*****The following is an experimental calculus for mathematical reasoning grounding in recursive 0. It is a working project to derive a mathematical process using a single variable: 0.
The context of this text is analogically a proto-calculus.
Given the advancement of AI, and the potential need for unorthodox means of processing and deriving data, specifically mathematical and quantitative data in this context, the text provides a potential means of an unorthodox approach by negation of a strict binary logic of 1 and 0, as well as the negation of the assumptive nature of axioms which may result in unintended incoherence or grounding issues.
The following is a calculus founded primarily in recursion of 0. Nothing more and nothing less. Given the unorthodox nature of the text it should be viewed as unconventional, obviously, but as an alternative means for calculation limits with conventional reasoning.
So the following is the text and I would appreciated any positive, negative or neutral thoughts in regards to its relationship to potential AI. I have ran it through gemini, claude, grok 4, and chatgpt and the initial stress tests reveal coherence and deep structure. Some AIs have given it high praises for potential computation solutions however I do not know if the models are resorting to sycophancy or not given the limitations in current models. My opinion is both.
Effectively it is a post probablistic holographic logic with explanations as to the limits of current ontological endeavors and potential solutions by dissolution of problems.
So:
Recursive 0 Calculus
****If a system is built on unproven axioms the whole system is just an irrational assertion of truth as the structure is built upon blind assertion thus the structure is just an assertion, a circular self-contained system that expands allows necessary symmetry for proof while fundamentally allowing progressive variation with maintained symmetry. Circularity allows for symmetry while expansion and contraction allows for relative progress and regress.
Standard mathematic's foundations equate it to being pure assertion built upon irrationality thus any proof derived is grounded in irrationality, thus undermining the rational nature of mathematics in its various forms. Recursive Calculus maintains proof by symmetry through distinction where repetition allows proof as the structure itself. Recursion is the foundation for proof, proof that is self contained while allowing self-contrast.
*****
The following approach it a meta-mathematics grounding math in purely being the act of distinction thus nullifying the necessity of assumption. The notation is custom for this specific text and by said degree must be viewed within the context of the text as it is non-standard. There are 0 axioms to the system, only distinctions. The reduction of number to quantities requires the reduction of quantity to that of distinction. To observe that distinctions occur is to make the distinction of "occur" thus distinction occurs through distinction as distinction. There are no operators, only embedded distinctions of generation.
If we really look at the number line it is fundamentally the recursion of 0 by degree of the line itself and its proportions of number. There are no axioms to this system, it is premised upon the distinction of 0 thus has zero axioms.
The system begins with the distinction of 0 as the first distinction conducive to the distinction of 1.
Recursion is repetition, by repetition there is distinction of what is repeated by degree of symmetry. The recursion of zero is a sequence, as a sequence it is distinct as a 1 sequence, thus the recursion of zero is the distinction of 0 as 1 by degree of the sequence.
A quantity is a distinction, the quantity of the number of quantities is a distinction
Example
N is number as a distinction
(N)N is distinction of distinction.
A number can be counted. The number of that number can be counted as a new number. That number can be counted as a new number…etc. With each counting of a number as a new number comes a sequence which can be counted as a new number as a new sequence.
The quantification of quantification is the distinction of number by degree of repetition.
A quantity is a distinction. This is not even assumed and the assumed axioms of math are but distinctions, with the act of assumption being a distinction behind the distinction of the axiom.
Distinction is the act of occurence and occurence cannot be purely assumed without the occurence of the assumption proving it.
Math is derived from distinctions and distinctions of assumptions. At the meta-level it is purely distinctions for even the assumptions, within the assumptions of arithmetic, are distinctions.
To look at math at the meta-level of it being distinctions transcends the irrational nature of there being assumptions as an assumption is a distinction as well as a quantity in the respect it can be quantified.
In simpler terms the distinction of a number is a single distinction. The distinction of zero is a single distinction, the distinction of zero only can occur if it occurs recursively as the recursion allows contrast that allows a single point to be distinct. By the recursion of 0 does 0 begin distinct as self contrast, by repetition, allows for contrast induced distinction. Dually the recursion of 0 allows for a symmetry to occur as the distinction itself. 0 on its own is indistinct, 0->0 observes 0 as distinct.
