How much freedom does math have?

[Eodnhoj7]

Math is based on logic. The paradoxes? Yeah, they exist, students learn them, but since it does not prevent math from carrying on, we simply ignore such things, unless one specializes in mathematical logic. If we could come up with better definitions in mathematics to avoid the paradoxes, we would. We are just presently stumped, and lucky that we do not need to resolve such issues for math to be useful.

What if the paradoxes should not be ignored but rather observed as foundational axioms for further mathematical theories? For example the most common form of paradox involves some form of circularity or rotation, however circularity and rotation do not necessarily limit the nature of knowledge specifically.

For example, one thing I am work on is a mirror function, that embraces your standard circular paradox of form while maintaining the linear characteristics necessary for progression.

Take for example, and you might have seen me write this else where, that standard 1 + 2 = 3.

If we view the (+) as inherent within the numbers themselves we can observe that number manifests through a mirroring process conducive to sets, while simultaneously maintaining their foundational premise as part of the answer.

So using the symbol of "⊙" as "mirroring" and "⧂" as "mirrors in structure" we can maintain both circular and linear forms without a fear of contradiction:

So 1 + 2 is approximately equal to:

⊙(+1,+2) where the positive nature of 1 and 2, as addition is fundamentally inseperable from the number itself.

⊙(+1,+2) ⧂ {+1,+2} the inherent premises being the foundation of the linear form are inherent and inseparable from the answer.

⊙(+1,+2) ⧂ {+1,+2,+3} at the same time the stand addition applies.

In a seperate respect, we can observe that since (+) is inherent within the number it also succumbs to the mirroring process. Hence the mirroring of addition results in multiplicaiton, with multiplication being the addition of addition. In these respects "multiplication", as a second degree positive value, is also inseperable from the number.

So the (+) value of 1 and 2 in turn mirror to form (*) 1 and 2 while the mirroring of +1 and +2 resulting in +3 also mirroring in structure *3.

⊙(+1,+2) ⧂ {+1,*1,+2,*2,+3,*3}

Hence not only are the foundation of number observed, but we can observe that numbers are dependent upon sets in themselves. In these respects sets are inevitable while retaining not just the individual dignity of each number but also providing the foundation of arithmetic functions as inherently inseperable.

That's the thing JD. There are different types of foundational math, some not relying on logic. Not all mathematicians accept logic as a foundation for math.
You are right about that for sure. We are still stuck we a basic problem though: We have math. We have logic. Where are they unified? Where are they seperate? Where can they be synthesized and what is the best methodology for synthesis?


Maybe better definitions will resolve the problem, but mathematicians have been looking hard for a long time (in fact there doesn't seem to be a universally acceptable definition for number).

I don't mean to suggest that math is useless, but it needs sharpening.

I agree...but a paradox occurs where the more one focuses on "x" the less they are able to observe "y". However "x" must be observed in order to observe "y"....the question of "clarity" comes to the for front as how clearly must I observe "x" in order to observe "y"?

It appears that what we observe as "clarity" is merely the ability to observe definitions approximately, or in simpler terms "connect" the definitions. The problem occurs when we observe "connections" we must also observe the connections as things in themselves hence "what originally connects' paradoxically causes a simultaneous division of properties also...hence a paradox again ensues and the only axiom, I believe, we cannot doubt is that symmetry as order is dependent upon an alternation.

To get back to my original point, if we view numbers from a perspective of "alternation" we may not only be able to gain a better universal foundation for understanding "what" they are, but simultaneously we may be able to observe universal functions that provide the foundations for other universal functions.

Even in basic math, and even logic to a degree, we are still stuck with universals of "positive" as a form of summation and "subtraction" as a form of absence...from this we get the further foundations of multiplication, division, roots, and powers. So we have six basic functions that provide the root for all mathes and linguistic logic (unless you see something in this premise I don't).

The question occurs to me are what are the foundations of these three duals? We take standard arithematic apriori, but why?

