WRT = with respect to

I think that the two biggest cruces are (1) whether the individual at hand can transcend their own context…

Context ≅ Environment. In my thinking, environment suggests state of affairs, which suggests reality.

In your usage here, is individual… can transcend their own context an action symbolic or literal?

(2) whether the idea of the essay preceding “logical languages” (or theories of logic)…

My thinking proceeds from foundational assumptions that bump up against some foundational assumptions employed herein by you.

• An essay is, at bottom, the logical language of argumentation

• The stuff of logic is a continuum of conditionals that unfold sequentially, thus implying a temporal process

• Although logical expressions can be conceptualized as atemporal mental objects, continuity is always empirical & temporal

…a sine qua non has no prepositions…

…a context that is universal I really wouldn’t constitute as a context…

If, as I interpret you to be saying with the above two claims, sine qua non is not of anything, and, moreover, is not at all contextual, then I get the impression the whereness of sine qua non is more mysterious than the position of an orbiting electron at any given moment. Is that the case?

Firstly, a sine qua non is “without which, not” (where “not” is an unbounded infinite negative) and, therefore, the possibility of “without PoR, not derivation” invalidates “derivation” as being a sine qua non. Secondly, this is exactly why, derivation not being a sine qua non, produces the possibility that someone can completely remove it within their derivation (no matter how irrational it may be, as someone else could easily mention that I just literally said “someone can remove derivation from their derivation”), whereas they cannot remove PoR without utilizing it.

The above section of paragraph is wonderfully clear and thus it makes me hope I’m beginning to get some real grounding within your essay.

I now have an impression of your essay’s essence via use of a helpful metaphor wherein your sine qua non holds status akin to the singularity that precedes the Big Bang.

If there’s even a particle of truth in application of my pre-Big Bang metaphor to your metaphysical claim, then hopefully I can proceed to an understanding you’re wrestling with the boundary ontology of origin.

Boundary Ontology of Origin – continuity via hyper-logic across the super-position of a non-localized QM event.

The above definition is my best-to-date exposition of a hairy beast of a concept that is one of my works-in-progress. I won’t elaborate it’s possible pertinence to your essay because that would entail an inappropriate digression from your work. I will say I expect it to inform some of my commentary upon your work henceforth.

Likewise, time is by no means something one can posit as sine qua non, as “without PoR, not time” and, honestly, there are many principles that are required for it to be affirmed in the first place (i.e., faculties of reason which allow one to determine that time is enveloping of oneself, or that there is a non-temporal true claim, or neither true nor false, etc.).

Since you reject time_sine qua non, I think it imperative you state (If you have not done so) whether PoR_sine qua non is temporal, or atemporal.

I’ve been understanding regulation in the everyday sense of a transitive verb that controls & shapes an object under its influence. I don’t presently see this function as being atemporal.

Answering the question of sine qua non’s relationship to time entails whether or not your universe is static or dynamic. Does a universe without motion make any sense?

Likewise, you may have also noticed that it isn’t logically…coherent…to claim multiple sine qua nons as true—for if there existed two then they are thereby not sine qua nons (that’s a contradiction). In other words, if a sine qua non is “without which not”, if we allow ourselves the importation of useful logical axioms, then only one can be true by definition (otherwise we have a situation where two principles are supposed to be negatable in relation to one another, but yet the source of an unbounded infinite of negations respectively).

Your above statement, speaking potentially, has a lot to say to the project to bring the rules of inference into congruence with QM.

Please elaborate how regulate & modulate compare.

By “modulate”, what are you referring to? I am not completely following.

My Apple Dictionary tells me regulate & modulate are synonyms. I’ll buy that. However, I sense that modulate, more so than regulate, gets into the deep interior of language.

Language -- a collective, or gestalt of the systematic boundary permutations of a context or medium; a record of the systematic boundary permutations of a narrative medium.

If the above claim contains a particle of truth, then your sine qua non, as presently perceived by me, embodies something akin to the Original Utterance, itself, in turn, akin to the pre-Big Bang Singularity, itself, in turn, akin to God’s “Let there be light!”

I hope you’ll forgive the tincture of theism_Jungian psychology pooling into my assessment of your essay.

Might sine qua non, per your essay, be your Logos?

I would say not quite. “existence” is indexical: is it awkward that I can refer to different people with the same word ‘he’? I personally don’t think so. I can posit, without contradiction, that the cup in my hand “exists” (by constituting, for example, its existence as phenomenal) while denying it as existing in an ontic sense. I gave a couple examples, such as Spinoza to illustrate this clear distinction: do you disagree with that distinction as demonstrated in the examples? — Bob Ross

Indexical means 'relative to context of utterance' - like 'he' or 'here', as you say. The term 'existence' does not seem relative to context in that way. You go on to say that it has different senses, which is different from indexicality. You may well be right. Hamlet exists as a character in a play and does not exist as a flesh and blood human being. So sure, there are different kinds of existence in that way. But to say for example that Hamlet exists but does not [open italics]actually[close italics] exist is confused and confusing. To choose Hamlet as an example of some entity which uncontroversially exists is equally confusing. You chose the cup in your hand as a straightforward example of something which exists, distinguishing it perhaps from the tiger in your hallway which (ex hypothesi) does not. That's a much better example. But it does not allow for the kind of apparent equivocation that Hamlet does. It's a useful example specifically because it won't let us wriggle away from its existence.

If I were to posit, for example, that the cup in my hand exists (contextually to phenomena), but really exists as one infinite substance, then, regardless to its truth, there is a distinction being made there within the concept of “existence”. Another example is that a cup may exist in the sense that I can interact with it, yet not exist sans my consciousness. — Bob Ross

The problem is that your cup doesn't exist sans your consciousness and the cup in your dreams also does not exist sans your consciousness. We are left with the problem of distinguishing a cup in the hand from a dreamed cup. That is, a real cup from an imagined cup. Or, in other words, a cup that exists from one which does not. Similarly, the cup in your dreams also exists contextually to phenomena and for all I know it may exist as one infinite substance as well. But at some time, possibly outside the philosophy laboratory, we are going to have to distinguish the cup of our dreams from the cup in our hands, the car that hit ours from the car that did not, the positive bank balance from the negative. I mean, while we enjoy this delicious atmosphere of confusion we must still keep a concept of 'existence' tucked in our back pockets for use when we actually need it and not just for when we are playing at metaphysics. And that, I submit - the concepts tucked away for use when we are serious - is our metaphysics.

The only valid, thus far, determined sine qua non is that the subordinate rules cannot be affirmed and denied in accordance to the superordinate rules within the given operation of derivation; the derivation of derivation, and its recursive utilization, is this principle—which shall be termed the principle of regulation.

If the terms mean something like the interpretation I gave them, then I can get little sense out of this - except perhaps that if we fail to follow rules of logical inference, then we will fail to make logical inferences.

By “rule”, I mean “a regulating principle”. Within the context of my derivation in the example, 1 and 1 being identical but not indiscernible was the superordinate rule guiding my conclusion that 1 = 1 (in part); in other words, a regulative principle determining the course of my derivation. — Bob Ross

It is true that 1=1 in the world defined by the definitions and rules of mathematics. The rest of us just accept this truth on blind faith based on the accomplishments and power of mathematics to be useful in the sciences.

I have brought up the pitfalls of 'true' in metaphysical reasoning. For metaphysics akin to mathematical reasoning, True is a binary value for evaluating dichotomies, any other use of truth is common but can be shown to be invalid or unsound. Since '1' is just like any other concept, it can not be true that '1' and '1' is anything other than '1'. Just as 'orange' and 'orange' are 'orange' and nothing else. However, instantiations of 'orange' are countable. 1 orange +1 orange = 2 oranges. And 1 apple +1 orange = 2 fruit.

* I have edited this comment *

Hello again Bob! A late reply, but I'll try to refresh where we were.

All possible numbers would be, with respect to the essay, a bounded infinite. — Bob Ross

We're in agreement then Bob! That's what I was trying to point out.

By true infinite, I think you are talking about something entirely different than me, but I could be wrong. — Bob Ross

I don't think so here. I was regressing through all numbers, and noted that all numbers themselves are a bounded infinite as well. I was trying to lead to the point that an unbounded infinite cannot be quantified or limited.

This is where it gets incredibly subtle, but equally incredibly vital: it is not “without it an unbounded infinity is negated” but, rather, without it there are an unbounded infinite of negations. Sounds kind of like the same thing, doesn’t it? I agree, but yet they are entirely different ideas. — Bob Ross

If we are in agreement that numbers are bounded infinites, then whenever we come up with an identity, we are creating some type of bounded infinite. If we use the word "negations" were are implicitly talking about bounds then. I don't think we can say an "unbounded infinite of negations". That's really, a "bounded infinite of negations". I can see an unbounded infinite negated, because an unbounded infinite is the base from which all bounded infinites are formed. But if we say that all possible bounded infinites are negated, isn't that the same as stating an unbounded infinite is negated? Can you give an example showing how they're entirely different ideas?

Is there a superordinate to 1? I'm not sure.

It is entirely possible to declare a particular derivation complete; that is, that it has been sufficiently justified and, therefore, can be put to rest. This doesn’t negate the principle of regulation’s truth: that assertion (i.e., that it has been sufficiently justified) is yet another conclusion which utilized the principle of regulation. — Bob Ross

Thank you, I re-read and realized you had covered that part. Also, you have not had the chance to show how sufficient justification works under your system, so I accept this for now.

In terms of the former (your version), I would have no choice but to concede that a sine qua non is simply a misapprehension; that is, not an unbounded infinite is to necessary conceive of it in toto to thereby flip its affirmation into a denial (i.e., negation): therefore, it would be nothing more than the masking of a bounded infinite under the name of an unbounded infinite. However, in terms of the latter (my version), it is simply the negation, sequentially, of everything (i.e., not …, not not {…}). I think this is potentially where you may be misstepping (or I may be simply incorrect). — Bob Ross

For myself, I think this is a crux of your argument that needs better explication. You are as usual, brilliant Bob, but I'm having a difficult time conceptualizing the latter as something real. In trying, the best I can come up with is that it is some conceptualization that is necessary for an unbounded infinite to be. The best I can think of is that we must be able to make conceptualizations out of/within the unbounded infinite. Because if something could not, then nothing could create any sort of differentiation between bounded, and unbounded. Does this somehow fit within your PoR?

Being in bounded infinites is not a bad thing however, as I believe its the only way we can have concepts. Perhaps we can simply reform your idea into, "A sqn is what is needed for concepts to exist." Basically try to find what is logically necessary for concepts to occur.

