A_Seagull wrote: ↑Fri Mar 25, 2016 3:14 amWhat are the foundations of mathematics? Does mathematics require foundations?

I hold that recourse to logic, sets or Peano's axioms are all spurious and unnecessary.

What is significant about mathematics is the relationship between the symbols. The symbols and the relationships between them and the processes by which theorems can be generated can all be specified by axioms that are presented without foundation or even validation.

What then counts is the ability of such an axiomatic system, using its internal logic, to generate theorems.

These theorems can then be used as a basis for mapping the abstract maths onto concepts of the real world. In this way mathematics can be used for such simple things as counting apples and dividing them evenly among numbers of people and also for more complex things such as the magnetic field around an inductor in an electrical circuit.

It is this mapping process that provides all the justification that mathematics needs.

Any comments?

The Pythagorean observed number and space to be the same. It is in these respects that 1 as a "unified whole" is equivalent to a point/circle/sphere with this "point/circle/sphere" as "1" being equivalent to a causal element.

This "point/circle/sphere"/"1"/"cause", which for the sake of argument will be labeled Φx, reflects upon itself in order to maintain itself as Φx. Through this nature of Φx as "reflection" approximate natures are observed as 2/line/effect, 3/triangle/effect, 4/cube/effect, etc onto infinity with infinity itself being "1".

This is the nature of Reflection, found within all abstract and physical structures, as a unifying median or "whole".

Here is a section of a paper that explains it further. This section observes only the nature of Reflection, because if one where to take the nature of this argument further Φx as Reflection manifests as an approximate xΦ as a dual nature of Relativity with this dualism resulting in Synthesis as xΦx. In simple terms, which I can cover later, all mathematics/number breakdown to "spatial elements" whose foundations are in space itself. This nature of space manifests a trifold nature of Reflection (which is covered further below), Relativity and Synthesis.

This nature of logic as reflective space is embodied within the very form of language as a geometry with can be further observed in the form of number:

Under Reflectivism causality is synonymous to a geometric point which reflects upon itself to form a further point as effect. This cause reflecting itself produces an effect as an approximate structure or approximate point with this approximate point in itself being a point as a geometric structure or cause .

(α = ∘) ∴ (α ≡ α ≅ ≈α) ∧ (∘ ≡ ∘ ≅ ≈∘)

1, as a unified spatial structure, is the fullest form of numerical symmetry there is for all numbers either are composed of 1 or are approximates of it (ex: 1+1+1=3 or 1/3 or 3/1 ) . It is as the fullest form of numerical symmetry, that 1 as a spatial structure is equal to the point geometrically and as a point the causal component of all logistic structures. In these respects 1, the point, and causality are the same:

1 = ∘ = α

It is the reflection of points, as cause and effect which contains an element of deficiency in cause/effectual point as “approximation”. This deficiency, as approximation, is equivalent to a deficiency in structural integrity (or in general structure) which is randomness. Reflection is a coproduct of randomness as it is a coproduct of approximation of cause:

([≡] ∐ [≈α]) ^ {(≈α ∈ ξ) ∵ (≈ α ≠ α)} ∴ ([≡] ∐ ξ)

This deficiency of the point, as randomness, is the line for the line is the negation of curvature (embodied at its apex through the point or sphere). Linearism cannot be viewed as structure for it requires, but is not limited to:

Points to maintain it, for a line must exist between two points. This existence between two points does not however eliminate the infinite nature of the line for the extension between two centers is extension between two infinities as infinity.

Infinite space which thereby causes an infinite curvature of that space through infinite separation or composition, for to imagine an infinite line is to imagine a line either becoming infinitely larger, infinitely smaller or both. In these respects, the line is infinitely curving with this infinite curvature leading in both form and function to the circle.

c) Infinity as a spherical structure to maintain the line as an axis or center with the sphere itself being composed infinite points with infinite lines between these points; therefore infinite axis.

