From the Vasardatta of Subandhu:
" And at the time of the rising of the moon with its blackness of night, bowing low, as it were, with folded hands under the guise of closing the blue lotuses, immediately the stars shone forth...like zero dots... scattered in the sky as if on the ink blue skin rug of the Creator who reckoneth the total with a bit of moon for chalk."
(Translated by Louis H. Gray)
In so far as I understand it, the zero was created as a place holder to perfect the decimal system of numerical notation. As the zero was created in South Asia there was very likely speculation upon the meaning and philosophy of a symbol, in early times a dot, that clearly existed as a symbol and was certainly useful, yet in itself designated a nullity or absence of being. In the further development of mathematics in the West such speculations have generally been held in disrepute and the zero entered deeper into the mathematical lexicon as a useful necessity that was more or less taken for granted.
As with all avoidance of essential questions, a high price has been paid for the merely pragmatic acceptance of the zero. Several elementary arithmetical operations cannot be used with the zero; this situation mars the unity and coherence of arithmetic and the higher mathematical forms erected upon it.
For example, one cannot divide a real number by zero. Beginning students will argue that since dividing a given positive real number by a small number gives a large result, and the smaller the divisor the larger the result, why cannot one just assign infinity as a result of dividing a positive number by zero. It is easiest for the teacher to point out that since dividing the same positive number by a small negative number gives a large negative result and that as zero is taken to be the dividing point between the positive and negative numbers, being itself neither positive nor negative, the resulting infinity of dividing by zero would be ambiguously either positive or negative.
To extend the example, one cannot even divide zero by itself. Although the division of any other real number by itself gives the result one, in the case of zero the result would be ambiguously either positive or negative one.
In another example, one cannot define the form 00 to which very similar ambiguities could be described.
As with all things which are, there is causality behind this situation. The general cause is the uncritical acceptance that a symbol perfectly useful for a place marker in the decimal notation can therefore accept several other meanings and mathematical applications.
Some of the contradictions involved in the multiple uses of zero are apparent from the above examples. Zero used as the dividing point between the positive and negative numbers makes division by the number zero impossible. Consider some of the common uses of zero. Zero is a place marker in the decimal system; in the number 103 the zero indicates that there are no tens, in a number such as 103 the zero has very little of the value of nullity, although there are no tens in the tens decimal position there are in fact ten tens in the number 103. In a sense, as a place marker in the decimal notation, zero can be seen more as a separator than as a nullity; the real function of the zero in 103 is to tell us that there are three ones to be added to the hundred, we do not have to interpret it as being a nullity of anything.
Zero is also used as an integer, as the integer before one, and as a number which counts none of something. Zero as integer and counting number are subtly distinct but closely related. Zero as an integer still does not quite require a sense of the nullity of zero, it could still be seen as a mere mathematical convenience. However, Zero as a counting number does seem to indicate some essential aspect of nullity. If you have seven sheep and you give away seven sheep then you have no sheep and you have zero sheep. There seems little difficulty in such an example but it does begin to indicate an essential voidness of zero. Looking at the example more closely, note that zero sheep means no sheep, not any of something. Although mathematically it is easy to count abstractly 0,1,2,... one begins to get into a little more philosophical difficulty when one considers what zero means when it is not applied to a specific category, although no worse than when one or two is similarly applied. Zero of nothing in particular is difficult but no more difficult than one of nothing in particular.
Let us look at another logical difficulty which arises when zero is used in elementary arithmetic. Although this difficulty is not commonly taught, it is in fact the beginning point for the substance of this essay.
Consider multiplying by zero. If we multiply zero times one, the arithmetical algorithm 0x1 = 0 corresponds to the null interpretation of zero: zero ones means no ones which can readily be taken as zero. However if we multiply zero times zero the common arithmetical algorithm 0x0 = 0 contradicts the nullity of zero: zero zeroes means no zeros while the arithmetic algorithm says that zero zeroes is zero. This is a clear and present contradiction.
If 0x0 != 0 (where for typographical convenience != is used for not equal) then what is 0x0?
A first response could be to simply state that 0x0 is another of those uses of zero which must be arbitrarily excluded from elementary arithmetic, but that is dull and cowardly. Let us see if something more valuable might come from a closer analysis of the situation.
It is the next step that is critical, everything else follows from it.
Consider that although zero is not generally considered a positive number, it is certainly not a negative number; thus multiplying an inequality by zero, although we are not supposed to do it in ordinary arithmetic, would not, if we were to do it, reverse the sense of the inequality.
As a notational convenience we could write 0x0 = 02 = 0exp(2), whichever is most convenient.
Now we get to the heart of the matter, to the door into a new realm of mathematics: Since 0 < 1 let us multiply this inequality by zero on both sides. We will later, from a new perspective, return to see that this is valid. Then we have the following sequence:
0 < 1
0x0 < 0x1
0x1 = 0
0x0 = 02
02 < 0
The second step in the above sequence would not be valid in ordinary arithmetic, since zero is not there a positive number. But, in the more extensive number system we are about to construct, zero is indeed a positive number and the step is valid. And since the above sequence is presented as a motivation for the initiation of the construction, rather than as a step in the construction, such circularity as may exist is psychological, which is permissible, rather than logical, which would be more suspect.
According to the sequence, we have the result 0exp(2) < 0. If we let 0exp(2) be negative, then 0 = (0exp(2))exp(.5) is imaginary; not an inconceivable possibility but more complicated than what we shall do here. So, excluding negative 0exp(2), we have a number, or something, than is smaller than zero yet not negative.