PhilX

[Science Fan]

That series does not add up to -1/12. There is a fundamental error in logic, by assigning a value to a non-converging series, which is relied upon for this so-called "proof." It's bogus as hell.

[Science Fan]

Eod: I'm not sure what you mean by stating that + is "inherent in the number." It's simply a defined operation, which is closed under the set of natural numbers. I don't see how that is then "inherent" within the number itself then, since if we add 5 and negative 15, we get negative 10, which is why subtraction is not closed under the set of natural numbers, although it can be closed under the set of integers. The same problems arise with respect to division, the search for roots, which ultimately requires us to come up with a complex number system; otherwise, we end up with non-closure for these basic operations.

[Philosophy Explorer]

That series does not add up to -1/12. There is a fundamental error in logic, by assigning a value to a non-converging series, which is relied upon for this so-called "proof." It's bogus as hell.

Calculus is big on convergent series and functions (the particular series in question is the triangular numbers series after you sum the terms). For calculus, the limit must exist for the function to be of any use.

PhilX

[Philosophy Explorer]

JD asks:

"You are right about that for sure. We are still stuck we a basic problem though: We have math. We have logic. Where are they unified? Where are they seperate? Where can they be synthesized and what is the best methodology for synthesis?"

Since math is a human creation, I would answer that logic and math unify where a human mathematician unifies them. They are separate where other human mathematicians separate them. Where can they synthesized and what is the best method would depend on what type of math you're talking about and how you define math.

PhilX

[Eodnhoj7]

Eod: I'm not sure what you mean by stating that + is "inherent in the number." It's simply a defined operation, which is closed under the set of natural numbers. I don't see how that is then "inherent" within the number itself then, since if we add 5 and negative 15, we get negative 10, which is why subtraction is not closed under the set of natural numbers, although it can be closed under the set of integers.

When looking at the origin of number I am not entirely sure we can observe natural numbers as the base foundation. While these numbers are useful for counting and ordering phenomena, we are still left with a relativistic view point in that we are observing the relation of parts. The problem occurs that these units, the natural numbers that is, in one respect must contain some consistent and never changing attributes yet what we observes are merely "relations" or "changes" through them, when applying them to physical phenomena.

The natural numbers exist because of the functions we apply to them. If I count three apples for instance, or even just count three abstract forms, I am observing a constant function of "addition" as a form of summation. The process of "unifying" the counted (and ordered) phenomena is inseperable from the addition function it entails. Hence we can observe that form not only follows function, and vice versa, but that they are inherently inseperable to a degree. Now this seperation between form and function works practically when counting and ordering phenomena considering that counting and ordering are dependent on observing the relations of parts through time as time. So the function may observe an active movement, and the number a passive movement (temporal form) and this works fine and well for understanding number through "time" as relations.

The problem occurs in the respect of viewing number, and hence quantifiable reality, from a unified perpsective. Now under standard counting we of course form "units" as parts of a whole, or as a gradation of "unity", but we never view "unity" for what it is...we observe relations. The question occurs in the attempt to look at the center of origin of not just number, but counting and ordering itself (hence reason and one could further imply logic), how do we observe this unity?

1) One axiom we may begin with is that in unity there is not seperation, everything is connected and totatlized as "one". This would require us however to step out of the present boundaries of what we can call logicistic atomization where the observation of logistic parts are observed in the manner through which they relate.

2) We cannot see the whole, only approximate it, however this approximate is an extension of the whole. Considering we can observe equations fundamentally as approximations of a larger set of equations, the standard natural numbers and the following functions which they relate through seems to "pop up" again as an issue. However the question of form and function remains....do the natural numbers determine the functions? Or do the functions determine the natural numbers? Does 1,2,3...etc. exist because of "+,-,*,/" or is it the other way around?

For example when counting 1,2,3,4 am I "summating" something into a unit of 1 then summating another unit into one, then both of those into 1 unit of 2...etc.? If I am summating something into 1 unit of 2, considering when we are observing "one" "2", am I simultaneously subtracting that unit of units from surrounding units during the act of quantification (and one could argue qualification in the respect we are forming units)?