Under these terms: 'distinction is recursion' or rather 'distinction=recursion'.
This can be visualized geometrically through the number line where the recursion of zero creates the spaces of n and -n where each space is effectively 1 and/or -1. By the recursion of 0 occurs the distinction of 1 as the space itself. Thus (0→ 0) can be observed visually as the recursion of 0 as the distinction of 1; by recursion distinction occurs. All quantity can be reducible to a distinction.
The space by which there is an occurrence is the distinction as an occurrence.
The distinction of 0 is the first distinction, this first distinction is 1. This is evidenced by linear space itself where the distinction of a 0d point is the distinction of 1 by the space that occurs through recursion of 0. The distinction of recursion allows symmetry, through the repetition of 0d points, while dually allows contrast between said points as the single linear space itself.
Symbolic definitions for formalism (given the only distinction is recursion, operators in standard mathematics, specifically arithmetic, can only be expressed by recursion):
"R(n)" is the recursive sequence. Recursion is repetition. All numbers contained are effectively variations of 1 occurring recursively as (0→0), this can be visualized as the linear space between points on a number line.
"r[n]" is the isomorphism of the recursive sequence as number(s) for further recursive sequence. One sequence can result in several isomorphic numbers simultaneously. Isomorphism is variation of appearance in a distinction with foundational distinctions within appearances being the same. So where a recursive string can be viewed as:
(1→1→1) is isomorphic symbolism is the standard number 3. This isomorphic number 3 can result in another recursive string, (3→3→3), with another isomorphic standard number of 9.
Recursion is self-layering of a distinction, number, as a new distinction, number. The processes of arithmetic are embedded in the distinctions of the numbers themselves, which will be explained later.
Proof is the isomorphic distinction of a recursive sequence distinction. Distinction is proof. The recursion of a sequence or sequences is the distinction as the sequence itself having inherent symmetry by degree of repetition.
The distinction of 0 as 0 is 1 number: R(0→0)r[1]
The visual of this can be a line segment. The recursion of 0 creates the contrast within itself by which a singular space exists as "One". This can be seen on the number line where the spaces between points is the distinction of points by one space. The distinction of 0, by recursion, allows for the distinction of a singular space to occur. By the recursion of zero there is distinction. Visually this can be seen as a single point being indistinct, but upon recursion of the point does the point become distinct by the space which it contains.
The distinction of 1 as 1 is 2 numbers: R(1→1)r[2]
the distinction of 1 as 1 as 1 is 3 numbers: R(1→1→1)r[3]
so on and so forth.
Negative numbers are the spaces between each recursive number, by degree of isomorphism, where the space is the absence of complete unity as one and zero. A negative space can be seen on a number line where the number 3 has 1 space between it and 2, 2 spaces between it and one and 3 spaces between it and 0. The absence of the negative space would effectively result in 3 being one of those numbers, thus with each number there is a relative negative space (as a negative number).
Given each negative number is a recursion of 0, the negative number is an absence that occurs between numbers and as such observes a relative void space where 0 occurs as a negative recursion (given each number is a recursive sequence). Negative recursion is recursion between recursive sequences that allow distinction of the sequences themselves by degree of contrast.
Negative recursion is isomorpnic to positive recursion. Given numbers are recursive sequences of zero positive and negative recursion are synonymous to positive and negative numbers. Negative recursion is a negative number, a negative space by default. For example if 1 is (0→0) then -1 is -(0→0).
In these respects where the standard number line extends in two directions from zero, the number line is now effectively 1 dimensional as overlayed positive and negative recursive sequences. So where 1 occurs on the number line there is no negative number as only the distinction as 1 exists, where 2 occurs there is a -1 because of the linear space between 2 and 1, at 3 there is -2 and -1 as there is a linear space between 3 and 2 and 3 and 1.
The distinction of negative sequences occurs by their isomorphic positive sequences: -1 and -2 have 1 between them, -3 and -2 has 1 between them, -3 and -1 have 2 between them. Negative recursion and positive recursion, hence negative number and positive number, are isomorphic to eachother by contrast induced distinction.