When we consider that 0exp(2) is a perfect square, the square of zero, and since the square of a real number is positive, and since 0 is a real number, we can take 0exp(2) to be positive. We have actually deduced the stronger result that 0exp(2) is positive
Let us rewrite our result as
0exp(2) < 0exp(1)
If we can perform the above sequence, then clearly we can repeat it by multiplying the resulting inequality by zero again, reaching
0exp(n) < 0exp(n-1) <...< 0exp(3) < 0exp(2) < 0
and thus we have a sequence of diminishing numbers smaller than zero. Since none of the even powers of zero can be negative without having an imaginary zero, and since each odd power of zero will have a smaller even power of zero, all of the powers of zero will be taken to be positive.
This situation raises some curious points. For one, zero is no longer the boundary between the positive and negative numbers, instead it is a positive number. In addition to implying the question of what, then, is the boundary between the positive and negative numbers, we also need to consider how the results obtained can be justified in light of the nullity aspect of zero.
In the first essential use of zero, as a decimal place holder, we have already seen how the nullity of zero is very little involved; it is merely required that zero designate no units in the unit place or no tens in the ten place, etc., but that actually zero here is not essentially null since it will already be preceding by another nonzero number symbol as in the example of 103 which does not really have no tens but in fact contains ten tens.
Zero as integer or counting number does seem to include more nullity but, as we have seen, as a counting number zero is practically used to count none of something: if one has no sheep then it is sheep that one has none of, which isn't the same as saying that one has nothing at all. In fact it would be both odd grammar as well as existentially incorrect to say that one has zero unless one meant to say thereby that one did indeed posses zero itself, the number, symbol, or concept. As an integer we have even less problem with a not entirely null zero, since the zeroth integer means just that one has no positive integers, or no positive integer number of something, but that is not necessarily nothing.
It is clear that the inequality sequence we have constructed implies: that zero is not entirely the number of nullity and that zero is not the dividing point between the positive and negative numbers and that there in fact exist, or at least can be constructed, smaller positive numbers than zero.
We have split the zero. We have retained the uses of the traditional zero as a decimal place holder, as the zeroth counting number, and as the zeroth integer; while we have rejected the use of zero to divide the positive from the negative numbers.
As a consequence of splitting the zero the ambiguities of sign involved in the use of zero in elementary arithmetic are avoided. Thus, at least as far as sign is involved, it becomes possible to define division of a nonzero number by zero, division of zero by zero, and exponentiation of zero by zero. We have, of course, also removed the contradiction involved in multiplying zero by zero.
In the case of dividing a nonzero number by zero, we will find it natural to set 0exp(-1) as a very large number; a number which at least has the ordinal aspects of an infinity.
We will also find it natural to set 0/0 = 1.
We will also, in constructing our new set of numbers, which we shall call the 'zero numbers', find that 0exp(0) becomes a number very close to one, although we will find that it is not exactly equal to one, being, as we shall see, slightly smaller than one.
We will note here, and later show, that our not entirely null zero does not have a finite absolute value, being smaller than any positive real number.
We have so far constructed our inequality of powers of zero by operating on zero with
different numerical exponents. We shall now look at this construction from a slightly different
point of view. Instead of numerical exponents operating on zero let us consider the zero as
operating on the numerical exponents to create the new zero numbers. Thus we define a new
operation, '0exp', read "zero exponent", which operates on the real numbers to generate the zero
numbers.
DEFINITION
(0exp)r = 0exp(r) = 0r
DEFINITION:
A zero number is positive if the square of that number has the same sign as the
number.
PRINCIPLE:
Any positive number is greater than any negative number.
We have seen that increasing the integral exponent of zero decreases the value of the result; that is, n2 > n1 implies that 0n1 > 0n2, which can also be written
0exp(n1) > 0exp(n2). We see that 0exp is an order reversing operation. It is a reasonable
extension to generalize from integral powers of zero to real powers of zero, giving:
ORDER PRINCIPLE
For positive real r2 greater than positive real r1:
0exp(-r2) > 0exp(-r1) > r2 > r1 > 0exp(r1) > 0exp(r2) >
-0exp(r2) > -0exp(r1) > -r1 > -r2 > -0exp(-r1) > -0exp(-r2)
DEFINITION:
Call the numbers of the form 0expr, where r is a real number, the 'zero numbers of the first tier'.
Denote
{0exp(r)} = {z1}
Call the numbers of the form 0exp(z1) the 'zero numbers of the second tier'.
Denote
{0exp(z1)} = {z2}
Continue through {zn}.
DEFINITION:
Call
Zn0 = {zn} U {zn-1} U ... U {z1}
the 'n-tier pure zero numbers'.
DEFINITION:
Call
Zn = Zn0 U R
the 'n-tier zero numbers'.
We shall use the phrase 'zero numbers' to refer to the members of any set Zn. We shall use the phrase 'pure zero numbers' to refer to the members of any set Zn0. Note that, in each case, while we allow n to have any finite value, we do not here address the extension of n to infinite values.
LEMMA 1:
For x and y any positive real numbers, then
x > 0y
Proof:
Suppose
0y > x
for some positive reals x and y. Raise each side of the inequality to the 1/y power (take the y th root). Since 0y, x and 1/y are all positive, we have
(0y)1/y > x1/y
but we also have
(0y)1/y = 0yx1/y = 01 = 0
implying that
0 > x1/y
which is false.
The same procedure removes the possibility of 0y = x.
LEMMA 2:
For x and y positive real numbers, then
0-x > y
Proof:
Suppose, on the contrary, there exist some positive real numbers x and y such that
0-x< y
Use
0-x= 1/(0x)
and multiply both sides of the inequality by (positive) 0x
1 = (1/(0x)* 0x < y* 0x
(We use '*' for the multiplication sign when the use of 'x' could be ambiguous.)