Is the simultaneous act of addition and subtraction what causes the form to exist through function? This is considering each "unit" in itself is both a summation of units and subtraction of other units. If that is the case is the function of the number inseperable from the form?

The next question occurs in the respect that since we can see the functions as inherent within the number itself (as the positive and negative values of numbers differ little than the addition and subtraction of numbers when viewed as inherent), what is the function which give precendent to these functions and forms?

In simpler terms where do addition (as summation), subtraction (as deficiency), multiplication (as summation of summation within units of time), and division (as the deficiency of a deficiency within units of time) come from?

Viewing numbers as inherently positive and negative fundamentally unites the function with the form, the question must occur what is the larger "framework" which determines this current framework, as the current framework is dependent upon other frameworks to give it justification?

3) As "unified" form and function are united. The question occurs as to the mirror function then, considering this in itself seems to be a function of "functions", or what according to some people (Hofstatder (Godel, Escher, and Bach)) might call a "meta-function", as to whether or not a unified foundation can be observed for mathematics considering all the answers we recieve are finite?

For example

1+2=3 is still a finite equation but it does not unify the form with the function in one respect, in another the premises of 1 and 2 while composing 3 only observe the relations. The premises are not included in the answer, yet the premises are the foundation of the answer. The answer in itself may be composed of various premises, but the 1+2 = 3 must be the whole answer in itself, not just the 3, considering the answer of 3 is really strictly observing the relations of 1 and 2...in reality there are infinite ways to observe 3...hence observing the numbers of 1 and 2, through the function of (+), is necessary to understand three. 3 itself cannot be the answer, because the answer must be one of both form and function.

The mirror function observes "less" of seperation of the inherent scenarios as it observes approximately all numerical scenarios happening as one inherent set as extensions of "1". Form and function are inseparable from the number when viewed from a higher framework, with "1" being the foundation of all number, from which all number extends itself through a mirroring process (maintenance of dimensionality) from a similar perspective to how the pythagoreans observed the number extending from a circle mirroring itself.

Considering I observe the mirroring of positive numbers in the prior post, and that the function is inseperable from the number, I will observe the mirroring of multiples:

⨀(*3,*2) Multiple three mirroring multiple 2.

Considering all multiples, as observed in the prior post, are composed of addition values we can first observe that *3 and *2 not only exist within the answer but simultaneously +3 and +2 as not only multiplication its a positive value but extends from addition as addition:
⨀(*3,*2) ⧂ (+2,*2,+3,*3)

So we can observe the mirroring of multiple three and multiple two results in:
***for the sake of simplicity, additive values will be replace with a "∙", multiplication as second degree addition will be "∶", and powers as multiplication reflecting multiplication will be "⁞".


∙2 ∙3 ∙(2+3) ∶2 ∶3 ∶(2+3) ⁞2 ⁞3 ⁞(2+3)
∶2 ∶3 ∙(2*3) ∶(2*3) ⁞(2*3)
⁞2 ⁞3


This leads to, with "∙:⁞1" always inherently being in the answer set as each number is an extension of one mirroring itself.

⨀(∶2 , ∶3) ⧂ (∙:⁞1 ∙:⁞2 ∙:⁞3 ∙:⁞5 ∙:⁞6)




The same problems arise with respect to division, the search for roots, which ultimately requires us to come up with a complex number system; otherwise, we end up with non-closure for these basic operations.

I will address that next dependent on whether you agree with the supplied premises.

[Eodnhoj7]

JD asks:

"You are right about that for sure. We are still stuck we a basic problem though: We have math. We have logic. Where are they unified? Where are they seperate? Where can they be synthesized and what is the best methodology for synthesis?"

Since math is a human creation, I would answer that logic and math unify where a human mathematician unifies them. They are separate where other human mathematicians separate them. Where can they synthesized and what is the best method would depend on what type of math you're talking about and how you define math.

PhilX

From this premise it may be implied that math is less about applying limits in terms of finiteness but more of a continual synthesis of dimensions from which structure arises?

In simpler terms we synthesize the measurements which form reality?

