Abstract
This is an attempt to axiomatise the natural laws. Note especially axiom 4, which is expressed in third order predicate logic, and which permits a solution to the problem of causation in nature without stating that “everything has a cause”. The undefined term “difference” constitutes the basic element and each difference is postulated to have an exact position and to have a discrete cause. The set of causes belonging to a natural set of dimensions is defined as a law. This means that a natural law is determined by the discrete causes tied to a natural set of dimensions. A law is defined as “defined” in a point if a difference there has a cause. Given that there is a point for which the law is not defined it is shown that a difference is caused that connects two points in two separate sets of dimensions.
Keywords
Natural laws
Axiomatisation
Causality
Objects
1. Undefined terms
1. ρ
2. σ
3. Difference
4. Dimension
5. Relation
6. Element
7. Cause
8. Point
9. Belongs to
10.Existence
2. Initial definitions
a set = df A specific existence of elements (in this extraction defined by occurrence within brackets ({})).
a complex of dimensions = a field of dimensions = df A set of dimensions.
D = df A specific and limited set of dimensions.
π = df The cause of ρ on σ.
θ = {σ, ρ, π} = df
1. A specific π that causes a specific ρ on a specific σ,
2. the specific ρ that is caused by the specific π in 1. and
3. the specific σ mentioned in 1.
Dkm = df A specific and limited field of dimensions; {dk, dk+1, …, dm}, in which d is a separate dimension and Dkm contains m-k+1 dimensions.
form = df A specific set of relations.
Ξ = df The form of θ.
elements of relation = df Parts of a structure of relations necessary to define a form.
Π, Ρ and Σ = df The elements of relation of Ξ; where Π represents the relations of π, Ρ the relations of ρ and Σ the relations of σ.
3. Axioms
Axiom 1: ρ is a difference
Axiom 2: σ is a difference
Axiom 3: ρ belongs to Dkm, a specific and limited field of dimensions
Axiom 4: In all points X belonging to an arbitrary D, Ξ is true.
4. The object Ω
Ω = df
1. {ρ1, ρ2, …, ρi},
2. in which each and every ρx (1 ≤ x ≤ i) constitutes a difference towards {ρ1, ρ2, …, ρx-1}, and where
3. ρx+1 constitutes a difference towards {ρ1, ρ2, …, ρx-1, ρx}.
5. π:s relation to D
θ implicates an unique cause π to each and every ρ. For a specific and limited field of dimensions Dkm therefore, a precise set of causes λ is tied to included ρ. This specific set causes the total set of ρ in Dkm. Each and every ρ in Dkm therefore can be explained with the set λ. Why ρx+1, for instance, is answered with πx.
Definition of the law λ
λ = df {π0, π1, …, πq}, in which each and every πx causes a ρx+1 belonging to the set {ρ1, ρ2, …,ρq,ρq+1} which constitutes the total amount ρ in a specific and limited field of dimensions (Dkm).
From the definition above follows theorem 5 and theorem 6.
Theorem 1: (Not part of this compilation.)
Theorem 2: (Not part of this compilation.)
Theorem 3: (Not part of this compilation.)
Theorem 4: (Not part of this compilation.)
Theorem 5: λkm causes all ρ in Dkm.
Theorem 6: Every ρ caused by a certain law λx exists in a limited and specific complex of dimensions Dx.
6. Inter-relations of laws λ
Definition of Dn
Dn = df The field of dimensions {d1, d2, …, df, …, dg, …, dn-1, dn},1≤f ≤g≤n that contains;
1. all ρx belonging to Dfg,
2. all ρy that can form Ω for ρx and
3. all ρz that ρx can constitute Ω for.
Definition of Λ of Dn
Λ = df {λ1, λ2, …, λP}, where Ρ is the total amount of laws applying in Dn and where {λ1, λ2, …, λP} causes all ρ belonging to Dn.
Another definition concludes this section:
initiating difference = df σ
7. Definition of “λ defined in a point X0”
With Λ and its part-laws λ each and every difference related to Ω (ρ) has a cause π belonging to Λ. Assume a point X0 belonging to Dkm belonging to Dn. What “λkm is defined in X0“ means is defined below.
Definition of λ defined
λkm is defined in a point X0 belonging to Dkm = df θ is true in X0.