Unfortunately, that would defeat the point of the essay, as that is not a foundation (unless we speaking of contextual foundations). Likewise, a sine qua non is not deriving what is necessary for concept to occur, as that is within the sphere of critique of derivation (as opposed to its higher form of performance of derivation). — Bob Ross

This again is where I have a hard time. Without a sqn, nothing can be. Which means without a sqn, concepts cannot be either. The way I read the essay and your explanation, it seems to imply without a sqn, the infinite, bounded or unbounded could not be.

This would be true if the principle of regulation pertained soley to explicated superordinate and subordinate rules. It’s quite literally being postulated as an unbounded infinite of such. — Bob Ross

Again, I think this is really where my issue resides. The unbounded infinite is the source of all explicated infinites. Negating the unbounded infinite, negates all explicated infinites as well. Without a sqn, the unbounded infinite would be negated. And I think we agree there is only 1 unbounded infinite, as more than one would be by definition, two bounded infinites. All explicated infinites are within the unbounded infinite. Which means a sqn is necessary for all explicated infinites to occur as well. If this is the case, then a sqn must stand without contradiction in all explicated infinites. Meaning that if it does not stand within even one explicated infinite, it cannot be a sqn.

As I noted earlier, the burden of demonstrating this is nigh impossible to meet. But this again, is through my interpretation so far that the sqn is a misapprehension. If you can demonstrate your version " it is simply the negation, sequentially, of everything (i.e., not …, not not {…})." somehow is not logically equivalent to my version, then there may be something to explore.

And I think we agree there is only 1 unbounded infinite, as more than one would be by definition, two bounded infinites — Philosophim

This essay might get a larger following if all this infinite stuff were in mathematically acceptable nomenclature. Just a thought.

I apologize for the belated response ucarr! I had a hefty week or so. With that being said, let me dive into your post.

WRT = with respect to

Thank you for the clarification: I was not familiar with that acronym.

Context ≅ Environment. In my thinking, environment suggests state of affairs, which suggests reality.

I wouldn’t quite go as far as to claim context is synonymous to environment (maybe that’s why you used a tilde equal sign?--to suggest an approximate equivalence). I would be hesitant to confine “context” to “reality” (which I would agree is implied with environment, at least to some degree). Therefore:

In your usage here, is individual… can transcend their own context an action symbolic or literal?

In terms of object relations, I would say that I can meaningfully produce contexts sans-my-body (e.g., I can see red, but they cannot).

In terms of consciousness, it is much harder to produce anything (or at least anything rational) without admitting the contingency of other objects on the sensations and perceptions of my object (i.e., the body). However, it is possible for a subject to posit things without their bodies (no matter how irrational/rational it may be), as it is possible to deny the contingency of something on such and, not only that, but people can produce meaningful predictions that involve positing their non-existence. So I wouldn’t holistically grant that I cannot posit sans-my-consciousness.

In terms of reason, I think that, although an individual can still assert its omission, the most rudimentary thereof is without omission (in possibility). This is the area of discourse which I am attempting to convey with the principle of regulation.

Now whether that is literal or symbolic, as I do not entirely know how you are utilizing those terms, I will leave up to you.

An essay is, at bottom, the logical language of argumentation

I think, in the sense you are positing it, I would agree. However, I would distinguish “logical language” (in the sense of a formal or informal theory of logic) from hyper-logic (maybe “meta-logic”?--I am not too sure as of yet). If you mean it in terms of the latter, then I agree. If in terms of the former, I disagree.

The stuff of logic is a continuum of conditionals that unfold sequentially, thus implying a temporal process

Within the scope of the essay, I would disagree (albeit incredibly reasonable to assume). Yes, it is reasonable to infer that the procedure and proof of the essay is necessarily that of temporal relations (sequences in succession of one another). The important thing is that, as of now, I find such a conclusion (i.e., derivation or the principle of regulation is temporal) to only be found by importation of other axioms (or, in my terms, superordinate principles which are not apart of the standard terminology nor proof explicated in the essay. My point here is not to completely discourage your conclusion here, but only to expose that it is by means of other superordinate rules other than what is required (I would argue) to prove PoR to be true. In other words, it is entirely possible for one to accept PoR as true and immediately thereafter assert PoR is in time, is time, is sans time, neither in or outside of time, etc.

Although logical expressions can be conceptualized as atemporal mental objects, continuity is always empirical & temporal

I would personally agree that derivation is always empirical and temporal; however, I don’t find, as of yet, that that is necessary to hold to prove PoR. Please let me know if you think I am wrong here.

If, as I interpret you to be saying with the above two claims, sine qua non is not of anything, and, moreover, is not at all contextual, then I get the impression the whereness of sine qua non is more mysterious than the position of an orbiting electron at any given moment. Is that the case?

I am not entirely certain that a stable methodological approach can be establish to examine the properties or existence of PoR, but that is something I am currently contemplating. I find compelling arguments to assert it is aspatio-temporal (because there is no where which would reasonably pertain specifically to PoR and any derivation of its temporal sequences of derivation are simply via it), but, in contradistinction, I find it compelling to argue for its spatio-temporality (because being sans time & space seems merely to be a conceptualization under space and with time); however, I think both arguments are within the real of critique of derivation (as they are both inheriting from this PoR meta-derivation if you will) and, therefore, I think that, with respect to PoR itself, the best way to conceive of it for the essay is neither true nor false of the former nor the latter. It just simply seems inapplicable, but correct me if I am wrong.

I now have an impression of your essay’s essence via use of a helpful metaphor wherein your sine qua non holds status akin to the singularity that precedes the Big Bang.

If there’s even a particle of truth in application of my pre-Big Bang metaphor to your metaphysical claim, then hopefully I can proceed to an understanding you’re wrestling with the boundary ontology of origin.

If I am understanding the analogy correctly, then I think I would more or less agree. PoR is simply what (I think) can be proven to be a sine qua non, which essentially means that it is the meta-derivation (so to speak). Although I think there’s a strong case to make that I am acting as if PoR is what ontologically exists, I can’t say I am able to place that within the essay itself (for the same reason as my previous elaboration on spatio-temporality).

One could, I suppose, think of PoR as a spark of derivations. Is that what you mean?

I would emphasize that the essay is not making reference to “reality” though, so it is not entirely analogous to the Big Bang.

Boundary Ontology of Origin – continuity via hyper-logic across the super-position of a non-localized QM event.

By QM, do you mean Quantum Mechanics? If so, the essay is not meant as an exposition of any quantum mechanical principles (nor is it meant to assert for or against any given scientific principle). Again, the essay is meant to be a inquiry into higher reasoning (or lower depending on how one wants to visualize it).

In terms of superpositioning, do you mean to reference its use from quantum mechanics? If so, I would say the same thing as previously, but please correct me if I am misunderstanding you.

The above definition is my best-to-date exposition of a hairy beast of a concept that is one of my works-in-progress. I won’t elaborate it’s possible pertinence to your essay because that would entail an inappropriate digression from your work. I will say I expect it to inform some of my commentary upon your work henceforth.

I appreciate and respect your effort to keep our conversation pertinent to my essay, but if it helps you elaborate on your views (in contrast to mine), then please feel free to discuss them!

Since you reject time_sine qua non, I think it imperative you state (If you have not done so) whether PoR_sine qua non is temporal, or atemporal.

I don’t find time to be a consideration necessary to prove PoR as a sine qua non and, furthermore, any assertion of atemporality, temporality, spatial references, etc. is via PoR (thereby dependent on it). As I alluded to earlier, I think for the sake of the essay it may be best to conceive of a sine qua non as neither in time nor not in time.

I’ve been understanding regulation in the everyday sense of a transitive verb that controls & shapes an object under its influence. I don’t presently see this function as being atemporal.

I agree, but by “regulation” I was meaning “to govern or direct according to rule”, which (I think) coincides with my construction of superordinate and subordinate rules. I am honestly not sure what “modulation” would entail beyond “regulation”, but if you think it is a better term for what I am trying to convey then please feel free to critique me!

Your above statement, speaking potentially, has a lot to say to the project to bring the rules of inference into congruence with QM.

If I am understanding correctly, I don’t think our ideas are perfectly aligning, but are similar. Maybe in due time we will determine them to, indeed, be equivalent.

If the above claim contains a particle of truth, then your sine qua non, as presently perceived by me, embodies something akin to the Original Utterance, itself, in turn, akin to the pre-Big Bang Singularity, itself, in turn, akin to God’s “Let there be light!”

I hope you’ll forgive the tincture of theism_Jungian psychology pooling into my assessment of your essay.

Might sine qua non, per your essay, be your Logos?

Although I don’t mind you invoking theology, I don’t think PoR is synonymous with the vast majority of conceptions of God, Logos, or the Original Utterance. The essay isn’t positing it as what created the universe, it is simply where I hit bedrock: it is the most rudimentary aspect of me (that is, me as reason: the subject).

If you think it is more alike to theological conceptions than I would grant, then I would love to hear why!

Bob

Indexical means 'relative to context of utterance' - like 'he' or 'here', as you say. The term 'existence' does not seem relative to context in that way. You go on to say that it has different senses, which is different from indexicality.

Upon further contemplation, I agree with you that ‘he’, as an indexical use of language, is not analogous to my use of different senses. However, what I was really trying to convey in terms of “indexical” is “of or relating to an index”, which I do still think applies (although my example of ‘he’ was fallacious).

You may well be right. Hamlet exists as a character in a play and does not exist as a flesh and blood human being. So sure, there are different kinds of existence in that way. But to say for example that Hamlet exists but does not [open italics]actually[close italics] exist is confused and confusing.

I still think that there is a meaningful distinction to be made the sense of existence of a thing and there are different terms for such. For example, ‘ontic’ (e.g., noumena and phenomena) usually refers to the being of things and ‘ontological’ is the discussion of being of being (e.g., dasein). In terms of ontical consideration, I would simply dividing it up further:

There is a ‘colloquial’ sense of ‘existence’ wherein laymen tend to denote an objects tangibility.
There is a ‘phenomenological’ sense of ‘existence’ wherein in a person denotes that an object exists as an appearance, with no immediate classification of what it may exist as otherwise.
And so forth…

Now, to be honest, I slightly blundered in my use of ‘ontic’, in hindsight, as what I really was trying to convey was what sense they were meaning the term more generically than that.

When you say it is a confused outlook, I would agree if what you mean is that it would be a contradiction to assert that, for example, something exists phenomenologically and does not exist phenomenologically. However, I would be hesitant to concede that making such contextual sense of “existence” are unwarranted or that they are confusing.