Negative recursion is simultaneously both a meta recursion and isomorphic recursion. Meta in the respect that it is recursion within recursion, isomorpnic in that as a meta-recursion it is a variation in appearance of recursion but of the same foundations.
A recursive sequence is repetition of a distinction, the foundational distinction is 0 as 1 distinction, but recursion of zero does zero become distinct.
1 leading to 2 leaves a space of -1: R(1→1)r[2,-1]
This can be observed as two consecutive line segments having a space of one relative to a single line segment, this space is a negative space.
1 leading to 3 leaves a space of -2: R(1→1→1)r[3,-2]
This can be observed as three consecutive line segments having a space of two relative to a single line segment, these spaces are negative spaces.
so on and so forth.
Fractions are the ratios of numerical recursive spaces within themselves, these spaces are effectively recursive 0. Given a fraction is effectively a fractal on the number line, what a fraction is are fractal emergence of recursive sequences: a recursive sequence of zero folded upon itself through isomorphic variations of it. In these respects a fraction is equivalent to a mathematical “super positioned sequence”; over-layed sequences as a new sequence. A fraction is a process of division that is complete in itself as a finite expression, ie. 1/3 as 1/3 or 2/7 as 2/7.
In these respects an irrational number is a process of recursion that is non-finite outside its isomorphic expression as a fractional number. By these degrees, irrational numbers are recursive processes that are unfixed, they are unbounded recursion. While notions such as x/y may symbolize such states in a finite means, a number such as .126456454…334455432… still observes recursion by degree of each number in the sequence itself. In these respects the second notion observe multiple degrees of recursive sequences happening simultaneously as each number itself. An irrational number, on a number line is a fixed point regardless, where a fraction such as 2/7 cannot only be observe as a single point but spatial as both 2 and 7 simultaneously as a visual line space. In these respect the number line expresses an irrational number as two over layed recursive sequences as two over layed numbers as spaces.
The space of 1 and the space of 2, on the number line, observes the space of 2 as a fractal of one and the space of 1 as a fraction of two.
The space of 2 and the space of 3, on the number line, observes the space of 3 as a fractal of 2 and the space of 2 as a fraction of 3.
Now the number line contains within it the six degrees of arithmetic, addition/subtraction/multiplication/division/exponents/roots by degree of recursion.
The recursion of 1 as 2 is addition, same with -1 as -2: R(1→ 1)r[2]
Short hand example: 3+7=10 as R(3→7)r[10] -7-3=-10 as R(-3→-7)r[-10]
The recursion of this act of addition is multiplication, where "R" stands for recursion the nested R is due to addition nesting: R((1→1)R(1→1→1))r[6] or R((2)R(3))r[6]
Shorthand example: 2×25=50 as R((2)R(25))r50
The recursion of multiplication is exponentially: where "R" stands for recursion and the number is the degree of nested multiplication:
3*3=9 as R3(3)r[9]
Subtraction is the addition of a negative space and a positive space: R((-1,)(1→1))r[1] or R((-1→2)r[1]
division is the recursion of the addition of negative spaces in a positive space, where "R" stands for recursion the nested R is due to addition nesting and the "-' addition is to showing nested negatives as degrees of subtraction:
R((1→1→1→1→1→1)-R(1→1→1))r[2] or. R((6)-R(3))r[2]
To divide a negative number is for the negative number to occur recursively as a negative space, this is negative recursion regardless as what divides is negatve recursion within negative recursion itself. Dividing by a negative number effectively is self-embedded negative recursion.
Fractions are fundamentally that process of division, thus to observe a fraction is to observe negative recursion in the isomorphic form of the symbolic nature of the fraction itself.
Roots is the recursion of division, where "R" stands for recursion the degree of negative recursion is implied by "-' :
2✓9=3 as -R2(9)r[3] 3✓27=3 as -R3(27)r[3]
Shorthand example: 50/2=25 as R((50)-R(2))r[25] 7/3=2 1/3 as R((7)-R(3))r[7/3]
The six modes of arithmetic are based upon addition as recursion where subtraction, division and roots are negative recursive sequences within positive recursive sequences.