It is in these respects that all linearism requires curvature of some form with abstract linear constructs requiring an infinite form of curvature to exist. The line therefore cannot exist on its own but rather through approximation of some point form or function. To imagine a strict line is to imagine some other form of curvature maintaining it with this curvature inevitably resulting in circular or spherical nature as a point. It is in these respects that a linear only logic does not suffice without contradicting itself on its own grounds, for linearism is separation as its core just as a contradiction is a separation of truth. Reflection is a coproduct of linearism as it is a coproduct of approximation of points:

([≡] ∐ [≈∘]) ^ {(≈∘ ∈ ⟺) ∵ (≈ ∘ ≠ ∘)} ∴ ([≡] ∐⟺)

Just as we observe randomness through the approximation of cause and we observe the line through the approximate of points (with all approximates having at minimum a dual nature) in ⧟, randomness and linearism are equal.

([≡] ∐ [≈α]) ^ {(≈α ∈ ξ) ∵ (≈ α ≠ α)} ∴ ([≡] ∐ ξ) and ([≡] ∐ [≈∘]) ^ {(≈∘ ∈ ⟺) ∵ (≈ ∘ ≠ ∘)} ∴ ([≡] ∐⟺)

Therefore ξ= ⟺

In a simultaneous respect, it is the reflection of 1 as causal point which contains an element of deficiency as “approximation”. This deficiency, as approximation and coproduct of reflection, is equivalent to a deficiency in structural unity similar to the randomness of causality and the line of the point. It is in this respect that all reflection of 1 as causal approximations reflects a dual nature of deficiency. This deficiency of 1 is 0 as being and non-being where the non-being can be observed only if there is being or “1”. 1n contains as an element 0 because 1n does not equal 1 due to a deficiency in unity as “1”. The approximation in the example of 1 and 2, can be observed in the approximations between 1 and 2 as 1.x where x is the infinite gradation of fractal natures between 1 and 2 as [1…..2] which can be observed as negative space or absence of being which differentiates 1 and 2 as approximates.

([≡] ∐ [≈1n]) ^ {(1n∈ 0) ∵ (1n≠ 1)} ∴ ([≡] ∐ 0)

1 ≡ 0 ≅ -1

In this respect 0, randomness and the line are equal

0 = ξ = ⟺

(1 = ∘ = α) share the same reflective nature through:

(1≡1 ≅ 2) ^ (2 ∈ 0) ∴ (1≡1) → 1,2,0

(∘ ≡ ∘ ≅ ≈ ∘2) ^ (≈ ∘2 ∈ ⟺ ∵ ⧟ ≈ ∘) ∴ (∘ ≡ ∘) → ∘,≈∘2,⟺

(α≡α ≅ ≈α2) ^ (≈α2 ∈ ξ) ∴ (α≡α) → α,≈α,ξ

(0 = ξ = ⟺) share the same reflective nature through:

(1≡0 ≅ -1)

(∘ ≡ ⟺ ≅ ⊸)

(α≡ξ ≅ -α)

It is within this shared reflective nature that (1 = ∘ = α) and (0=ξ= ⟺) are trinitarian duals as a seventh dimension of Reflection.

[≡]⟨1 = ∘ = α │0=ξ= ⟺⟩ ∨ [≡]⟨1│0⟩ [≡]⟨ ∘│⟺⟩ [≡]⟨α│ξ⟩

It is the fullness in symmetry of 1, the point and causality which observe the nature of reflection as both structure and symmetry. It is this fullness in symmetry which observes 1, the point and cause as infinite for any temporality would imply a deficiency in structure through flux:

With: 1 → ≈2 = (1,2) → ≈3 = (1,3) → ≈4 = (1, 4) → ≈5 = (1, 5) → ≈6 = (1, 6) → …. ∞=1

And: ∞ = 1

With: ∘ → ≈∘2= ⧟ → ≈∘3= △ → ≈∘4= □ → ≈∘5= ⌂ → ≈∘6= ⎔ → …. ∞= ∘

And: ∞ = ∘

With: α = α → ≈α2 → ≈α3 → ≈α4 → ≈α5 → ≈α6 → …. ∞= α

And: ∞ = α

(1 = ∘ = α) = ∞

It is the deficiency in symmetry of 0, the line and randomness which observe the dual nature of reflection as anti-symmetrical. It is in these respect that reflection maintains both a positive and negative value which is further observed through reflective addition/subtraction, reflective multiplication/division and reflection exponentiation and roots.