You chose the cup in your hand as a straightforward example of something which exists, distinguishing it perhaps from the tiger in your hallway which (ex hypothesi) does not.

A cup does not uncontroversially exist, unless you are referencing a colloquial use of the term (e.g., it is tangible). Many philosophers, some of which I already mentioned, would not state a cup exists ontologically (or they may have other distinctions such as a cup as a noumena verses a phenomena).

It's a useful example specifically because it won't let us wriggle away from its existence.

If by “tiger in your hallway”, you mean an imaginary tiger, then I would say an imaginary tiger exists in the imagination: that’s still qualified as existing. Again, if you are thinking of colloquial “to be”, then, yes, an imaginary tiger is not tangible, therefore it does not exist (and that is a meaningful distinction in its own right).

The problem is that your cup doesn't exist sans your consciousness and the cup in your dreams also does not exist sans your consciousness. We are left with the problem of distinguishing a cup in the hand from a dreamed cup. That is, a real cup from an imagined cup.

This is true, and the subject can do so however they so please. In a colloquial setting, I typically denote a tangible cup (i.e., non-imaginary) as “a real cup” because I know most situations that is how everyone is thinking about it (they aren’t philosophers); however, I can easily and reasonably (I would say) assert that both the imaginary and non-imaginary exist in their own respects. Therefore, an imaginary cup is not the same as a non-imaginary cup, but nevertheless they both certainly exist.

Similarly, the cup in your dreams also exists contextually to phenomena and for all I know it may exist as one infinite substance as well.

I think you are committing an equivocation: just because two things are phenomena doesn’t mean they are equivalent. Just because I conceive of a chair and a table under the concept of “object” does not mean I can thereby assert them equal to one another. An imaginary and non-imaginary cup are both phenomena, but are nevertheless distinguishable in many meaningful ways. Likewise, one can most certainly reject the “phenomena” vs “noumena” distinction, as my main point pertained to possibility and not favoring one over the other.

But at some time, possibly outside the philosophy laboratory, we are going to have to distinguish the cup of our dreams from the cup in our hands, the car that hit ours from the car that did not, the positive bank balance from the negative.

Practicality is a worthy consideration, but I genuinely don’t see, as of yet, how one cannot pursue such within what I have posited hitherto. Nothing about what I stated necessarily determines imagination and non-imagination indistinguishable, an inability to distinguish different cars, nor an inability to discern bank balances.

I mean, while we enjoy this delicious atmosphere of confusion we must still keep a concept of 'existence' tucked in our back pockets for use when we actually need it and not just for when we are playing at metaphysics. And that, I submit - the concepts tucked away for use when we are serious - is our metaphysics.

I understand: it seems as though you are arguing for the practical over what you would deem the philosophical; however, I think the deeper issue is that they aren’t incompatible with one another whatsoever.

If the terms mean something like the interpretation I gave them, then I can get little sense out of this - except perhaps that if we fail to follow rules of logical inference, then we will fail to make logical inferences.

If there’s any way you think I can provide better clarification on the essay, then please let me know! What I can say is that it is not about “failing to make logical inferences”, as that is a contemplation of logic, which is not determined nor argued for in the essay.

Bob

It is true that 1=1 in the world defined by the definitions and rules of mathematics. The rest of us just accept this truth on blind faith based on the accomplishments and power of mathematics to be useful in the sciences.

I have brought up the pitfalls of 'true' in metaphysical reasoning. For metaphysics akin to mathematical reasoning, True is a binary value for evaluating dichotomies, any other use of truth is common but can be shown to be invalid or unsound. Since '1' is just like any other concept, it can not be true that '1' and '1' is anything other than '1'. Just as 'orange' and 'orange' are 'orange' and nothing else. However, instantiations of 'orange' are countable. 1 orange +1 orange = 2 oranges. And 1 apple +1 orange = 2 fruit

I would like to clarify that the essay pertains to the higher principles involved in the example given therein, as opposed to a critique of the derivation itself (i.e., of the example derivation of 1=1). Therefore, within the scope of the essay, nothing about it is meant to prove that you must accept 1 = 1 as true. This seems to be what you are contending with: am I correct? If not, please correct me where I am wrong.

I am not entirely certain that a stable methodological approach can be establish to examine the properties or existence of PoR, but that is something I am currently contemplating — Bob Ross

Since this seems to lie at the foundations of your paper it would be good to make it a bit clearer what you are talking about. I'll ignore the infinity stuff, that itself is puzzling from a mathematical perspective.

Another member of TPF has in the past submitted a lengthy and sophisticated essay on a theory of everything (or roughly that), starting with an assumption every fact in the universe can be encoded for use in Turing machines. But doesn't explain how.

When one doesn't explain clearly at the outset what the fundamentals are or how they can be attained, readers may not be enticed to go further.

But that's just how I see it. Others here may differ. OK :cool:

In isolation, I agree that your quote of mine makes little sense; however, in the context of the entirety of the response where it is contained I think it makes sense. I would suggest reading this post in its entirety (if you haven't already) and then feel free to ask specific questions about it.

If I had to extract certain parts of the linked response, then I would provide further context:

Within the scope of the essay, I would disagree (albeit incredibly reasonable to assume). Yes, it is reasonable to infer that the procedure and proof of the essay is necessarily that of temporal relations (sequences in succession of one another). The important thing is that, as of now, I find such a conclusion (i.e., derivation or the principle of regulation is temporal) to only be found by importation of other axioms (or, in my terms, superordinate principles which are not apart of the standard terminology nor proof explicated in the essay. My point here is not to completely discourage your conclusion here, but only to expose that it is by means of other superordinate rules other than what is required (I would argue) to prove PoR to be true. In other words, it is entirely possible for one to accept PoR as true and immediately thereafter assert PoR is in time, is time, is sans time, neither in or outside of time, etc

…

I am not entirely certain that a stable methodological approach can be establish to examine the properties or existence of PoR, but that is something I am currently contemplating. I find compelling arguments to assert it is aspatio-temporal (because there is no where which would reasonably pertain specifically to PoR and any derivation of its temporal sequences of derivation are simply via it), but, in contradistinction, I find it compelling to argue for its spatio-temporality (because being sans time & space seems merely to be a conceptualization under space and with time); however, I think both arguments are within the real of critique of derivation (as they are both inheriting from this PoR meta-derivation if you will) and, therefore, I think that, with respect to PoR itself, the best way to conceive of it for the essay is neither true nor false of the former nor the latter. It just simply seems inapplicable, but correct me if I am wrong.

Nothing about it has to do with the fundamentals being proven in the essay (i.e., that the principle of regulation is a sine qua non--being a true statement) I would say. If it is still confusing, then please let me know and I can elaborate further.

Another member of TPF has in the past submitted a lengthy and sophisticated essay on a theory of everything (or roughly that), starting with an assumption every fact in the universe can be encoded for use in Turing machines. But doesn't explain how.

I am not familiar with the essay you are referring to, but it seems disanalogous to mine. It seems as though the point of their proof was to show that their claim is true, yet their proof was vague: I don't find that my proof is vague, but correct me if I am wrong.

In terms of your other post:

This essay might get a larger following if all this infinite stuff were in mathematically acceptable nomenclature. Just a thought.

What exactly would you suggest in terms of mathematically acceptable nomenclature? What about our discussion of infinities is confusing within the apperception of mathematics?

Bob

Yes, it is reasonable to infer that the procedure and proof of the essay is necessarily that of temporal relations (sequences in succession of one another). The important thing is that, as of now, I find such a conclusion (i.e., derivation or the principle of regulation is temporal) to only be found by importation of other axioms (or, in my terms, superordinate principles which are not apart of the standard terminology nor proof explicated in the essay. — Bob Ross

I don’t find time to be a consideration necessary to prove PoR as a sine qua non and, furthermore, any assertion of atemporality, temporality, spatial references, etc. is via PoR (thereby dependent on it). As I alluded to earlier, I think for the sake of the essay it may be best to conceive of a sine qua non as neither in time nor not in time. — Bob Ross

I asked my questions about time vis-a-vis PoR because I want to know who does PoR as sine qua non have as his neighbors? I was conjecturing that time is one of PoR's neighbors. As such, time does not prove PoR as sine qua non. Instead, time is one of PoR's neighbors, which is to say time & PoR are a matched set. One always implies the other.

I haven't forgotten your explanation to the effect that, by definition, two sine qua nons are mutually exclusive and thus cannot both belong to one set.

Some other candidates for neighbors of PoR might be superordinate & subordinate rules? This would mean PoR's neighborhood is divided by class, with the superordinates as elites & the subordinates as commoners? PoR, totally aloof, sits on high & reigns over both? Even if this is the case, the lowliest subordinate rule nonetheless stands a comrade alongside PoR. This is so because, by force of the premise> universe is the limit of system, without which, not is, in fact, bi-directional. In other words, no object inhabits absolute isolation. Therefore, speaking extremely broadly, all things are equal.

With the following, I will try to show why I suspect sine qua nons cannot be mutually exclusive.

If PoR has no neighbors, then for me a fundamental question arises pertaining to relationship. Derivation & meta-derivation, as I presently understand them, imply inter-relatedness i.e. relationship.

If PoR has no neighbors, how can it fulfill the role of sine qua non in total isolation?

Moreover, if PoR has no peers, that is, no other sine qua nons as neighbors, how can he be an unbounded infinite? Haven't you established a causal agent confined to a single set as a bounded infinity?

As I alluded to earlier, I think for the sake of the essay it may be best to conceive of a sine qua non as neither in time nor not in time. — Bob Ross

I'm wondering if the above assertion (that sine qua non WRT temporality is undecidable) raises a question of foundational metaphysics> When the temporality of an object is undecidable, is not the location of said object also undecidable?

If the answer is yes, then how can sine qua non fill the role of foundational cause of derivation?

My underlying premise here is that even a purely cognitive "object," holding a priori status, by force of causality (inter-relatedness) obtains location. In this example, location of sine qua non is first member of a sequence.

I'm starting to suspect that sine qua non, as absolute solitary, without neighbors covering peers & subordinates alike, in parallel to the singularity of the Big Bang, cries out for conceptual revamping that addresses the deeply problematical boundary ontology of origins.

Hello Philosophim,

Wonderful points as usual! Let me try to respond adequately.

