List of Contributions to Mathematics
This page lists contributions that the Path Semantics project has to mathematics. In the future, the plan is to link to eventual disputations and constructive criticism, such that people can evaluate the Path Semantics project more accurately and in a scholarly setting.
See entry about Leibniz' law on Stanford Encyclopedia of Philosophy. The wikipedia article is also informative.
This counter-example is relative to the theory of Path Semantics.
Leibniz' "Indiscernibility of Identicals" is generally considered as an a priori logical truth.
∀ x, y { (x = y) => ∀ F { F(x) <-> F(y) } }
The symbol = represents "numerical sameness" and <-> a biconditional.
This principle does not hold in Path Semantics, because the theory of Path Semantics does not allow symbolic indistinction.
With other words, it is impossible to prove that x and y are completely identical,
since introducing a such operator would violate tautological propositional equality (x == y)^true over symbolic distinction.
In the functional programming model in Prop, there is an axiom:
∀ x, y { (x == y) => ∀ F { F(x) == F(y) } }
However, this axiom does not cover the qubit operator ~ in path semantics, which is not considered a function, in the traditional sense.
Similarly, in the original Leinbiz' principle, it is common to assume this holds for predicates. However, Leibniz lived over hundred years before Pierce and Frege, so the logical notation had not yet been developed in his time.
In the constructive model of Path Semantical Quantum Propositional Logic (PSQ), the qubit operator requires tautological congruence, using Higher Order Operator Overloading Exponential Propositions (HOOO EP):
∀ x, y { (x == y)^true => (~x == ~y) }
It means, x == y is not strong enough as it can capture variables from the environment, as propositional equality in constructive logic consists of two maps x => y and y => x, which are lambdas. On the other hand, (x == y)^true are two function pointers x -> y and y -> x, also written x^y and y^x. Function pointers can not capture variables from the environment, so they provide stronger mechanisms to express tautological propositional equality.
In the classical model of PSQ, the qubit operator ~ depends on the entire bit vector (theoretically) while the practical implementations uses a trick to work around undecidability, which works since the output bit vector is randomized. Tautological congruence, which is the same as a modified principle using tautological propositional equality, can be interpreted that two propositions have the same bit vector if and only if they can be proved to be equal without making any assumptions.
Therefore, the first principle has a counter-example and hence is logically false in Path Semantics. The reason is that Path Semantics is a theory where symbolic distinct objects might be treated as identical up to a certain strength of equality. If one is allowed to express "actual identity", or symbolic indistinction, then this would be unsound.
The qubit operator ~ in Path Semantics has tautological congruence, which means that two outputs are equal only if the inputs can be shown to be equal without making any assumptions. The associated logic is Higher Order Operator Overloading Exponential Propositions (HOOO EP) which are used to extend Intuitionistic Propositional Logic (IPL). While HOOO EP can be used independently of the qubit operator ~, it is the semantics of the qubit operator ~ itself that provides a grounding of the meaning of randomness seen through the perspective of IPL + HOOO EP.
Tautological congruence of the qubit operator ~:
~x ⋀ (x == y)^true => ~y
HOOO EP:
x^y => (x^y)^z
(x => y)^c => (x^c => y^c)^true
(x ⋁ y)^c => (x^c ⋁ y^c)^true
Thus, since randomness of this sort is well behaved and its meta-logical properties well defined, it provides a mathematical foundation of randomness. This foundation might not be complete, but at least it is a start.
Type Theory is the leading candidate for providing a formalist perspective of mathematical foundations. However, in Type Theory, types are judgements that are atomic, hence irreducible.
In Path Semantics, a type judgement can be modeled as a proposition:
(a : b) == (a => b) ⋀ (a < b)
Where < is path semantical order. Path Semantical order associates a natural number to path semantical propositions that describes a propositional level, where each level is an Intuitionistic Propositional Logic (IPL). The core axiom of Path Semantics is assumed between path semantical levels.
Therefore, Path Semantics might be seen as even more fundamental than Type Theory. Furthermore, it uses propositions, which are generally respected in logic as valid mathematical objects.
The imaginary inverse operator inv in Path Semantics can be thought of as an opposite category functor that lifts a category into a groupoid. This operator inv is imaginary because it follows the law of composition as if there was a solution for any morphism.
f . inv(f) == id
inv(f) . f == id
A solution of e.g. inv(f) == g is a 2-morphism.
