are you contending that I ought to remove the unbounded vs bounded distinction because it is not highly disputed amongst mathematicians? — Bob Ross

Not at all. I just pop in now and then whenever math is mentioned and provide my perspective. Most in my profession are not in foundations. In the most recent 24 hour period only 1 in 58 papers submitted to ArXiv.org were in that subject (logic, set theory, etc.).

I appreciate your friendly attitude! :cool:

It is really a question of whether derivation is arbitrary (i.e., axiomatic) or grounded in a sine qua non. I wasn’t stipulating one as supreme over the other: I simply wanted to derive if there is. If not, then my subsequent essays would have been derived from axiomatic principles for “foundations”. Is that what you are asking? — Bob Ross

I asked what difficulties would be caused by denying everything you wrote - for example, supposing there never is such a thing as a principle of regulation, never has been and never need be. Does that cause a problem in any way? The answer to that might give me an idea about the value of the theory - that is, why it might be needed.

It is really a question of whether derivation is arbitrary (i.e., axiomatic) or grounded in a sine qua non — Bob Ross

With a view towards answering the above question, I'm making an attempt to get my general bearings within your project by elaborating the overview below. Let me know if it's sufficiently accurate to be helpful.

Schematic of Foundational Metaphysics of Derivation

A scheme to establish an algorithm for expressing & establishing a causal chain of derivatives culminating in a conclusion. This algorithm will be expressed in terms of the widest generality.

Some key elements that hold priority within the scheme:

• The principle of regulation

• The sine qua non

• Superordinate rules

• Subordinate rules

• {Infinite Series} bound, unbound, indeterminate

• {Ground} not subjective, not objective

By convention, the derivatives are configured in accordance with the established rules of inference.

The upshot of the scheme is elaboration of a plan applicable to the entire edifice of derivation to a conclusion.

Successful execution of the scheme will, by design, entail the establishment of a foundational metaphysics of derivation to a conclusion.

This foundational algorithm will embody a logical imperative for all derivations to conclusion.

So “without which, not” is meant as an unbounded infinite negative (i.e., if not A, then an unbounded infinite of negative judgments). It is not meant to negate only one particular derivation. — Bob Ross

Could you clarify with an example here? When you mean infinite, do you mean "All possible derivations in total/tota"? To compare again to the bachelor, we could derive another term called a bachelum, which is an unmarried man that is about to be married. Again, we could not derive the term bachelum without the superodinate "unmarried man". As such, there are an infinite derivations we could not create without the concept of "unmarried man", many which we do not directly know or have been invented yet.

A sine qua non is not denoted by being the anchor of a biconditional statement (such as D IFF A); for that could entail that it is only valid within one or a finite set of contexts. For example, it’s possible that A IFF D is true of context C1 but not true of context C2. — Bob Ross

Could you give an example of what you mean by context here? If it is unbounded context, I cannot see a sqn forming simply by the fact language and thinking can change. Lets look at the principle of regulation. A fine principle, but can it be proven that its a sqn in unbounded context? What if something does not think in a derivative manner? This may be due to low intellect, or simply a brain that does not process in such a way. Does a plant think in terms of the principle of regulation for example?

The problem with an unbounded infinite is we can always come up with a situation that negates another.
To your end, I believe you are implying a bounded context. For example, in individuals who have the capacity to only think in superordinate and subordinate manners, we could say the principle of regulation holds. Because people in this context have no other way of possibly thinking, it is impossible to think differently. Among creatures that had alternative thinking processes, the principle of regulation does not apply to them.

Again, please correct me where I am incorrect Bob.

Not at all. I just pop in now and then whenever math is mentioned and provide my perspective. Most in my profession are not in foundations. In the most recent 24 hour period only 1 in 58 papers submitted to ArXiv.org were in that subject (logic, set theory, etc.).

Oh, I see! I would love to have a conversation about infinities (e.g., set theory) if you are interested: my knowledge of it is by no means expert level and would love to hear what you think of it. However, for this discussion board, if the discussion of infinities doesn’t pertain to the essay in the OP, then I would like to respectfully ask if we could shift that discussion to occur somewhere else (that is, if you would like to continue that discussion). Feel free to DM me on this forum whenever you would like! Or maybe you would like to open up a discussion board specifically pertaining to infinities (or maybe there already is one that I am not aware of)? Otherwise, no worries.

I appreciate your friendly attitude! :cool:

Same to you my friend! (;

Bob

I asked what difficulties would be caused by denying everything you wrote - for example, supposing there never is such a thing as a principle of regulation, never has been and never need be. Does that cause a problem in any way? The answer to that might give me an idea about the value of the theory - that is, why it might be needed.

I think I understand what you are asking (but correct me if I am wrong), and, unfortunately, I anticipate that my answer will not be satisfactory. You see, for me, I consider an axiological evaluation outside the scope of the essay itself. If I were to respond with any given problem, then it would necessarily presuppose some set of values (which I am not attempting to argue for nor against in the essay): that is kind of my dilemma. In other words, a “problem”, to me, entails necessarily an underpinning value (e.g., if I value being well fed, then something which contradicts that, at least prima facea, may be a problem for me—but it is no problem if I reject the underpinning valuation).

It is simply an inquiry into how the process of derivation operates as opposed to critique of a derivation itself. I think it is and will be useful for my subsequent essays, but I am not arguing for its value in the essay.

With that being said, I feel like I didn’t really answer your question. I think that its usefulness is found in after it is found to be true (regardless of the value of endeavoring on a such a journey in the first place). For example, albeit outside the scope of the essay, I think that the principle of derivation (being a higher order, so to speak, than derivation itself—or at least, for all intents and purposes, can be visualized that way), once it is affirmed, proves the relativist nature of any particular derivation. That will play a significant role in my epistemology, for example, which will not assert its construction as built on any absolute conclusion from my derivation (assuming I am not persuaded to change my mind by the time I start writing that essay). Maybe that will help you understand its “value” to some degree (or maybe not).

If that wasn’t a satisfactory answer, I apologize and please feel free to grill me harder on it and I will do my best to provide something which I think isn’t extending into the sphere of speculation (on my end).

Bob

With a view towards answering the above question, I'm making an attempt to get my general bearings within your project by elaborating the overview below. Let me know if it's sufficiently accurate to be helpful.

Sounds good my friend: hopefully I can provide a substantive and helpful response here.

Schematic of Foundational Metaphysics of Derivation

If I am understanding this title correctly, then I actually think it is a really good title! It could be thought of as a scheme of the foundations of the metaphysics of derivation (kind of a bit wordy, but I think is true nevertheless).

A scheme to establish an algorithm for expressing & establishing a causal chain of derivatives culminating in a conclusion. This algorithm will be expressed in terms of the widest generality.

