AJ:In other words, at a sufficiently deep level of analysis, the only thing we can meaningfully intend when we say "X is true" is "X follows from the premises." (Note what this does to Goedel's incompleteness thereom!)
A small aside question:
Let X be any statement-interpretation. Do you believe that if X follows from your premises, then X?
Zavoi:
The final "...then X" in your statement is shorthand in logical/mathematical parlance for "X is true," is it not? So to me that statement says, "If X follows from my premises, X is true." So no, I am not saying that. I am saying that "X follows from my premises" is the only meaningful/coherent (and non-trivial) way to interpret the statement "X is true."
Note: I am essentially arguing for a first-person version of the coherence theory of truth. The non-first-person version discussed in the link would render "X is true" as really meaning "X follows from the premises," whereas the first-person version I believe is the only appropriate one at this level of analysis specifies "X follows from my premises."
The standard view most people seem to have is based on the correspondence theory of truth, in which statements of "truth" are rendered as "X corresponds with reality" or "X is a matter of fact." I believe the correspondence theory of truth to be incoherent (pun only partially unintended) at sufficiently deep levels of analysis (such as the level we are now discussing), although it is generally useful and convenient for everyday discussion.
Why anarchy fails
AJ:Note: I am essentially arguing for a first-person version of the coherence theory of truth. The non-first-person version discussed in the link would be "X follows from the premises," whereas the first-person version I believe is the only appropriate one at this level of analysis specifies "X follows from my premises."
http://en.wikipedia.org/wiki/Private_language_argument
"I'm not a fan of Murray Rothbard." -- David D. Friedman
Thanks for the interesting link, Neoclassical.
I am not saying that "X follows from my premises" is a private thought or private language. I don't believe we think in words, so I don't believe we think that sentence when we draw such a conclusion. I am just writing that in words because, well, I have to explain the idea here somehow.
Could you just say a word about what relation you believe the link has to the passage you quoted?
AJ:The final "...then X" in your statement is shorthand in logical/mathematical parlance for "X is true," is it not?
On the contrary, I avoided saying "...is true" in my statement because "X is true" is just shorthand for "X." The only reason why we have the word "true" at all is so that such sentences can be made grammatically correct. Debating the meaning of "truth" is like insisting that we define "it" in "It is raining." See the deflationary theory of truth. Although, that is not so much a theory of truth as it is an anti-theory-of-truth theory of truth.
AJ:I am saying that "X follows from my premises" is the only meaningful/coherent (and non-trivial) way to interpret the statement "X is true."
Strictly speaking, we still have, at least from your perspective, "X is true" ↔ "X follows from my premises", which holds by your definition.
But more to the point, to put it in more concrete terms: Suppose upon reflection you realized that "Snow is white" follows from your premises. Would you become convinced that snow is white? If you realized that "Santa Claus exists" follows from your premises, would you become convinced that Santa Claus exists? And can we say the same substituting "Snow is white"/"Santa Claus exists" for any other statement (or statement-interpretation, if you prefer)?
Zavoi:But more to the point, to put it in more concrete terms: Suppose upon reflection you realized that "Snow is white" follows from your premises. Would you become convinced that snow is white? If you realized that "Santa Claus exists" follows from your premises, would you become convinced that Santa Claus exists? And can we say the same substituting "Snow is white"/"Santa Claus exists" for any other statement (or statement-interpretation, if you prefer)?
Exactly why the private language argument is relevant.
AJ, do you believe insane people arrive at truth, simply because their beliefs occur within their own first-person subjectivity?
Zavoi: AJ:The final "...then X" in your statement is shorthand in logical/mathematical parlance for "X is true," is it not? On the contrary, I avoided saying "...is true" in my statement because "X is true" is just shorthand for "X." The only reason why we have the word "true" at all is so that such sentences can be made grammatically correct. Debating the meaning of "truth" is like insisting that we define "it" in "It is raining." See the deflationary theory of truth. Although, that is not so much a theory of truth as it is an anti-theory-of-truth theory of truth.