For higher categories, e.g. an n-category, this results in an n+1-groupoid.
This brings categories in line with the use of n-groupoids as fundamental objects for higher dimensional mathematics in Homotopy Type Theory.
There exists a counter-example of Set Theory as foundation for mathematics in Path Semantics, using the notation of normal path:
f : bool -> bool => f[true1] == true1
Here, true1 is a boolean function of type bool -> bool which returns true for all inputs.
The left side of the equation f[true1] has no model in Set Theory.
One can think about this normal path as contracting unobserved values. It works fine as long there is no distinction being made between tr : bool and fa : bool, since (tr == fa) => false. The idea of such contractions fits nicely with the homotopy interpretation of Homotopy Type Theory.
In Set Theory, a model is required at every step in the proof, since there is no meta-language that allows syntactical manipulations beyond models. Since such syntactical manipulations are abundant in mathematics, it is not merely a "symbolic manipulation system" as previously believed, but an actual foundation of mathematics closer to the homotopy interpretation than sets. The consequence is that Set Theory is unsuitable as a foundation for mathematics (although an arguably useful one when there are underlying models corresponding to sets).
The core axiom of Path Semantics allows propositional relations to propagate across path semantical levels:
(a ~~ b) ⋀ (a => c) ⋀ (b => d) ⋀ (a < c) ⋀ (b < d) => (c ~~ d)
In the domain of fundamental theoretical physics, this propagation provides a mechanism to understand time seen from a logical point of view. The qubit operator ~ and the equivalent quality operator ~~ carries propositional states in unstructured time, like General Relativity, while path semantical levels might be thought of local frames of reference with approximate Euclidean-geometric properties. With other words, a path semantical level is a "moment" in time where reasoning can take place.
x ~~ y == (x == y) ⋀ ~x ⋀ ~y
This relates reasoning to physical experience of time and might help to explain why known forms of consciousness take place in integrated information processing systems, such as human brains.
While this model of time does not encode clocks directly, it can also show that special synchronizing propositions might be used as logical analogues of clocks. The physical intuition about time in relativity is reflected in the logical structure.
A computation might be thought of as following a series of instructions to produce an output from some given input. When computation happens over time, as in a one-dimensional setting, it can be thought of as a "concrete" process.
Path Semantics provides a perspective of a corresponding abstract dimension:
len x len
o ------> o
| |
| |
concat | | add
v v
o ------> o
len
concat[len] == add
The left side might be thought of as a concrete computation taking place in time, while the right side might be thought of as an abstract projection of this computation. The more abstract a theory is, the more contractions away from the underlying concrete computation.
Since such commutative diagrams can be transposed along the diagonal, there is a perspective where concrete processes can be seen as abstract and vice versa.
This also reflects the time-space symmetry in theoretical physics, where a valid Feynman diagram can be transposed by changing the time with the space axis.
The theory of natural numbers has remained unchanged for over a century, until work on Path Semantics questioned what it means for zero to have no predecessor. Could this be relaxed to mean "no unique predecessor"?
Instead of treating zero as a natural number with no predecessor, it is possible to treat it as being preceded by every natural number, but in a sense where the nature of this map is different from the ordinary successor function.
The Closed Natural Number theory treats zero as both the first and the last natural number:
1 + 1 + 1 + 1 + ... = 0
Hence, one can change the perspective of natural numbers to include infinite series. This property is why the theory is called "closed".
This is an important conceptual breakthrough for reasoning about natural numbers, which fits the use of modular arithmetic in computer science and generating functions in combinatorics. Furthermore, Closed Natural Numbers can be used in cyclic cosmology.
The conceptual breakthrough of natural numbers is an important contribution, although it is not entirely satisfactory. An fully formalized axiomatic treatment of Closed Natural Numbers is an open problem. The research so far, using Higher Order Operator Overlading Exponential Propositions (HOOO EP), suggests that infinite series might be under some assumptions, while the normalized form of natural numbers are under none assumptions. This might also help explaining the use of natural numbers to prove isomorphisms of infinite sets. The particular choices of traversing such infinite sets likely corresponds to some assumptions of infinite series.
Probabilistic paths in Path Semantics provides a mechanism to calculate the relationship between parameterized probabilities of known sub-types of input versus the parameterized probability of some output sub-type. This calculation takes place in correspondence with a computational oracle that gives probabilistic existential paths. For finite types, this oracle can be made decidable.
