The Universal Theorem

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The Universal Theorem

Introduction

I am a full time philosopher, in that I spend all my time thinking about truth. But I am not a professional philosopher because I don’t get paid for it. On the surface, I am just another insane homeless guy wandering around Santa Monica. But in my unhumble estimation, the truth is that I am at once one of the deepest and most logically minded thinkers the world has ever known. So let’s get started!

On the Use of the Term, “Theorem”

In mathematics, the term “theorem” is used to denote the existence of a proven proposition. This simply means that, from a set of “reasonable” assumptions, a certain conclusion has been logically deduced. My goal is to deduce a system of thought—from the most basic of principles—that will stand the test of time. In this sense, it must remain as free from empirical speculation as humanly possibly.

In service of this, we must come to understand what we mean by the terms, “thesis” (ie theory, or proven theorem) and “hypothesis” (ie conditioned statement). We can say that a thesis necessarily follows from the assumptions, while the truth of a hypothesis depends on the truth of its axioms.

However, there is no difference between the two once we realize that they both must necessarily assume their axioms to be true in order to arrive at deduced truth. We can call the act of formulating one’s axioms the act of discovering manifest (simple) truth.

The only reason why the appendage “hypo” is added to the beginning of the word “thesis” is that those of an empirical bent are given to seek for final confirmation of a theory from their immediately given sensations. This thinking, however, reverses the true course of events. The first thing that we are given as infants is a jumble of sense data. It is only after repeated exposure to the same sorts of data that we organize it into concepts. That is, at no point can our concepts be falsified by future data, because data is what made them possible in the first place. Instead, we must say that future concepts result from the addition of new sense data into our present concepts, but the present concepts, all in all, are simply not subject to falsification.

And so it is with our theories, which are just systems of thought that use concepts as their constituents. A theory, that is, comes to us when the faculty of reason tries to unify a heretofore disparate collection of concepts. This can all, in so many words, be found in Kant’s major Critique.

So in reality, there are two major problems with the notion that sense data has the last say over the validity of any theory.

  1. Sense data necessarily temporally precede theory.
  2. Sense data are one level removed from theory (with concepts being in the middle).

On the Burden of Proof

Our burden of discovering that a statement has been proven to be true will simply be this:

“A proposition, X, will be considered true if it is not unreasonable to assume that X is not necessarily untrue.”

Another (albeit incorrect) way of saying this is:

“A proposition, X, will be considered true if it is reasonable to assume that X is possibly true.”

This second form is nothing but circular reasoning. In addition, it makes the error of placing the burden on reason to take an active role in making assumptions, rather than simply filtering out those assumptions that appear to be against its [reason’s] own nature. In other words, we must not think of reason as an old West gold panner, always trying to increase its prospects, but rather as the pan itself, which is the tool that extracts the wealth from the rubbish.

It also might seem rather farfetched to have the level of burden so low as to allow in any proposition that is “not necessarily” untrue. But we must do it this way because we can’t speak in the language of probabilty when it comes to reaching a conclusion that has theoretical (rather than empirical) value. Furthermore, it is only by way of demonstrating the neccessary untruth of a proposition that we can then say that the statement’s logical complement (ie the proposition that we are trying to prove) is, in fact, true.

We must further distinguish the notion of a proposition from the notion of an utterance in general. That is, the sense of a proposition must be perfectly clear, such that its logical complement can also be clearly stated. The question of whether a proposition is sensible in this way is utterly distinct from the question of its truth value. It is for this reason that, for the present purposes, we can think of the vast majority of possible utterances as being senseless, and therefore, as not being susceptible to the discovery of truth value.

It is for the above reason that our theorem must have the character, wherever possible, of objectivity. This just means that the language of quantity and proportion (ie mathematics) will be used whenever it serves a demonstrative purpose.

