The journey of discovery begins with the notion of differentness spontaneously and dependently interacting and changing as first principle in contrast to regarding a mathematician with pencil and paper as first principle.
2.1. Differentness of condition
Condition is a common term representing differentness. Each differentness is represented as a condition that can interact its differentness with one or more other different conditions and change condition differentness. A condition is different from all other conditions in its propensity to spontaneously interact with specific other different conditions and to not interact with all remaining different conditions. Interacting conditions of differentness symmetrically and equally appreciating each others differentness by changing is the essence of computational interaction. The different after conditions are the appreciation the differentness of the interacting conditions. A differentness of condition that never interacts and never changes is not appreciable. A differentness of condition that changes spontaneously without interacting is not appreciated.
2.2. Sameness of persistence
Persistence is a common term representing sameness. The sameness of a persistence spans and relates a specific after differentness as the appreciation of a specific before differentness of a specific interaction change. Without a spanning persistence a newly appearing after differentness is not an appreciation of any specific before differentness.
Sameness and differentness are codependent and collaborative. Codependent in the sense that sameness presents in relation to differentness and differentness presents in relation to sameness. Collaborative in the sense that the spanning sameness of persistence links the changing differentness conditions of interaction and the interaction propensities of differentness of condition links the sameness of persistences mutually and indefinitely renewing each’s expressivity.
For this narrative persistence will be characterized as a carrier of condition differentness that asserts one at a time of two or more mutually exclusive condition differentnesses each with a specific interaction propensity. When an interaction occurs the persistence changes it asserted condition to the interaction result condition linking the after condition with the before condition through the sameness of the persistence and ensuring the beforeness and afterness of the interacting conditions with its mutually exclusive one at a time assertion of condition.
2.3. Place of common association
Persistences indiscriminately associate in a place of common association which is itself a spanning persistence and which might be a gravity well, a cell membrane, a concentration gradient, a tide pool, a community and so on. When two or more asserted conditions encounter they either interact or do not interact. If the encountering conditions have no propensity to interact then nothing happens. Each persistence with its asserted condition moves on with no lingering consequences of the failure to interact. If the encountering conditions have a propensity to interact the conditions interact with the asserting persistences spontaneously changing their asserted condition. A molecule, for instance, is a persistence that asserts a chemical condition that may spontaneously interact with and change its chemical condition only in combination with specific other chemical conditions.
The place of common indiscriminate association for this discussion is represented by a shaking bag, illustrated in Figure 2.1. The bag prevents contained persistences and their asserted conditions from wandering off and prevents external persistences from intruding. The shaking ensures that each asserted condition encounters all the other asserted conditions in the bag.
Figure 2.1 Persistences and their conditions agitated in a place of common association.
2.3.1. Interaction dependency
As conditions interact and change the interacting conditions disappear and different result conditions with different interaction propensities appear which interact further and change into further different result conditions forming progressions of dependent interactions within the bag represented by the different conditions and their specific interaction propensities.
2.3.2. Interaction coordination
Interactions coordinate their dependency relations with completeness relations. An interaction begins with completeness of input, when all of the interacting conditions are sufficiently proximally associated. An interaction is completed when the interacting conditions disappear and the interaction result conditions appear. The appearance of an interaction result condition is sufficient to imply that the interacting conditions were completely present and that the interaction has completed.
2.3.3. Pure condition differentiation
Because all specificity of interaction is in terms of differentness of conditions and their interaction propensities this form of interaction is called pure condition differentiation.
The familiar exemplar of pure condition differentiation is proteins in the cytoplasm of a biological cell. Each protein molecule is a persistence and the folding of each protein molecule manifests two or more unique configuration conditions determining its interaction with other molecular conditions. The warm cytoplasm and the cell membrane form the agitating place of common association.
2.4. A familiar example of pure condition interaction: Roman numeral addition
Pure condition interaction is illustrated with Roman numeral addition considered without the subtractive principle: 9 is VIIII instead of IX, 40 is XXXX instead of XL and so on. A Roman numeral is represented by the possible conditions: I, V, X, C, D, M. Without the subtractive principle Roman numerals are association independent. Place in relation does not contribute to differentiation. The subtractive principle which may have been invented by stone carvers as an economy measure is not intrinsic to Roman numerals. For example:
XLII = LXII = ILXI = IILX = LIIX = LIXI = IXIL = XILI = XIIL = IIXL = … = 62
Roman numeral addition is expressed with the interaction propensities shown in Figure 2.2.
