EXTRAVAGANT
CONCEPTS (A TENTATIVE START)
by F Richard Singer III edition date 11/07/07
website: www.conceptualstudy.org
email: richardsinger3@sbcglobal.net
SECTION 0 INTRODUCTION
Perspective: Being told that God could do anything, a child asked if God could make a rock so heavy that he could not lift it. One theological answer is that he could, but if he did, he would still be able to lift it. This paradoxical answer may indicate theological awe at the mystery of omnipotence, but it probably would not satisfy a child. I would simply say that doing anything just means doing anything we can imagine as possible. My concept of an omnipotent goes beyond this simple response, but it is still a fairly mundane extravagant concept.
For X to be an omnipotent being, X must have the ability to transform any state of affairs we can imagine as possible to any other state of affairs we can imagine as possible. X would also be able to bring about an unlimited number of states of affairs that we might never imagine as possible.
As conceptualized, omnipotence still points beyond what I can imagine, but in an open ended manner that does not seem paradoxical. However it is not a concept that I encounter very often, since I only use it is when communicating with someone who think of God as omnipotent. I mention it to illustrate the idea of an extravagant concept and to illustrate a way to avoid any seeming paradox.
In general, an extravagant concept for a person P points beyond what P can imagine. The word used in connection with the concept often has emphatic as well as cognitive purposes, suggesting an absence of any limitations. They are also used in a way that would seem to give closure to something which has no limitations.. Words used in some contexts that may suggest extravagant concepts include: absolute, omniscient, impossible, eternal, indestructible, immovable, indisputable, perfect, hopeless, worthless, inflexible, irrefutable, irresponsible. Terminology alone is not sufficient to classify a concept as extravagant. For example, there is nothing extravagant about the absolute value concept of algebra.
Some Background Concepts: The concepts sketched below are developed in more detail in the paper Propositions and Queries. A personalized statement IS any instance of a declarative sentence that is used, or at least appears to be used, with the intent to propose or focus on information. A personalized proposition IS a statement in which such information is clear enough for the purposes at hand. It is common for a statement by one person to be received by others giving rise to personalized statements for each of them. The result S of this IS a shared statement, and S IS a shared proposition to the extent that the information in S is clear and is essentially the same for both. The word ‘proposition’ may also refer to either the personalized information or shared information being proposed.
A personalized question IS any individualized instance of an interrogative sentence that is used, or at least appears to be used, with intent to request information. A personalized query IS a question requesting information, where the person knows what type of information would satisfy the request, at least clearly enough for the purposes at hand. The concept of shared applies to questions and queries in the same manner as it applies to statements and propositions. For a simple query, the type of information requested can easily be given by a single proposition. For more complicated queries it is usually more convenient to provide the information by using a number of propositions, altho in principle we could always conjoin these to form a single proposition. Thus I conceptualize appropriate answers to queries as propositions. A more detailed analysis of this concept is given in Section 2.
Overview: Section 1 focuses on the concept of absolute certainty. Some of the concepts developed there resemble concepts more fully developed in my paper Plausibility Concepts. Section 2 is written with no serious purpose in mind, other than to play with some extravagant concepts, and to suggest a paradigm case formulation of the concept of an extravagant concept. Section 3 is about extravagant concepts in mathematics, indicting how the concept of the set of all sets caused a temporary problem in the foundations of mathematics. These sections are independent of each other. They can be read in any order.
SECTION 1 ABSOLUTE CERTAINTY
In a 10/17/98 FOM mail list communication Harvey Friedman asks about absolute certainty.
Q: Is there a concept of absolute
certainty, beyond which one cannot go?
I do not know if the question sent is a query for him, but the one I received was too vague to be a query. I posed Q to a several mathematically competent persons. To enhance clarity, I also gave them the further questions he asks in connection with Q. Even so none of them found Q to be a query. I must admit before starting that my interest in his question and in these related questions per se is minimal. However focusing on them is allows me to play with and exemplify some of the conceptual tools I enjoy using.
- Is absolute certainty ever realized in mathematics?
- If it is sometime realized in mathematics, then where in mathematics is it realized?
- Do you know of a candidate for a piece of mathematics that is not absolutely certain?
- If so, what is the oldest and/or most elementary example of a piece of mathematics that is not absolutely certain?
- Is the "fact" that there is no one-one correspondence between {1,2,3} and {1,2,3,4} absolutely certain?
- Is the "fact" that,
for all positive integers n, there is no one-one correspondence between
{1,¼ ,n} and
1,¼ ,n+1} absolutely certain?