On the Use of the Term, “Universal”

I seek to deduce a theorem that has universal validity, that is, it must not be applicable merely to certain moments in time or regions of space. It must on the other hand, be applicable to any sense in which the a universe of entities can be given. That is, we must always have in mind the idea of a universal set within which subsets can be created and relationships between subsets can be characterized. (From now on, a “subset” will just be called a “set”; otherwise, the phrases “the universal set” or simply “the universe” will be used.)

Let it be hereby asserted that a large number of questions that deal in foundational mathematics (eg number theory) suffer from the fatal flaw of refusing to define a universal set by way of appealing to, for example, “all real numbers”. It will be shown in the sequels to this essay that any such appeals to mathematical infinitude are simply the result of failing to define a domain against which questions can be sensibly posed. That is, to say that a set contains “all numbers” is nothing other than saying that the domain (ie the universe) under investigation is undefined; ie., a concept, within the context of a logical deduction that does not have a definition therefore fails to have any sense. For all intents and purposes, the given concept is null and void: it simply fails to exist.

On the Method of Discovery of the Universal Theorem

While it is true that much of the work has already been done in terms of realizing the basic principles at work in any possible universe, the fact still remains that it takes much collaboration in order to translate it from its indigenous state (ie the way that it exists in the author’s mind and notebooks) into a way that it will be generally accepted by present and future generations of critical thinkers.

It is therefore necessary to take serious regard of any reasonable questions that may be raised by any of my readers. But of course, there must always be a limit placed on what qualifies as a reasonable question, first and foremost is that it must have sense. It is necessary, therefore, that any flaws in logic be exposed in a method that is at least as rigorous as the method that is used by the author. In service of this, the following are forms in which counterexamples are encouraged to be presented:

  1. Symbolic equations, using either accepted mathematical or logical notation
  2. Computer programs, preferably to be compiled on an open source platform such as Linux
  3. Graphical representations of data

Final Thoughts (for the moment)

In the end, we must not take ourselves too seriously in attempting to develop a theory of the universe, for the simple fact that open hostility will tend to hamper our creative impulses. Furthermore, we might be more justified in regarding all of this in terms of an essential human process that must be lived through, rather than some sort of end that is always yet to be finally reached.

Through my efforts, I hope that I will be able to improve the quality of the lives of my readers, and in return, I hope that the quality of my life will be improved as well.

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I am glad that nobody has replied yet because it gives me a chance to lay out a few “suggestions” (ie ground rules).

First and foremost, since I want this thread to be as naturally pleasing to our eyes and minds as possible, please do not interrupt the flow of the conversation with massively long quotes of preceding posts! And if at all possible, instead of using the quote feature at all, please instead formulate, in your own words, your understanding of the concepts contained within the previous posts in question, and respond to them accordingly. The ability to reformulate that which has already been said, after all, lies at the very heart of the understanding process.

Also, I am not quite sure to what extent I want to take this topic, but I was considering the possibility of using this particular thread only for the most general levels of discussion while using other threads for more detailed discussions of possible points of contention. And particularly, if you want merely to give an opinion concerning all of this, without attempting to contribute any words of substance, I think that a separate thread in Mundane Babble would very much suffice for that purpose.

I am no rush to get any of this done. There might only be a day or two between posts, or it might be a week or month or more. What I want from myself, more than anything else, is to create a body of work that is of sufficient quality to be fully appreciated by my current audience and by all of posterity. And from you, I request that you be as vigilant as possible when it comes to keeping me—and keeping each other—as intellectually honest as is humanly possible.

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So your intention is to just restate everything that was said in Kant’s Critique of Pure Reason but in a way that is universally applicable?
First of all, why? It is tedious and unnecessary. The world of empirical science doesn’t need a foundation (opposed to what Kant assumed) - the only reason we might feel like it should have a foundation is because of people who deny logic to uphold tradition; and even if you were to formulate some way of assuring “universal proof” for a given theory, there are still going to be unconvinced imbeciles around who are willing to deny it, no matter how much countless evidence stands against them.

Kant already attempted the same exact thing - however, since you mentioned Kant, it seems like you only wrote all of that out of the inspiration you received from reading the style of an outdated philosopher.