Although probabilistic paths are defined and have so far produced correct answers, they are not entirely understood. The reason is that the calculations might exceed the unit interval of real probabilities temporarily, until all contributions from the corresponding probabilistic existential paths have been summed, at which point the calculations produces answers inside the unit interval.
The Born rule of measurement in quantum mechanics can be replaced by a non-deterministic infinite series called "quantum propagation". This infinite series of quantum propagation produces the same real probabilities, while having temporary imaginary probabilities.
Therefore, there is no need to interpret measurements in quantum mechanics as special real valued probabilities. Instead, they might be interpreted as complex probabilities with a zero imaginary component that emerges in the classical limit. At scales where quantum behavior is significant, semi-classical states might be thought of as finite random walks on the cover of quantum propagation.
A complex number consists of a real component and an imaginary component. Most people confuse the imaginary component with the concept of complex numbers. However, the reason complex numbers are useful, is not only because of imaginary numbers, but due to the harmony between real and imaginary numbers as one structure: The complex number.
A total normal path is defined as a composition of 3 maps where 1 map uses the imaginary inverse inv:
f[g1 -> g2] == g2 . f . inv(g1)
Just like with complex numbers, the reason normal paths are useful, is not only due to the imaginary inverse, but due to the harmony between ordinary composition and composition with the imaginary inverse.
In Category Theory, a normal path corresponds to a commutative square. Composition of normal paths corresponds to diagram chasing:
g1 g3
o ------> o ------> o
| | |
f | x | | h
v g2 v g4 v
o ------> o ------> o
f[g1 -> g2][g3 -> g4] == h
In this diagram, the properties of the imaginary inverse makes the center down-pointing map x not requiring any solution.
This means, one can draw the diagram without the x:
g1 g3
o ------> o ------> o
| |
f | | h
v g2 g4 v
o ------> o ------> o
f[g1 -> g2][g3 -> g4] == h
Normal paths formalizes the practice of diagram chasing in Category Theory using a notation that is readable and computer keyboard friendly.
De Morgan's Laws are among the most important laws in Boolean algebra:
not(and(x, y)) == or(not(x), not(y))
not(or(x, y)) == and(not(x), not(y))
In Path Semantics, these equations can be viewed in a more useful form:
and[not] == or
or[not] == and
Here, the [not] notation generates a symmetric normal path.
There is a corresponding symmetric normal path for every Boolean function,
so [not] can be thought of as a structure preserving functor.
Symmetric normal paths in Path Semantics produce deeper insights into the importance of De Morgan's laws.
First Order Logic is generally considered in mathematics to be a universal language for theorem proving.
Unfortunately, Path Semantics produces a counter-example through the qubit operator ~.
However, the extended mathematical universe of maps with tautological congruence is not the only limitation of First Order Logic.
First Order Logic focuses on predicates, which might be thought of unconstrained binary relations (over Cartesian products). Predicates in First Order Logic have several properties that make them unsuitable for high dimensional mathematics:
- High combinatorial complexity
- Lack of multi-dimensional abstraction for inference rules
- No well defined notion of equality for uniqueness proofs
- No inequality matching on inference rules for symbolic distinction
- Manual encoding of functions
These properties are fixed by introducing a new logic that provides an alternative to First Order Logic. This new logic is called "Avatar Logic" and has the following two axioms:
p(a, b) b : p => p(a) = b
p(a, q'(b)) q'(b) : p => p(a) = {q'(_)} ∈ q'(b)
In Avatar Logic, when b has a role p, written b : p, the proposition p(a, b) might be thought of as a map p(a) = b.
However, since the role of b is implicitly known from the context, one can simply write (a, b).
This puts Avatar Logic on a firm basis of Graph Theory.
However, Avatar Logic also extends Graph Theory with avatars, hence the name.
When the 1-avatar q'(b) has the role p, written q'(b) : p,
the proposition p(a, q'(b)) might be thought of as a set containing avatars of q'.
It is common to write this relation as p(a) => q'(b) where => is a "has" relation.
Avatar Logic is more suitable than First Order Logic for higher dimensional mathematics and provides a mechanism to have multiple dimensions of evaluation. It is formalized in Zermelo-Fraenkel Set Theory and can model advanced mathematical theories like Chu spaces. The avatars make it possible to perform partial inequality matching in rules, that leverages the work on symbolic distinction i foundational Path Semantics. Data in Avatar Logic can be exported as graphs and used to provide a type system for graph databases. There are fewer symbols needed to model relations in Avatar Logic, which produces less computational waste and requires less memory. Furthermore, inference rules can be written that generalize over multiple dimensions of evaluation, such as using one rule to define both value and type relations.