Right, and my conception is deflationary in that sense of being an anti-theory-of-truth. Note that I'm not debating the meaning of "truth"; I'm recommending that we avoid it at this level of analysis in favor of more useful terms. I didn't mention the deflationary theory of truth above, not because my conception isn't deflationary in that sense, but because there are many other deflationary theories of truth that I want nothing to do with.
ETA: I believe the reason deflationary theories of truth came about is because of a deeper problem: verbal propositions are not evaluable as notions, only interpretations are. Once we ditch the impossible endeavor of trying to analyze verbal propositions as verbal propositions (OK at shallow levels of analysis, problematic at deep levels), none of these "theories of truth" or even "anti-theories-of-truth" will be needed, because the word itself is an artifact of language.
Zavoi: AJ:I am saying that "X follows from my premises" is the only meaningful/coherent (and non-trivial) way to interpret the statement "X is true." Strictly speaking, we still have, at least from your perspective, "X is true" ↔ "X follows from my premises", which holds by your definition.
I'm not defining the word true, I'm recommending that we get rid of it entirely (at least at this level of analysis).
Yes, that's generally what it means for people to convince themselves of things: to realize (correctly or incorrectly) that those things follow from their premises.
Neoclassical, you're have to clarify. I'm not talking about a private language as it's defined in the link.
Also, given what I wrote above, yes, provided their logical reasoning faculties are intact, insane people indeed arrive at conclusions that follow from their premises (can we stop using the word "truth" in this discussion?). Not "truth" - as I deem that term meaningless unless it means "follows from my premises." Their premises probably include things like, "The voices in my head must be trusted!" And of course they can make logical errors just like the rest of us, concluding things that also don't actually follow from their premises.
AJ: Zavoi:But more to the point, to put it in more concrete terms: Suppose upon reflection you realized that "Snow is white" follows from your premises. Would you become convinced that snow is white? If you realized that "Santa Claus exists" follows from your premises, would you become convinced that Santa Claus exists? And can we say the same substituting "Snow is white"/"Santa Claus exists" for any other statement (or statement-interpretation, if you prefer)? Yes, that's generally what it means for people to convince themselves of things: to realize (correctly or incorrectly) that those things follow from their premises.
There we go. You have just asserted the completeness of your own axiomatic system. And that makes you certifiably insane.
Methinks you misunderstand: I'm simply clarifying definitions. Do you believe anything that doesn't follow from your premises, Zavoi?
EDIT: Ah, I see what you're probably thinking. Notice I didn't say you couldn't revise your premises if they lead to silly things...but note how the process works: "Santa Claus exists" would not necessarily seem silly at all to someone for whom it actually followed from their premises. It only seems silly to us because it (radically) doesn't follow from ours. More likely, someone will find that "Santa Claus exists" follows from their premises, but later sensory input (new data! new premises!), if they bother to consider their old beliefs in light of the new data, will result in something different following from their premises. So it's nothing too insane in that respect anyway =)
Regarding the Liar's Paradox, Curry's Paradox, and Godel's Incompleteness Theorems, you'll find that an entirely visual conception of thought, for instance, undoes most or all of their significance. These paradoxes rely on the notion that words and symbols are evaluable rather than their actual mental interpretations. Hence these cannot be used as evidence against my conception, which rejects that notion. I realize this seems pretty out there, and if you don't want to look at it, that's fine.
The base question underlying "all of this" (including the link you gave) is whether words and symbols are evaluable, or whether only mental interpretations of them are. If you'd like to discuss that, I would be glad to.
AJ:Do you believe anything that doesn't follow from your premises, Zavoi?
No, unless my deductive process is mistaken. But I can still entertain the notion that some of my beliefs may be incorrect, and this notion is not obviously contradictory. Just because I can’t understand the limit of my understanding, that doesn’t mean that I can’t understand in the abstract that my understanding is limited.
AJ:EDIT: Ah, I see what you're probably thinking. Notice I didn't say you couldn't revise your premises if they lead to silly things...but note how the process works: "Santa Claus exists" would not necessarily seem silly at all to someone for whom it actually followed from their premises. It only seems silly to us because it (radically) doesn't follow from ours.