The first problem I found with this question was the concept of certainty. I use the word certain as a relation between a person and a proposition rather than as an attribute of a proposition. The persons I asked about Q also used this relational concept of certainty. For example, a theist might say he is certain that God exists. I might ask him if he is absolutely certain, but in a sense this seems redundant, since normally the word certain alone indicates a complete lack of doubt. However asking if he is absolutely certain, gives an emphasis which leaves room for a revision. A response of almost certain would use the word certain for a concept that is not absolute. Certainty can also change, a person may have doubts about something which at other times seems absolutely certain.
In My Net for Understanding, the primary concept of absolute certainty is a relation involving a person P, an instance in time, and a proposition.
A person P is absolutely certain
of a proposition at some point in time
MEANS
At that time P believes this
proposition and even if seriously challenged admits no doubts
This concept of absolute certainty is not directly applicable to Q, since the intent of this question seems to be independent of time and person. Absolute certainty seems to be intended as an attribute of a proposition. Later I will see if I can modify my relational concept of absolute certainty to obtain a useful attribute concept, but first let me use the relational concept and indicate why even with this concept I do not yet find Q to be a query. One minor problem is to interpret the phrase ‘is there a concept of’. A greater difficulty is indicated by the phrase ‘beyond which one cannot go’.
A concept IS a state within some net N and asking about a concept in relation to a net might have various intents. P might ask if there is a concept X in N, where X is some partially descriptive phrase indicating the concept and P’s expectations. P might also be asking if N had the potential to formulate X in a way that was a fairly conservative extension of N.
Example: I encountered a diophantine equation that suggested a concept that I called consecutive factorial sums. Consider the question "Is there a concept of consecutive factorial sums?" Defining consecutive factorial sums as a pair of natural numbers of the form [k!+ j!, (k+ 1)!+ (j+ 1)!], my net for number theory now has such a concept, but the standard net for number theory does not. This involves interpreting the question as a query about the actual state these nets. Interpreting the question as a query about the potential of the standard net, the answer is that a very minor extension that net would include such a concept. In this sense there is such a concept for the standard net.
In the case of Q the phrase X that indicates the concept and expectations is ‘absolute certainty, beyond which one cannot go’. Using Q to ask about the actual status of X, I would need more clarity about X in order to receive Q as a query. For any reasonable clarification of X that I can currently imagine, this concept does not already exist in any net I use. Since I find much of PNCP (our public net for contemporary philosophy) extremely vague, it is hard for me to judge if X already exists in this net, but I would be surprised if it did. I suspect Q was sent to ask about the potential of PNCP to formulate X rather than to ask if such a concept already exists in PNCP. Without considerable effort I do not know how to interpret this as a query. Being much more familiar with MNP (my net for Philosophy), I turn to a clarification of X that allows Q to be used as a query. I begin with the relational concept indicated earlier. Since the query I formulate is not likely to be anything intended by Q, this may seem like a way to avoid Q rather than clarify it. This is not my purpose. Instead it is a prelude to an attempt to formulate queries that might fit with the intent behind Q.
I shall first interpret ‘beyond which one cannot go’ as meaning beyond which I cannot go in a way that would serve the ordinary use I might make of the concept of absolute certainty. With this interpretation Q is query for me, and I would currently answer Q by yes. Using absolutely certain as admitting of no doubts regardless of being seriously challenged, I cannot imagine going beyond this for any purposes for which I would use this concept. I am absolutely certain about that a large number of the mathematical proofs that I have encountered are correct.
To use (2) and (4) as a queries I would need to decide what I meant ‘where in mathematics’ and by ‘oldest and/or most elementary example of a piece of mathematics’. All the other questions are queries to which I would answer yes. In answering yes to (6) I have no doubts that this is a correct conceptual result in PNCM and that I could also deduce such a result in any reasonable net involving such concepts. In respect to (3) I would say that while I am fairly certain that there is no way to color every possible map with 4 colors, I am not absolutely certain of this. My doubt depends on the elaborate nature of the proof, and while I trust the competence of those who have examined it I cannot be absolutely certain they did not make any error. Part of my doubt arises from hearing that part of the proof involved using a computer program, and have seldom been able to say without doubt that any elaborate program contains no bugs. Another result that I find almost certain, rather than absolutely certain, is the consistency of first order number theory. I have formulated a version of a proof, and I have no doubts that my proof is correct. Of course since the proof is given in set theory, its application to number theory depends on the adequacy of set theory and the way number theory is modeled in set theory. Since the consistency of set theory is less certain than the consistency of number I cannot regard this proof as establishing the consistency of number theory beyond any reasonable doubt.
