Philosophy Now Forum

Philosophy Explorer wrote: ↑Thu Mar 08, 2018 7:58 pm

Eodnhoj7 wrote: ↑Thu Mar 08, 2018 7:48 pm

Science Fan wrote: ↑Thu Mar 08, 2018 7:29 pm 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 wrote: ↑Thu Mar 08, 2018 8:48 pm 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 wrote: ↑Thu Mar 08, 2018 8:54 pm 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.

Philosophy Explorer wrote: ↑Thu Mar 08, 2018 10:38 pm 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 wrote: ↑Fri Mar 09, 2018 5:56 pmIn simpler terms we synthesize the measurements which form reality?

wtf wrote: ↑Fri Mar 09, 2018 6:47 pm

Eodnhoj7 wrote: ↑Fri Mar 09, 2018 5:56 pmIn simpler terms we synthesize the measurements which form reality?

We integrate the dimensional modalities to recontextualize the gestalt.

Eodnhoj7 wrote: ↑Fri Mar 09, 2018 6:56 pm
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 wrote: ↑Fri Mar 09, 2018 7:01 pm

Eodnhoj7 wrote: ↑Fri Mar 09, 2018 6:56 pm
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 wrote: ↑Fri Mar 09, 2018 7:10 pm In simpler terms a "mirror function" must be inevitable.

wtf wrote: ↑Fri Mar 09, 2018 8:00 pm

Eodnhoj7 wrote: ↑Fri Mar 09, 2018 7:10 pm 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 wrote: ↑Sat Mar 10, 2018 6:00 pm ӫ(∙y,∙x) ⧂ (x+y=z) → ∸((z-1)*z(1/2))

wtf wrote: ↑Sat Mar 10, 2018 7:15 pm

Eodnhoj7 wrote: ↑Sat Mar 10, 2018 6:00 pm ӫ(∙y,∙x) ⧂ (x+y=z) → ∸((z-1)*z(1/2))

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