I don't think we can say an "unbounded infinite of negations". That's really, a "bounded infinite of negations"

By “negation”, to be more precise, I mean a “complete negation”; that is, that the entirety of what is negated is completely obliterated (so to speak). Therefore, I do not mean a “partial negation”. Consequently, I am in agreement with you that “not X” necessarily entails that X is bounded (which is what I noted as “being conceived in toto”) because to negate it I must implicitly treat it as completely not (as opposed to “partially not”).

However, as far as I am understanding you, you seem to be asserting that an “unbounded infinite of negations” (which, we are in agreement, is an “unbounded infinite of nots of bounded concepts”) is somehow entailed to be equivalent to “a bounded infinite of nots of bounded concepts”. In other words, it seems as though, from my point of view, you are rightly identifying the bounded nature of the contents of the unbounded infinity and, in virtue of that, extending it (or maybe misassigning it) to its form.

I can see an unbounded infinite negated, because an unbounded infinite is the base from which all bounded infinites are formed.

Categorically, I think it would be a contradiction in terms to posit the negation of an unbounded infinite: that is actually a bounded infinite. Let me try to explain:

But if we say that all possible bounded infinites are negated, isn't that the same as stating an unbounded infinite is negated?

I don’t think these are the same concepts that you just described and I think it is the root of our dispute. Firstly, it is important to note that I am not, at this point, attempting to prove that there is an unbounded infinite but, rather, I am merely trying to prove there is a valid concept of such that is at your disposal—for, as of now, it seems as though your contention lies in the denial of an “unbounded infinite” as a valid concept (i.e., it is really a “bounded infinite” assigned a new name).

To simplify it down for all intents and purposed for now, it seems as though, to me, you are essentially stating: an unbounded infinite of nots = not an unbounded infinite.

I would describe it as a difference between conceiving in toto and in total. Bundling up all those negations found within the unbounded infinite into a “complete concept” is to necessarily contradict the very concept (that is, attempt some operation which necessitates it to be conceived in toto). By “concepts” I am not entailing that it have a bounded form, which is what I suspect you are at least partially committing yourself to.

You are conceptually performing a different task to completely negate an unbounded infinite. What you seem to have done is analyzed the content of an unbounded infinite in terms of the sum of its parts to derive what it approaches (i.e., in total) and, thereafter, conflated that with in toto--thereby considering “not an unbounded infinite” valid; However, nothing about the sum of the parts of a concept entails that it can be conceived as a whole (that is, nothing about being conceivable in total entails that it is conceivable in toto).

This is the exact issue that required of me to explicate that, in the essay, a sine qua non is “without which, not” not “without which, none”; that is, the natural and swift leap from the sum of the parts of the content of a concept cannot entail its form in any way whatsoever (and nothingness is a great example of that). The unbounded infinite of nots does not necessitate nor prove a complete concept of nothingness (i.e., in the essay: “none”). The essay itself, consequently, does not even attempt to prove that “without the principle of regulation, there is nothingness”, because the PoR is also valid of the statement “without PoR, not nothingness” and “without PoR, not everythingness” (and so on): there is nothing which “escapes” it, so to speak.

The issue you may be having conceptualizing it is quite understandable, as with everything else we tend to swiftly conceive of finites and bounded infinites in toto based off of in total: but I have separated the two modes of thinking. For example, if one were to postulate what an infinite of empty sets would exist as in space, then they are more than likely going to quickly derive the summation of the parts to conclude that it would be nothing. This, however, in the sense of separating the two modes of thinking, is only valid by positing the form of the infinite as bounded. Therefore, I am noting the two different modes of thinking involved in the assessment:

First, the individual determined in total the infinite of empty sets, which is 0. This mode of thinking requires simply the ability to conceive what a sequence approaches (e.g., I cannot actually perform 0 + 0 forever, but I can nevertheless reasonably conclude it results in 0).

Second, the individual implicitly shifts their mode of thinking to in toto to “package” and “bundle” their conclusion into a wholly conceivable concept (e.g., 0 in space is nothing); that is, they assume that their evaluation in total of the infinite of empty sets warrants the ability to substitute the infinite for a bounded counterpart equal it in total (e.g., the total of the infinite is 0, therefore where ever I utilize the infinite I can substitute 0 for it). But, there is a distinction here, I think, in that they are not necessarily equivalent and, thusly, they are not always guaranteed to be valid of substitution: performing substitution of a bounded for an unbounded necessarily means that the form of the concept has been reshaped (by means, I would say, of utilizing a different mode of thinking). This substitution is only valid if the intents have no bearing on the form of the infinite: if the form matters, then 0 cannot be a valid substitution for an infinite of empty sets. Admittedly, a vast vast majority of the time I think the form is dismissable; however, my essay (I would argue) is an example where the distinction is vital.

The best I can think of is that we must be able to make conceptualizations out of/within the unbounded infinite. Because if something could not, then nothing could create any sort of differentiation between bounded, and unbounded. Does this somehow fit within your PoR?

I may be misunderstanding you here, but PoR can be utilized to make distinctions, but the very concept of PoR is also via itself. Therefore, the indifferentiation or differentiation of PoR from other things is via PoR. In that sense, PoR cannot be separated from anything, including nothing.

This again is where I have a hard time. Without a sqn, nothing can be. Which means without a sqn, concepts cannot be either. The way I read the essay and your explanation, it seems to imply without a sqn, the infinite, bounded or unbounded could not be.

Another great point Philosophim! This, I would say, is your other major contention with my work (which is not the same as what was previously mentioned). Let me explain it back to you (to ensure that I am understanding correctly) and then I will attempt to adequately address it. Here’s what I think you are essentially saying:


#1
p1. A sine qua non is “without which, not”
p2. Therefore, by definition of #1p1, a sine qua non contains “without which, not “concepts””

#2
p1. A unbounded infinite is a concept
p2. By substitution property and #1p2, a sine qua non contains “without which, not “unbounded infinite””
p3. By #2p2, it is a contradiction in terms to assert an unbounded infinite as a sine qua non.
p4. Therefore, an unbounded infinite cannot be a sine qua non.

The issue I would have is:

1. #1p2 is only partially true: I am allowing a “concept” to be incomplete in form and, therefore, a sine qua non only contains “without which, not “concepts with a complete form””. The mode of thinking matters to me.

2. Therefore, a sine qua non cannot contain a concept with an incomplete form.

Note: by “incomplete form”, I do not mean a concept merely conceived as incomplete in content (e.g., an incomplete apple) as that is complete in form (e.g., a complete concept of an incomplete apple).

Now, at face value, my response seems to be a contradiction: if a sine qua non is an unbounded infinite of negations, then it seems as though I ought to be able to negate it within itself, otherwise it cannot be deemed true. The answer is that one can subvert a sine qua non through itself; however, that is to necessarily erode its form to that of being bounded (i.e., to shift my mode of thinking to in toto), which is not really a sine qua non: I am thereafter dealing with an imposture so to speak and not the real thing.

Likewise, if only complete forms are allowed, then it seems as though there is something which persists with the negation of a sine qua non: concepts with incomplete forms, which contradicts the idea of it being a sine qua non in the first place. However, again, an incomplete form is an unbounded infinite or a contradiction in terms (e.g., either unbounded with infinite content or a contradiction wherein it is unbounded with finite content). In terms of the latter, it is invalid. In terms of the former, we must contend with it: does an unbounded infinite persist, as a concept with incomplete form, without a sine qua non? The answer is no, because, again, to posit an unbounded infinite as without a concept necessarily shifts the mode of thinking in toto, which contradicts the term itself. If this operation is permitted, then the unbounded infinite simply (1) is not an unbounded infinite and (2) does not persist without a sine qua non because its eroded bounded conception is wholly within the jurisdiction of a sine qua non (“without which, not”). Therefore, I submit to you that an unbounded infinite is not out of nor within wholly and, therefore, it stands not outside (without) a sine qua non and, in turn, it does not pose a threat to the concept thereof.

Again, I think that our dispute first lies in whether an “unbounded infinite” is valid as a concept, which hopefully I have proved herein, and, after that we can then discuss whether there is such a concept. As always, I could merely be wrong about the concept itself.

Bob

I think that our dispute first lies in whether an “unbounded infinite” is valid as a concept — Bob Ross

Yes, I think this is really the issue. Lets see if we can put this in terms of math.

You already mentioned that the infinite X is bounded if we use actual numbers. The only way to really capture an unbounded infinite is not to use numbers at all, but the relation itself, where is is not limited by any number or dimension. I have no problem with this. What I will attempt to demonstrate is that there is only one unbounded infinite, and the X "without numbers" is it.

Your original bounded infinite could be represented as
X = Y with limit 5. Here we have X is fine as long as it doesn't equal 5. But if X is bounded as soon as numbers are used, then as soon as a number is used in the equation, it is also bounded. So X = Y with a limit of 5 is a bounded infinite by the limit.

But lets go further. X = Y is really a limit of "Whatever Y is, X is. We can say we won't assign actual numbers to X, but there is a number, a bound within the formula itself that acts exactly on a limit. That limit is that Y will always be X, and Y cannot be anything but X.

The above may be confusing, so let me add another detail. 2X = Y. Now we explicitly have a number in which Y will always be double X. Even if we don't use actual numbers in X or Y, this double explicitness is a limit, or a bound. Referencing the previous X = Y, lets change it to 1X = 1Y, which is equivalent.

Ok, if X, unnumbered is an unbounded infinity, while all the rest are bounded, can we have multiple unbounded infinities. Can I just say Y without using actual numbers and have that different from saying X without actual numbers? Besides the symbol itself, they are both identical. X is unbounded, and Y is unbounded. They are not bounded in relation to one another. If they are not bounded in relation to one another, they are not different from one another. Neither has any limits, so they are both the same.

Lets now translate that to words, context, and meaning. As soon as you put a limit in words, context, or meaning, you are no longer talking about an unbounded infinite. You are talking about a bounded infinite.

Now, this still doesn't convey the whole idea fully. We now have to change it to words, meaning, and context. To represent X, we need unstated words, unstated meaning, and unstated context. The moment we state anything, any "number", we are now within a bounded infinite limited by the expression of that word, meaning, and/or context.

Can we have a sine qua nons for an unbounded infinite. Yes, but there is only one. That would be "not X". If not X were true, then X would not follow. Anything more specific may be a sqn for a bounded infinite, but it cannot be a sqn for an unbounded infinite.

The same applies to the principle of regulation. Within X words, Y meaning, and Z contexts we are still bound by words, meaning, and context. Let simplify this further. W = { X, Y, and Z } all without "numbers" or explicit individual representations. W is still bound by X, Y, and Z. The only way for W to be unbounded is just "W".