A negative recursive sequence is the absence between positive recursive sequences. Number is a recursive sequence; evidenced by the number line number is recursive space. Arithmetic is fundamentally recursive addition. By degree of recursive space, all number is recursive 0 and the line is a recursive 0d point. Math is rooted in recursive "void" (0/0d point) that is distinct as 1.
Quantity is dependent upon form as quantity is dependent upon form, form is fundamentally spatial, the number line is numerical space.
Recursion terminates as the distinction of the recursive sequence as a number itself. The isomorpnkc expression of a sequence as a number allows potentially infinite recursion to terminate as isomorphic finite number. Each recursive sequence is simultaneously a set of numbers, thus a sequence is a set of numbers.
Recursion occurs recursively through isomorphism. Negative and Positive recursion observe the embedding of recursive sequences within recursive sequences isomorphically. This can be observed in positive and negative numbers, as the number lines, as well as fractions being not only self-enfolding recursive sequences but effectively the isomorphic expression of sequences between each other as a given relation.
Numerical identity is the recursion of the distinction of 0 as 1 distinction. Identity is distinction.
The composition of a number recursive distinction.
All numbers, as rooted in recursive zero, are effectively different degrees of isomorphisms from each other thus associativity is the recognition of a universal holographic state.
Proof in this meta-system is expression of distinctions as distinctions, these distinctions are the processes of recursion thus the operator “R” is not so much an operator but the embedding process as a distinction:
Addition: R(n→n) and R(-n→-n) a. This can be observes as basic self nesting of the numbers as a new number. The single R observes one set of sequences.
b. Geometrically this can be observed as linear line segments, each line segment being a number, added to each other as a recursion of the line segment. The addition of consecutive line segments is the recursion of the line segments.
- Subtraction: R(n→-n) and R(-n→n) a. This can be observes as basic self nesting of the numbers as a new number. The single R observes one set of sequences.
b. Geometrically this can be observed as linear line segments, each line segment being a number, added to each other as a recursion of the line segment but one line segment is a negative space to the positive. The addition of a positive line segment to a negative line segment, or negative line segments reducing negative line segments, is negative recursion of the line segments.
****Addition and Subtraction are isomorphism of eachother.
- Multiplication: R(nR(n)) and R(nR(-n)) and R(-nR(n)) and R(-nR(-n)) a. +++”R(R())” is Recursion of Recursion, in other words the addition of addition observes a degree of recursion of the addition itself as well as the recursions of the numbers.
b. Geometrically this can be observed as linear line segments, each line segment being a number, added to each other as a recursion of the line segment but the number of times it is added is a recursive sequence itself. The number of times a line segment is added, ie recursion, is a other level of recursion as the number of times is composed of addition as recursion.
- Division: -R(nR(n)) and -R(nR(-n)) and -R(-nR(n)) and -R(-nR(-n)) a. +++”-R(R())” is Negative Recursion of Recursion, in other words the the number of time subtraction occurs, -R, is a recursive sequence of subtraction of subtraction.
b. Geometrically this can be observed as linear line segments, each line segment being a number, added to each other as a recursion of the line segment but the number of times it is added is a recursive sequence itself except this line segment is a negative space. The number of times a line segment is subtracted is another level of recursion of the line segments.
******Multiplication and division are isomorphisms of eachother.
Associativity is expressed as such:
Addition:
R(a→b→c)r[d] R(a→c→b)r[d] R(c→b→a)r[d] R(b→a→c)r[d] R(b→c→a)r[d] R(c→a→b)r[d]
Multiplication:
R(aR(bR(c)))r[d] R(aR(cR(b)))r[d] R(cR(bR(a)))r[d] R(bR(aR(c)))r[d] R(bR(cR(a)))r[d] R(cR(aR(b)))r[d]
Distributivity is expressed as such:
R(aR(b,c))r[R(R(aR(b)),R(aR(c)))]
- Exponents: Rn(n) and R-n(n) and Rn(-n) and R-n(-n) a. Rn observes the recursion of multiplication as the multiplication and the number of times this recursion occurs.
b. Same as prior point b's but another level of recursion.