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This thread immediately distinguishes itself as the work of an authentic philosophical worker. Hostile quips can hardly distract anyone from its universally pertinent substance. Furthermore, the author is a gentleman, showing due concern and noble intent to please not only the reader’s abstract mind, but also the eye and its visual-tactile sense. The author’s proposition to forego using automatic quotation and the clear statement of what form reader responses are expected to take is itself worth thinking about and emulating.

-WL

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You two, and whoever else, can sit around all day formulating “universally applicable theorems” and feel like you’ve accomplished something all you want, but no one else is going to feel that way about it.

Out of the small amount of people who end up reading it, only a fraction of them aren’t going to immediately stop reading because they don’t understand what you’re talking about.
And of the remaining people who do understand what you’re talking about, they’ll still probably stop reading because they don’t see any sort of reason for reading such a strenuous piece of material that has done nothing more than over-analyzed common sense. Hence, why no one replied.

And even of the remaining people who don’t stop reading, they will either:
A) Think it is a blatant rip-off of Kant
B) Be an amateur philosopher who still holds philosophy up on a pedastal, and thinks that restating common sense in a pragmatic way is an achievement of some kind.

Furthermore, Kant was horribly deluded in how he thought his writings would make a significant impact in the world of science – and it was even more tragic that so many scholars and professors of the age bought into it.

Not only is the original post stating a load of useless information, but the concepts are copied from Kant no less. And even worse, the writer of the original post acknowledges Kant and his philosophy, yet fails to recognize that his concepts are so similar to Kant’s that they are practically restating them.

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I don’t mean to seem offensive or hostile, and I do feel bad for it - but if I hadn’t told you, somebody else would have had to eventually.

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These are valid criticisms, only one can’t shake the distinct feeling that they belong in your thread “My Final Philosophy”, and not here. Perhaps you mistakenly posted them here instead of there. If I hadn’t told you, somebody else would have had to eventually!

-WL

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Now that was just a bad attempt to retort, and it didn’t even fully make sense.
If you aren’t going to welcome criticism (even though you acknowledged it as valid), what exactly are you expecting people to do? Agree with you unconditionally?

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Haha! That’s the thing about WL…

He/she is obviously a ‘serious’ intellectual, and no doubt something of a word-smith, but one can’t help but question the intent of his/her replies…almost always.

– WL – the philosophical content is there, I can decode some of it from your posts. You seem to understand what you speak of, probably to some great length. But, I get a feeling that your perspective regarding the philosophy of a respective subject often takes a back seat to criticism. That is not inherently bad, and even beneficial in some contexts, but I find that your brand of criticism often holds implications of animus toward the author; not just the subject of conversation. Of course, this is well disguised in your talent for writing and sarcasm, so I may be jumping to conclusions. I just figure it this way – discussion, debate, criticism, and even arguments are great tools (even luxuries) for a thinker. But, why attack the thinker for attempting to think? That is admirable, if anything, in my humble opinion.

You are intelligent. We get it.

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As I have read much of WL over the years, Statik, I can tell you that he is among the most cheerful of philosophers, much like a famous thinker that he is often in agreement with.

He does have a perspective, yes.

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Then perhaps it is better that I say WL’s talent for writing exceeds my talent for reading comprehension.

I was jumping to conclusions I suppose. My B.

Frowny face.

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Our first goal for this essay is the definition of certain foundational concepts that currently exist in a state of philosophical ambiguity, thus allowing for an environment within which mathematical logic finds itself ill-equipped to resolve certain of its most profound concerns. We will then seek to understand how these concepts can best be organized within a system of thought that takes as its goal the objective representation of any possible physical universe.