So I do not think it can be shown the Principle of Regulation is a sqn. There are specific words, such as principle, regulation, of, that are understood within a particular bounded infinite meaning, and in particular bounded infinite contexts. Can thinking things within this limit form and use conclude the logic of the principle of regulation is necessary. Absolutely. But can this be concluded from "W" alone? No, I don't believe it can.

To clarify on

p1. A unbounded infinite is a concept — Bob Ross

No, I'm not stating this. I'm stating an unbounded infinite is not a concept. The moment we create a concept within it, we are now within a bounded infinite. As such, there is only one unbounded infinite. Anytime any explicit infinite is proposed, it is by nature bounded.

That being said, this does not mean you should give up on the principle of regulation as a basis for a theory. I think it is a fine starting point, and I know I, and probably many in this discussion would love to see where your mind takes this. I would hate it to be stopped by something as trivial as a debate over infinity.

You shouldn't need sqn's to prove the principle of regulation to logically thinking minds. And even if you do, perhaps its something you could come back and show later? Is the concept of a SQN within an unbounded infinite absolutely needed to continue your line of thought from the PoR proposal? If you just started the sentence with, "If we have the ability to derive, the principle of regulation logically arrives," would that hamper what you want to do? I feel you have so much more to say, and possibly introduce greater thoughts that I would hate to see stopped over focusing on what may be a technical, and perhaps unnecessary detail to show us what you have planned.

In my experience in philosophy, it is easy to get stuck on approaches that seem necessary to us when first formulating the idea, but as we evolve the idea, were perhaps not as necessary or important as we thought to those who are reading our papers. Consider your readers so far. Very few have argued against the PoR, but almost everyone has a problem with your views of infinity. Now we may all be wrong, and you may be correct. But is it necessary at this time to focus on the infinite as such, or can this be shelved or stated another way that allows your readers to focus on the first premise they can readily accept?

I realized, to keep it brief, that even if I concluded that there was no foundation to derivation, or no derivation, it is all by means of the principle of regulation (or whatever one wants to call it). — Bob Ross

This right here is where I think you should go into detail. Prove not only to yourself, but that none of us can conclude anything differently. If you do this, I don't think anyone is going to need the infinite. How in the absence of derivation must we all necessarily have the principle of regulation? If I am not a being able to derivate, could I conclude I could not derivate?'

I look forward to your work Bob.

. . . almost everyone has a problem with your views of infinity. Now we may all be wrong, and you may be correct. But is it necessary at this time to focus on the infinite as such, or can this be shelved or stated another way that allows your readers to focus on the first premise they can readily accept? — Philosophim

:up:

I asked my questions about time vis-a-vis PoR because I want to know who does PoR as sine qua non have as his neighbors? I was conjecturing that time is one of PoR's neighbors. As such, time does not prove PoR as sine qua non. Instead, time is one of PoR's neighbors, which is to say time & PoR are a matched set. One always implies the other.

If I am understanding you correctly, then I would say that there is no “neighbor” to PoR. I think, and correct me if I am wrong, you are essentially arguing that time and PoR are biconditionally related. However, an argument for time (and, I would say, space) encompassing all one’s cognitive faculty requires the use of axioms which do not lie within the essay. The axiom, so to speak, for the essay is two-fold: (1) an engagement in the attempt to derive a sine qua non and (2) implicitly the use of the definitions without contradiction.

With that being said, I agree that “derivation” implies “time” (and space): it is just that that would require axioms not granted nor required to accept PoR.

I haven't forgotten your explanation to the effect that, by definition, two sine qua nons are mutually exclusive and thus cannot both belong to one set.

I think that, upon further contemplation, I can prove, without the importation of other axioms, that two or more sine qua nons cannot be true. For example, if there were two true sine qua nons, then they would, by definition, have to independent, but, also by definition, they would be, as independent “without which, not”, biconditionally dependent on one another. In other words, for one to be a sine qua non, the other would have to be dependent on it; but, since likewise for the other, they would really be two biconditionally dependent concepts, which would not be sine qua nons by definition. Therefore, there can only be one true sine qua non. I think I will interpolate that into the essay here shortly.

Some other candidates for neighbors of PoR might be superordinate & subordinate rules?

PoR is the principle that the subordinate rules cannot be affirmed and denied in accordance to the superordinate rules within the given operation of derivation (as a recursive principle). The terminology of “superordinate rule”, “subordinate rule”, “rule”, “in toto”, “in total”, etc. are merely a means of describing PoR itself and, therefore, I, as of now, do not see them as “neighbors” of PoR but, rather, more like a means of explication.

If PoR has no neighbors, how can it fulfill the role of sine qua non in total isolation?

I apologize: I may have lead you astray by not acknowledging that only one sine qua non can be true. A sine qua non is not isolated but, on the contrary, it is that which everything in total (as opposed to in toto) is contingent upon (including “contingency” as a primitive faculty of reason). Therefore, it is ever present (in total that is, not in toto).

When the temporality of an object is undecidable, is not the location of said object also undecidable?

PoR is what we utilize to determine what “time”, “undeterminancy”, “location”, etc. is. It is what we utilize to determine what is is, etc. It doesn’t have a location like a physical object in space (I would argue).

My underlying premise here is that even a purely cognitive "object," holding a priori status, by force of causality (inter-relatedness) obtains location. In this example, location of sine qua non is first member of a sequence.

I don’t think it is a priori or a posteriori because both require PoR to classify anything therein or to even construct the terms themselves. I don’t think it would make sense to classify it as a priori, for example, because that is merely something else we derive, given a set of implicit or explicit superordinate rules.

I'm starting to suspect that sine qua non, as absolute solitary, without neighbors covering peers & subordinates alike, in parallel to the singularity of the Big Bang, cries out for conceptual revamping that addresses the deeply problematical boundary ontology of origins.

Although I may just be merely misunderstanding you, I do not view a sine qua non as isolated whatsoever.

Bob

Can we have a sine qua nons for an unbounded infinite. Yes, but there is only one. That would be "not X". If not X were true, then X would not follow. Anything more specific may be a sqn for a bounded infinite, but it cannot be a sqn for an unbounded infinite.

My issue would be that “not X” is a bounded infinite and so is “without numbers”: they both must conceive of the concept in toto to negate it. Therefore, “without numbers” really is an assertion that negates itself: for I am forced to conceive of “numbers” in toto, which is a complete whole (i.e., a traditional numeric “one”) which entails I have not, in fact, omitted numbers thereby.

In saying this, I think (although correct me if I am wrong) I am, at prima facie, not only agreeing with your examples but also supplementing them here.

That is why a sine qua non is not “one” in a traditional, numeric sense: it is an infinite in total as opposed to in toto. I can posit as many in toto conceptions I want, systematically, (i.e., numerical wholes—one) within a sine qua non (as being negated) without contradiction.

Likewise, I also agree that two unbounded infinites is a contradiction in terms and, therefore, I will interpolate that into the essay (as I believe I can prove it without further axiomatic importations).

In other words, “one” sine qua non is not “one” in the sense of a numerical whole but, rather, in total; that is, the analysis of what it approaches without the ability to encapsulate it. Perhaps a distinction of a “numerical one” (i.e., “in toto one”) and a “in total one” would be useful in the essay?

The same applies to the principle of regulation. Within X words, Y meaning, and Z contexts we are still bound by words, meaning, and context. Let simplify this further. W = { X, Y, and Z } all without "numbers" or explicit individual representations. W is still bound by X, Y, and Z. The only way for W to be unbounded is just "W".

Again, I would say that all of this, including any conceptions “without numbers”, for all intents and purposes, is being conceived numerically (that is, in toto). “just ‘w’” could very well, depending on how one is noting its form, be “one” (in toto) or “one” (in total)--I am unable to discern given the context thus far.

In terms of the arguments for our contingency on words (as an example), that argument is ultimately contingent on PoR. A set of implicit or explicit superordinate rules were utilized to derive the conclusion (reasonable and rational conclusion I must add) that a human being is contingent on words to explicate their messages (or at least, heavily contingent, as body language is a thing technically). However, the argument from the essay is that without PoR not “a human being is contingent on words”.

Can thinking things within this limit form and use conclude the logic of the principle of regulation is necessary. Absolutely. But can this be concluded from "W" alone? No, I don't believe it can.

I am not quite following what you mean by “’W’ alone”: could you please elaborate a bit? The point of noting the form of an infinite is for the expression that one cannot omit it without absurdity in relation to the definitions provided: it, simply put, cannot be bounded. I feel like we are more in agreement then we may have realized.

No, I'm not stating this. I'm stating an unbounded infinite is not a concept. The moment we create a concept within it, we are now within a bounded infinite. As such, there is only one unbounded infinite. Anytime any explicit infinite is proposed, it is by nature bounded.

I view this as you defining “concept” as having an essential property of “bounded form”; however, it is possible to define it as “bounded or unbounded in form” and I do not see, as of yet, why this would be a contradiction or an absurdity. Therefore, I think, so far, we are merely semantically disagreeing. If by “concept” you mean something with necessarily bounded form, then I simply would have to come up with a different term. I think what you are trying to convey is that that term I create, whatever it may be, is necessarily bounded in form: but why? At the bare minimum, I can conceive of a concept which prohibits boundaries as a continual process (therefore, not merely within a context of space or time or what have you).

You shouldn't need sqn's to prove the principle of regulation to logically thinking minds. And even if you do, perhaps its something you could come back and show later? Is the concept of a SQN within an unbounded infinite absolutely needed to continue your line of thought from the PoR proposal? If you just started the sentence with, "If we have the ability to derive, the principle of regulation logically arrives," would that hamper what you want to do? I feel you have so much more to say, and possibly introduce greater thoughts that I would hate to see stopped over focusing on what may be a technical, and perhaps unnecessary detail to show us what you have planned.

I agree! I think that I am going to begin building off of the essay and, once I am done, I will post them all together (as I think doing it in segments has only produced more confusion). I think that this essay will be readily available to reference if a conversation requires it, but there’s no need to squander time on it right now. With that being said, I still would love to hear what everyone thinks, so continuing to contend with infinities is totally fine with me: I will just be allowing myself to continue my adventure whilst that is occurring.

This right here is where I think you should go into detail. Prove not only to yourself, but that none of us can conclude anything differently. If you do this, I don't think anyone is going to need the infinite. How in the absence of derivation must we all necessarily have the principle of regulation? If I am not a being able to derivate, could I conclude I could not derivate?'