It’s not that “Santa Claus exists” is a universal reductio ad absurdum to bring down any belief system. After all, it could be us that are mistaken. But you have stated that Santa Claus exists if it turns out that the belief “Santa Claus exists” pops out from your premises; and furthermore, that very statement itself pops out from your premises. Therefore, according to the theorem, upon reflection you will realize that you already (i.e., not just in some hypothetical world, but right now, from your current premises) believe that Santa Claus exists – and any other statement, for that matter. P⊢(P⊢S → S) → P⊢S.
AJ:The base question underlying "all of this" (including the link you gave) is whether words and symbols are evaluable, or whether only mental interpretations of them are. If you'd like to discuss that, I would be glad to.
I don’t want to debate this issue too protractedly, because it’s so far removed from any political implications, and this is, after all, a politics forum. But I’ll bite a little bit.
Is your problem with the part where they go
If the result of substituting “If the result of substituting x for ‘x’ in x is provable in PA, then C” for ‘x’ in “If the result of substituting x for ‘x’ in x is provable in PA, then C” is provable in PA, then C.
?
Zavoi:But you have stated that Santa Claus exists if it turns out that the belief “Santa Claus exists” pops out from your premises
I'm saying there's nothing I can usefully consider other than my belief, at a sufficiently deep level of analysis. In other words, I'm speaking from an agnostic stance about the existence of the real world (which changes the meaning of "exist" in the sentence as well, so it'd be something more like, "It follows from my premises that it's useful to believe the perception that I might, for instance, one day be able to meet Santa Claus or to get presents from him if I'm good.").
If you'd like, lets talk about the liar's paradox "This sentence is false" that Goedel (which I believe Lob's theorem requires) is essentially a formalization of, as that will be a lot simpler to discuss and illustrate the difficulty I see. Only thing is, I am saying, as above, that we can only coherently talk about what follows and doesn't follow from the premises. So the sentence becomes "This sentence doesn't follow from the premises." (Or in the deflationary theory perhaps "Not- this sentence.")
Even in its original conception, though, my main concern here is that words are tools to express thoughts, yet it's not altogether clear that it's possible to think this thought (or a thought corresponding to a Goedel sentence for that matter), and certainly no one can show or explain what this thought would be (at least I haven't seen clear to imagine this alleged thought).
I'm saying we can't evaluate sentences, only sentence-interpretations (thoughts), but does anyone really interpret this sentence completely into thoughts before evaluating it, as they would with "Cats are carnivores"? No, they seem to analyze it entirely formally, as if the words and the thought are equivalent (or in fact that "thought" is not even relevant). This seems a case of the tail wagging the dog. I suspect this is because many have believed that we think, or at least logically reason, in words. I completely reject this notion that it's even possible to think fundamentally in words, although they certainly play a useful role in our thought process. Since I reject "thoughts=words," I find both the natural language analysis of the liar paradox and the formal symbolic analysis of Goedel sentences to be hollow and trivial, if not outright nonsense.
If you disagree with me that we don't think in words, that would be a better thing to debate - and that would also have more relevance to, for example, praxeology. (If we don't agree on this, I suspect nothing else on the above will work as a discussion.)
AJ:I'm saying there's nothing I can usefully consider other than my belief, at a sufficiently deep level of analysis. In other words, I'm speaking from an agnostic stance about the existence of the real world (which changes the meaning of "exist" in the sentence as well, so it'd be something more like, "It follows from my premises that it's useful to believe the perception that I might, for instance, one day be able to meet Santa Claus or to get presents from him if I'm good.").
The statement doesn’t have to be about Santa Claus; it could be about mathematics, or even your own mind.
If “X” is shorthand for “X is true,” and “X is true” can only be meaningfully interpreted as “X follows from my premises,” then, in your view, “X” is logically equivalent to “X follows from my premises.” Isn’t this what you’ve been saying?
AJ:If you'd like, lets talk about the liar's paradox "This sentence is false" that Goedel (which I believe Lob's theorem requires) is essentially a formalization of, as that will be a lot simpler to discuss and illustrate the difficulty I see. Only thing is, I am saying, as above, that we can only coherently talk about what follows and doesn't follow from the premises. So the sentence becomes "This sentence doesn't follow from the premises." (Or in the deflationary theory perhaps "Not- this sentence.")