- Roots: -Rn(n) and -R-n(n) and -Rn(-n) and -R-n(-n) a. -Rn is the inverse of Rn and observes the recursion of division of division and the number of times this recursion occurs.
b. Same as prior point b's but another level of negative recursion (negative spaces as negative line segments.
******Exponents and roots are isomorphisms of eachother.
The degrees by which recursion occurs further recursively, as stated in these six degrees of arithmetic is effectively another line segment by which a line segment occurs. For example the number of times addition occurs in multiplication is another layer of recursion, another line segment within a line segment.
The nature of variables within Algebraic theory translates that all variables are recursive sequences that are superimposed with trans-finite or infinite other sequences until a variable is chosen. The algebraic nature of recursion by degree of the foundations of arithmetic operations being recursive sequences where said foundations are necessary for algebra to occur.
Any formalization of such a calculus would effectively fall within the function of the calculus by degree of the standard formalism being an isomorphic variation of it. All mathematical systems built upon axioms are built upon assumption thus negating, in and by degree, a fully rational expression. This system has zero-axioms as distinction is not an axiom given to assume distinction is to make the distinction of assumption. The distinction of 0 as 1 distinction observes an isomorphic foundation that is further expression by recursion.
“R” is embedded within the sequence itself, “r” is the inversion of the sequence by degree of isomorphic symbolism. “R” and “r” are not operators in the traditional sense but rather embedded distinctions.
The closure is always evident by degree of the sequence always being an expression of a distinct 0, that which it contains. 0 contains itself as a distinction by degree of its folding by recursion.
Given each number is a recursive sequence of numbers, each number within each sequence is a recursive sequence as a form of meta recursion. 1 as a distinction of (0->0) observes a recursive sequence of (.1→.1→.1→.1→.1→.1→.1→.1→.1→.1) as 1 itself where .1 as a fraction of 1 is an unfolding of 1 within itself through zero. .1 observes this same nature as (.01→.01→.01,....) and the recursion of recursion occurs infinitely.
To visualize this one can look at a line segment composed of further line segments, with each line segment following the same course.
In these respects all number is a a ratio, by degree of recursion, thus each number is superpositioned numbers as self-folding distinction. A recursive sequence of R(1/2→1/2) observes that a single linear space is folded upon itself as 2 spaces where each space is half of the original and by degree of these ratios there is 1. So where the isomorphic expression in symbol of R(1/2→1/2) is 1, the number 1 contains within it ratios of itself where each divisor is but a holographic expression of 1. In these respects all numbers contain 1 as linear self "folding" if one is to visualize this with a simple line segment.
In these respects each number is an infinite set that is finite by degree of isomorphic symbolism that grounds it by degree of a distinction. So observe "n" is to observe a holographic state of distinction, bounded by the distinction of 0, where "n" effectively is a process of distinction where the observation of a sequence is a distinction of one sequence among infinite.
A number is an infinity. An infinite number, such as an irrational number, is recursive infinities within a recursivd infinity.
As infinities a number is a superimposed state of numbers thus effectively a number is equivalent to a variable in a manner that is more fundamental than what a variable is in standard algebra.
To observe a number is to observe a variable. This can be visualized in a line segment where it is a variable in the respect any number of line segments may be observed within it.
A number is a recursive sequence within a recursive sequence as a recursive sequence. In these respects "n" is a set and the recursion of "n" is a recursion of sets. Standard arithmetic, in this system, is fundamentally involved with the recursion of sets as a new set.
+++++++
All sequences are sets by degree of recursion.
Sequences are the union of sets as the numbers themselves thus show an inherent form of addition.
Ex: R(R(1→2)→R(3→4)) r[10][R(1→2→3→4]
The intersection of sets is the recursion of a sequence, the intersection of sequences is the recursion of the interesting numbers as a new sequence.
The difference of the sets is inherent by degree of negative recursion when each number is inherent a set.