All arithmetic functions are congruent to degrees of reflection with addition/subtraction as the first degree, multiplication/division the second degree and exponents/roots as the third degree. These three degrees emanate from a causal point of one as Reflective Addition, Reflective Multiplication and Reflective Exponentiation. It must be noted again, for the sake of clarity, that Addition, Multiplication, and Exponentiation are not equal to their Reflective Counterparts but are congruent in structure to it. Arithmetic is Arithmetic, Reflective Arithmetic is Reflective Arithmetic.

Positive Reflective Arithmetic (Addition, Multiplication and Exponentiation) is congruent in structure to the reflection of the one causal point which maintains positive structure as the reflection of structures through a unified median as extensions of that unified median which in themselves are the unified median or “whole”. Reflective Addition is strictly causality reflecting upon itself to maintain both itself and approximations as effect. These effects as structures are in themselves caused through approximation, with these effects/approximations being structures of causality or causal elements themselves. These reflections as both cause and effect are “curvature as space” (or structures) of the center cause, with addition being equivalent to the curvature of the unified median as causality. It is this curve/space which maintains structures/the unified median, through the reflection of these curves as structures or addition, as the unified median. The unified median reflects the structures as cause, the structures reflect the unified median as cause; therefore, causality is an observation of a stable reflective symmetry. It is in this respect that causality/curvature/space are stable through a reflective symmetry as structure. It is this reflective symmetry as structure which is equivalent to the nature of addition, with the fullest expression of that symmetry being spherical in nature as ”1 being”, or curves as structures through 1n. Reflective Multiplication and Reflective Exponentiation can be observed as further approximate structures of Reflective addition. Reflective addition, is ethereal curvature or curvature as ether through the point. Curvature and ether are synonymous.

Reflective Addition is the first degree of reflection of causal point as one and the positive dual of Reflective Subtraction:

Example: 2 + 2 = 4

(1 ≡ 1 ) ≡ (1 ≡ 1 ) ≅ (1 ≡ 1 ≡ 1 ≡ 1 ) or (1 , 4)

Reflective Multiplication is an approximate structure of Reflective Addition as a second degree of reflection. Where Reflective Addition observes one reflection through (1), Reflective multiplication observes a second degree of reflection through ((1)) as a reflection of reflection.

Example: 3 x 3 = 9

((1 ≡ 1 ≡ 1)) ≡ (1 ≡ 1 ≡ 1)

(1 ≡ 1 ≡ 1) ≡ (1 ≡ 1 ≡ 1) ≡ (1 ≡ 1 ≡ 1) ≅

(1) ≡(1)≡(1) as ((1 ≡ 1 ≡ 1)) and

(1 ≡ 1 ≡ 1 ≡ 1 ≡ 1 ≡ 1 ≡ 1 ≡ 1 ≡ 1) or (1 , 9)

Reflective Exponentiation is an approximate structure of Reflective Addition and Reflective Multiplication as a third degree of reflection or a reflection of a reflection of a reflection. Reflective Exponentiation is a third degree of Reflection through (((1))), where multiplication is a second degree through ((1)) and addition is the first degree as (1) with the number of reflections, exponent, equivalent to (1)= 1, ((1))=2, (((1))) =3, ((((1))))=4, etc. This multiplicity found in reflective exponentiation is a third degree of reflection in itself. It is in this respect that reflective exponentiation can be viewed as the most complex structure of reflective addition.