But since
0x < z
for any positive real z, if we set
z < 1/y
we arrive at a contradiction.
With the lemmas we can now order the real numbers with the zero numbers.
LEMMA 3:
1] The number '0' is the only real pure zero number.
2] '-0' is a zero number distinct from '0'.
3] 0 > -0
Proof:
1] Any pure zero number has the form 0z; if z is 1 then 0z is equal to 0; if z is not 1 then 0z is not real.
2]&3] In the zero numbers there exist nonegative numbers, such as 02, which are smaller than 0; -0 is separated from 0 by these numbers and -0 is negative, therefore -0 is both distinct from and smaller than 0.
LEMMA 4:
For any positive real number e
1 + e > 0-0 > 1 > 00 > 1 - e
Proof:
We have
0 > -0
so
0-0>00
For any real r, not equal to zero
r0 = 1
and
r-0 = 1/r0 = 1
Also
0-0x00 =00/00 = 1
So
0-0 > 1 > 00
Suppose
0-0 = 1 + e
then a sequence such as (1/n)-0 with n approaching infinity appears convergent for any finite n,
indeed it is always equal to 1, but it jumps by e in the infinite limit. This would contradict the
established notion of limit. The other side of the proof is essentially the same.
We can now construct ordered sequences of zero numbers.
We shall begin with a simple ordered set of real numbers:
n+1 > n > 1 > 1/n > 1/(n+1) > 0 > -1/(n+1) > -1/n > -1 > -n > -(n+1)
Applying 0exp to the above sequence we get, in Z10
0exp(-(n+1)) > 0exp(-n) > 0exp(-1) > 0exp(-1/n) > 0exp(-1/(n+1)) > 0exp0 > 0exp(1/(n+1)) > 0exp(1/n) > 0 > 0expn > 0exp(n+1)
Using the order lemmas we can combine the Z10 sequence with the real sequence. We also extend the resulting Z1 sequence antisymmetrically to negative values, giving
0exp(-(n+1)) > 0exp(-n) > 0exp(-1) > 0exp(-1/n) > 0exp(-1/(n+1)) > n+1 > n > 1 > 0exp0 > 0exp(1/(n+1)) > 0exp(1/n) > 0 > 0expn > 0exp(n+1) >
-0exp(n+1) > -0expn > -0 > -0exp(1/n) > -0exp(1/(n+1)) > -0exp0 > -1 > -n >
-(n+1) > -0exp(-1/(n+1)) > -0exp(-1/n)) > -0exp(-1) > -0exp(-n) > -0exp(-(n+1))
In order to simplify the synthesis of real numbers in zero number sequences, we introduce
the following lemmas
LEMMA 5:
In Zn ordered sequences (e.g. 41)), '1' is always located between the adjacent pair
0exp(-z) > 0expz
as in
0exp(-z) > 1 > 0expz
Proof:
Note that these pairs are always located as
0exp(-0) > ... > 0exp(-z) > 0expz > ... > 0exp0
The same balancing procedure which locates '1' between 0exp(-0) and 0exp0 now locates '1' more exactly as
0exp(-z) > 1 > 0expz
We can now use the lemmas and 0exp to raise the Z1 sequence 41) into a Z2 sequence
0exp(-(0exp(-(n+1))) > 0exp(-(0exp(-n))) >
0exp(-(0exp(-1))) > 0exp(-(0exp(-1/n))) >
0exp(-(0exp(-1/(n+1)))) > 0exp(-(n+1)) >
0exp(-n) > 0exp(-1) > 0exp(-(0exp0)) > n+1 > n >
0exp(-(0exp(1/(n+1)))) > 0exp(-(0exp(1/n))) >
0exp(-0) > 0exp(-0expn) > 0exp(-0exp(n+)) > 1 >
0exp(0exp(n+1)) > 0exp(0expn) > 0exp0 >
0exp(0exp(1/n)) > 0exp(0exp(1/(n+1))) > 0exp(0exp0) >
0 > 0expn > 0exp(n+1) > 0exp(0exp(-1/(n+1))) >
0exp(0exp(-1/n)) > 0exp(0exp(-1)) > 0exp(0exp(-n)) >
0exp(0exp(-(n+1))) > -0exp(0exp(-(n+1))) >
-0exp(0exp(-n)) > -0exp(0exp(-1)) > -0exp(0exp(-1/n)) >
-0exp(0exp(-1/(n+1))) > -0exp(n+1) > -0expn >
-0 > -0exp(0exp0) > -0exp(0exp(1/(n+1))) > -0exp(0exp(1/n)) > -0exp0 > -0exp(0expn)) >
-0exp(0exp(n+1)) > -1 > -0exp(-0exp(n+1)) >
-0exp(-0expn) > -0exp(-0) > -0exp(-0exp(1/n)) >
-0exp(-0exp(1/(n+1))) > -n > -(n+1) > -0exp(-0exp0) >
-0exp(-1) > -0exp(-n) > -0exp(-(n+1)) >
-0exp(-0exp(-1/(n+1))) > -0exp(-0exp(-1/n)) >
-0exp(-0exp(-1)) > -0exp(-0exp(-n)) >
-0exp(-0exp(-(n+1)))
Clearly the above process may be used to construct further Zn sequences. Different
choices of an initial real sequence will lead to different but similar Zn sequences.
DEFINITION:
If the whole real number system is used as the initial sequence to construct a Zn sequence, then
call the resulting sequence Zn+.