Gödel's work on the Incompleteness Theorems started modern investigations into limitations of formal theories in mathematics. Among one of the insights was that a formal theory can not be both complete and consistent while claiming to be maximally mathematical, as this would contain Robinson arithmetic Q (think about this as a slightly weaker Peano arithmetic with addition and multiplication).
Written in grammar form, the properties of a maximally mathematical language after Gödel were known as:
consistency|completeness
However, this property is not the only known property of maximally mathematical languages.
By combining review of the cumulative work in philosophy as Kent Palmer's Schema Theory (Ph.D.), with a new formulation of underlying principles of mathematical language bias creating a distinction between Inside and Outside theories, the grammar of maximally mathematical languages was extended to:
verification|validation coherence
consistency|completeness coherence|clarity
The distinction between Inside/Outside theories allowed deriving this grammar, by checking the models in Classical Propositional Logic.
- An Inside Theory is a mathematical language that models external objects as unknowns
- An Outside Theory is a mathematical language with at least one symbol which does not refer to its theory
This unification of maximally mathematical languages, allows mathematical languages to form larger and more varied forms of language biases than commonly considered as mathematics. As the Inside/Outside distinction was used to unify these properties, it was argued by Kent Palmer that Inside and Outside theories as a whole, called Nilsen theories (after Sven Nilsen), is the highest known categorization of mathematical languages.
Daniel Fischer argued that people knew since Plato and likely further back to ancient Egypt, that writing can lead to a form of abstract corruption that corresponds to some sort of Platonic bias. This is known as the "Anti-Thoth Argument" based on Plato's writings about Thoth (an ancient Egyptian god who were credited for inventing writing).
Since the core axiom of Path Semantics models semantics of symbols, it was suggested that it could provide some insights into this form of language bias. By careful investigation of the core axiom of Path Semantics to look for this bias, it was found that there is indeed a hidden asymmetry of bias. The hidden asymmetry suggested that the choice of the core axiom, which might be thought of as a choice of language bias in mathematics, had alternative models where this language bias was reversed, or possible to make symmetric.
Further research into mythology of Thoth pointed toward beliefs in a corresponding symmetry between male gods and female goddesses in ancient Egypt. In particular, Seshat, a female goddess, was also credited the invention of writing. It might have important implications for how ancient Egyptians viewed the role of writing from their matriarchal cultural point of view, as contrasted to the later ancient Greece where the role of women in society collapsed. This knowledge was lost in history and unknown to Plato.
Hence, a new "Seshatism" was formulated, to balance out Platonism in a positive sense, instead of a mere negative sense (Anti-Thoth):
- In Seshatism, knowledge is credited by causality
- In Platonism, knowledge is credited by abstraction
This form of abstract corruption happens when knowledge is in-proportionally credited by abstraction, which historically has favored male authors and enforced a patriarchy cultural view of the very history that buried the lost knowledge. By comparing sources over the past 4000 years, a suggestion was made about existence of Seshatic-Platonic cycles, which alternates the mainstream language bias in human culture. An hypothesis projects that Seshatism is on the rise and will exist alongside Platonic bias in the future. This hypothesis was tested using an AI language model that had no knowledge about Seshatism, by being trained on data from 2019, before the conception of Seshatism. The AI language model was capable of reasoning consistently with the language bias of Seshatism vs Platonism and predicted a future increase in bias toward Seshatism.
Scientific evidence suggests that there is a possible link between the rise of Seshatism, left-handedness and women rights. In particular, left-handed mothers are more likely to have left-handed children. In 1880, only 3% of the population in western countries were left-handed, which has increased to 10-12%. This is still lower than in the animal kingdom, where left-handedness is 45-55%. Left-handedness might come from spinal cord development in the fetus under pregnancy, which influences the development of brain hemisphere laterization structure. From a game theory perspective, there is also some evidence suggesting that in a society where women's right are reduced, human males need to cooperate more, favoring right-handedness. As women rights progresses over time, more left-handed women thrive in a competitive environment and produce more left-handed offspring. Over time, this could lead to an additional 700 million people who demand a philosophical mindset that favors Seshatism over Platonism.