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[I,I,I,I,I] => V [V,V] => X [X,X,X,X,X] => L [L,L] => C [C,C,C,C,C] => D [D,D] => M |
Figure 2.2 Roman Numeral interaction fulfillment relations.
The conditions embody the interaction propensities. When two Vs encounter they fulfill the interaction relation [V,V] and spontaneously change into X. Given two Roman numerals these interaction propensities will reduce them to a minimal single numeral representing their addition. The numbers 1878 and 122 are used as example.
MDCCCLXXVIII + CXXII = MM
The two example numerals are placed into a shaking bag as in Figure 2.3 below. The five Is will fulfill relation [I,I,I,I,I], and change into a V. There are then two Vs that will change to an X resulting in five Xs which will change to an L resulting in two Ls which will change to a C resulting in five Cs which will change to a D resulting in two Ds which will change to an M. What remains in the bag are two Ms. There is no interaction propensity with fulfillment [M,M] so no more interactions can occur and the addition is completed. Conditions associate, interact according to their propensities and the Roman numeral sum appears.
Figure 2.3 Roman numeral addition in shaking bag.
A fully determined numerical interaction occurs as a progression of discriminate interactions of different conditions according to interaction propensities while conditions indeterminately associate inside the shaking bag
2.4.1. Interaction incompleteness
But the addition interaction is not complete in itself. It cannot, itself, determine when its interaction is completed. As with a Boolean logic network the only way to determine the progress of the addition is with an external agency which, in this case, must open the bag and perform a complicated count of the conditions. When there are four or fewer of I, X, C and one or fewer of V, L, D the addition is completed.
2.4.2. Completeness of expression
Intrinsically determining when a interaction is done requires that there be a necessarily last interaction propensity fulfilled. With the present form of the expression there might not even be a first interaction. VI + XII = XVIII is done with no interaction behavior at all. There must be a completeness of behavior at each stage of interaction to insure that every interaction propensity is fulfilled in an orderly progression to the necessarily last interaction.
Completeness of behavior requires a completeness of representation. The first question of an addition is, how many Is are present in the bag. The interaction must count the Is and somehow determine that it has considered all the Is and has not missed any Is in the bag. This counting can be accomplished if the number of Is in the bag to be counted is always the same. This constancy of quantity of Is is arranged with buffer conditions such that there is always the same number of each condition and its buffer condition in each Roman numeral. The corresponding lower case letter will be used as the buffer condition for each numeral condition as shown in Figure 2.4.
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Numeral condition I V X J C D M |
Buffer condition i v x l c d m |
Figure 2.4 Roman numerals with their associated buffer conditions.
The buffer condition i is used such that there is always exactly four of I and/or i in a Roman numeral: iiii, Iiii, IIii, IIIi or IIII. The buffer conditions serve a role similar to zero in place value numbers. When two numerals are added there will always be exactly eight of I and/or i. The criterion for completeness, the counting of I/i, can now be represented by the completeness fulfillment of each I/i Interaction propensity, shown in Figure 2.5 below, which is exactly eight I/i conditions. Each completeness of association fulfillment is an unordered association of conditions. There are no order relationships in the bag. The Interaction propensity fulfillment will occur only when the proximal association of all eight conditions is completely formed.
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Interaction propensities for I,i [i,i,i,i,i,i,i,i] => [i,i,i,i,v] [I,i,i,i,i,i,i,i] => [I,i,i,i,v] [I,I,i,i,i,i,i,i] => [I,I,i,i,v] [I,I,I,i,i,i,i,i] => [I,I,I,i,v] [I,I,I,I,i,i,i,i] => [I,I,I,I,v] [I,I,I,I,I,i,i,i] => [i,i,i,i,V] [I,I,I,I,I,I,i,i] => [I,i,i,i,V] [I,I,I,I,I,I,I,i] => [I,I,i,i,V] [I,I,I,I,I,I,I,I] => [I,I,I,i,V] |
Interaction propensities for V,v [v,v,v] => [v,x] [V,v,v] => [V,x] [V,V,v] => [v,X] [V,V,V] => [V,X] |
Figure 2.5 Interaction propensities for I,i and V,v.
