dash b). In fact the inverse of a - b is b - a since (a - b) + (b - a) = 0. Note also
that -(-(a - b)) = a - b. Hence we have the following theorem.
Theorem. (Z,+, ·) is a commutative ring.
The map j : N ! Z : n 7! (b + n) - b is a bijection from the natural numbers to
the positive integers. Suppose that j(n) = a - b and j(m) = c - d. Then j(n) + j(m) =
(a+c)-(b+d) = (b+n+d+m)-(b+d) = j(n+m). Similarly, j(n) · j(m) = j(n ·m) and
n  m if and only if j(n)  j(m). Since the map n 7! b - (b + n) is also a bijection from
the natural numbers to the negative integers we have that any integer may be represented
by either 0, j(n), or -j(n) when n is a suitable natural number. These considerations show
that N may be treated as a subset of Z. Hence we will drop the usage of j and just identify
the numbers n and (b + n) - b. Also we will use -n instead of b - (b + n). Finally, if we
introduce (as a luxury)

I wanted to add here (in order to clarify my argument) that the added note above "as a luxury" wasn't overlooked.
The practical meaning of this statement is that once having established N as a subset of Z, or IOW, after having established that subtraction is unary, we no longer need to include the binary operation of subtraction, since we can proceed without it. No longer needing to include an operator is not however equivalent to not needing it.

a - b =/= b - a when b=/=a, doesn't reflect in incompatibility between subtraction and commutation, but rather it reflects a failure of logic, since these expressions represent operations on completely different sets. On the left we have a set of a elements from which we are removing b elements, and on the right we have a set of b elements from which we are removing a elements. It's an error in notation rather than a logical problem.

a - b should read (be interpreted as saying): From a given set we are to add a elements, and then subtract b elements. It doesn't matter in which order we perform these two operations, we will get the same result. Thus the commutative property is already inherent.
-b + a is the actual inverse of the expression, and reads: From a given set we are to subtract b elements and then add a elements, which gives precisely the same result.

The - symbol was thus a unary operator all along and as noted below, it represents subtraction. The only thing rendered redundant by group theory is the notation.

IOW, "as a luxury" means that we can script ((a) + (-b)) in the form (a - b) as long as we understand that this means
(a + (0 - b)) and that the + operator doesn't really mean to add (0 - b), but rather to simply proceed to subtract b.



the binary operation of subtraction on Z as addition of an additive
inverse we obtain that the symbol - (dash) represents subtraction."

ALGEBRA, Lecture notes for MA 630/631 -- Rudi Weikard

Now that you have the proof in hand that the addition of an additive inverse is in fact just subtraction in disguise, how is it that you say subtraction isn't implied by the defintion of additive inverse, when it is painfully obvious that the equivalence of these operations follows directly from Group theory? To top it off, the - symbol in the form of a binary operator must be introduced early on as an yet undefined operation, i.e. the inverse of addition, but without name or substance initially, in order to prove the theorem of additive inverses, which in turn leads to the definition of said unnamed operation as subtraction. Not required indeed! The - symbol isn't required, but there is a vast distinction between symbols and operations.

Here is a somewhat different, and I hope refreshing perspective: Let a, b, and c be natural numbers.
In the equation a + b - c = d, what we are expressing here is that we are to perform a series of sequential operations on the elements of a given set. We are to add a elements, then add b elements, then subtract c elements to obtain a final cardinality of d elements.
We are of course assuming that our set is initially empty.

If we perform these operations in any other sequence we will again get precisely d elements. Thus both subtraction and addition are commutative wrt the natural numbers. It is thus only the notation that isn't commutative, that is:
c - a + b =/= d, not because of the noncommutative nature of subtraction, but because we didn't commute the operations correctly.
We are now subtracting a elements whereas before we were adding a elements. In short, there really are no binary operations, these were simply a mistake in judgment. OTOH, it follows that addition isn't a binary operation either, that is, we can do without it as well, but we will not have done away with addition or subtraction per se, since they will both be retained in the form of unary operators.

In order to illustrate this idea let's adopt a completely new notation scheme, replacing both + and - binary operators with a single neutral command (as opposed to an operator) that we'll call "and then". The above equation becomes:

(+a) -> (+b) -> (-c) = (+d)

In this form of notation our statement reads exactly the same as the sequence listed out above. The unary operators are monadic instructions, in reality, to simply either add or subtract a given number of elements to or from the set. In this form of expression we no longer have the awkward b + (-c) of group theory to interpret, let alone draw conclusions about, as if you could have actually ever added a negative number to a set anyway. That's pure double-speak. The manner in which this should've been read all along is: Add b and then subtract c, or IOW, the + symbol has been used improperly in the classical form of notation, since *adding* and *and-thenning*<sic> are not exactly equivalent expressions of thought, after all. I mean really, how complicated is this? The expression "Adding the additive inverse of a" is little more than gobbledygook.

The entire derivation of Z (in the quoted article) amounts to nothing but obfuscation, and is based upon an almost complete lack of understanding of the simplest of concepts, namely the relationship between numbers and logic to everyday experience and the inherent mental processes with which we perceive it. IOW it is a load of absolute crap disguised as advanced theory, that proves nothing about numbers, but only shows that at least one form of notation was redundant. Nashism by the book. Schizophrenia by numbers. An idiotic venture into an abstract abyss. Addition and subtraction aren't really quite that complicated. Nor do we need to derive the negative numbers as a subset of natural numbers, they ARE the natural numbers, precisely, albeit with an instruction attached telling us to subtract them rather than add them to the total. The + and - operators are completely independent of the set of numbers to which they are attached. 5 steps is still five steps, regardless of which direction you move.

Richard Perry










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