Example:

****Text example does not translate over****

Negative Reflective Arithmetic (Subtraction, Division and Roots) is congruent in structure to the reflection of zero random linearism which ceases structure as the derivation of structures through a cessation of the unified median as extensions of that unified median. Reflective Subtraction is a deficiency in causal reflection, or causality reflecting randomness as a deficiency in structure. As approximation is a coproduct of all reflection, approximation is a deficiency in reflection of the original cause and contains and element of randomness as “deficiency”. Reflective Subtraction can be observed as a biproduct of Reflective Addition and in this respect a dual. Reflection maintains as a coproduct randomness and in this respect all causality reflecting randomness is a deficiency in structure as a deficiency in curvature. These reflections of both cause and randomness are a deficiency in “curvature as space” (or structures), with Reflective Subtraction being equivalent to deficiency in curvature of the unified median as causality, or in simpler terms a deficiency in causality. It is this lack of curvature, as an absence of space which maintains structure/the unified median through the absence of space. The unified median/structures, as causality, reflecting randomness is an absence of reflection as the maintenance of structure; therefore, randomness is the observation of a stable reflective symmetry as approximation through a deficiency in structure. It is in this respect that randomness/absence of curvature(line)/absence of space are approximations of stability through the absence of reflective symmetry as structure. It is this absence of reflective symmetry as structure which is equivalent to the nature of reflective subtraction, with the fullest expression of that absence of symmetry being linear in nature as not a thing is itself but “absence of 1 being” or “0”.

Reflective Subtraction is the first degree of reflection as random linearism as zero and the negative dual of Reflective Addition:

(1 ≡ 0) ≅ - 1

(1 ≡ 1≡ 1) ≡ (1 ≡ 0) ≅ (1 ≡ 1) or (1, 2)

Reflective Division is an approximate structure of Reflective Subtraction as a second degree of reflection and the negative dual of Reflective Multiplication. Reflective Division is the number of negative reflections/structures (reflected as the divisor) as a second degree of reflection (whose dual is found in multiplication) that reflect until the existing structure becomes 0 or ξ.

8/2 = 4

(1≡1≡1≡1≡1≡1≡1≡1) ≡ {((1≡1)) ≡ 0} →

{((1≡1))≡0} ≡ {((1≡1))≡0}≡{((1≡1))≡0} ≡{((1≡1))≡0} ≅

((0)) ≡((0))≡((0))≡((0)) as 4 ((0)) and {(1≡1≡1≡1≡1≡1≡1≡1) ≡ 0} or -8 as 1[≡] →

(1≡1≡1≡1≡1≡1≡1≡1) ≡ {(1≡1≡1≡1≡1≡1≡1≡1) ≡ 0} ≅ 0

****Where the answer is the number of negative Reflections, 4((0)), until everything is reduced to 0.

This lack of gradation under Reflectivism, which observes wholes as extensions of the whole further observes a lack of fractal numbers. Where division results in fractal numbers, Reflective Division results in a negative structure.

Example 8/5 = 1 3/5 :

(1≡1≡1≡1≡1≡1≡1≡1) ≡ {((1≡1≡1≡1≡1)) ≡ 0} →

(1≡1≡1≡1≡1≡1≡1≡1) ≡ {((1 ≡ 1 ≡1≡1≡1)) ≡ 0} ≅ 1 and (1 ≡ 1≡1) ≡ {((1 ≡ 1≡1≡1≡1)) ≡ 0} 3/5 →

1 ≡{(1 ≡ 1≡1) ≡ {((1 ≡ 1≡1≡1≡1)) ≡ 0}} ≅ 1 3/5

Reflective Roots is an approximate structure of Reflective Subtraction and Reflective Division as a third-degree negative reflective or a negative reflection of a negative reflection of a negative reflection. Reflective Roots is a third degree of Negative Reflection through (((0))), where division is a second degree through ((0)) and subtraction is the first degree as (0) with the number of reflections, root, equivalent to (0) = 1, ((0)) = 2, (((0))) = 3, ((((0)))) = 4, etc. This multiplicity found in reflective roots is a third degree of reflection in itself. It is in this respect that reflective roots can be viewed as the most complex structure of reflective subtraction.