I wouldn’t use this terminology, but I think it suffices. My concern is that it isn’t quite narrow (or precise) enough; but for intents and purposes that is fine. I would say to keep the concepts very the most generic senses of the terms. For example, “causal chain” would be incorrect if it were to assert for or against it being physical, mental, or where this causal chain is originating from. But in the sense that there are a series, if one wills, of connectives: that’s basically derivation.

If you would like to use that kind of terminology, then I think that is fine (for all intents and purposes): I will do my best to interject when I think your use deviates from what is meant by a sine qua non.

Some key elements that hold priority within the scheme:

If by “key elements” you mean key terms being used in the essay, then I think that most of your list is fine. Except:

{Infinite Series} bound, unbound, indeterminate

There is no “indeterminate” category proposed for infinities: it is indefiniteness—which I wouldn’t hold means the exact same thing (but if you just mean that in the sense that the bounds in undetermined, as opposed to indeterminate, then I think that is fine). For me, I am defining “indeterminate” as not able to be determined, whereas “undetermined” simply means it hasn’t yet been determined.

{Ground} not subjective, not objective

Since you included this as an element in the list but not others (like how it is not indubitable), I feel inclined to ask if there is some sort of significance you are extrapolating from this particular claim that isn’t found in the other (such as the aforementioned example)? By “Ground”, I am thinking of an ontic claim: is that what you mean? In that case, I am not saying it is or is not grounded in the subject or an object (or objects or what have you): I am claiming it is simply not addressed, purposely, in the essay. Attempting to argue for any sort of ontic claim, even its own ontic grounding, is beyond the scope of the essay.

By convention, the derivatives are configured in accordance with the established rules of inference.

By “established”, I would like to clarify that they can be either implicit or explicit.

By “derivatives”, I am thinking of “conclusions produced via the process of derivation”. Please correct me if that is not what you meant.

By “rules of inference”, I would agree if you mean the relation between superordinate and subordinate rules. They can be, for intents and purposes, thought of as “inferred” in the sense that they are “intuited”, which I mean in the sense that we don’t necessary explicate them.

The upshot of the scheme is elaboration of a plan applicable to the entire edifice of derivation to a conclusion.

If I am understanding correctly: exactly!

Successful execution of the scheme will, by design, entail the establishment of a foundational metaphysics of derivation to a conclusion.

If I am understanding correctly: yes! I would slightly refurbish to be “establisment of a foundational metaphysics of derivation itself” or “of derivation to any given conclusion”: just to clarify that it is not just a particular conclusion. That might just be knit picky though (:

This foundational algorithm will embody a logical imperative for all derivations to conclusion.

I would have a problem with the use of the term, semantically speaking, “logic”. However, I see this as one of the potential weaknesses of the essay; that is, it could be considered a “logic”, but as the nth-order logic where n is actually an unbounded infinite series. You see, in that sense, it could be considered a higher-order logic (maybe that’s what you meant in your previous post?), but I don’t think it is constituted as a second-order or third-order logic (as it is an actually abstraction to an unbounded infinite use). If you have any ideas on this, then I would love to hear them!

Other than that, yes it is an imperative (the principle of regulation) of all derivations to a conclusion. Well said ucarr!

Bob

Could you clarify with an example here? When you mean infinite, do you mean "All possible derivations in total/tota"? To compare again to the bachelor, we could derive another term called a bachelum, which is an unmarried man that is about to be married. Again, we could not derive the term bachelum without the superodinate "unmarried man". As such, there are an infinite derivations we could not create without the concept of "unmarried man", many which we do not directly know or have been invented yet.

I want to clarify that I think you are thinking of it conversely to what I was proposing. When you say:

As such, there are an infinite derivation we could not create without the concept of “unmarried man”

That proves that “unmarried man” is not a sine qua non, which I believe (and correct me if I am wrong) you are thinking it would prove it if there’s an infinite amount of things that could be postulated without “unmarried man”. It isn’t that “if without A, there’s an infinite of other things we could postulate, therefore A is a sine qua non”: it is “if without A, we are met with an infinite amount of negations (nots)” (e.g., “without A, not this”, “without A, not that”, etc. until we realize by proof that it can be abstracted to infinity as “without A, not ...” if you will).

One of the reasons I made the distinction between “bounded” and “unbounded” infinities is because one (not necessarily you) may be incentivized otherwise (that is, if it is just postulated as “infinity”) to counter the validity of the idea of a sine qua non by means of asserting that they can conceive of and derive infinities of negations (as a concept) as without another concept. For example, one may be inclined to determine that they can categorically define that Y is without X and Y is an infinite of negations. That can most certainly postulate it, but I wanted to clarify that that is by no means a counter to the validity of a sine qua non (as an idea). You see, as I would argue, that concept of Y, valid as it is by means of derivation, is a bounded infinite because I can abstract further by questioning the grounds of that very concept of Y, thereby invalidating it as an unbounded infinite.

Could you give an example of what you mean by context here?

By “context”, I just loosely meant an idea that is sandboxed. So it is perfectly possible that “A IFF D” is only true within a sandbox, so to speak, and not true universally.

If it is unbounded context, I cannot see a sqn forming simply by the fact language and thinking can change. Lets look at the principle of regulation. A fine principle, but can it be proven that its a sqn in unbounded context?

I am not entirely sure if I am understanding, but by “unbounded context” I am envisioning a sandbox which has no bounds in form. That seems kind of like a contradiction in terms to me: a context, by definition, has a limited form, otherwise it is not a context.

If you are trying to inquire how it can be proven sans context (i.e., unbounded context), then I can provide further detail: I continually performed abstraction to its highest point, whereat I could not longer abstract higher and, thereafter, determine what (if anything) produces the negation of those abstractions if removed (or it could be thought of as the negation of particulars too, if you will). The principle of regulation was the only thing that remained. Now, at this point in my thinking, it was not so discernible whether it was (1) an unbounded infinite, (2) a bounded infinite, or (3) indefinite.

To keep it brief, I determined it not to be #2 by virtue of the proposed definition of sine qua non the negations cannot be conceived in toto and, therefore, it is not possible to prove a in toto conception without the utilization of the principle of regulation in the first place (as a separate, out of scope of the essay, derivation). I determined it not to be #3 because the derivation of without the principle of regulation was recursive (thereby solidifying its infinite nature as opposed to be undetermined bounds in content).

What if something does not think in a derivative manner? This may be due to low intellect, or simply a brain that does not process in such a way. Does a plant think in terms of the principle of regulation for example?