Theorem 7: If λkm is defined in X0, Λ is defined in X0
Theorem 8: If Λ is defined in X0, λkm is defined in X0
A special case is at hand when for a point X0 holds {¬σ, ¬ρ, ¬π}. Is in this case λkm defined in X0? Since λkm does not exist in X0 (¬π is true and π is λ:s representative in X0), λkm is neither defined nor not defined in X0. Thus the next theorem applies:
Theorem 9: If for a point X0 holds {¬σ, ¬ρ, ¬π} λkm for the point is neither defined nor not defined.
Before going further some new concepts are introduced:
effect = df ρ
a point of effect = df A point X in which ρ is true.
From the two definitions above follows:
Theorem 10: In a point of effect θ is true.
8. Beyond θ
Either the state of things is such that it is not possible that θ does not apply in each point where π apply, or it is not impossible. If the latter is the case something not of Λ bound can emerge in a point. Arbitrariness though, in that case, is not imminent, nor chance, due to axiom 4: “In all points X belonging to an arbitrary D, Ξ is true” (Ξ = df The form of θ). This implies that if a law for a point is defined in that point Ξapply and if the law is not defined Ξapply:
Theorem 11: Ξ is true in all points X0 whether or not λ(X0) is defined.
Ξ,”the form of θ”, does not include chance because the form implicates a cause to each difference. Therefore the following is valid:
Theorem 12: It is not true for any point that effect can occur by chance.
9. Derivation and definition of ρ’ and ~ρ
Λ not defined in X0
Assume Λ is not defined in a pointX0. This implicates according to the definition of ”λ defined” that θ is not true in X0. For X0 then the following is true:
(1) ¬θ
θ has three elements for which thus apply “not”:
(2) ¬{σ, ρ, π}
(2) implicates that at least one element of θ is negated:
Theorem 13: ¬θ ⇒ i) {¬σ, ρ, π}∨ ii) {σ, ¬ρ, π}∨ iii) {σ, ρ, ¬π}∨ iv) {¬σ, ¬ρ, π} ∨ v) {¬σ, ρ, ¬π}∨ vi) {σ, ¬ρ, ¬π} ∨ vii) {¬σ, ¬ρ, ¬π}
According to theorem 9 Λ is neither defined nor not defined in a point X0 where vii) is true, therefore vii) is not true in X0.
Again ¬π implicates a cause-less difference [iii) and v)] and also a cause-less negation of difference [vi)]. Furthermore ¬σ implicates that a cause of a difference has emerged at random [i)] respectively a cause of a negated difference emerging at random [iv)]. When a cause-less difference or negation of difference is equal to chance i), iii)-vi) implicates chance. Since axiom 4, by theorem 12, does not permit chance i), iii)-vi) are not true in X0. ¬ρ finally implicates negation of difference [ii)].
¬θ then implicates seven alternatives of which six are not possible. Then the seventh, ii) {σ, ¬ρ, π}, is true:
Theorem 14: If Λ is not defined in a point X0 {σ, ¬ρ, π} is true in that point.
10. Of Ρ in X0 where Λ is not defined
Theorem 14, though, does not show how Ξ:s elements of relation are fulfilled when it is lacking a fulfilment of Ρ. Axiom 4 implicates that Ρ is fulfilled in X0. Thus Ρ is fulfilled in X0.
Theorem 15: If Λ is not defined in a point X0 then holds for X0: {σ, ¬ρ, π} ∧ Ρ is fulfilled.
Ρ is not fulfilled by the ρ that is negated (ρ), nor by the negation of it (¬ρ). That which fulfils Ρ in X0 can be called ρ’.
Definition of ρ‘: ρ‘ = df That which fulfils Ρ in a point X0 for which Λ is not defined.
11. Dimensionality
In X0 ¬ρ is true. Since X0∈Dn ρ’ can not belong to Dn, nor is it possible that the point which ρ’ belongs to, belongs to Dn.
Theorem 16: The point that ρ‘ belongs to, does not belong to Dn.
Definition of X’0 = df The point that ρ‘ belongs to.
Here a hypothesis will be introduced, in which it is assumed that ρ’ exists in the dimensions Dn symbolises with the addition of some more, separating it from Dn:
Hypothesis 1: ρ‘ exists in a complex of dimensions with the n dimensions of Dn plus ω numbers of dimensions, ω∈N, ω>0.