We will here take finitude to be regarded as a necessary condition for [objective] mathematical existence. As a result, the term infinitude will signify the condition whereby mathematical existence is impossible. In other words, for an object to be “finite” means that it is “definite” or “defined”. For an object to be “infinite” means that it is “indefinite” or “undefined”, and a strict mathematical definition cannot be attached to the object, and it therefore cannot exist, in a mathematical sense. This sort of logic is in keeping with taking the prefix, in-, to have a negative connotation (i.e. “ingracious” or “incapable”) rather than denoting excessive amounts of the characteristic in question (as is usually the case with the term, “infinite”). In order to describe what “infinitude” is typically meant to express, we will use the phrases “limitlessness” or “without bound”, which should both rightly be understood in the sense of “never-ending process” rather than in the sense of “enormous scale”. In order to refer to finite objects that have the characteristic of [relative] enormity, we will use phrasings like, “of arbitrarily large magnitude”.

We will take the phrase self-limited to mean any mathematical object that, qua its own dimensional context, is without boundaries. This is to say that a circle, in terms of the one-dimensional entity that constitutes it, is limited only by its own nature, and that the term, “boundary”, can only be applied to it when considering the nature of the two-dimensional plane that it surrounds. Self-limitation, together with the equidistance of several parts from an original point (i.e., the “origin”), are the necessary characteristics of any possible sphere. Let us now distinguish between an onto-logical sphere, as an unrepresentable (i.e. purely subjective) object that exhibits constant curvature, and a logical sphere, as a mathematically defined, many-faceted (where a “facet” can be understood as an n-dimensional, limited form) object whose facets are of identical form and size. In aid of clarity, we will simply denote the various possible dimensionalities of spheres with a prefixed number, such that:

-A 0-sphere is an object that circumscribes a line
-A 1-sphere is an object that circumscribes a plane
-A 2-sphere is an object that circumscribes a volume
-An n-sphere is the general term for an object that circumscribes an n+1 dimensional space

We will use the term structure to refer to the local rigidity (e.g. the joints of a scaffolding) that allows Euclidean space to retain its integrity. The term form, however, will denote a complete mathematical object, structured by Euclidean space, that is necessarily limited, regardless of whether this limitation is self-imposed.

Any sphere is, by definition, concave. By this term, we just mean that there exists only a singular endpoint on the line of any vector that begins from the original point and also that the endpoints of the vectors must be joined in angular sequence. We can now take the term, spheroid simply to refer to any self-limited, concave object.

We shall now explore the conceptual distinction between the point-set, or Cartesian, notation of mathematical objects and the vector-sequential, or polar, notation. In the case of the former, there is not given to us any necessary order that determines how the points are to be connected. It is simply given to us that there exists an n-tuple of points, and the way that the final object is constructed is left to the common sense of the interpreting audience. And while it is possible to speak in terms of an ordered n-tuple of points, we must understand that the language of succession is not native to the 2±dimensional Euclidean spatial structure. By thinking in terms of polarity, however, it is automatically understood that directionality is an integral part of the logical structure of space. But even though Euclidean space should not necessarily be taken as the proper choice when it comes to the immediate representation of mathematical objects, it should be apparent to us that it is nonetheless the only possible starting point when it comes to developing an idea of variable dimensionality, through which radially defined objects can be dependably classified.

As a final point of order, it is crucial to realize the distinction between analytical mathematics—which primarily makes use of the Cartesian system during the course of its purely logical investigations—and constructive mathematics—which is concerned with how the onto-logical forms that are immediately presented to our sense organs can be most efficiently represented. Or more to the point, analysis is an attempt to isolate a certain fact from an otherwise chaotic onto-logical system; whereas the endgame of construction is to deduce, from first logical principles, the necessary conditions that allow such a system to be possible.

From all of the above, we have attempted to comprehend a notion of natural constructive mathematics, with the descriptive, “natural”, meant to denote the set of conditions that forces any theoretical system to be unambiguous in terms of the essential form of its domain (i.e. self-limiting finitude) and the method whereby this domain is investigated (i.e. radial directionality). In the sequel, we will take our first steps towards the derivation of a mathematical object that can be understood as a “universal form”, and furthermore lay the groundwork for the manner in which certain essential questions may be more properly framed.

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That’s just silly.