Firstly, I would say that the essay is meant only to prove in relation to the subject at hand (or more modestly, me as the subject). I by no means disproved or even mentioned solipsism. However, with that being said, I think it is easily arguable that this principle extends to the vast majority of non-brain-dead human beings: the principle is observable without postulating an actual subject as the originator. In terms of low intellect, I think they still exhibit the principle, just not as rationally as we do. Even the most primitive use still counts to me.

In terms of animals, I think most would fit the bill and maybe plants. To be honest, I haven’t contemplated that aspect enough to give a substantive response. However, I do think, off the top of my head, plants, for example, exhibit the abidence of such a regulatory principle (e.g., a plant does make binary decisions, which does require superordinate/subordinate rules, albeit it primitive).

The problem with an unbounded infinite is we can always come up with a situation that negates another.
To your end, I believe you are implying a bounded context. For example, in individuals who have the capacity to only think in superordinate and subordinate manners, we could say the principle of regulation holds. Because people in this context have no other way of possibly thinking, it is impossible to think differently. Among creatures that had alternative thinking processes, the principle of regulation does not apply to them.

Maybe if you provide an example I could respond better: what about the principle of regulation do you think doesn’t hold for a plant that demonstrates it reacts to its environment (which, I would argue, pretty much happens in virtue of them being alive—no?). I certainly don’t think a plant would be able to affirm the principle of regulation, but I think I can affirm that they use it (which is a different claim, I would argue).

I look forward to hearing from you,
Bob

That proves that “unmarried man” is not a sine qua non, which I believe (and correct me if I am wrong) you are thinking it would prove it if there’s an infinite amount of things that could be postulated without “unmarried man”. — Bob Ross

No, that wasn't my intention. What I was trying to note was there are an infinite number of things I could postulate with "unmarried man", that I could not postulate without "unmarried man". If unmarried man did not exist, then the defintions of bachelor and bachelum would not exist. Perhaps the words could still exist, but their meaning could never be "unmarried man", because "unmarried man" does not exist. If we disregard all possible synonyms for "unmarried man" in all possible contexts, would this be a sqn?

You see, as I would argue, that concept of Y, valid as it is by means of derivation, is a bounded infinite because I can abstract further by questioning the grounds of that very concept of Y, thereby invalidating it as an unbounded infinite. — Bob Ross

I think I understand this. Words like bachelor and bachelum all rely on the concept of "unmarried man". Again, it is not the words we are really referencing, but their meaning. Without "unmarried man", any derivations from the concept of "unmarried man" cannot exist. Of course, there are a potentially infinite number of derivations we can establish from "unmarried man" that we could not without the concept of "unmarried man". From the finite springs the infinite, though this infinite is bounded by the finite superordinate.

By “context”, I just loosely meant an idea that is sandboxed. So it is perfectly possible that “A IFF D” is only true within a sandbox, so to speak, and not true universally. — Bob Ross

The problem I see you running into is when you note a "universal" infinite. Having worked with infinite before, its very easy to lose the real consequences of true infinity. Real infinity has no limit. Which means practically any formation within that infinite can also be negated.

Think of numbers for example. Numbers are bounded limitations within true infinity. One such measurement is discrete data versus continuous data. Continuous data is a bounded infinite, such as "height". In theory, there is no limit to how high we can measure. A discrete data point would be 5 feet high.

As noted, continuous data is still a bounded infinite. Without the context of dimension, height just dissolves into the true infinite. There are an infinite amount of potential dimensions that we can create within that true infinite. To your point, I'm noting that the rule of regulation too would dissolve into the true infinite without certain bounded contexts. If a sqn must be true universally, then it must be true in the unbounded infinite.

If you are trying to inquire how it can be proven sans context (i.e., unbounded context), then I can provide further detail: I continually performed abstraction to its highest point, whereat I could not longer abstract higher and, thereafter, determine what (if anything) produces the negation of those abstractions if removed (or it could be thought of as the negation of particulars too, if you will). — Bob Ross

This is still within your own bounded context. I take no objection to there existing a sqn within a bounded context. It is completely true that you thought everything you did, and could only come to one conclusion. But is that true of all contexts, of the true infinity? That has not been proven. Let me give you an example. There are some people who cannot visualize in their mind. As in, they cannot think of images like most people can. They close their eyes, and the world is completely dark for them. Think of the host of conclusions and thinking you've done with your ability to visualize in your head, and then try to imagine the conclusions one can or cannot make if they cannot visualize.

In the same manner, a personal conclusion of thought within your own bounded context does not prove a universal context. In the same manner, we can imagine a creature that can think without the rule of regulation. Its difficult for those of us who use the rule of regulation on a daily basis to imagine this, but we already know that some things think differently from ourselves. This is what I was noting earlier. If you personally think using the rule of regulation, and nothing else, then yes, its a sqn for you. But that doesn't mean its a sqn for something that does not think like you do.

what about the principle of regulation do you think doesn’t hold for a plant that demonstrates it reacts to its environment (which, I would argue, pretty much happens in virtue of them being alive—no?). I certainly don’t think a plant would be able to affirm the principle of regulation, but I think I can affirm that they use it (which is a different claim, I would argue). — Bob Ross

First, there's the idea that we're assuming our own basis of thought applies to all other thinking things. We cannot conclude that just because you and I think in the terms of the principle of regulation, that every other thinking thing does as well. All it takes is one thinking thing that does not, and then we don't have a universal sqn anymore. I'm not saying you can't come up with a universal sqn, but it must be provably true within the true infinite. I don't see the PoR doing that currently.

Second, we can speculate that a plant, or any other creature thinks with the PoR, but we have to prove that. The burden of proof is not on me within the true infinite, the burden of proof is on yourself. And even if we prove that, we must prove it for all plants of that type, then all plants, all creatures, etc. The PoR is not something provable, because it is a bounded idea that relies on certain bounded infinites thinking in a particular manner.

Now, if we have a bounded sqn, we avoid the problems noted above. That of course, brings about new problems. If sqns are bounded to contexts, which context should we choose? I think you know this, which is why you wanted to note a sqn is universal. The ultimate problem is that I believe you have not shown that the PoR is something true universally. As noted above, I'm not sure its something you can either.

That being said, it may be that there are things I still don't understand, so please correct me if I'm in error. I also think the PoR is a fine principle within bounded contexts, and see nothing overtly wrong with it within these bounded contexts. I just don't think at this time that you've provided what is needed to show it is true universally, and not just within the contexts you've been thinking in.

If by “key elements” you mean key terms being used in the essay, then I think that most of your list is fine. Except:

{Infinite Series} bound, unbound, indeterminate

There is no “indeterminate” category proposed for infinities: it is indefiniteness—which I wouldn’t hold means the exact same thing (but if you just mean that in the sense that the bounds in undetermined, as opposed to indeterminate, then I think that is fine). For me, I am defining “indeterminate” as not able to be determined, whereas “undetermined” simply means it hasn’t yet been determined. — Bob Ross

I differentiate indefinite from indeterminate thus, the former means not specified whereas the latter means cannot be specified.