Firstly, Gödel’s statement is not “This statement is false,” but “This statement is not provable.” So, in fact, Gödel’s sentence is precisely what you just said: “This sentence doesn’t follow from the premises.” Indeed, if you try to do this with “true” rather than “provable,” then you end up with a contradiction – which shows that you cannot define truth without resorting to a meta-language (Tarski’s theorem). Hence the need to discard “truth” as an attribute.
But Gödel’s theorem (and Löb’s theorem) does not require a definition of truth; they only require a definition of provability, which you seem willing to accept as a coherent, definable notion.
AJ:Even in its original conception, though, my main concern here is that words are tools to express thoughts, yet it's not altogether clear that it's possible to think this thought (or a thought corresponding to a Goedel sentence for that matter), and certainly no one can show or explain what this thought would be (at least I haven't seen clear to imagine this alleged thought).
A Gödel sentence is ultimately just a statement about numbers; although it may be too long and complicated for a person to understand, it is in principle understandable.
AJ:I'm saying we can't evaluate sentences, only sentence-interpretations (thoughts), but does anyone really interpret this sentence completely into thoughts before evaluating it, as they would with "Cats are carnivores"? No, they seem to analyze it entirely formally, as if the words and the thought are equivalent (or in fact that "thought" is not even relevant). This seems a case of the tail wagging the dog. I suspect this is because many have believed that we think, or at least logically reason, in words. I completely reject this notion that it's even possible to think fundamentally in words, although they certainly play a useful role in our thought process. Since I reject "thoughts=words," I find both the natural language analysis of the liar paradox and the formal symbolic analysis of Goedel sentences to be hollow and trivial, if not outright nonsense.
Of course we must use words to talk about thoughts, but that doesn’t mean that the theorems apply only to words and not to thoughts.
The syntactic substitution operation (e.g., what goes on in “The result of substituting ‘Cats’ for ‘X’ in ‘X are carnivores’”) is just a convenient way of talking about (via isomorphism) a particular mental operation. You have a mental concept of “cats,” and a mental concept of “being carnivores,” and you combine the two concepts to see what you get. For example, if you have a bunch of animals and some meat to feed them, you ask yourself (in thoughts, not in words) “Which of these animals are carnivores?” You answer this question (again in thoughts, not words) by substituting each animal-concept into the being-carnivore-concept.
When we perform the clever syntactic “quining” maneuver to get a sentence to refer to itself, we are attempting to evoke an analogous thought that refers to itself, via an analogous mental substitution operation:
Santa Claus exists if the result of substituting the “Santa Claus exists if the result of substituting the concept into itself is provable”-concept into the “Santa Claus exists if the result of substituting the concept into itself is provable”-concept is provable.
The letters on your screen are not really the object of inquiry; they are merely a guide towards it.
AJ:If you disagree with me that we don't think in words, that would be a better thing to debate - and that would also have more relevance to, for example, praxeology. (If we don't agree on this, I suspect nothing else on the above will work as a discussion.)
I do agree with this. I’m sure we’ve all had the experience of having trouble expressing our thoughts.
Zavoi: So, in fact, Gödel’s sentence is precisely what you just said: “This sentence doesn’t follow from the premises.”
So, in fact, Gödel’s sentence is precisely what you just said: “This sentence doesn’t follow from the premises.”
Yes. (Or if we end up needing to use the Goedel sentence, G = "G doesn't follow from the premises of the theory T.") Now above you wrote:
"X is true" is just shorthand for "X."
When you wrote that, I assume you meant as an underlying assumption that "truth is a norm of assertion." Or in other words, there's no need for the word "true" except as a grammatical device or clarifier, because it is inherent in the act of uttering a statement as a statement that it is purported by the speaker to be accurate.
Now it still seems there should be no objection to including the word "true" or similar terms in some instances, as long as it is fully understood by all parties that this is nothing more than a clarifying device, and adds no actual information or meaning. Hence we can say "X follows from the premises" is just a roundabout way of saying "X". Please stop me here if this is incorrect.