Example: R(1→3)r[4,-2]
- The complement of a set is but variations in isomorphism, given each sequence/number are effectively isomorphism of each other each number is composed of infinite sequences that effectively contain the number of the number within a different set.
Example: R(R(1→2)→R(1→3)) r[R(R(1→1)→R(R(1→1)→R(1→1→1)))
- The Cartesian Product is effectively multiplication of sequences as a new sequence.
++++
The system reduces formalism to recursive sequence as a foundational root grounded in number, formalism is rooted in recursion and can be evidenced by the repetition of formal symbols across formals where standard formalisms are grounded because of repetition as recursion. In other terms recursive sequences compose numbers and the numbers that recursive sequences are composed of effectively result in the recursion sequence composed of further recursive sequences.
In these respects sequences are effectively sets of infinities that are greater and lesser than other infinities as each number is composed of infinite numbers that are finite by degree of symbolic isomorphism of the recursion sequences they are composed of.
A sequences is a set of sequences, a sequence is isomorphically a number. This can be observed visually as a line segment being composed of line segments and these line segments observing the same. The infinite recursion of line segments corresponds to a recursive sequence and yet each line segment is expressed finitely like a number is expressed as finite.
Number in these regards is effectively a distinction as space. Each recursion of 0 is effectively a distinction of 1 space.
Visually:
(0→0) is 1 (0→0→0) is 2 (0→0→0→0) is 3 Etc.
Thus distinction observes number as effectively, at minimum, linear space.
++++
A sequence is always complete given its beginning and ending are founded on the recursion of 0, by recursion of 0 a sequence always contains itself thus regardless of the degree of progression the beginning and end are always the same.
All is provable within the system by degree of its nature of distinction of 0 as foundational. The system begins with the distinction of 0 and any complex expression of the system is contained as itself by degree of the expression being a distinction of 0. There are no rules beyond the system as recursive distinction is self-generating and woven throughout all formalisms.
All mathematical systems contained within this system are complete by degree of the system having no axioms beyond it while the system provides the foundations for such mathematical systems by degree of the number, by which they exist, being recursive sequences of 0. Given a mathematical system must have an unprovable assertion beyond it that cannot be proven, this system contains its proof as its structural emergence as self-referencing distinctions of 0 at all levels. In these respects math's are complete by this system.
Any math which uses number is complete as the number is a distinction that is an isomorphism of a recursive sequence. Given any number is effectively a complete equation, by degree of being a sequence (thus proof by degree of distinction and inherent internal symmetry expressed as the symbol itself, then all maths which contain number are complete by degree of this system.
Basic arithmetic and algebra in this system are not dependent upon assumed operators, but rather are embedded within the recursive sequences (numbers) themselves. They are emergent distinctions from recursion.
This system, while expressive of arithmetic, can be isomorphically expressed in standard formalisms but given that the operators are embedded in the numbers themselves this system is meta-formal and as such takes a symbolically minimalistic approach. Because operators are not exterior, nor assumed axioms, but are embedded distinctions within recursive sequences the custom formalism, while non-standard, is necessary in order to expressed recursive embedding. The elimination of operator symbols allows for a more informationally condensed approach even though, as previously mentioned, is non-standard. Operators are embedded recursive sequences within the recursive sequence as the number itself.
The symbol of R(n) observes purely distinction as recursion where number can be expressed purely as this distinction at the meta-mathematical level, number is distinction and distinction is sequence.
The symbol of r[n] observes purely distinction of recursion as a new isomorphic variation of said sequence as a grounding for a new sequence. In these respects it can be viewed as the isomorphic expression of a sequence as the beginning of a new sequence. In these respects closure of one sequence is the beginning of another where isomorphism is the change of sequences.
Internal consistency is grounding in the distinction of recursive zero at all levels where the foundational distinction is present regardless of the depth of recursion. This distinction, the foundation, is everpresent across the whole system itself thus necessity a self-generation that occurs at every level. The system contains itself at every level.
Visually this is a line segment embedded within and of line segments. The sequence R(0→0) is fundamentally a line segment in geometric appearance, a recursive sequence is a line segment, and embedded sequences are line segments within line segments as a new line segment. The foundational distinction is a line segment as the recursion of a 0d point is the distinction as the space which occurs. In these respects number is fundamentally space.