It may lie outside the scope of your project, but I want to broach the topic of infinity as it is conceptualized by you vis-a-vis how it is conceptualized by me.

I define infinite as volume unspecifiable. This is a way of saying infinities cannot be made explicit. I believe this truth persists even in the instance of hierarchies of infinities.

If {x_?} = infinity and P = {x_?} and IFF_not = if and only if negated and if {x} = bounded set, then P ⇒ IFF_not for {x}.

The above argument is predicated upon boundary = territorial limit.

Positing an infinite value (unspecifiable volume) within bounds is tricky because, in my opinion, territorial limit takes on a special meaning such that limit transforms into asymptote.

Perhaps curiously, an infinite value "warps" a (conceptual) boundary into a "curved space" that functions as an unspecified boundary in that it is a boundary that is never reached.

Is an unreachable boundary really a boundary?

In the instance of a bounded infinity, whose unspecifiable volume is quite free to expand forever, can we truthfully claim that it is contained?

It occurs to my visualization that a bounded infinity is a configuration wherein an unspecifiable volume has PoR as a neighbor who speaks another language and thus, there is no dialogue between the two. In this situation, can we truthfully say PoR acts as modulator of unspecifiable volume?

The ultimate problem is that I believe you have not shown that the PoR is something true universally. As noted above, I'm not sure its something you can either. — Philosophim

That being said, it may be that there are things I still don't understand, so please correct me if I'm in error. — Philosophim

I also think the PoR is a fine principle within bounded contexts, and see nothing overtly wrong with it within these bounded contexts. I just don't think at this time that you've provided what is needed to show it is true universally, and not just within the contexts you've been thinking in. — Philosophim

To far greater extent than Philosophim, there's much I neither know nor understand, thus I might be egregiously wrong when I use my argument above to expand Philosophim's doubt to include bounded contexts.

Having said that, I admit I do, now, have the audacity to entertain nascent doubt about the PoR's ability to modulate a bounded infinity.

Note – The core logic of my argument is the following premise,

Premise – the inherent unspecifiability of an infinite volume implies its expansion towards a boundary is necessarily asymptotic.

Hi Bob,
In your reply to Cuthbert's germane remarks you seem to me to have replied, (and pardon my extraction thereof what might appeal to me)

It is simply an inquiry into how the process of derivation operates as opposed to critique of a derivation itself. ...
... I think that its usefulness is found in after it is found to be true ...
For example, albeit outside the scope of the essay, I think that the principle of derivation ..., once it is affirmed, proves the relativist nature of any particular derivation. — Bob Ross

My impression is that by reducing the process to what is 'true' you have already relinquished your quest in favor of strictly realist binary meta-possibilities. For example, there is no truth in science! In science true is replaced by correct or more likely or most likely the case. In most aspects of personal life the only truth is death (and not even life according to our faithful judges). If this is so given that the process is not the same as its derivations, then you might limit yourself to closed objective identity and the PNC everywhere. Since I am a radical metaphysical pluralist I hope I am wrong in this.

No, that wasn't my intention. What I was trying to note was there are an infinite number of things I could postulate with "unmarried man", that I could not postulate without "unmarried man"

You are correct and this is why the form, as opposed to mere content, of an infinite is incredibly important. Let’s take my Y example again: without Y, there’s an infinite of postulations that cannot be proposed anymore (let’s call that infinite X).

There’s two ways (pertinent to this conversation, that is) we could conceive of this X: in toto or in total. In terms of the former, it is conceived of as bounded in form (i.e., complete in form). In this case, if Y can be omitted and there are still concepts which remain intact, then Y is not a sine qua non (although without it there is X). In terms of the latter, it is conceived as without bounds in form (i.e., never complete in form) and, in this case, there must be no exception to the negations (that is, every concept is being negated).

Therefore:

If we disregard all possible synonyms for "unmarried man" in all possible contexts, would this be a sqn?

It would not, because fundamentally we would have a situation where we are positing “without Y, there’s a bounded infinite of negative judgments”. That claim is not coherent if posited as an unbounded infinite because the omission of “unmarried man” leaves many concepts intact.

Of course, there are a potentially infinite number of derivations we can establish from "unmarried man" that we could not without the concept of "unmarried man". From the finite springs the infinite, though this infinite is bounded by the finite superordinate.

Exactly! It is still an infinite negative judgment, and valid at that, to assert the withouts of “unmarried man”, but it is bounded and not unbounded. Same is true of chains of infinities, cyclical infinities, etc.: they are only conceivable by means of eroding the infinite to a bounded one (that is, by means of conceiving it in toto: complete). It’s subtle, but an incredibly vital distinction (I think at least).

The problem I see you running into is when you note a "universal" infinite. Having worked with infinite before, its very easy to lose the real consequences of true infinity. Real infinity has no limit. Which means practically any formation within that infinite can also be negated.

…

As noted, continuous data is still a bounded infinite. Without the context of dimension, height just dissolves into the true infinite. There are an infinite amount of potential dimensions that we can create within that true infinite.

As you noted, continuous data (and height) is a bounded infinite; that is, must be conceived as in toto, which is not a sine qua non. I don’t have any contention with the idea that, in content, we can measure height infinitely. Maybe I am misunderstanding you, as I see nothing wrong with this.

However, I want to clarify that I do not use the language of “true infinite” (although you certainly can), and so I am interpreting that as an infinite (which is simply defined, generically, as limitless in content, which no specification of its form). Correct me if I am misunderstanding you here.

To your point, I'm noting that the rule of regulation too would dissolve into the true infinite without certain bounded contexts. If a sqn must be true universally, then it must be true in the unbounded infinite.

That is what my essay is arguing for: by being a sine qua non, it is an unbounded infinite.

This is still within your own bounded context. I take no objection to there existing a sqn within a bounded context. It is completely true that you thought everything you did, and could only come to one conclusion. But is that true of all contexts, of the true infinity?

Whenever I even attempt to derive other contexts sans my own, it is contingent on my own. I never once escape out of my context, not even in terms of conceiving of “escaping my own context”. This is why the principle of regulation is an unbounded infinite (more precisely, sine qua non).

The best I can do is postulate that, via my own derivation contingent on me, that if there were a duplicate reason out there of mine (to any degree in its manifestation), then this principle would apply. But that will always be inevitably self-referencing.

There are some people who cannot visualize in their mind. As in, they cannot think of images like most people can. They close their eyes, and the world is completely dark for them. Think of the host of conclusions and thinking you've done with your ability to visualize in your head, and then try to imagine the conclusions one can or cannot make if they cannot visualize.