Zero Number Regions
We need one more stage of development of our above Zn sequences. In particular, we need the Z3 sequence
0exp(-0exp(-0exp(-n))) > 0exp(-0exp(-0exp(-1))) >
0exp(-0exp(-0exp(-1/n))) > 0exp(-0exp(-n)) >
0exp(-0exp(-1)) > 0exp(-0exp(-0exp0)) > 0exp(-n) >
0exp(-0exp(-0exp(1/n))) > 0exp(-0exp(-0)) >
0exp(-0exp(-0expn)) > 0exp(-1) > 0exp(-0exp(0expn)) >
0exp(-0exp0) > n > 0exp(-0exp(0exp(1/n))) >
0exp(-0exp(0exp0)) > 0exp(-0) > 0exp(-0expn)) >
0exp(-0exp(0exp(-1/n))) > 0exp(-0exp(0exp(-1))) >
0exp(-0exp(0exp(-n))) > 1 > 0exp(0exp(0exp(-n))) >
0exp(0exp(0exp(-1))) > 0exp(0exp(0exp(-1/n))) >
0exp(0expn) > 0exp0 > 0exp(0exp(0exp0)) >
0exp(0exp(0exp(1/n))) > 0 > 0exp(0exp(-0expn)) >
0exp(0exp(-0)) > 0exp(0exp(-0exp(1/n))) > 0expn >
0exp(0exp(-0exp0)) > 0exp(0exp(-n)) >
0exp(0exp(-0exp(-1/n))) > 0exp(0exp(-0exp(-1))) >
0exp(0exp(-0exp(-n))) >
-0exp(0exp(-0exp(-n))) > ... > -0exp(-0exp(-0exp(-n)))
The above sequence, and similar sequences, divide naturally into several regions.
1. Region 1 is illustrated by the sequence
0exp(-exp(-exp(-n))) > ... > 0exp(-1)
We call region 1 the 'region of infinities'.
2. Region 2 is illustrated by the sequence
0exp(-0exp(0expn))) > ... > 0exp(-0exp0)
We call region 2 the 'region of seminfinities'.
3. Region 3 is illustrated by the sequence
0exp(-0exp(0exp(1/n))) > ... > 1 >
... 0exp(0exp(0exp(1/n)))
We call region 3 the 'sublimation of 1'.
4. Region 4 is illustrated by the sequence
0exp(0exp(0exp0))) > ... > 0exp(0exp(0expn)))
We call region 4 the 'region of suprazeroes'.
5. Region 5 is illustrated by the relation
0exp(0exp(-0expn)) > ...
We call region 5 the 'region of subzeroes'.
6. Region 6 is illustrated by the relation
-0exp(0exp(0exp(-n) > ...
We call region 6 the 'region of negative zero numbers'.
Region 6 can of course be divided into subregion exactly analogous to the positive regions.
Sublimation of 1
The structure
0exp(-z) > 1 > 0expz
surrounding '1' can be used as a tool for constructing additional zero number sequences. Whenever the real number '1' appears in a zero number we can expand that number into an ordered sequence of zero type numbers by replacing the '1' in the original number by each of the ordered sequence of zero numbers in the region 3, the sublimation of 1. For example, consider the sublimation of 1 sequence
0exp(-0) > 0exp(-0exp(0exp(-0))) >
0exp(-0exp(0exp(-1))) > 1 > 0exp(0exp(0exp(-1))) >
0exp(0exp(0exp(-0))) > 0exp0
and consider the zero number 0exp(-1). Replace the '1' in
0exp(-1) by the above sublimation of 1, 54), arriving at
0exp(-0exp(-0)) > 0exp(-0exp(-0exp(0exp(-0)))) >
0exp(-0exp(-0exp(0exp(-1)))) > 1 >
0exp(-0exp(0exp(0exp(-1)))) > 0exp(-0exp(0exp(-0))) >
0exp(-0exp0)
The direction of sequential ordering in such a new sequence can be calculated by counting
the levels of 0exponentiation, which we will call the 'tier number', in which the '1' appears in the
original zero number, and adding to that number the count of negative signs appearing in the
various levels of 0exponentiation leading up to the '1'. If the sum is even, the new sequence will
have the same direction of order as the sublimation of 1 sequence, if odd, opposite.
Phi
In our constructions of sequences of zero numbers we find that there is always a gap between the positive and negative numbers. This gap can no longer be filled by zero since there are positive zero numbers smaller than zero. Clearly the gap will remain in all Zn. The structure of the gap is analogous to the structure in the sublimation of 1, but at alternate tier levels.
We designate phi, for which the corresponding Greek letter, f, may be used, as the symbol for a number marking the dividing point between the positive and negative zero numbers.
By construction the zero numbers are symmetric about phi.
THEOREM ONE:
0f = 1
Proof:
The theorem is shown by the equivalent constructions, at alternating tier levels, of the sublimation of one and the structure surrounding phi.
It would be possible to interpret Theorem One as generating the real number one from the
0exp operation and phi. It is as if we have always been looking at the real numbers through a lens.
Now we find that then lens can be focused more sharply and then the zero numbers appear along
with and among the reals. It may be that the zero numbers are the more fundamental system; the
real numbers exist as a simplification, or condensation, of the zero numbers.
Unitary Zero Numbers
In the first construction of zero numbers above we assumed the existence of the real numbers. From theorem one we see that we can construct a subset of the zero numbers, that which is limited to the real initial '1', without recourse to the prior existence of the real numbers, using theorem one to generate '1'.
Since much of the intrinsic structure of the zero numbers appears in this more limited case,
we will now focus our attention on this subset of the zero numbers.
DEFINITION:
The 'unitary zero numbers' names the subset of the zero numbers which can be constructed
using only '1', from the real numbers.
LEMMA:
'0' can be constructed from phi and 0exp.