If you agree with the above, it seems that "This statement doesn't follow from the premises" can be rewritten in longhand form as "That this statement doesn't follow from the premises, follows from the premises." Converting, we get: "This statement follows from the premises, and this statement doesn't follow from the premises." However, this is just a garden variety contraction.
Possible objection: I made the above substitution with the promise that the change could not be construed as adding any actual information or meaning, just clarification, and yet now the result is that a Goedel sentence is merely a trivial self-contradictory statement. Therefore I must have added actual information or meaning after all, because the conclusion reached has changed.
Answer: The issue of whether Goedel sentences are trivial or not is the very issue in question. Worded more relevantly, the issue of whether the Goedel sentence will turn out to be trivial once clarified is the very matter in question. Hence even if the above substitution is only clarifying and does not add any actual information or meaning, it can still change our conclusion as to the triviality of Goedel sentences if we weren't seeing the matter clearly before.
Now I can imagine that the above might be wrong. Even if so, I believe the following criticism of Goedel sentences is correct:
Zavoi:A Gödel sentence is ultimately just a statement about numbers; although it may be too long and complicated for a person to understand, it is in principle understandable.
Can you truly conceptualize it, other than formally? Strange as it may sound, I am saying that it is not possible (or no one ever seems to explain how it is possible) to think the thought that corresponds to "This sentence doesn't follow from the premises."
Unless I'm forgetting some other odd sentence forms, all other types of sentences that are well-formed and non-contradictory seem to be conceptualizable in some not-purely-formal way, at least to anyone who claims to understand them. Since this is the case for all other types of sentences, the underlying obvious fact that "words are just devices for communicating thoughts" is normally not referenced at all, as it simply doesn't matter because we are otherwise always dealing with statements whose thought correspondents are clear to all parties. It normally does not need to be pointed out.
However, in these rare cases where the thought correspondent of the statement is not obvious, it seems the burden is on the speaker to clarify or add some explanatory remarks as to how to form or identify the corresponding thought, since the thought is tacitly purported to be both actually thinkable and obvious from the words as stated. (If you say the same applies to concepts like Riemann's "point at infinity" or just the idea of infinity itself, I would agree - what I am saying may support mathematical constructivism/intuitionism to some degree or other.)
Zavoi:When we perform the clever syntactic “quining” maneuver to get a sentence to refer to itself, we are attempting to evoke an analogous thought that refers to itself, via an analogous mental substitution operation: Santa Claus exists if the result of substituting the “Santa Claus exists if the result of substituting the concept into itself is provable”-concept into the “Santa Claus exists if the result of substituting the concept into itself is provable”-concept is provable.
I am very glad you see this, Zavoi. This is more or less where I'm at. And here is my central claim: a thought cannot refer to itself, or if it can, no one seems to be able to tell me how. For example, the first obvious way in my mind, since I seem to think primarily in pictures, would be to conceptualize it as follows:
Note this is an infinite regress in pictures with no end in either direction. There is no defined point of evaluation, as far as I can tell, even if we can really say it's possible to think it at all (depends on whether it's possible to really think "infinity" other than formally).
Of course, a reasonable objection to my posting the above picture is that it's a strawman: "That is not what I meant by 'This sentence does not follow from the premises'." But then I must ask, "What did you mean?"
And naturally, no one said we had to conceptualize it in pictures, but my question remains as far as how is one supposed to conceptualize it. In the absence of guidance, I can only try and fail to interpret these words into thoughts so many ways until I say, "All right, although it may appear to be obvious what you mean by that because it's a well-formed sentence, it's actually not obvious what you mean when I take it to the level of thought. So I can only await clarification." Again, for most statements that I claim to understand, the understanding itself gives the answer to my concern. Here is a rare exception, so it seems the speaker of the statement would need to clarify.
AJ:Or in other words, there's no need for the word "true" except as a grammatical device or clarifier, because it is inherent in the act of uttering a statement as a statement that it is purported by the speaker to be accurate.
Agreed so far…
AJ:Hence we can say "X follows from the premises" is just a roundabout way of saying "X".
Is this your view? Is this your impression of my view? This is the view that I’m “accusing” you of having.