If they react to their environment to any degree other than 0, then I think it is provable that it applies to them. By “conclude”, I mean it in the most affirmative & negative sense (affirmation and denials). It’s not that they have to explicate this process of the principle of regulation or that they have the exact same faculties of consciousness as me: it’s that their conclusions are regulated by this very principle.

In the same manner, a personal conclusion of thought within your own bounded context does not prove a universal context. In the same manner, we can imagine a creature that can think without the rule of regulation. Its difficult for those of us who use the rule of regulation on a daily basis to imagine this, but we already know that some things think differently from ourselves. This is what I was noting earlier. If you personally think using the rule of regulation, and nothing else, then yes, its a sqn for you. But that doesn't mean its a sqn for something that does not think like you do.

I cannot think of a creature that can think without utilizing (implictly at least) superordinate and subordinate rules. I can conceptually the omission of the concept of “principle of regulation” from my mind and attempting, thereafter, to derive what is left, but that inevitably utilizes it. If you can conceive of such a thing without its utilization, then I would be genuinely interested to hear more! How are you able to do that?

I will say that my approach, or argument, is that it regulates me and all human beings (and most likely creatures) that exhibit any form of life. Prima facea, I can trick myself into thinking sans “my context” to see what someone who wasn’t constrained to PoR would exist as, but that inevitably utilizes it as well! If there is a way to break the cycle, I would love to hear about it!

So, I guess, it depends on what you mean by “universal”. I cannot constitute it as not universal (in virtue of being my context) because “sans my context” is still contingent on PoR for me. Furthermore, I could postulate an if conditional of how it would be for an individual who didn’t have such a regulatory principle guiding their derivation, but that, again, is contingent on it.

Perhaps you have found the root dilemma of my essay!

I do want to clarify that I am able to derive other contexts sans me, but what do I mean by me? Can I posit a context sans PoR? No, and that is my point.

First, there's the idea that we're assuming our own basis of thought applies to all other thinking things. We cannot conclude that just because you and I think in the terms of the principle of regulation, that every other thinking thing does as well. All it takes is one thinking thing that does not, and then we don't have a universal sqn anymore. I'm not saying you can't come up with a universal sqn, but it must be provably true within the true infinite. I don't see the PoR doing that currently.

I would ask, in light of what I previously stated, for you elaborate on how you are able to omit the PoR when deriving conclusions of its omission. I would like to know how it is not recursively utilized therein. If you can demonstrate that, then that would be really helpful for me!

Second, we can speculate that a plant, or any other creature thinks with the PoR, but we have to prove that. The burden of proof is not on me within the true infinite, the burden of proof is on yourself. And even if we prove that, we must prove it for all plants of that type, then all plants, all creatures, etc. The PoR is not something provable, because it is a bounded idea that relies on certain bounded infinites thinking in a particular manner.

All derivation is subsumed under this principle for me, as a subject, and thus nothing escapes it (not even that idea of escaping). Therefore, it really is an unbounded infinite, universal if you will, but maybe I am missing something. Maybe it is possible to conceive of its omission without recursively utilize the principle itself: but, yet again, I just used it to postulate that possibility. Maybe you mean something entirely different than me by “universal”?

That being said, it may be that there are things I still don't understand, so please correct me if I'm in error. I also think the PoR is a fine principle within bounded contexts, and see nothing overtly wrong with it within these bounded contexts. I just don't think at this time that you've provided what is needed to show it is true universally, and not just within the contexts you've been thinking in.

If this is the case, I would love to know how. I wouldn’t say I am in agreement with you: I think it is proven to be universal, but I could be wrong as always!

Bob

I define infinite as volume unspecifiable. This is a way of saying infinities cannot be made explicit. I believe this truth persists even in the instance of hierarchies of infinities.

Positing an infinite value (unspecifiable volume) within bounds is tricky because, in my opinion, territorial limit takes on a special meaning such that limit transforms into asymptote.

Perhaps curiously, an infinite value "warps" a (conceptual) boundary into a "curved space" that functions as an unspecified boundary in that it is a boundary that is never reached.

I think I understand. Basically (and correct me if I am wrong), an infinite within bounds actually simply approaches the limit as opposed to actually reaching it. So, for you, I would image that the contents of a line connecting two dots does not actually reach the two endpoints: it approaches them infinitely. Is that correct?

I think, if that is what you are saying, then this objection makes sense (in those terms as you proposed):

Is an unreachable boundary really a boundary?

In the instance of a bounded infinity, whose unspecifiable volume is quite free to expand forever, can we truthfully claim that it is contained?

To me, this still produces the same in toto claim (that is, a complete infinity): even if it asymptotes, every single point within the infinite of points of the finite line exists. For every possible point, it exists between the two dots on the line and the two endpoint dots exist for the line: therefore, the concept can be thought of as in toto (that is, complete, which is an encapsulation within a finite).

To be honest, I actually think that you are right, as you are discussing the rightful method of conceiving of a bounded infinite (I would say): the infinite content quite literally does approach as opposed to arrive at the ends. If it reached the endpoints of the line, then it wouldn’t have an infinite content.

Assuming I am not misrepresenting your view (which please correct me if I am), I would like to clarify that a bounded infinite is just like that line connecting two dots: there’s a form that is conceivable in a finite concept (i.e., the line from point X to point Y), but the content quite literally is infinite and, therefore, the points approach the endpoints which actually achieves a complete concept when combined with the endpoints themselves (i.e., an infinite approaching, asymptoting at by endpoints + the endpoints themselves is conceivable as a complete concept).

By unbounded, I meant that it cannot be conceived of as this sort of complete, in toto, concept: try to imagine a line that just continued forever and, with respects to its ends, never ended a particular point. For all intents and purposes, that would be unbounded in form and that cannot be conceived of as in toto. The best I can do is formulate its in total by means of a summation of its parts (potentially, that is, depending on what the parts are).

It occurs to my visualization that a bounded infinity is a configuration wherein an unspecifiable volume has PoR as a neighbor who speaks another language and thus, there is no dialogue between the two. In this situation, can we truthfully say PoR acts as modulator of unspecifiable volume?

I am sorry, I don’t quite follow what you are asking here: could you please elaborate further? What do you mean by neighbor? PoR, as I was postulating it, is, as opposed to a kin to, a unbounded infinite (or unspecificable volume that cannot be conceived in toto).

the inherent unspecifiability of an infinite volume implies its expansion towards a boundary is necessarily asymptotic.

I honestly have no problem with this premise: I think you are misled to a wrong conclusion by it though. A infinite volume that asymptotes as two finite endpoints is a bounded infinite, as it can be conceived in toto. I might just be misunderstanding you thought, so please correct me where I am wrong!