Proof:
0 = 0exp1 = 0exp(0exp(phi))
Strings
In the construction of the unitary zero numbers we begin with the initial 'f'. Applying the operation 0exp to f we arrive at '1' and then '0'. Further zero numbers are constructed from 1 and 0 by a series of operations of 0exp and negation.
Negation here is not mysterious but simply a mirroring of the construction across f.
This simple construction of the unitary zero numbers may be readily symbolized if we denote the 0exp operation by 'a' and the negation operation by 'b'. We may then construct each zero number by the 'string' of operations which construct it, according to the following rules:
1] Operators operate to the right.
2] f appears as the initial symbol on the right.
3] a can appear to the left of any string symbol.
4] b can appear to the left of any a.
Since the unitary zero numbers can be defined by this construction we have:
LEMMA 6:
There exists a natural one to one mapping between the set of unitary zero numbers and the set of strings.
Proof:
We should demonstrate that the natural mapping is one to one.
Note that 4] is required to make the mapping one to one.
We shall list the first few unitary zero numbers and their strings:
f
a f = 1
ba f = -1
aa f = 0
baa f = -0
aba f = 0exp(-1)
aaa f = 0exp0
abaa f = 0exp(-0)
Clearly the 'f' at the end of each string is mathematically redundant. If one were to actually find an application in which one used such strings in multiple calculations, one might be well justified in dropping it. Since this is a foundation paper, and rather short, I prefer to leave it in for now.
If we construct and order a sequence of unitary zero numbers according to our standard order principle, and translate that sequence into string notation, we have:
ababababa f > abababa f > abababaa f > ababa f >
ababaaa f > ababaa f > ababaaba f > aba f > abaaaba f >
abaaa f > abaaaa f > abaa f > abaabaa f > abaaba f >
abaababaf > af > aaababaf > aaabaf > aaabaaf >
aaa f > aaaaa f > aaaaaa f > aaaa f > aaaaba f >
aa f > aabababa f > aaaba f > aabaaa f > aaba f >
aababaa f > aababa f > aabababa f > baabababa f
(Such strings are pronounceable, and distinguishable, if multiple 'a's are used as extensions of the sound. Try speaking the above sequence in front of small children.)
Examination of the above sequence and a little reflection upon the order principle will demonstrate the following
LEMMA 7:
All strings with an 'a' on the left are greater than all strings with a 'b' on the left. (Positive
numbers are greater than negative numbers.)
LEMMA 8:
For any string z, if n > 1 and m > 2. then
1 > (an)z
and
1 > (am)z > 0
In the above an means a concatenated n times.
Proof:
For any z, az is always positive.
Since az is positive
1 > aaz
Since
1 > aaz
then
a3z> 0
If
1 > (a3)z > 0
then
1 > 0exp0 > (aexp(n+1))z > 0
by the fundamental order principle.
LEMMA 9:
For all zero numbers z
abz > z
Proof:
If z is positive, then
z > bz
and
abz > az
If
z !> 0
then
az > 0 > z
If
z !< 1
then abz is an infinity greater than z.
If
1 > z > 0
then
-0 > -z
and
0exp(-z) > 0exp(-0) > 1 > z
and
0exp(-z) = abz
If z is negative the result follows from the positivity of 0expz for all z.
LEMMA 10:
For all unitary zero numbers z
0 > aabz
LEMMA 11:
For all strings of the given length or less
1] The maximum positive z of string length 2(n+1) is ((ab)expn)af.
2] The furthest negative z of length 2(n+1) is ((ba)exp(n+1))f.
3] The minimum positive z of length 2n+3 is a((ab)expn)af.
4] The maximum negative z of length 2n+4 is ba((ab)expn)af.
5] The nearest z below 1 of length 2n+4 is aa((ab)expn)af.
6] The nearest z above 1 of length 2n+5 is aba((ab)expn)af.
Trees
The string construction of the zero numbers may be represented as a graph tree. The
initial branching of the tree is
f
.
. 0exp
.
1 ......negation......... -1
. .
. .
. .
0 .................-0 0exp(-1) .................... -0exp(-1)
. .
. 0exp(-0)
.
0exp(0) ......... -0exp(0)
.
.
.
0exp(0exp(0))
Condensation
There exists an order preserving map which maps many of the zero numbers into the real
numbers.
DEFINITION:
If {z} is a set of zero numbers, such that there exists a real number r, such that for any real number e > 0,
r+e > z > r-e
for all z in {z}, then we say that {z} 'condenses' to r, and write
[{z}] = r
For a single z we also write
[z] = r
LEMMA 12:
If
[z1] = r = [z2]
and {z} consists of zero numbers z such that
z1 > z > z2
for all z in {z}, then
[{z}] = r
LEMMA 13:
For all z such that z is nonnegative and z < 0
[z] = 0
LEMMA 15:
For all z such that
0exp(-0) > z > 0exp0
we have
[z] = 1
DEFINITION:
The 'complete sublimation' of a real number r consists of the set of all zero numbers which
condense to r.
DEFINITION:
A sublimation of a real number r consists of a subset of the complete sublimation of r.
T Numbers.
We have constructed Zn numbers by the operations 0exp and negation applied to any set of
real numbers, and to f. We now construct a larger class of zero numbers.
DEFINITION:
'Replacement' is an operation which removes any '0', 'f', or real number symbol from a zero number and replaces that symbol by a zero number.
Even though the original number may be a Zn number the result of applying the
replacement operation to that Zn number generally does not give a Zn number.
EXAMPLE:
In the unitary Z2 number 0exp(0exp(-0)), if we replace the middle '0' ny '0exp0' we have
0exp((0exp0)exp(-0)) which is not a Zn number, since it cannot be produced by just the operations
0exp and negation.