The result of the interaction of eight I/i is four I/i and one V/v. The one V/v is the carry to the V/v interaction. There is one of V/v in each Roman numeral and there will always be a carry condition of V or v so there will always be exactly three of V/v present after the carry. The Vv Interaction propensity fulillments require three of V/v in Figure 2.5 above ensuring that the V/v interaction occurs strictly after the I/i interaction. The V/v interaction produces one of V/v and one of X/x.
There will be four of X/x in each Roman numeral so adding two Roman numerals will involve eight X/x conditions and the carry X/x condition making exactly nineX/x conditions to form the X/x Interaction propensity fulfillments shown in Figure 2.6. Again, because of the carry, the X/x interaction will occur strictly after the V/v interaction.
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Interaction propensities for X,x [x,x,x,x,x,x,x,x,x] => [x,x,x,x,l] [X,x,x,x,x,x,x,x,x] => [X,x,x,x,l] [X,X,x,x,x,x,x,x,x] => [X,X,x,x,l] [X,X,X,x,x,x,x,x,x] => [X,X,X,x,l] [X,X,X,X,x,x,x,x,x] => [X,X,X,X,l] [X,X,X,X,X,x,x,x,x] => [x,x,x,x,L] [X,X,X,X,X,X,x,x,x] => [X,x,x,x,L] [X,X,X,X,X,X,X,x,x] => [X,X,x,x,L] [X,X,X,X,X,X,X,X,x] => [X,X,X,x,L] [X,X,X,X,X,X,X,X,X] => [X,X,X,X,L] |
Interaction propensities for L,l [l,l,l] => [l,c] [L,l,l] => [L,c] [L,L,l] => [l,C] [L,L,L] => [L,C] |
Figure 2.6 Interaction propensities for X,x and L,l.
After the X/x interaction there will be three L/l conditions fulfilling the L/l Interaction propensities strictly after the X/x interaction. The L/l interaction produces one L/l condition and one C/c carry condition.
There will be four of C/c in each Roman numeral so adding two Roman numerals will involve eight conditions and the carry condition will make exactly nine C/c conditions to form the C/c interaction propensity fulfillments as shown in Figure 2.7. Again, the C/c interaction will occur strictly after the L,l interaction.
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Interaction propensities for C,c [c,c,c,c,c,c,c,c,c] => [c,c,c,c,d] [C,c,c,c,c,c,c,c,c] => [C,c,c,c,d] [C,C,c,c,c,c,c,c,c] => [C,C,c,c,d] [C,C,C,c,c,c,c,c,c] => [C,C,C,c,d] [C,C,C,C,c,c,c,c,c] => [C,C,C,C,d] [C,C,C,C,C,c,c,c,c] => [c,c,c,c,D] [C,C,C,C,C,C,c,c,c] => [C,c,c,c,D] [C,C,C,C,C,C,C,c,c] => [C,C,c,c,D] [C,C,C,C,C,C,C,C,c] => [C,C,C,c,D] [C,C,C,C,C,C,C,C,C] => [C,C,C,C,D] |
Interaction propensities for D,d [d,d,d] => [d,m] [D,d,d] => [D,m] [D,D,d] => [d,M] [D,D,D] => [D,M] |
Figure 2.7 Interaction propensities for C,c and D,d.
After the C/c interaction there will be three D/d conditions fulfilling the D/d Interaction propensities strictly after the C/c interaction. The D/d interaction produces one D/d condition and one M/m carry condition.
The M/m interaction propensities, shown in Figure 2.8 pose a difficulty because M does not have an intrinsic maximal form. One can put as many Ms as one likes in a Roman numeral and the only way to pre-determine how many Ms there are is to limit the number of M/ms allowed in a Roman numeral. In this discussion the number of M/ms will be limited to five so that, with the carry, there are always exactly eleven of M/m. The M/m interaction occurs strictly after the D/d interaction and is the necessarily last interaction of the Roman numeral addition. The interaction produces five of M/m and the condition Z which indicates the completion of the addition.
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Interaction propensities for M,m [m,m,m,m,m,m,m,m,m,m,m] => [m,m,m,m,m,Z] [M,m,m,m,m,m,m,m,m,m,m] => [M,m,m,m,m,Z] [M,M,m,m,m,m,m,m,m,m,m] => [M,M,m,m,m,Z] [M,M,M,m,m,m,m,m,m,m,m] => [M,M,M,m,m,Z] [M,M,M,M,m,m,m,m,m,m,m] => [M,M,M,M,m,Z] [M,M,M,M,M,m,m,m,m,m,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,m,m,m,m,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,M,m,m,m,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,M,M,m,m,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,M,M,M,m,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,M,M,M,M,m] => [M,M,M,M,M,Z] [M,M,M,M,M,M,M,M,M,M,M] => [M,M,M,M,M,Z] |
Figure 2.8 Interaction propensities for M,m.