To far greater extent than Philosophim, there's much I neither know nor understand, thus I might be egregiously wrong when I use my argument above to expand Philosophim's doubt to include bounded contexts.

Having said that, I admit I do, now, have the audacity to entertain nascent doubt about the PoR's ability to modulate a bounded infinity.

In terms of Philosophim’s argument, I think it reveals a more fundamental dispute I will have to contend with: is it possible to extend this beyond the sphere of my individual context. I think it has been proven to be true, but there’s much to discuss (so I could be wrong). The problem, I think, is that one is perfectly capable of omitting themselves (i.e., their context) to see what remains and, thereby, if it is constrained to my context then that is self-defeating. However, I think it is a facade of sorts: one cannot actually conceive of sans themselves (in sense of derivation). When someone validly conceives of someone else’s context sans their own, this is all contingent on their derivation and does not transcend themselves whatsoever (I would argue). If you would like to discuss Philosophim’s objection as well, I would be more than happy too! I will leave it there for now and let you navigate where you would like to go from here.

Bob

Nice to meet you magritte!

My impression is that by reducing the process to what is 'true' you have already relinquished your quest in favor of strictly realist binary meta-possibilities.

For clarification, are you saying that defining what ‘true’ I have thereby restricted myself to some set of realist binary (so true or false) meta-possibilities? By “binary meta-possibilities”, do you mean that I am rejecting non-binary ones (e.g., like fuzzy logic)? By “meta-possibility”, what exactly are you referring to?

For example, there is no truth in science! In science true is replaced by correct or more likely or most likely the case.

I have no problem with using cogency as opposed to an absolute truth in relation to derivation and the conclusion produced therefrom (if that is what you are getting at: I am not too sure yet). But then we need to define what you mean by “true”? I do not mean it in an absolute sense at all.

In most aspects of personal life the only truth is death (and not even life according to our faithful judges)

Can you define “truth” for me (in terms of what you mean)? I was anticipating you would have claimed the opposite, as death is an induction and thusly “more likely” the case.

If this is so given that the process is not the same as its derivations, then you might limit yourself to closed objective identity and the PNC everywhere.

The principle of regulation by no means necessitates the principle of noncontradiction. Furthermore, I would argue that process of the performance of derivation abides by the principle of regulation but, more importantly, not that the performance of derivation is always exactly the same. One person could use PNC, another could not. Nothing about my essay favors either one (at least that I am aware of).

I may just be misunderstanding you, so please feel free to correct me!

Since I am a radical metaphysical pluralist I hope I am wrong in this.

I am by no means an expert on “metaphysical pluralism”, so if you could elaborate a bit on what you mean that would be much appreciated!

Bob

I would love to have a conversation about infinities (e.g., set theory) if you are interested: my knowledge of it is by no means expert level and would love to hear what you think of it — Bob Ross

Tones-in-a-deep-freeze is more an expert in this area. I'm from the generation of naive set theory. Your use of infinite is a philosophical excursion beyond my experience.

Positing an infinite value (unspecifiable volume) within bounds is tricky because, in my opinion, territorial limit takes on a special meaning such that limit transforms into asymptote. — ucarr

The set [0,1] is uncountably infinite with no asymptotes. Clueless what you mean.

Perhaps curiously, an infinite value "warps" a (conceptual) boundary into a "curved space" that functions as an unspecified boundary in that it is a boundary that is never reached. — ucarr

Give an example, please.

I fail to see where any of this is going. I got lost somewhere around a principle of regulation. Some time back Philosophim gave a short but clear example of where he thought this was going. Bob, I recommend you do the same.

Tones-in-a-deep-freeze is more an expert in this area. I'm from the generation of naive set theory. Your use of infinite is a philosophical excursion beyond my experience.

Oh I see: no worries my friend!

Bob, I recommend you do the same.

There is an example in the essay which I think explains it well: did you find it to be confusing as well?

Bob

There is an example in the essay which I think explains it well: did you find it to be confusing as well? — Bob Ross

Sorry, Bob. See if you can parallel what he did in a short paragraph. A clear example with less abstraction. Give a clear example of the principle of regulation as well. Or just ignore me and continue on - I would not take offense. :smile:

I fail to see why anyone in his right mind would want to use a highly controversial concept such as infinity as the bedrock of his/her thesis (on metaphysics)?

That said, such risky ventures are not without precedent - Gödel used a spin-off of the liar sentence to torpedo Whithead & Russel (vide Gödel's incompleteness theorems).

Nevertheless, from the posts I read, the OP gets points for being systematic, a quality that I respect (a lot). Bonam Fortunam OP.

P.S. A quick question: Why, o why $\infty$?

Thank you for your reply Bob, I believe I'm beginning to see what you're going for more clearly. First, lets cover what I mean by the true infinite versus the bounded infinite. Lets look at the true infinite as all possible numbers. Within that infinite, you can have bounded infinites. For example, all numbers that end on the tenth's place is a bounded infinite within the true infinite. A bind is a limit. To speak of an unbounded infinite, is to speak to something without limits.

Within the infinite, I can create many bound ways of comparing numbers. I can create bounded ways of adding, substracting, etc. But does the negation of one of these comparisons negate the true infinity of numbers? No. But if we think about numbers for a second, we realize they are bounds as well. Each "number" is a bounded concept. So we get rid of numbers as well, and we are finally left with true infinity.

When you say a sqn is needed, because without it an unbounded infinity is negated, I'm not sure that's possible. The unbounded infinite is a total, and we can only represent it with a toto, or a bounds of some kind. For the most basic of bounds, we create a number, 1. To your principle of regulation, we can then create the number 2 as a subordinate to the idea that its a 1 and a 1 together. Is there a superordinate to 1? I'm not sure.

The point though, is that all ideas are bounded within unbounded infinity. Unbounded infinity is the stream from which all identities and relations are pulled from. Unbounded infinity is where all bounded infinities are created. An unbounded infinity is something we can never understand in total, but only in toto as well.

So when you declare a sqn is that without it, unbounded infinity cannot exist, it something that I'm not sure can ever be proven. The best you can do is use bounded infinity. But at that point, that seems to defeat the purpose of the sqn. The best we can do is re-create our "bachelor" example repeated among several different contexts. The PoR is no exception.

If we disregard all possible synonyms for "unmarried man" in all possible contexts, would this be a sqn?

It would not, because fundamentally we would have a situation where we are positing “without Y, there’s a bounded infinite of negative judgments”. That claim is not coherent if posited as an unbounded infinite because the omission of “unmarried man” leaves many concepts intact. — Bob Ross

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.