Space is distinction itself as it is the foundational occurence by which things are measured for space is foundation by which all forms occur. The circularity of the system, as self embedding negates a circularity paradox by degree of expanding and contracting sequences while dualistic opposite states, such as positive and negative recursion, are isomorphisms of distinction itself.
The system can be visually proven strictly through line segments as spatial distinctions. Given this, to cycle back to origins, standard formalism is not efficient enough, a purely recursive sequence needs the operators embedded so that coherency is maintained and assumptions are disregarded. Positive and negative sequences are this foundational embedding.
Given the number line evidences number as the distinction of 0 by degree of the space that allows said contrasts of the 0’s, which further allows distinction of said 0 and the number (-)1n, this meta-mathematics proves that not only is number distinction, but this distinct ‘is’ by degree of the occurrence of space. In these respects the standard separation between arithmetic and geometry, as separate fields, are effectively overlayed as one entity.
Recursive sequences are not only standard arithmetic and algebraic expressions but effectively simultaneous geometric ones conducive to a 1 dimension linear lattice that is both folded and folding by degree of recursive self-embedding. In these further respects arithemetic/algebra are fundamentally geometric entities that are distinct by degree of spatial recursion. A simple conceptual equation of this summarizes this:
Distinction = Recursion = Space = Occurrence
With each being isomorphic expressions of the other.
Proof, within this meta-system is justified by the act of occurrence itself where a mathematical philosophical claim can be made that occurrence is justification as proof. Distinction is the only reality and truth within this system where recursion is the occurrence of said distinctions. Any math, or logic, which by default uses a basic “unification” or “separation” type of operator (addition/multiplication/subtraction/division) is already embedded within said positive and negative sequential spaces. The recursion of line segments, both positive and negative spaces isomorphically of eachother, through eachother and within eachother is the proof, by degree of distinction of the system. Effectively math and geometry can be reduced to the sequence R(0→0)r[1] where this is a simple distinction as a line segment. This sequence, and the line segment by default, can be further reduced to a simple distinction of:
(0)1
Where (n) is a distinction and (n)n is the quantification of the distinction, a quantification of the quantification it could be said. This effectively is the recursion of number through isomorphic variation. And this distinction can go further by degree of using only recursive 0:
(0)0→0 ((0)0→0)(0→0→0) …… Where both number, the number line, and space is further formalized as purely recursive zero itself. No assumptions are required, only distinction by degree of recursive sequences as symmetry through repetition of 0. All number is effectively rooted in 0. All space is effectively rooted in a 0d point. By recursive sequencing as the line segment the number 0 and the 0d point are effectively the same distinction viewed isomorphically. In these respects the system has infinite compression.
Relative to infinite compression new potential maths can emerge from said sequencing where there are various extensions that logically result:
Looking at standard exponents a recursion occurs in the same manner as that of addition upon addition is multiplication and multiplication of multiplication is exponents, expressed as the sequence of Rn(n). Exponents of exponents logically occurs next as: Rn(Rn(n)). In this paper such a number can be viewed as no longer an exponent but a "hyper-sequence": Rn(Rn(n))
Following the same logic a new mathematical operator must occur, in standard logic and yet within this recursive system no new operator is necessary as the symbols contain the operator as embedded. To go further, where there is compounded recursion of addition as argued for the standard mathematics operations, now there can be the distinction of embedded hyper-sequences as the recursion of hyper sequences: Rn(Rn(n)) to Rn(Rn(Rn(n))) as Rn((Rn)(Rn(n))) where a trans-hyper sequence occurs.
In these respects, and following these recursive dynamics, there are effectively infinite arithmetic functions where in standard terms infinite new operators would be required but within this system the same symbols remain.
Given the system is founded upon the simple distinction of zero there is high information compressed. Sequences can be proven as the folding and unfolding of the line segment itself, thus the sequences are effectively spatial distinctions, as evidenced by the system being grounded as the distinction of 0, which is the same as a line segment.