DEFINITION:
The set of numbers produced from the Zn numbers by the operation of replacement,
including a finite number of iterations of the operation, and including the identity replacement, is
called the set of 'T numbers'.
DEFINITION:
If a T number construction originates with the set of unitary Zn numbers, and all of the
replacements are by unitary Zn numbers, the result is called a set of 'unitary T numbers'.
T Strings
We can form strings for T numbers. It is natural to again generalize the T numbers into a
larger set of numbers, when working with the strings.
DEFINITION:
The numbers constructed by the following process form the Tx numbers. We remove a contiguous set of string symbols, excluding f, from a Zn string, then replace those symbols by the string of a, generally different, Zn string.
The original T numbers form a subset of the Tx numbers wherein only a single string symbol at a time is replaced from the original Zn number.
For example, if in 0exp(0exp(-0)) we wish to replace the middle '0' by '0exp0', the string replacement is as follows
aabaa f ...R(a,aaa)...> a(aaf)baa f
where '...R(a,aaa)...' denotes the replacement of 'a' by 'aaa', although it does not say which 'a' is replaced by 'aaa'.
We may also construct T and Tx strings directly. The following rules construct the unitary T strings.
1] All of the rules for Zn strings, except as in 4] below.
2] An 'a f)' may be placed just to the left of any 'a' or 'b'.
3] A '(' must be placed somewhere to the left of any 'a f)'.
4] A 'b' may be placed just left of a '(' provided that that '(' is not just left of a 'b'. (We
avoid 'b(b'.)
Lo Operation
We may consider the inverse operation for 0exp. The inverse operation for 0exp is to take
the logarithm to base zero. Both 0exp and Lo are one to one by construction on the zero
numbers, but neither is necessarily one to one on the T numbers.
DEFINITION:
Loz = logz to base 0
EXAMPLES:
Lo(0exp0) = 0
Lo(0exp(0exp(-1))) = 0exp(-1)
Lo(Lo(0exp(0exp(-1)))) = -1
LEMMA 15:
Lo(1) = f
WHY ZERO NUMBERS
USES
The zero numbers resolve contradictions in arithmetic.
The zero numbers are an exploration of a new realm of thought and being.
The zero numbers are potentially a tool for exploring the void.
The zero numbers are new mathematics and point to much more new mathematics, and perhaps new physics or developments in other sciences and humanities.
Because it is fun to explore them and I hope it is pleasurable to recognize them.
Because the night is delightfully dark and the day is full of light and the border times are beautiful beyond compare and it is good to extend the boundaries of knowledge and perception.
Because the zero number theory leads to the Staley equation.
Because it is always nice to have new numbers to play with.
APPLICATIONS
One of the potential applications of zero numbers is mathematical. Every mathematical equation can be put into a form where one side of the equation is zero.
A = B is equivalent to A - B = 0
One could replace the '0' in the last form by any zero number which condensed to zero; thus every mathematical equation becomes a countable set of equations. This opens up a considerable new avenue for exploration.
And consider the possibilities when the new spectra of zero number equations are applied
to physics and other mathematicized sciences.
Another potential mathematical application of zero numbers is in the set theory of measure. Often sets are said to be of measure zero when in fact the sets are distinct. Measure theory condenses the measure zero sets, but the zero numbers potentially provide a tool for discriminating between such sets or classes of such sets.
For example, suppose the set {1/n} is said to be of real measure zero( = 0exp(1)), then would it
not make sense to say that the set {1/n^2} is of 'real' measure 0exp(2)? This gives a finer
distinction than mere cardinality allows.
PHI
In the zero numbers phi is the symbol of the boundary between the positive and negative zero numbers. This is its formal and algorithmic meaning. Just as we interpret zero as being both a place marker for no units but also as the symbol for none, we can also look at phi as standing in place for the void. Thus the duality of interpretation of zero is repeated for phi. I hope that this time at least I can make that duality explicit and recommend it to notice.
Phi is the aim, the mark for the arrow into the void.
The equation
0f = 1
strikes me as extremely beautiful. It connects the three fundamental numbers 0, 1 and f; it generates the real numbers, based on 1, from the zero numbers; it creates the unit from zero and the symbol for the void.
I would like to call it 'Joe's equation'.
Joe's equation can serve as the root for a new metaphysics that relates the interaction
between being, existence and the void. More speculatively, Joe's equation may be found to serve
as the root of a new physics.
ZEROES OF THE VOID
There are levels of the void. We can experience them differently and interact differently with the levels. I may describe a linear structure but of course it is not linear except perhaps in the nearer levels. Nearer to what? Let us say: nearer to the surface of the void, to the surface interaction wherein we begin to recognize the void.
Although the most important things are to learn to recognize and utilize these levels, it is also possible to represent them symbolically. We might use the zero numbers. In a simple model we could take '0' as representing the nearest of the void. Then we might take '02 ' to represent the second level of the void. And, to get richer, we might take '00 ' to represent the void of the void (which is almost, but not quite, one).
0 = the entrance to the void
02=in the void
03=deep in the void
04=deeper in the void
It is apparent that the zero numbers can count the void. If we consider zero, '0', as the first number of the void, then our next natural construction 0x0 = 0^2 is, in a sense, the second number of the void. In any case 0^2 denotes a number smaller than zero, and so a number which penetrates deeper into the void than zero.
Then the construction of 0^2, which is less than zero, implies the existence of 0^(1/2), which is greater than zero.