[wtf]

In simpler terms we synthesize the measurements which form reality?

We integrate the dimensional modalities to recontextualize the gestalt.

[Eodnhoj7]

In simpler terms we synthesize the measurements which form reality?

We integrate the dimensional modalities to recontextualize the gestalt.

But that means their is an inherent self-mirroring process in the Gestalt that maintains it through a continual recontextualization...hence the movement as of a mirroring process (intradimensional reflection through continual mirroring) is a constant form of identity in one respect (we maintain and form the dimensions which form us) and in a seperate respect we dually project past the point of origin (as integretation implies an outside set of influences we project through).

In these respects number must have constant intradimensional properties through 1 as a self-mirroring unity and extradimensional properties where 1 as unit is defined through relations.

To summate the argument above, natural number must have a prerequisite set of mirroring properties to form them considering the continual synthesis is dependent upon a constant form of intradimensional self-forming and extraddimensional self-projection...if that makes sense.

[wtf]

To summate the argument above, natural number must have a prerequisite set of mirroring properties to form them considering the continual synthesis is dependent upon a constant form of intradimensional self-forming and extraddimensional self-projection...if that makes sense.

Yes, you understand perfectly.

[Eodnhoj7]

To summate the argument above, natural number must have a prerequisite set of mirroring properties to form them considering the continual synthesis is dependent upon a constant form of intradimensional self-forming and extraddimensional self-projection...if that makes sense.

Yes, you understand perfectly.

Unless you see something I do not see, the premises of the mirror function I present (I have not shown all the possible algebraic expression because it is 120+) should in theory provide a non-contradictory standard as to the foundation form and function of natural numbers.

In simpler terms a "mirror function" must be inevitable.

[wtf]

In simpler terms a "mirror function" must be inevitable.

Only if the bivalent adjacency matrix splits over the base field modulo a Fermat prime.

[Eodnhoj7]

In simpler terms a "mirror function" must be inevitable.

Only if the bivalent adjacency matrix splits over the base field modulo a Fermat prime.

I had to do a little research in the above points, considering I am not well versed with this form of mathematics, but upon multiple glances I do not think an argument can be presented that observes the origin of number from these premises....bear with me...

The reason is:

1) Standard arithemetic is required as apriori knowledge for the fermat numbers to work. The question occurs as to the nature of the arithmetic functions themselves, and a function must be required to "view" this function from the outside. The matrices and Fermat Primes require a form of symmetry to exist, hence they cannot be viewed as the origin of symmetry.

2) Mirroring, as a replication of symmetry, must be viewed as a universal binding median, considering what we understand of all form is symmetry. The replication of dimensions is inseperable from dimension itself, considering how we observe dimensions are merely boundaries which exist through "direction", primarily that of an intra-dimensional (self-relfecting) and extradimensional (self-projecting) nature.

Considering the perspective that number exists as a dimension in itself, and relative to other dimensions, I do not think it is entirely possible to separate a number from the direction it "moves".

In these respects number must maintain a dual intradimensional and extradimensional nature. The foundation of number "intradimensional" would fundamentally have to begin with "1" considering "0" is not only an absence of existence but can be observed only if viewed relativistically to "1".

"1" with one existing as "direction", from a perspective of universal unity all number and function must inherently extend from "1" itself and in these respects "1" must contain all these functions. "1" as dimensionality must intradimensionally mirror itself in order to exist. Considering "1" is dimensionality we cannot observe "1" as seperate from a positive value with that positive value being "unity" as a form of summation.

1 as pure "direction" cannot be seperate from a dual to a 0d point, and in these respects "1" may be the "cause" of number and may be inseperable as an intradimensional point which glues reality together through itself, as itself, and what we understand of logistic atoms (the various numbers) are merely approximations of it.


Using an excerpt from a paper I am working on to argue my third point:

3) Approximation observes the connections of multiple 1's and in itself is a negative dimension in the respect that if all existence, through number, is unified what we understand of as "lines" can be observed as negative dimensions (or not dimensions in themselves but rather the connection between them, hence as negative in value the line strictly observe the points as "point" by approximating unity through multiplicity of parts.)