Definition of D’: D’ = df The complex of dimensions that ρ‘ belongs to.
Theorem 17: Dn ∈ D’.
12. New laws
Λ does not apply in X0. In spite of that ρ’ is caused for X0 (in X’0). With this, one could say that Λ’ determines ρ’. The specific law that applies in X0′ can be called λ’1. Also π did not cause ρ’. The cause of ρ’ can be called π’.
Definition of π’: π‘ = df The cause of ρ’.
Definition of λ‘1: λ‘1 = df The law that the cause of ρ‘ belongs to.
Definition of Λ‘: Λ‘ = df The law-domain that contains λ‘1.
13. The cause of ρ’
Since ρ’ does not belong to Dn it cannot exist in X0. Therefore there are two points to be considered though they are connected. For the pair of points X0-X0′ holds:
#1 {σ, ¬ρ, ρ’, π}
σandπ on the other hand cannot belong to X’0, since they belong to Dn.
In X’0 there is ρ’. According to axiom 4 in X0′ there also has to be more elements. Axiom 4 states that the cause and condition of effect have to be found in the point of effect. Therefore cause and condition of effect is part of #1. Since only ¬ρ is not occupied as an element of relation it has the quality of the two missing elements of X0′. Thus ¬ρ is part of X0′. For not violating logical rules of dimensions, namely that what is part of Dn cannot be identical to that which is part of D’ ≠ Dn, ¬ρ in Dn is not identical to that of D’. ¬ρ in X0′ can be called ~ρ (“denied” ρ).
14. ~ρ as a set
Because π’ and ~ρ are elements, not for instance numbers, the relation between the two can be formulated as a relation between sets. Then the one is an element of the other. Since π’ definitely is one:
Definition of ~ρ: ~ρ = df The representation of ¬ρ in X’0
Theorem 18: In X’0 ~ρ is cause and condition of ρ‘.
Theorem 19: (Not part of this compilation).
Theorem 20: σ‘ ∈ ~ρ
Theorem 21 : π‘ ∈ ~ρ
Definition of σ‘: σ‘ = df What fulfils the relations of Σ in X’0
Therefore:
X0: {σ, ¬ρ, π}
X’0: {σ’, ρ’, π’}
Theorem 22: (Not part of this compilation.)
Theorem 23: (Not part of this compilation.)
Theorem 24: (Not part of this compilation.)
Theorem 25: (Not part of this compilation.)
Theorem 26: (Not part of this compilation.)
Theorem 27: (Not part of this compilation.)
Finally a theorem that sums up some aspects of the theory so far:
Theorem 28: If Λ is not defined in a point X0 {σ, ~ρ, ρ‘, π} is true.
15. The concept Θ
If Λ is not defined in a point X0 belonging to Dn, ΡforX0 is shifted to D’, a complex of dimensions separated from Dn. Ρ in D’ is called ρ’. This implicates an existence of something with association to ¬ρ, ~ρ. The cause of ρ’, π’, in turn, belongs to ~ρ.
For X0-X’0 holds according to theorem 28: {σ, ~ρ, ρ’, π}. In a point X’1, separated from X’0, and belonging to D’, the case is: {σ, π, ρ}, that is, θ. Between Dn and D’ {σ, ~ρ, ρ’, π} is true, a state of facts below symbolised Θ.
Definition of Θ: Θ = df {σ, ~ρ, ρ’, π} .
That Θ can be true is the result of the present study.
Theorem 29: (Not part of this compilation.)
Theorem 30: (Not part of this compilation.)
Theorem 31: (Not part of this compilation.)
16. Axiom(s) of existence
Axiom of existence 1: There is at least one point for which Θ is true.
17. Conclusion
Given this extraction something exists in two separate sets of dimensions. Extrapolating this finding we have a new perspective on quantum entanglement (Bub 2020). If a set of quantum particles pair wise are joined by what has been labelled “Θ:s” they would be entangled. It would also be interesting to investigate “interfaces” between separate sets of things (Gamper 2017) using the concept of “Θ“.
References
Bub, Jeffrey, “Quantum Entanglement and Information”, The Stanford Encyclopedia of Philosophy (Summer 2020 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2020/entries/qt-entangle/>
Gamper, J. On a Loophole in Causal Closure. Philosophia 45, 631–636 (2017). https://doi.org/10.1007/s11406-016-9791-y
Investigations 