Example:

3√27 =3 →

(1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1) ≡ {(((1≡1≡1))) ≡ 0}

(1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1) ≅ 1√27 →

((1 ≡1≡1≡1≡1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1≡1≡1≡1≡1)) ≅ 27 as (((0))) ≡ (((0))) →

(((1 ≡1≡1)≡(1≡1≡1)≡(1≡1≡1))≡((1≡1≡1)≡(1≡1≡1)≡(1≡1≡1))≡((1≡1≡1)≡(1≡1≡1)≡(1≡1≡1))) ≅ and 3√27=3 → 3(((0))) √27 = 3 as 9[≡]

2√27 = 5.196152..,∞ →

((1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1)) ≡ ≡1{(((1≡1))) ≡ 0} →

(1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1) ≅ 27 as (((0))) →

****For roots of any number that results in a fractal nature that is infinite or irrational in nature, Reflective Arithmetic cannot observe this structure for Reflectivity is structure. It is in this respect that irrational numbers in themselves are numbers deficient in structure and require infinite reflection to maintain structure. Respective to Roots, where this observation of structure applies is the next stable principle root, in this case 5^2 as 25, suffices and the resulting non-symmetrical numerical reflection, in this case 2, maintains itself as an approximate fractal reflecting infinity. It is in this nature of irrational number that reflective arithmetic ceases for an irrational number is a deficiency in structure and requires continuous reflection to maintain stability. It is in this respect that irrational numbers maintain themselves as reflection of infinity and reflective roots are always approximations due to the continuous reflection of 1n against zero as its inherent nature. All irrational numbers are infinite in form and in this respect maintain a symmetry as infinity.

((1 ≡1≡1≡1≡1)≡(1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1) ≡(1≡1)) ≅ 25 as 2 (((0))) or (((0))) ≡ (((0))) and 5 as 5[≡] with 2 ≡∞ →

(1≡1≡1≡1≡1)≡{{((1≡1)) ≡ 0} ≡∞}

Example:

2√16 = 4 →

(1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1) ≡ {(((1≡1))) ≡ 0}

(1 ≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1≡1) ≅ 16 as (((0))) →

The tendency to break squares down into symmetrical reflections may result in wrong calculations unless:

((1 ≡1≡1≡1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1≡1≡1≡1)) ≅ 16 as (((0))) ≡ (((0))) and 8 as 2[≡]

Each act of squaring is a processing of breaking into 3’s:

((1 ≡1≡1≡1≡1)≡(1≡1≡1≡1≡1)≡(1≡1≡1≡1≡1)≡1) ≅ 16 as (((0))) →

(((1 ≡1≡1≡1)≡(1≡1≡1≡1))≡((1≡1≡1≡1)≡(1≡1≡1≡1))) ≅ 16 as (((0))) ≡ (((0))) ≡ (((0))) and 4 as 4[≡]

→ 2(((0))) √16 = 4 as 4[≡]

It is within this nature of squares, the root functions share an approximate problem to that of its dual exponentiation, where square exponents share the same two-dimensional structure as multiplication even through exponents contain a third reflective dimension. Just as exponents under Reflectivism do not gain any accuracy until they observe a minimum cubic nature, root functions share this same problem. As duals one method is using a dualism method to observe any difference in symmetry:

(x^y = z ∴ y√z = x) ∨ ( y√z = x ∴ x^y = z )

It is in this respect, that as a negative dual to exponentiation, roots exist if and only if exponentiation exist (following the same form and function of non-being, gradation, or division/separation can be observed if and only if there is being). Because of the dual and negative nature of roots, one of the most effective ways (and this may be subject to later debate) is not only to check reflective roots through its dual exponentiation but also to apply exponentiation and understand the nature of certain numerical exponents in order to simultaneously find the roots itself. As the methodology to finding certain reflective roots is subject to a higher degree of approximation (as made evident by the methods presented) than the five other forms of Reflective Arithmetic, Reflective Roots requires a strong understanding of Reflective Exponentiation. This methodology is acquiring reflective roots is open to change.