Regardless of how one puts it, whatever they derive utilized the principle that the subordinate rule(s) could not contradict the superordinate ones. However, the tricky and slightly confusing aspect is that, they can most definitely utilize PoR in a manner where they are convinced that it is not true or the case.

For example, if I were to postulate a concept of “a being that cannot derivate”, then I am doing so by means of deriving something which cannot derive. Consequently, whatever concept I derive for “non-derivation” is contingent on derivation to have been manifested.

As another example, if I were to postulate a concept of a derivation that “is not the use of subordinate rules not being able to contradict the superordinate rules”, then I can readily identify the superordinate rules I utilized to even derive that very concept itself. As a particular example:

I concluded X.
Y was the sole superordinate rule within the derivation of X (i.e., I determined it solely off of X, let’s say).
X is and is not true of Y.
Therefore, PoR is not applicable to this derivation.
Therefore, PoR is not true.

If I take it as granted that each is true (e.g., Y actually was the sole producer of X), then I can dissect this further to realize it is being derived by PoR (as it is a mere facade that it is not applicable by means of the possibility of negating it). For example, for me to have concluded X, I abided by the rule that the following could not be true and false within the derivation: X can be true and false of Y and Y can still determine X, but thereof cannot be true and false lest “I concluded X” is indeterminate as is. Likewise, to determine something as the “sole contributor” requires its own set of derivation with this same exact relation (and PoR being not applicable by some deductive or inductive argument, and PoR not being true by some deductive or inductive argument, etc.). Likewise, my argument that it is required requires one, and so forth. I can quite literally keep abstracting, particularlizing, and more generally deriving this infinite nest whereof it never rests upon an absolute foundation, so to speak.

This even functions for irrational or absurd derivations, such as:

I think 1 = 2 is true and 1 = 1 is true

We can perform analysis on this in any direction, with any goal, in mind—but one of particular interest is that they seem to be committing to the idea that 1 is and is not 1 (i.e., the absence of the law of noncontradiction): that would be an implicit superordinate rule.

In explicating that to them, they may reject that notion as well:

I accept the law of noncontradiction and “1 = 2 and 1 = 1” is still true.

Assuming, for all intents and purposes, that they aren’t merely conceptualizing things differently or semantically refurbishing well known concepts, they are still abiding by some set of superordinate rules to be able to possibly derive it in the first place. It could be that they hold the law of noncontradiction as exempt from itself or some other goal motivating their derivation, but the point is that we never get to a point where we can rest metaphorically upon absolute grounds, so to speak. Even if I conclude that “they simply derived it arbitrarily”, then the rule of arbitrary randomness governed their derivation. And my conclusion thereof and its justification will be by superordinate rules as well.

Bob

This essay might get a larger following if all this infinite stuff were in mathematically acceptable nomenclature. Just a thought. — jgill

Or be revealed as poetry that can't be combed into a formal system ?

Likewise, I also agree that two unbounded infinites is a contradiction in terms and, therefore, I will interpolate that into the essay (as I believe I can prove it without further axiomatic importations).

In other words, “one” sine qua non is not “one” in the sense of a numerical whole but, rather, in total; that is, the analysis of what it approaches without the ability to encapsulate it. Perhaps a distinction of a “numerical one” (i.e., “in toto one”) and a “in total one” would be useful in the essay? — Bob Ross

I may have been focusing too much on bounded vs unbounded when I think toto and total are really the focus in your essay. I think what I'm trying to note is that no matter how you shake it, toto and total are both bounded infinities. But I honestly don't think that's important to your overall concepts and where you want to take the essay.

So with this, let me make sure I understand your definitions of toto and total without the use of bounded and unbounded infinities, but just infinities. Instead, let me relate it to concepts if I could.

Lets look at the concept of "trees". A tree can be imagined an infinite number of ways. In toto seems to be close to "realized".

"In toto, on the contrary, cannot be conceived for a given concept without admitting of that concept bounds (in form). " - Foundational Metaphysics

So if I were relate this to trees, perhaps we could say its the realized number of trees for just one person. But, just because we have a realized a limited number of trees, it does not negate the fact we could keep realizing more. In fact, an infinite amount of trees if we so desired.

To my mind, the words total and toto is more like potential vs. actual. If I imagine the total amount of trees I can conceive of, its infinite. But if I imagine the tota number of trees I can conceive of, this seems to require a form of some sort, like trees. But, when speaking in total, I require some word like "trees" as well. There's no real difference in this instance, because both are still the unrealized concepts of trees themselves.

Instead of using both tota and total as representatives of infinity, perhaps one should represent infinity, while the other represents what is realized within the potential infinite. Infinity after all, can never be fully realized by any being. It is a concept of an unending pattern. I think this is also where you're implicitly intending to go, but feel free to correct me if I'm wrong.

So for example
1. The total number of trees I can realize is the unformed potential of all possible trees. As they are unformed, we cannot establish them all. It is an unending pattern.
2. The toto number of trees I can realize is the actual number of trees I realize (perhaps through my life? Or X time?). Perhaps in your original conception we could say if you lived an infinite time, the toto number of trees would be all the trees you actually conceived of during your infinite life.

The point that I want to note is that there is no actual infinity, only a potential infinity. As we are limited beings, the actual of what we are cannot be noted in terms of infinity.

I also don't think this hurts your essay. If we go to the principal of regulation, we can then apply the concept similarly. The total number of derivations I'm able to make is infinite. The tota number of derivations I have made are X. We can derive from concepts in two ways. I can derive a concept post, or subordinate, that follows from my current concept. Or, I can also derive a concept pre, or superordinate, that creates a concept that one could use to lead to the original concept.

As an example I could create the concept of a man on a moon. Then I could create the subordinate concept that, "The man traveled there from Earth". Taken without the consideration of derivation, one could say, "Ah, the man traveled to the moon from Earth, that's why they're on the moon." While the order of time or logical consequence might indicate it as the "beginning", in order of derivation, it is actually the second concept conceived of.

As such, we could say the toto number of concepts would be the derivation chains I've conceived of, but in total, there are an unrealized infinite I could conceive of. Is this along the lines of your thinking, or am I still missing or confusing something?

This leaves the sqn. What I feel you are trying to imply is that a sqn is what is required for the potential of derivations to exist at all. Because the total number of derivations I can make is unrealized, we're not going through and cancelling a "set" of all unrealized concepts I would actually make, but the total potential of what I could make. Because this is unrealized infinity, there are no "numbers" or actuals to negate, only the potential itself. Does this work?

If this is the case, you're noting that the principle of regulation is a sqn, because without the principle of regulation, there can be no derivation in potential. If derivation could only be done with subordinates, it would miss the picture of the superordinate. If derivation could only be done with superordinates, it would miss the picture of the subordinate. And if a being did not consider anything subordinate or superordinate, there would be no derivation at all.

For me, this is where I think the essay runs into problems. Noting that derivation has both superordinate and subordinate concepts is fine. But those are simply definitions we can realize. What is to prevent a person from defining derivation as something that is only subordinate? What if they made a different word for constructing a superordinate, and did not find that was a derivation at all? What if something has a completely different thought process than ourselves?

For example, if I were to postulate a concept of “a being that cannot derivate”, then I am doing so by means of deriving something which cannot derive. — Bob Ross

Yes, you are doing so, but you didn't negate the fact that the being could not derivate. And this being may be a highly intelligent being, even another human. Such a human could not use the the PoR. But this is basically because we have defined it as such right? If something cannot conceive of both superordinate and subordinate ideas, by definition, it cannot derivate. The PoR is not a universal concept that can be used or understood by all thinking things. It is a descriptor of certain logical processes of some beings.

But here is where I don't see a problem. The PoR is a concept that can be used and understood by many thinking things. I don't think you need a sqn to assert the PoR as a concept to derive other concepts. I think its a fine proposal that can be demonstrated, used effectively, and agreed upon by most people. Is it a necessary concept to thought itself? No. But is it a fine concept that I believe you will use to derive and explore other interesting and possibly useful concepts? Yes! So please continue Bob.

The terminology (i.e., potential and actual) are misleading (as a potential infinite is not merely potentially limitless in content) and, therefore, this distinction shall be hereon depicted as “bounded” vs “unbounded” infinity.

Here are commonly accepted concepts and notations:


$S=\left\{ {{s}_{n}}:{{s}_{n+1}}={{s}_{n}}+\frac{1}{n},\text{ }{{s}_{1}}=1,\text{ }n=1,2,3,4,... \right\}$


$T=\left\{ {{t}_{n}}:{{t}_{n+1}}={{t}_{n}}+\frac{1}{{{n}^{2}}},\text{ }{{\text{t}}_{1}}=1,\text{ }n=1,2,3,4,... \right\}$


$I=\left\{ r\in R:0\le r\lt1 \right\}$


$Y=\left\{ q\in Q:0\le q \right\}$


$X=\left\{ 1,2,3 \right\}$


S is countable, infinite, unbounded above but bounded below.
T is countable, infinite, bounded above and below.
I is uncountable, infinite, bounded below by its greatest lower bound, which it includes, and above by its least upper bound, which it does not include.
Y is countable, infinite, bounded below by its GLB, which it includes, but unbounded above.
X is finite and bounded above and below.

I suggest either using "bounded" and "unbounded" in their proper sense or defining other words to express what you mean. For example, S and T could be called something like scripted infinities, whereas I and Y would be unscripted infinities. But other mathematicians might disagree. If one has any math training it is disconcerting to read your definitions.

Nice to meet you Pie!

Or be revealed as poetry that can't be combed into a formal system ?

Could you please elaborate on what you mean by this? At prima facea, I don’t see how the essay would be poetry, as I am thinking of “writing that uses rhythm, vivid language, and often rhyme to provoke an emotional response “. In the essay, I am attempting at defining, clearly, a sine qua non and, thereafter, that the principle of regulation can be regarded as a such: I am not sure what emotional response can be derived therefrom, but I would be interested to hear what you think!

Bob

Wonderful analysis as always Philosophim: let me try to adequately respond.

To my mind, the words total and toto is more like potential vs. actual. If I imagine the total amount of trees I can conceive of, its infinite. But if I imagine the tota number of trees I can conceive of, this seems to require a form of some sort, like trees. But, when speaking in total, I require some word like "trees" as well. There's no real difference in this instance, because both are still the unrealized concepts of trees themselves.

For clarification, in toto and in total are meant to describe totality in relation to forms vs. contents of concepts (as a distinction between them) and are not infinites themselves. In other words, I would like to clarify that neither “in toto” nor “in total” are concepts that directly entail an infinite: the former is a conception which is conceived (i.e., defined) as holistic, whereas “in total” is the conception of the summation of its parts (i.e., in content).