Effect acts an approximation of cause with this approximation as a limit to “unity”. Approximation as limit is akin to the same definition of randomness through Chaos Theory in which approximation is merely a deficiency in observable structure In these respects the 1 as a causal structure, manifests randomness as a dual limit as absence of being. Considering nothingness is not a thing in itself, randomness as a deficiency in causal unity observes multiplicity as a deficiency in structure.

Considering the causal ethereal point is equivalent to 1 in quantity and quality what we observe as randomness, or its limits, is strictly the approximation of the ethereal point as points, or a perceived multiplicity where there is none. Multiplicity, in these regards can be observed as synonymous to approximation, as the limit of unity or a deficiency in structure through the changing relations of particles. The approximation of this ethereal point as points observes a connection, as not a thing in itself but rather a deficiency synonymous to randomness.

This is embodied through a negative dimensional line which, synonymous to an imaginary dimension in Euclidian terms, is equivalent to a deficiency in dimension and not a strict dimension in itself. Considering approximation, or randomness, is a deficiency in structure and not being in itself but rather a deficiency in being as “connection” which implies separation. What we observe as randomness is the negative dimensional line as -1 or -1d. This -1d line connects the ethereal points when viewed from a localized space/time or from the perspective where “x” is a cause for “y” with “y” being the effect of “x”.

This connection, in itself is the limit of ethereal unity as approximation observes multiplicity or the connection of unit-particles. Approximate points form -1d lines to connect them with these -1d lines existing as limits through approximation as randomness, or "∸x". ? Considering the mirroring process observes the maintainence of "∙x", it appears all "∙x" must be approximate to itself as all cause is simultaneously an effect. In these respects the "∸x" stretches through the point with the point acting as a field for it. “∙x", as spatial “no-limit” through center, acts as a field in its own respect.??
???
The -1d line will be observed as "∸x" and approximation as "ӫ". "ӫ" in turn will be observed as a dual to "⨀" in the respect that mirroring observes the unity of a spatial reality, through summation. The approximate function observes the limits of that very same unity through the lines which connects the points as point.

The approximator, ӫ , observes the approximation of point as points through the dual set of points above the circle while simultaneously observing the inherent linear characterisitics of approximation embodied by the line running through the circle.

The approximation of addition, through subtraction, observes the inherent linear characteristics stemming from the single positive point as “∸” or a subtractive negative line. The approximation of multiplication, as a dual point, observes the linear structure extending from it as division through “∹”. The approximation of powers, as a quad point, observes the linear structure extending from it as roots through ∺.

Approximation as individuation through "∸x" can be observed as a multiplication function resulting in negative numbers regardless of whether the values are positive or negative. The approximation of a positive additive values results in a subtractive value. This is considering subtraction is the deficiency of addition and what we are observing are -1d lines.


Furthermore the approximation function observes that each number, as a set of points in itself requires a connection for those points to exist as point. The equation
(α-1)*α(1/2)= ε_α

Observes this where "α" equals the number of points and “ε_α " is equivalent to the number of lines as approximate points. In these respects we observe the approximation of a set of points by multiplying points “x” and “y” and adding the equations of
(x-1)*x(1/2)= ε (y-1)*y(1/2)= ε

where the respective values of “x” and “y” take the place of "α". The resulting number is always negative.

A simpler equation of (x+y=z) → ∸((z-1)*z(1/2)) also suffices.



ӫ(∙y,∙x) ⧂ (x+y=z) → ∸((z-1)*z(1/2))

[wtf]

ӫ(∙y,∙x) ⧂ (x+y=z) → ∸((z-1)*z(1/2))

Almost. But you forgot to mod out by the Fermat prime.

[Eodnhoj7]

ӫ(∙y,∙x) ⧂ (x+y=z) → ∸((z-1)*z(1/2))

Almost. But you forgot to mod out by the Fermat prime.

Elaborate your point further, I do not seem to see what you are observing.