The PoR is a logical way of relating concepts. But can a being have a concept without conceiving of superordinate and subordinate concepts? Yes, we can prove this. To have a subordinate or superordinate concept, one must have two concepts. By the nature of a concept being a derivation, one must be formed before the other. If one cannot conceive of a single concept without the PoR, how does one conceive of the first concept?

One must conceive of that first concept prior to the second according to the PoR. That means one must be able to conceive of a concept without the PoR, because prior to the first concept, one has no concepts. If one can conceive of a concept prior to the PoR, than the PoR is not necessary to conceive of concepts. If this is the case, one could also conceive of a second concept that had no relation to the first concept. The ability to create concepts does not necessarily mean one will create derivated concepts, or use the PoR.

Thus we've shown that while the PoR is a way to view derivation itself, it is not necessary to hold or create concepts. Meaning that the PoR cannot be a sqn as the idea of "concepts" itself can still be conceived without it.

Overall, I think the true problem is trying to include unbounded infinity. Perhaps there is a sqn for unbounded infinity, but I don't think the PoR is it.

RE: Foundational Metaphysics
SUBTOPIC: Fundamental Issue
※→ Agent Smith, et al,

I fail to see why anyone in his right mind would want to use a highly controversial concept such as infinity as the bedrock of his/her thesis (on metaphysics)? — Agent Smith

(COMMENT)

The concept of that which is boundless, with no beginning and no end, that which is beyond the termination of any set, or larger than anything imaginable (∞) is not all that controversial.

If one either argues for the belief in a deity or against a deity, one must have some idea as to what "infinity" means. The degree of that understanding is altogether different.

Nevertheless, from the posts I read, the OP gets points for being systematic, a quality that I respect (a lot). — Agent Smith

(COMMENT)

Acknowledging a presentation that is assembled as a coherent body of ideas or principles → even that which is considered a balanced equation, is what it is... Once you grasp that a "Fibonacci Sequence" has the same number of elements in its set as does the "Set of All Numbers" THEN you begin to understand "infinity."

"infinity as the bedrock of his/her thesis (on metaphysics)" — Agent Smith

(COMMENT)

Certainly, both the set of all physical laws of the universe → and the set of all that is beyond the limits in the physical laws of the universe → are the same in size and magnitude.


Most Respectfully,
R
Saturday, July 16, 2022

Felicitations Rocco Rosano. Great to see you're still alive & well!

To get straight to the point, I'd like to bring up for discussion two kinds/types/varieties/strains of ideas (using an entomological analogy):

1. Instar ideas (immature/half-formed): Infinity is one such; we have a very rudimentary understanding of what it is despite the fanfare and celebrations of the early 1900s. I wouldn't recommend it as a foundation for any thesis.

2. Imago ideas (mature/fully-formed): Think of one and post it. One example is the concept of numbers as abstractions of "similar" sets.

Lets look at the true infinite as all possible numbers. Within that infinite, you can have bounded infinites. For example, all numbers that end on the tenth's place is a bounded infinite within the true infinite. A bind is a limit. To speak of an unbounded infinite, is to speak to something without limits. — Philosophim

This seems arguably non-mathematical. "all numbers that end . . ." is still unbounded in the traditional sense. If, on the other hand, you mean positive numbers less than or equal to 1, this set contains 10 elements. I think the real problem here is in not using existing math concepts for infinities, boundedness and unboundedness The notions of things being derived from preceding things and participating in further derivations, some of those preceding things being essential and others not, is pretty simple.

I admit, I did not read the entire essay. :sad:

Note: had to correct myself. Easy to put foot in mouth.

The set [0,1] is uncountably infinite with no asymptotes. Clueless what you mean.

Perhaps curiously, an infinite value "warps" a (conceptual) boundary into a "curved space" that functions as an unspecified boundary in that it is a boundary that is never reached.
— ucarr

Give an example, please. — jgill

Regarding {0,1} graph {x = 0.1 + 0.01 + 0.001 + 0.0001 + 0…nth.1 < 1}.

Above is my attempt to show a counting series from 0 towards 1 for values of x that graphs as an asymptotic progression.

The “gravitational” force of an infinite volume curves its own graphic progression to such an extreme it never achieves “escape velocity” to the next whole integer.

Regarding {0,1} graph {x = 0.1 + 0.01 + 0.001 + 0.0001 + 0…nth.1 < 1}.

Above is my attempt to show a counting series from 0 towards 1 for values of x that graphs as an asymptotic progression — ucarr

An asymptotic relationship requires a function g(x) where Lim f(x)/g(x) =1 as x becomes infinite. Or something similar.

The “gravitational” force of an infinite volume curves its own graphic progression to such an extreme it never achieves “escape velocity” to the next whole integer. — ucarr

A reference for this would help an awful lot. :roll:

Sorry, Bob. See if you can parallel what he did in a short paragraph. A clear example with less abstraction. Give a clear example of the principle of regulation as well. Or just ignore me and continue on - I would not take offense.

I must concede that, at this time, I do not have readily available a short and simple explanation for the principle of regulation (that doesn’t erode some of the meaning thereby). However, with that being said, let me attempt to give you a brief, overly-simplified example.

Imagine I were to postulate that “A is true” wherein “A” is a statement (whatever you would like it to be, let’s say). “A is true” is the affirmation that “A is true”. This affirmation abides inevitably by superordinate rules (that is, it could not have been affirmed otherwise by the utilization of other implicit, in this case, principles that are affirmed). Let’s say, for example, I were to postulate that my affirmation of “A is true” (i.e., A being true) by means of mere whim (i.e., completely arbitrary and baseless—just a random thought that manifested in my mind). Now, that justification can be the shifted focus: I affirmed that the affirmation of “A is true” is by means of mere whim. This postulation, likewise, inevitably abides by superordinate rules, of which I can explicate (or simply move on and thereby they remain implicit). I can do this forever.


I want to emphasize that the above example is incredibly over-simplified, but let me know if that at least partially helps you understand. If not, I can try again.

Bob

Nice to meet you Agent Smith!

I fail to see why anyone in his right mind would want to use a highly controversial concept such as infinity as the bedrock of his/her thesis (on metaphysics)?

…

P.S. A quick question: Why, o why ∞?

I understand that many fields of study are still of yet formulating the behavior and kinds of infinite (e.g., mathematics), but I don’t see its controversy with respect to the essay: could you please elaborate specifically on what within the essay is controversial (with respect to its use of infinities)?

Nevertheless, from the posts I read, the OP gets points for being systematic, a quality that I respect (a lot). Bonam Fortunam OP.

I appreciate that my friend! If you could please elaborate on your contention with its grounding in an unbounded infinite, then I would love to explore that issue.

Bob

Lets look at the true infinite as all possible numbers. Within that infinite, you can have bounded infinites. For example, all numbers that end on the tenth's place is a bounded infinite within the true infinite. A bind is a limit. To speak of an unbounded infinite, is to speak to something without limits.