Therefore, one can have a conception, A, of which they conceive in toto (i.e., as complete in form) and be able to formulate a conception, B, that is the summation of the parts of A (which would be in total). For example, I can manifest a conception of a set of integers {1, 2, 3} and determine that the summation of the parts as 6: the former is a conception in toto, and the latter is a conception of that conception in total. Firstly, I would like to clarify that by “summation of parts”, I am purposely leaving it vague, like that of a protocol, which is merely meant as any sort of combination of entities (e.g., {1, 2, 3} could be concatenated or mathematically summed for all intents and purposes or even a limit: 123 or 6 or what have you).

Secondly, it is important, as you probably immediately noticed, to note that my previous example is of a concept of finite form and content: now, we must properly determine the possible permutations of both to provide further exposition into such a distinction. For both, I think it is reasonable to conclude that there are three options for each:

1. Indefinite
2. Infinite
3. Finite

And, thusly, we can analyze each permutation of such, like so (briefly speaking):

Indefinite form and indefinite content

Indefinite in content dictates we cannot determine it in total but, rather, only the scope which we currently have.

Indefinite in form is a bit trickier to imagine, but it is something which the individual at hand has neither asserted its finitude nor its infinitude. Arguably, this is simply a state of confusion; that is, I am fairly confident, given a confused example, that I could expose whether it is finite or infinite.

An example of this would be if I were to know that there is a function, f(x), which has a point (3.3, 4.27) and I were to conceive of its form as simply undetermined in bounds (i.e., I am simply confused or, at least, refraining from judgment). The knowledge of the point is a scope, so to speak, of which I know of the content of f(x) and my undetermined boundaries of the concept of f(x) is form. In terms of the former, I cannot determine in total, but I can meaningfully assert that whatever it is it must involve that point. In terms of the latter, regardless of how mislead I may be, I simply have not asserted a form (even though, as I stated previously, I think both of us would probably be able to tell which one it really is).

In terms of the essay, this kind of conception (in form and content) is simply a state of confusion or ignorance and, therefore, is not relevant to the principle of regulation.

Indefinite form and infinite content

An infinite content can be determined in total.

An indefinite form, as per the last example, is simply a state of ignorance, so I won’t linger on it any further.

An example would be a function, f(x), where I know it’s absolute minimum is y = 2 and that the limit as x approaches infinity from the right side is infinity (no rhyme or reason to those numbers, I just made them up). I can conclude, in total, that the sum of its parts (i.e., y values, let’s say) is infinity: for an absolute minimum at y = 2 indicates that f(x) is never negative ys and the limit from the right being infinity tells me that even if the limit to the left is a finite number that the summation of the ys will be infinity.

Again, I would say the concept is finite (that is, bounded), but technically I could be in a state of ignorance or confusion, thusly determining it as indefinite.

Indefinite form and finite content

This is really just ditto but with finite set of numbers (for example), so I won’t linger on this either.

Infinite form and indefinite content

So this is interesting, because it is incoherent: if I assert that a concept is infinite in form, then its content must be infinite. If I assert the content is finite, then I must, conceptually, either implicitly or explicitly, fill the remaining parts with voids (or a filler of some sort). Otherwise I am admitting the concept to be unable to be negated, for example, yet have a totally negatable finite content, which isn’t coherent. The only way to repair this conception is to admit of it void filler parts.

Infinite form and infinite content

This is the realm of sine qua nons (and, in virtue, the principle of regulation): a concept which is repetitive affirmation of negations would be an example of it. This kind of form entails, I must add, only one of its kind as a conception (and not just merely in existence).

Infinite form and finite content

Same situation as infinite form and indefinite content: I must either (1) fill with voids to ensure the form is coherent with the content or (2) strip the form to a finite.

Finite form and indefinite content

This would be a concept which we have limited scope of its content, but we do conceive of it in toto; that is, as a finite form. This is perfectly coherent in itself.

Finite form and finite content

There are ample examples of this one that I feel you are well aware of, so I will not linger.

Finite form and infinite content

This would be like the whole set of natural numbers.

Besides providing the aforementioned as hopefully a means of better explication on my end, the other main point here is that nothing about knowing “in total” grants anything “in toto”, and vice-versa.

So, let me finally address your trees analogy:

So for example
1. The total number of trees I can realize is the unformed potential of all possible trees. As they are unformed, we cannot establish them all. It is an unending pattern.
2. The toto number of trees I can realize is the actual number of trees I realize (perhaps through my life? Or X time?). Perhaps in your original conception we could say if you lived an infinite time, the toto number of trees would be all the trees you actually conceived of during your infinite life.

The concept of “trees” is a bounded concept because it is conceived in toto (that is, a holistic concept) and I can determine in total the summation of its parts. In other words, for example, I can negate your entire concept of “the toto number of trees ...” just as much as I can for your “total number of trees ...” because I can bundle them up into a holistic concept (i.e., both are actually in toto). Likewise, I would like to stress that nothing about me asserting it as conceivable in toto entails what the state of “in total” is (other than what one could infer from my explication of the permutations of form and content).

The point that I want to note is that there is no actual infinity, only a potential infinity. As we are limited beings, the actual of what we are cannot be noted in terms of infinity.

Arguably, I would say there is one which could be valid but if it is then there cannot be more.

As such, we could say the toto number of concepts would be the derivation chains I've conceived of, but in total, there are an unrealized infinite I could conceive of. Is this along the lines of your thinking, or am I still missing or confusing something?

If I am understanding you correctly, then I would say that you are noting something distinctly different from “in toto” vs “in total”; they refer to the same conception and, therefore, to convert your idea here, it would be more like: the number of concepts that you have conceived of would be finite in content and bound in form; or, if you wanted to attempt it, infinite in content and bounded in form; or, infinite in content and unbounded in form; etc.

This leaves the sqn. What I feel you are trying to imply is that a sqn is what is required for the potential of derivations to exist at all. Because the total number of derivations I can make is unrealized, we're not going through and cancelling a "set" of all unrealized concepts I would actually make, but the total potential of what I could make. Because this is unrealized infinity, there are no "numbers" or actuals to negate, only the potential itself. Does this work?

I believe so: a sine qua non (specifically denoted as the principle of regulation) is what is required for the potential of derivations to occur. A sine qua non is sort of like a procedure wherein one negates all concepts in total, but obviously by means of strategic elimination and not brute force (as that is impossible).

What is to prevent a person from defining derivation as something that is only subordinate? What if they made a different word for constructing a superordinate, and did not find that was a derivation at all?

Thinkers can most certainly (and arguably will) construct their own derivations that omit, in definition, the concept of superordinates, subordinates, PoR, sine qua nons, etc. To me, this is not a problem: I am not attempting to argue that it is impossible for one to miss this principle. By means of it, it is entirely possible to never realize it. In your philosophy, I would view this as analogous to “discrete experiencers”, which do not, for your argument to work, have to ever realize they are discretely experiencing. Nothing about this, to me at least, is wrong nor a contention with the essay. But please correct me if I am wrong!

Yes, you are doing so, but you didn't negate the fact that the being could not derivate. And this being may be a highly intelligent being, even another human. Such a human could not use the the PoR. But this is basically because we have defined it as such right? If something cannot conceive of both superordinate and subordinate ideas, by definition, it cannot derivate. The PoR is not a universal concept that can be used or understood by all thinking things. It is a descriptor of certain logical processes of some beings.

I guess I am a bit confused here: what, in terms of mere possibility, could be defined as a “thinking being” which necessarily does not derivate? I would argue, upon further reflection, that all life can be classified as using PoR. PoR itself holds no inherent necessity of the degree by which it can produce superordinate/subordinate rules: a plant grows towards light, a bee operates by means of calculated movements (albeit not necessarily self-aware), etc. By my lights, I am having a hard time thinking of anything that would be constituted as “thinking” yet cannot derivate (to any degree, more specifically).

I look forward to hearing from you,
Bob

Thank you jgill for the elaboration! I am most definitely not an expert mathematician and I most certainly do not want to come across as disconcerting. Let me attempt to adequately respond to your post and you correct me where you deem fit.

Unfortunately, I do not know how to properly format mathematical equations on this discussion forum, so for now I will have to write it in less pretty formatting (dearest apologies in advance).

S is countable, infinite, unbounded above but bounded below.

I think that I understand: there is a set, S, where S is the outputs of the function equal to:

f(n) = n + 1 / n

Where n is constrained to be positive integers (i.e., natural numbers).

It is infinite because the limit as n approaches infinity is infinity, i.e.:

limit n + 1 / n = ∞
n → ∞

Because 1 / ∞ is equal to 0 and thus we have ∞ + 0 = ∞

T is countable, infinite, bounded above and below.

This one is confusing me a bit, as I don’t see how it is bounded above. By my lights, since we are speaking of natural numbers, then the negative n values do not exist and, therefore, are omitted from our consideration. Therefore, although taking the limit of n → - ∞ is - ∞, it holds no relevance if we are speaking of only positive integers for n. Therefore, the limit one really ought to care about is n → 0:

limit n + 1 / n^2 = ∞
n → 0 +

limit n + 1 / n^2 = ∞
n → 0 -

They equal each other, therefore:

limit n + 1 / n^2 = ∞
n → 0

And, also, it is important that as n approaches infinity it also equals positive infinity:

limit n + 1 / n^2 = ∞
n → ∞

Both, in quadrant one, approach infinity and, consequently, I do not understand why the related function, f(x), that is the values contained in set T, would be “bounded above and below”: are you referring to the x = 0 asymptote (i.e., that it is constrained to natural numbers)?

I is uncountable, infinite, bounded below by its greatest lower bound, which it includes, and above by its least upper bound, which it does not include.
Y is countable, infinite, bounded below by its GLB, which it includes, but unbounded above.
X is finite and bounded above and below.

Makes sense.

I think that “bounds” in mathematics is simply asymptotes, limits, restraints to X or Y, and any finite segments (e.g., T would be bounded on the left and restrained to natural numbers—contrary to my previous contention--, and a line segment from the interval [0, 3] inclusive would be a bounded finite).

To explain my form vs content, take set T that you defined and, more specifically, take note of my previous contention/confusion (i.e., the limit exists for 0 even though 0 is an asymptote because the left and right converge to the same value, which is valid technically). The content of f(n) would be the y outputs and the form would be, with respect to the left in quadrant one, its bounds to the asymptote x = 0.