In the construction of the zero numbers, the first level of development is to expand zero into a spectrum of numbers descending from zero and ascending from zero; they descend below zero but not into the negative realm, they ascend upward from zero but not into the finite realm; they are in between, and such sequences might be called the 'sublimation of zero'.
How does the construction of the zero numbers help one to understand being? Both the sublimation of one and the sublimation of zero give an expansion and resonance to the thoughts of unity and nothingness, which now can be mathematically expressed, or at least analogized.
There are deeper levels in the further construction of the zero numbers. They describe realms of being for which we have had no effective symbolic notation before the zero numbers. In this regards it is important to point out both the linear ordering of the pure zero numbers and the even more surprising linear ordering of the zero numbers. These linear orderings form a very powerful gift to help us find our way about complex being.
With the discovery or creation of phi, the number of the void and the boundary between the positive and negative numbers, we also have found a pointer deep into the void.
In the zero numbers phi is the symbol of the boundary between the positive and negative zero numbers. This is its formal and algorithmic meaning. Just as we interpret zero as being both a place marker for no units but also as the symbol for none, we can also look at phi as standing in place for the void. Thus the duality of interpretation of zero is repeated for phi. I hope that this time at least I can make that duality explicit and recommend it to notice.
Phi is the aim, the mark for the arrow into the void.
From the distance Mr. Zero appears as the sentinel of the center. He stands alone, solidly empty and round, the single occupant of the vanishing space between the positive and real numbers.
I see him exactly one unit away from me, facing both ways and neither. If I stretched myself out horizontally I could just reach him, as I could just reach two if I stretched out in the other direction. I am Mr. One.
Ms. -1, my identical but reversed twin, is just as far from Mr. Zero as Mr. Zero is from me.
I am curious about Mr. Zero. He is both there and not there. He is even more unlike me than Mr. 2. Mr. 2 is subtly incomprehensible. He is both two things and one, which should not be, yet he is there, definitely there. Mr. Zero is both there and not there. His figure goes all around but is empty in the center. He is transparent. I can look right through him to the negative numbers. Indeed until, he came into place a thousand years ago or so, I could not clearly see the negative numbers, if they were even there. So his transparency seems more effective than mere nothingness, for only through his empty center can I and my brothers see our twins.
Yet in his thousand years of silent duty Mr. Zero has never spoken. Mr. 2 speaks; Mr. 3 chatters away, our more distant brothers harmonize with complex subtlety, but Mr. Zero never speaks.
Perhaps Mr. Zero never speaks because he has never been properly spoken to. I speak to Mr. 2 with the clear tones of an elder and singular brother. Mr. 2 controls the destinies of half of our younger brothers who live in the shadow of his resonance. I speak with all of them, although I can hear the answers of the primes most clearly. Sometimes I can hear the distant call of the outer primes who live at incomprehensible distances from the center, and hear them with surprising clarity.
I can also see the great finite ramparts of googol and googolplex. Mr. 2 can also recognize his distant associates by the last figure of their symbols. Mr. 3 has only slightly more effort in recognizing distant members of his tribe. We all communicate in different modalities, but the communication is of the same general essence.
None of these forms of communication seems to work with Mr. Zero. He stands aloof, but not entirely so; that is the difficulty. If we attempt to add him to a summation nothing is changed; it is as if he were not there. If we attempt to factor him into a product he destroys all, or absorbs it into himself. Woe upon us if we attempt to divide by him.
This situation has long concerned me. There he stands as close as my closest brother, powerful yet silent; its spooky. Thus one fine dark night in autumn I took up my courage and spoke out to Mr. Zero. I had thought, if instead of trying to call Mr. Zero into our positive converse, rational or otherwise, I would ask him to speak for himself as he is, or is not. Then, perhaps, he would answer me.
"Mr. Zero," I said, "you have stood your place a thousand years in silent power. Who are you?"
Mr. Zero did not answer.
"Mr. Zero," I asked, "is it true that you are the guardian between the positive and negative numbers and none may pass you except by becoming the other?"
Mr. Zero nodded.
"So, Mr. Zero," I concluded, "it is your essential duty to be the guardian of the gate between positive and negative numbers."
Mr. Zero laughed as long and loud as silence permits. Then he answered.
"It is my duty, but it is as much a part of my essential nonbeing as it is of a man to occupy night and day in succession."
I didn't know what to make of that, but as Mr. Zero was finally speaking, I tried to keep the conversation going.
"Then if that is not your true place in being, what is?"
"I am not in being."
"But if not in being, then where are you?" I asked, and what is that circular figure I see around you?"
"My figure is a mask of being to denote my absence."
"Yet, Mr. Zero, you function among the numbers. Strangely, it is true, but it is only through you that I can see the negative numbers."
"That is true," replied Mr. Zero, "and I have many other functions among the numbers."
"Then if you act in being how is it that you are not in being?"
"My external figure acts in being, but from my point of view I am rather acted on by being."
"Yes," I answered, "I can understand that about your figure, but what are you besides your figure?"
"Am I not nothing?" replied Mr. Zero.
"Then you are merely the figure cast about nothing in order to extend being. By this figuration being creates a gate to further being, both a separation and a connection between the positive and negative numbers."
"I, myself, would not use the word 'merely', for I am also the gate connecting the scalars, vectors, tensors and higher number forms. Also, I play a figure in many abstract forms. Yet more: I am the geodesic, the path of light in space time. Further, I can be one side of any equation."
"And this then is your true and complete essence, to be a figure cast about nothing by being for being's own purpose." I concluded.
"No, that is not my essence, rather my present role in being." Said Mr. Zero with great solemnity.
"What is this?" I asked. "You say you are nothing except the figure cast by being, and now you claim that this is not your essence."