Although that is fine if you would like to use that kind of distinction, I would like to note that that is not what a “bounded infinite” is defined as in the essay. All possible numbers would be, with respect to the essay, a bounded infinite. I can abstract, for example, its contingency on distinction which thereby erodes it to a bounded infinite (i.e., conceived in toto); for I cannot posit the omission of that hypothetically “unbounded infinite” of possible numbers without conceiving it as in toto, of which its omission is entirely possible if everything was oneness. That’s just one example of many principles that it presupposes. Perhaps my essay, on that section, was misleading—as I did mention that the regards of postulating natural numbers can be possibly either “unbounded” or “bounded”, but what I was meaning to say is that it has no direct relevance to the point I was making therein (i.e., in toto vs in total). Maybe I should refurbish that paragraph if you think it was misleading.

Within the infinite, I can create many bound ways of comparing numbers. I can create bounded ways of adding, substracting, etc. But does the negation of one of these comparisons negate the true infinity of numbers? No. But if we think about numbers for a second, we realize they are bounds as well. Each "number" is a bounded concept. So we get rid of numbers as well, and we are finally left with true infinity.

Again, I would argue that the concept of all possible numbers as an unbounded infinite can only occur by means of the misapprehension of thinking it is such while actually conceiving it in toto. The idea of an unbounded infinite of possible numbers is contingent on many principles and faculties of reason (e.g., possibility, necessity, spatiotemporality, etc.).

Now, to your point I think, it is entirely possible to posit hierarchical structured infinities. For example, you and I can most certainly posit a bounded infinite wherein each element contains a bounded infinite and so on. I could postulate that there is a contingency structure (wherein the lower is contingent on the higher) that looks like so:

{All possible numbers}

{n + 1, …} {n + 2, …} {n + 3, …} etc.

The infinite sets of the iteration over each possible number + 1, + 2, +3, etc. is, as you said, contingent on the concept of there being an infinite set of possible numbers; however, this is a bounded infinite of bounded infinities: any permutation you choose, I would argue, is bounded—which I don’t think we are agreeing on as of yet.

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

When you say a sqn is needed, because without it an unbounded infinity is negated, I'm not sure that's possible.

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.

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).

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. This then can be further abstract to question its validity, which inevitably utilizes PoR. Likewise, it is entirely possible for it to remain implicit, which still utilizes PoR.

So, whether 1 has a superordinate or not, in the sense that you are asking, I think has not relevance PoR directly: you can posit whatever you want, which will be via PoR.

An unbounded infinity is something we can never understand in total, but only in toto as well.

Although I think this may be just that we are defining the terms differently, I want to clarify that the essay proposes the converse: a unbounded infinite is never understood in toto, but can be in total.

So when you declare a sqn is that without it, unbounded infinity cannot exist, it something that I'm not sure can ever be proven.

This is not, if I am understanding you correctly, what the essay defines as a sine qua non. To omit something is to thereby conceive of it in toto (as opposed to in total). So if you try to omit an unbounded infinity by any means it is thereby eroded to in toto. One cannot without a unbounded infinite, I don’t think at least.

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). Also, “existence”, to me, oversteps the bounds of the essay, as I am not trying to get into ontology therein.

Yes, we can prove this. To have a subordinate or superordinate concept, one must have two concepts. By the nature of a concept being a derivation, one must be formed before the other. If one cannot conceive of a single concept without the PoR, how does one conceive of the first concept?

One must conceive of that first concept prior to the second according to the PoR. That means one must be able to conceive of a concept without the PoR, because prior to the first concept, one has no concepts. If one can conceive of a concept prior to the PoR, than the PoR is not necessary to conceive of concepts. If this is the case, one could also conceive of a second concept that had no relation to the first concept. The ability to create concepts does not necessarily mean one will create derivated concepts, or use the PoR.

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. One can most certainly conclude something without explicating or even understanding how they were able to do so. Furthermore, all concepts are derived (that is, produced from the process of derivation): I am not positing that one can only formulate “derivated concepts”.

Thus we've shown that while the PoR is a way to view derivation itself, it is not necessary to hold or create concepts. Meaning that the PoR cannot be a sqn as the idea of "concepts" itself can still be conceived without it.

It cannot be conceived without its implicit use. However, it can most certainly be something the individual at hand has no clue about; nevertheless, concepts cannot be conceived without the principle of regulation.

Bob

An asymptotic relationship requires a function g(x) where Lim f(x)/g(x) =1 as x becomes infinite. Or something similar.

The “gravitational” force of an infinite volume curves its own graphic progression to such an extreme it never achieves “escape velocity” to the next whole integer.
— ucarr

A reference for this would help an awful lot. — jgill

There's no reference for my line above. It was produced by my act of imagination that attempts to parallel your elaboration of asymptotic relationship in line one above with Einstein's General Relativity. I'm trying to say that infinite volume, like infinite gravitational field, warps neighboring spacetime (in this metaphor spacetime = the number line) into a field so curved neighboring objects cannot achieve escape velocity from its grasp. "Material" evidence of this warpage herein is the asymptotic graph of numerical progression rendered as "curved numbers."

Re: g(x) where Lim f(x)/g(x) =1 as x becomes infinite. Let's suppose this to be numerical time dilation, with the progression of the value towards infinity being "time." This numerical dilation grounds an intentional maneuver that makes an equation start rendering an infinite value. We might think of this maneuver as the act of dropping a graviton into an equation in order to intentionally make it go infinite.

Foraging around for an application of the numerical graviton maneuver, I come up with using the graviton maneuver to effect a numerical time dilation that facilitates topological examination of boundary equations for First Causes.

Note -- Curved numbers have some type of relationship to imaginary numbers. I sense this because imaginary numbers, being displaced from the set of real numbers, exhibit something in common with curved numbers, extreme warpage.

If someone can further distill this relationship into clarity, I trust it will prove to be mathematically lucrative.

To found a system/theory of metaphysics on a concept we know very little viz. infinity about feels wrong to me. It violates a cardinal rule in philosophical argumentation to wit that axioms should be at best, self-evident or at worst, least controversial. Infinity is exactly the kind of concept that fails to fulfill both these conditions. In short, your system/theory is based not on knowledge but on ignorance.

Being new to the game of philosophy, I could, of course, be way off the mark.

Read Spinoza (re: substance / natura naturans which is both eternal and infinite – the only real, everything else that exists are merely ephemera necessarily dependent on substance). Or read Epicurus / Lucretius (re: the void which is both eternal and infinite ...) There are many other "infinite foundations" – the absolute, god, ground of being, the one, dao, xaos, etc – throughout the history of metaphysics.