At a deeper level, though, the problem would be that any “unbounded” f(n) one could provide is bounded to the, in toto, concept of that two-dimensional spatial graph. Nothing about it is an unbounded infinite; although I understand the confusion now, as I am not refer to a contextual usage of the term “unbounded” as in n → ∞ = ∞.

Bob

I will have to write it in less pretty formatting (dearest apologies in advance). — Bob Ross

Alas, for reasons good or bad I have no clue, you've compromised on what Robert M. Pirsig calls quality! No matter, I understand!

I would be interested to hear what you think! — Bob Ross

Hi ! I sometimes talk with people on forums who have various intuitions and suggestions about infinity and other mathematical concepts. Often enough they've had no experience writing mathematical proofs, which is like not knowing quite what math is, at least to a mathematician (even if, in fact, one can go a long way in applications without ever learning proofs.) 'Poetry' is a just a metaphor for these ideas that are not yet in mathematical shape. It's fair to expect some mastery of real analysis from an innovator. (Algebra and topology are natural mentions, but real analysis is the serious theory of the numbers we all are somewhat familiar with.)

The OP is probably trying to mathematize the great Anaximander's apeiron, in Matrix terms The Source. Bonam fortunam!

T is countable, infinite, bounded above and below.
This one is confusing me a bit, as I don’t see how it is bounded above. — Bob Ross

Bob, when one expands the sequence:

${{t}_{2}}=1+\frac{1}{{{1}^{2}}},\text{ }{{t}_{3}}={{t}_{2}}+\frac{1}{{{2}^{2}}}=1+\frac{1}{{{1}^{2}}}+\frac{1}{{{2}^{2}}},\text{ }{{t}_{4}}=1+\frac{1}{{{1}^{2}}}+\frac{1}{{{2}^{2}}}+\frac{1}{{{3}^{2}}},\cdots ,\text{ }{{t}_{n}}=1+\sum\limits_{k=1}^{n-1}{\frac{1}{{{k}^{2}}}}\lt\infty$

a line segment from the interval [0, 3] inclusive would be a bounded finite — Bob Ross

The real numbers constituting [0,3] are uncountably infinite, but the set of these numbers is obviously bounded above and below. This would of course be a finite line segment.

the problem would be that any “unbounded” f(n) one could provide is bounded to the, in toto, concept of that two-dimensional spatial graph. — Bob Ross

No. S is unbounded above, and if one plots a graph of the terms of S (vertical axis) vs n (horizontal axis) one would need a piece of paper having infinite dimensions. However, the sum of that series diverges so slowly that the sum of the first 6,000,000 terms is less than 21 !

'Poetry' is a just a metaphor for these ideas that are not yet in mathematical shape. It's fair to expect some mastery of real analysis from an innovator. (Algebra and topology are natural mentions, but real analysis is the serious theory of the numbers we all are somewhat familiar with.)

I appreciate the elaboration: thank you! I understand what you are conveying and I think it is perfectly fair and reasonable. However, I would like to note that my essay is not within the actual sphere of mathematical discourse (in other words, it is not a paper intended within the context of formal mathematics): it is a philosophical work pertaining to metaphysics (which I am understanding your metaphorical use of 'poetry' to be synonymous with 'non-formal-math' so to speak). Although there is always much to read, I can say that the vast majority of philosophical works pertaining to metaphysics that I have read do not provide formal mathematical proofs because, quite frankly, it isn't meant to do so. With that being said, if you think I ought to provide a formal mathematical proof of something within the essay, then please feel free to let me know! I would love to hear your critiques.

Bob

Bob, when one expands the sequence:

Awe, I see! I thought you were outlining a set because T was encapsulated in brackets, which I thought meant 'a set'. Correct me where I am wrong, but T = { } seems to be a set and not a summation (∑). To me, nothing about the notation of T = { ... } entailed that one is summing each t of n. My question to you would be, assuming I am simply misapprehending, what about your previous notation entailed (symbolically) the summation of T's elements? I understand that your use ∑ in your most recent post does, but I am failing to see how the T = {} does.

In terms of summing t of n, I totally agree and understand that starting at 1 will result in the limit approaching infinity to equate to 1.

The real numbers constituting [0,3] are uncountably infinite, but the set of these numbers is obviously bounded above and below. This would of course be a finite line segment.

Agreed.

No. S is unbounded above, and if one plots a graph of the terms of S (vertical axis) vs n (horizontal axis) one would need a piece of paper having infinite dimensions. However, the sum of that series diverges so slowly that the sum of the first 6,000,000 terms is less than 21 !

That is true; however, the entirety of mathematics is bounded philosophically speaking. For example, S is unbounded above because it approaches infinity, but S is a bounded concept of which I can negate: not S. This is where we start the philosophical inquiry which has no bearing on formal mathematics. If S were philosophically "unbounded" as a concept, then it would bleed into everything, so to speak: S would also encompass a jurisdiction over my apple I am eating right now, which we both completely understand this is by no means the purpose of formal mathematics whatsoever.

I think, and correct me if I am wrong, we are simply contextually utilizing the term differently for different contexts of inquiry (one of philosophy and the other of formal mathematics). For example, in mathematics, a line segment is bounded and yet has an infinite amount of points in between (due to intermediate value theorem), while a line that approaches negative infinite and positive infinity that is constrained to asymptotes x = 1 and x = 5 is also bounded with an infinite amount of points in between those asymptotes; for math, there's is a meaningful distinction between the two, but, in terms of what I am noting, they are both bounded infinites.

If you believe this to be a confusing conflation of mathematical terminology, then I am more than willingly to consider what you think would be better terms!

Bob

Both S and T are indeed sets of numbers that are generated by the formula to the right of the ":" symbol. The summation symbol is just shorthand for plus after plus .....

Perhaps you might just stick with the archaic terminologies of infinity (potential & actual) and avoid "bounded".

This is an essay in philosophy, not mathematics. Don't worry about it. It's your project. I'm still unsure of what PoR is, other than a sort of linear process that goes from previous to following. I'm not your target readership. I have enough trouble trying to keep up with a very minor PoR in a certain low interest math topic. :cool:

Wonderful analysis as always Philosophim: let me try to adequately respond. — Bob Ross

Likewise Bob! Despite my points against your essay, I am always impressed by your creativity, open mind, and thought process. Lets dive in again.

I would like to clarify that neither “in toto” nor “in total” are concepts that directly entail an infinite: the former is a conception which is conceived (i.e., defined) as holistic, whereas “in total” is the conception of the summation of its parts (i.e., in content). — Bob Ross

For me, the confusion about toto came because your previous paragraph talks about infinity. You then mention toto is a concept without bounds, which implies infinity. But if I'm understanding correctly, we're really talking about form, vs what makes up that form. So for example, a tree is a form. All the indeterminate encompassing (possibilities?) which can make up the form of a tree are toto, where as if we could know all the possibilities, we could summate those in total.

But then there seems to be a contradiction here:

For example, I can manifest a conception of a set of integers {1, 2, 3} and determine that the summation of the parts as 6: the former is a conception in toto, and the latter is a conception of that conception in total. — Bob Ross

If a set of integers is 1,2, and 3, aren't the total number of integers 3? If we're listing the set, then we can say the collection is made up of 1,2, and 3, and we don't need the word toto.

for an absolute minimum at y = 2 indicates that f(x) is never negative ys and the limit from the right being infinity tells me that even if the limit to the left is a finite number that the summation of the ys will be infinity.

Again, I would say the concept is finite (that is, bounded), but technically I could be in a state of ignorance or confusion, thusly determining it as indefinite. — Bob Ross

Here is where I also think there is a conflation of words. Bounded does not mean finite. You can have an infinity for example that is bounded by whole integers. All meaningful infinities are bounded. An unbounded infinity, is everything without any defined concept. In the past I've called it "the sea of existence".

Bounded can also refer to the finite. So I could have a set of integers bounded between 1 and 4. Those integers would be 2 and 3. My point in the earlier post was to note that when we speak of meaningful infinity, it is always bounded. It may be bounded by the idea of, "All integers". So that would be an infinite set of numbers that precluded any fractions. This is the same as stating, "All integers less than y=2".

Thus if the following is true:

An infinite content can be determined in total. — Bob Ross

then all meaningful infinities can be determined in total. This again leaves me wondering where toto falls. Again, the overall feeling I get from your essay is more that toto describes the indefinite, or the unknown. The toto number of trees a person can conceive of is an indefinite concept, but one we can conceive of with some type of limit. It is not infinite due to a person's limited life span, but one cannot actually count the number of possibilities.

We can also imagine the idea of an infinite amount of possibilities one person could potentially imagine, even though in reality there are only a limited amount they actually imagine. I could see this as the infinite version of toto that expresses indefinite form. The problem is provability, which of course one can never do with the indefinite.

But again, I'm still not sure we have a clearly defined and applied term of "toto". I think it loses its use in your paper because I believe there is a misunderstanding of infinity and the term bounded. With the idea that all meaningful infinities are bounded, and that we can also bind finite sets, how can toto be used clearly without any ambiguity?

Infinite form and infinite content

This is the realm of sine qua nons (and, in virtue, the principle of regulation): a concept which is repetitive affirmation of negations would be an example of it. This kind of form entails, I must add, only one of its kind as a conception (and not just merely in existence). — Bob Ross

So back to this then, infinite form and content would be all possible forms one could give within all of infinity. This again is the unbounded infinite, or all of existence that one person could form. If you remember our conversation, a form as I'm seeing it would be a "discrete experience". I have largely avoided referencing the terms in my previous paper, as I do not want to distract from yours. But I feel this accurately communicates my intentions easiest, so I will do so here. This is the form within the infinite. So all possible forms would be all possible discrete experiences.

So within all possible forms, you propose a concept that if this concept does not exist, neither can any other concept. Without this discrete experience, no other discrete experience can exist for example. While I am ok with the idea of this, I still am having difficulties seeing how the principle of regulation is this sqn.

Only repeating myself once, you still have the problem of a thinking thing that does not derive. I've mentioned before that you would have to analyze other thinking things besides humans to show that all thinking things derive. At best, you can claim that all humans of a certain intellect derive. I have no problem with this.

At a conceptual level beyond all of this, I would still assert that one must come up with a concept first before one can derive from the concept. The sqn to me would be that one must be able to conceive some form within the unbounded infinite. Without this, no other form can exist within the infinite. From there you could derive the principle of regulation, but I do not see it as a sqn itself. So if you are to assert that the PoR is a sqn, how do you deal with the above concept? Is it not true that the real fundamental is the ability to first conceive of a concept, before one can derive from that concept?

Great work again Bob, I look forward to hearing from you!