Suppose I do not restrict the idea of ‘beyond which one cannot go’ by the qualification about ordinary use of a concept of absolute certainty. After all being an extravagant concept indicates that limitations of this type are unlikely to meet the expectations for the concept. Allowing for a wide-open idea of ‘beyond which one cannot go’, I have no concept of absolute certainty beyond which I cannot go. Consider the claim that there is no a one to one correspondence between {1} and {1,2}. Altho I have no ordinary doubts, There is no limit on the extraordinary doubts I can imagine. Suppose that I am mistaken about the basic reliable knowledge I have about my ability to understand mathematical concepts because a supernatural being has placed a curse on me.
My first interpretation used certainty as an attitude of an individual without regard to whether the attitude was warranted or whether the belief involved was correct. Since it is easy to imagine someone being absolutely certain about something without sufficient evidence, this interpretation would not fit the intent of Q. It would be more appropriate to treat certainty as a warranted abstract attitude towards some claim that was true. To use Q as a query in this sense I would need to clarify what I meant by an abstract attitude being warranted. As a start I consider the concept of being inter-subjectively warranted.
An abstract attitude of belief towards X is
inter-subjectively warranted to the extent that the persons most qualified to
evaluate X would consider this attitude as appropriate if they knew what they
could reasonably know about X.
I AM UNCERTAIN OF WHERE I AM GOING
FROM HERE
Since being warranted is a matter of
SECTION 2 MISCELLANEOUS EXTRAVAGANT
CONCEPTS
immutable, irresistible, indestructible, immovable
the whole truth and nothing but the truth
absolute value
Perfection and Forever: It has
been said that Dante Inferno seem more realistic then his
Forever: My problem The problem is the ‘forever’. Physical pleasure can pale even after a short time. Lasting pain just tends to get worse. This is not the main problem however. Heaven must involve spiritual growth. Heaven could be would be an unlimited time in which to develop infinite understanding, to become infinitely creative, to become and become and become. The vision of this is more intense that even the vision of unlimited torment. However I do not expect to bring such a vision into focus. That is what makes it so significant. I want no vision of perfection, for perfection implies no where left to go. In fact I find the concept of an absolute state of perfection to vague to be of any utility It would mean a state so good that no better state is possible. Why would we need such a concept to think about heaven. We certainly need no concept of a state so bad that no state can be worse to think about hell. I am not surprised that I find the concept of absolute perfection incoherent. This is like the concept of a set so large that no set can be larger. At one time it was thought that the set of all sets would be such a set. This lead to a contradiction, and we found that our intuitive concept of the set of all sets could not be precisely formulated However ordinary concepts of perfection are fairly clear, and can be somewhat useful. Someone can bowl a perfect game, and thinking of it that way can add spice to the activity. Furthermore after the game is over this does not leave one with nowhere to go.
Paradigm Case of an Extravagant Concept: Since concept of an extravagant concept is too complex to be formulated by using an analytic definition, I will use a paradigm case formulation. In this formulation E denotes of an extravagant concept, W denotes the word or phrase commonly used for E, and S denotes some set that must be implicitly imagined in order to understand E.
Q The literal meaning of W leads to a contradiction.
(2) W has an emphatic as well as a cognitive purposes, suggesting an inability to go beyond E.
(3) S is partially ordered and with a maximal element e.
(4) Neither S nor e can be effectively described in a finite manner.
(5) Each element of S except e can be effectively described as an actual state.
(6) Understanding E depends directly on e.
Allowable transformations
A. Change ‘leads to a contradiction’ to ‘leads to a paradox’.
B. Change ‘leads to a contradiction’ to ‘suggests something that staggers the imagination’.
C. Change ‘maximal element’ to ‘minimal element’.
Note: The concept of a partially ordered set includes the concept of a totally ordered set as a special case.
Example (Omnipotence): The set S of states needed to imagine omnipotence includes is the set of all possible abilities. A set of abilities is greater than another set
Below are a few examples of abilities:
x: to lift a 150 lb. rock y: to lift a 9 lb. rock z: to play center in the NFL
Altho x as greater than y, I do not regard x as either greater than or equal to or less than z.
However a being A whose powers include only one of x or z is less powerful than a being B whose powers include all the powers of A along with both x and z Using the concept of possible indicated in the next example, I see no way to conceptualize the set of possible abilities as finite. In fact, using only rational number there is no end to the number of abilities one could insert between x and y. I also ??
Example (Impossible): A state of affairs IS possible if it has actually occurred or if it will occur sometime in the future. Some might
Appendix A: Modeling Extravagant
Concepts in Set Theory
Appendix B: The Set of All Sets