"I do. My essence is the combination, or integration, of the figure cast by being and the nothingness of my own 'being'. My own self is the flux and resonance induced by that duality combined into unity and zeroed in upon my name."
"I can sense something of what you say. Yet it appears to me that you pretend to draw mystery upon yourself where you might better content yourself as a subtle, but comprehensible, device of being to order and extend itself."
"There is more to me than being knows," said Mr. Zero with complete certainty.
"Will you tell me of this more?" I asked.
Mr. Zero paused with a stillness that seemed to stretch deep into eternity, although it might have been only an instant in the time of being. As he paused I was able to return consciousness to my own unity and re-erect myself as one. There stood Mr. Zero, a round gate at the center. Far and away stretched my numerical brothers, beyond Mr. Zero extended our sisters in similar array. Between each of us integers rested the calm faces of our rational children, while our irrational nieces and nephews flickered about forever escaping our exact perception. Far away the distant peaks of the infinities rose forever ascending. Finally Mr. Zero spoke.
"I cannot communicate to you the full truth of my essence. Indeed, as it is finally incomprehensible to being, it is therefore not entirely knowable to me, as my external figure also partakes of being. I can show you how I am also a further gate to an as yet undiscovered realm of being, or near being, a realm nearer to my heart, and nearer to your own identity as unity, than any other being you now know. If you will follow my guidance with care and patience, I will show you a vast and deep new realm of numbers which is both nearer to my own self, nearer to the true gateway between the positive and negative numbers, and even nearer to you than even that elusive nearest real number to one that may or may not exist but that you can never see nor touch."
"This is too much," I said, "You offer me some mystical consolation that I feel I will never accept into the true and clear realm of the numbers of being."
Mr. Zero answered, "I will show you by clear and easy steps, entirely constructive, a new order of numbers, an extension (although I would prefer to call it an intension) of the real numbers. Furthermore I will show you how your very own finiteness, and particularly your personal unity of being, is grounded upon the order and relations of these new numbers. Even more, since I have spoken of order, I will show you that these new numbers, at least in their primary constructive set, entirely fit into the perfect linear order of the real numbers, both of themselves and as they are mapped by a clear and definite finite sequence of constructions into the same order space as the real numbers, of which you are the one truest and most exact member."
This answer was so extreme that I returned to my own identity. All about me I felt the flowering of disunion. As I stood, perfect in my own self representation, where my figure was also an exact instance of my being, I wondered at Mr. Zero's temerity in stating, or rather claiming, that I would find my own perfect unity supported by his dreams of nothingness. This was not the subtle contradiction of my closest brother 2, whose single figure is supposed to represent two units. His contradiction is that the two units that he purports to enclose can never be exactly identical but must always differ to some degree and thus cannot be two of the same thing but rather one thing plus another thing. His ordinality, if not his cardinality, is clearly two, and to that extent at least he is clear and distinct. And, whatever diffuseness of exact designation he may have, Mr. 2 at least displays a clear identification with being, for all his uncertainty arises from the fact that he represents not only being, but still another being.
Mr. Zero presents a deeper problem. Not only is he a duality himself between his being of circular figuration and the nonbeing of his essence, he claims that by that duality he can substantiate my own unity. How can I accept, or even listen, to this? Nevertheless, by my own form of definition, or definition of form, Mr. Zero as near to me as Mr. 2, so neighborliness compels me to attend to his unlikely story, as well as the simple justice of the fact that I started the conversation.
"If you wish me to accept these stupendous claims, Mr. Zero, then you must show me, with clarity and logic, the validity, or at least the cause and consequence, of what you say."
"I do wish your acceptance," answered Mr. Zero. "As for validity, that is too abstract and metaphysical for my tastes. After all, we are here as constructs of the human imagination, which is biological, volitional, and as finally uncertain as it is creative. I will, as I said before, show you the construction of my statement."
"Yes," I said, if you can construct me from you that will be a fitting adornment to our being, as well as cementing our so near and yet so distant relationship."
"Do you object to circularity?" asked Mr. Zero.
"Of course," I answered, "if what is to be proven is already accepted, what is the gain?"
"But if the circle of reason and attention does so construct that a new reality is laid before your eyes, then what objection remains?"
Here I was given pause.
"If reason," continued Mr. Zero, "taking each step by rational and constructive means takes leave of priority and wings itself around to reproducing itself, is that a vicious circle?"
This statement plunged me deeper into introspection.
"No," I finally answered, "unless there is some pretension that the circularity goes beyond its roots and flowering."
xxx
"And yet," I continued, "it has seemed only natural to designate you as the gate between positive and negative real numbers, along with your several other properties."
"I have been given too many aspects: Mr. Zero, do this, do that! You can see for yourself that it leads to elementary contradictions."
"That is true," I answered Mr. Zero, who seemed quite despondent. We cannot divide by you, a/0 is undefined, I cannot even find myself in division of yourself by yourself, 0/0 not being defined; neither can we exponentiate you by yourself, 00 being also not defined. Your presence, while necessary, mars the completeness of the real numbers."
"Yes, you ask too much of me," said Mr. Zero.
"How," I asked, "did this situation come about?"
"Through genius, you, or rather your human creators, cast one circle about the void and expected that single creative action to provide all of your mathematical facilities needed from the void."
"Then were we wrong?" I asked.
"Not entirely wrong, but rather imperceptive and lazy," answered Mr. Zero.
"How so?"
"The original perception was just and correct. Yet for a thousand years, or rather more, the only improvement made upon the original thought was to replace the dot figure of my early representation by a circling of the void."
"Then your humans loaded me with properties real and abstract far beyond my original
service as a place marker for the null digit."
Frederick Joseph Staley