2.4.3. The new Roman numeral format
A modified Roman numeral is always exactly twenty conditions: four of I or i, one of V or v, four of X or x, one of L or l, four of C or c, one of D or d, and five of M or m. The number one is mmmmmdcccclxxxxviiiI. The number zero is mmmmmdcccclxxxxviiii. This is analogous to a 32 bit 2s complement binary number which is always 32 bits regardless of its magnitude.
The Roman numerals.
MDCCCLXXVIII and CXXII
now become
mmmmMDCCCcLxxXXViIII and mmmmmdCccclxxXXviiII
and the addition becomes:
mmmmMDCCCcLxxXXViIII + mmmmmdCccclxxXXviiII = mmmMMdcccclxxxxviiii.
The addition process accepts two completely represented numerals and produces one completely represented numeral preserving the numeral representation convention.
The above numerals are formatted for readability but there is still no intrnsic association differentiation.
mmmmMDCCCcLxxXXViIII = mmimVxIMDxICXmXCICcL = … = 1878
The only association relation is that all the conditions are collectively associated at a single place of common association.
2.4.4. Order from chaos
As shown in Figure 2.9 the indiscriminately associating conditions of two Roman numerals, autonomously add themselves in a dependent progression of proximity of association completeness fulfillments and interactions to a necessarily last fulfillment and interaction which completes the sum and produces the condition Z singularly indicating the addition is completed. The Z condition might open the bag and spill the result or it might perform a coordination duty within the bag.
Figure 2.9 Modified Roman numeral addition in shaking bag.
2.5. Interlude: Pure condition differentiation
A pure condition interaction is persistences asserting differentness of conditions with different propensities of interaction indiscriminately associating within a single place of common association (the shaking bag). Inside the bag there is no agency of central influence and there is no agency of explicit control. The persistences and their asserted conditions behave individually and independently. The only influence on their behavior is the confinement of the bag and its shaking. There is no ambiguous behavior that needs to be isolated or hidden. There is no need of any assistance or influence from outside the bag.
2.5.1. Search
The bag is not shaking to perform searches and realize interactions yet its shaking enables searches and realizes interactions. A persistence and its asserted differentness condition with its specific interaction propensity has no intrinsic agency to perform a search to fulfill its interaction propensity yet inside the shaking bag the condition will experience a journey of encountering many other conditions with which it does not interact and then find one condition with which it does interact.
2.5.2. Dependency and completeness
Dependency of interaction relations are expressed by the differentness of the conditions and their specific propensities to interact with other different conditions. An interaction occurs only with complete proximate association of interacting differentnesses which coordinates the interaction behaviors. The appearance of the result differentness and the disappearance of the interacting differentnesses indicates, that the different interacting conditions were completely proximate, that the interaction has completed and that the result differentness is the correct resolution of the interacting differentnesses. This is the completeness of input criterion.
2.5.3. Concurrency
There can be a multitude of pure condition interactions with mutually disjoint condition sets and interaction propensities concurrently realizing independent progressions of dependent condition interactions in a single frothing sea of conditions.
2.5.4. Real pure condition expression
Pure condition is the expressional form of a biological cell which is a single place of common association filled with thousands of different protein conditions with specific interaction propensities supporting the intermingled expression of thousands of different interactions proceeding concurrently and continually without ambiguity. The cell as a whole in complete control of itself.
2.5.5. Place and time in the shaking bag
There is no referential where or when inside the bag. There is no consistent or coherent metric of temporal or spatial relations among the persistences, their conditions of differentnesses or among their interactions within the chaos of the shaking bag yet fully determined interaction occurs. Trying to relate the bag to an external metric of space or of time contributes nothing to either understanding or to realizing pure condition interaction.
2.5.6. A first principle fulfilled
In a pure condition expression, differentness spontaneously and dependently interacting and changing (first sentence of chapter) is a complete and sufficient accounting in itself of its interactions with no ambiguous behavior in need of being isolated or hidden and in no need of any extrinsic support from outside the bag.
The journey continues in Chapter 3 exploring the static association of persistences and their asserted conditions.