AJ:And here is my central claim: a thought cannot refer to itself, or if it can, no one seems to be able to tell me how.
All right, make yourself comfortable.
Part 1: Mental substitution
Refer to what I wrote earlier about the animal-feeding example. The thought that “The result of substituting ‘Cats’ for ‘X’ in ‘X are carnivores’ follows from my premises” is a thought about a thought, not a thought about a sentence. It only appears to be about a sentence because we have to use sentences to talk about thoughts. We use syntactic substitution as an analogue for mental substitution. But your mind can perfectly well use mental substitution to form thoughts by combining concepts, as you do when you feed the animals.
Part 2: Gödel’s theorem in math
(Part 2 doesn’t depend on Part 1, but Part 3 depends on both.)
Let’s leave minds and thoughts to the side for now and concentrate just on Gödel’s theorem as it applies to math – it will come in handy later. (I don’t know if you know this already, so I’ll be relatively brief.)
Gödel’s goal is to construct a statement asserting “This statement is not provable in Peano Arithmetic.” It’s fairly easy to construct a formula template ¬Provable(#φ) that asserts that the formula φ is not provable. But if you naïvely try to construct the Gödel sentence by replacing φ with “¬Provable(#φ)”, then you end up in an infinite regress, and you can never finish writing the formula.
But Gödel figured out a way around this, by defining a series of formulas:
Notice that “The self-substitution of Penultimate(_)” is just Ultimate() itself, so Ultimate() asserts that Ultimate() is not provable.
This trick is known as diagonalization.
Part 3: Mental self-reference
AJ:For example, the first obvious way in my mind, since I seem to think primarily in pictures, would be to conceptualize it as follows: [Picture] Note this is an infinite regress in pictures with no end in either direction. There is no defined point of evaluation, as far as I can tell, even if we can really say it's possible to think it at all (depends on whether it's possible to really think "infinity" other than formally).
[Picture]
This corresponds to the naïve method of constructing the sentence. I.e.: “The statement ‘The statement “The statement … is not provable” is not provable’ is not provable.” You can never finish making that statement, just as you can never finish thinking that thought.
AJ:Of course, a reasonable objection to my posting the above picture is that it's a strawman: "That is not what I meant by 'This sentence does not follow from the premises'." But then I must ask, "What did you mean?"
We can instead use the diagonalization method to construct the thought. All the ingredients that were required in the math version, are also available in the mental version:
The resulting thought is something like: “The result of self-substituting the ‘The result of self-substituting the _-concept does not follow from the premises’-concept does not follow from the premises.” Does this make sense?
From this, it is a short step to Löb’s theorem.
Unless someone is dying to know the ending, we should move this to PM if this goes on much longer.
Zavoi: Unless someone is dying to know the ending, we should move this to PM if this goes on much longer.
Why would you move it to PM?
If I wrote it more than a few weeks ago, I probably hate it by now.
Because it has nothing to do with politics? But you're a moderator, so I'll defer to your judgment.
Zavoi: AJ:Hence we can say "X follows from the premises" is just a roundabout way of saying "X". Is this your view? Is this your impression of my view? This is the view that I’m “accusing” you of having. That was my impression of your view, or if you don't like "follows from the premises" then just use the word "true" and substitute it above to dissolve the Liar Paradox. In any case, I see that this section may be moot with respect to Goedel sentences because they are not strictly equivalent to the Liar Paradox (although I still think it applies to the philosophical controversy over such paradoxes - Curry's Paradox, etc.). Note: I am not using the "a thought can't refer to itself" argument to prove that my "follows from the premises" argument is correct. I am saying that it is only from the perspective of the former that I can coherently say the latter, so I would like to discuss the former claim irrespective of the latter, as we are doing presently. Zavoi:The thought that “The result of substituting ‘Cats’ for ‘X’ in ‘X are carnivores’ follows from my premises” is a thought about a thought, not a thought about a sentence. I can picture "Unknown thing is a carnivore" (possible mental image: fuzzy/shadowy creature-thing eating meat on a regular basis), but is that really the same as picturing "X is a carnivore"? I ask not because I know the answer off hand - I'll have to think more about this - but to stimulate discussion or other reader's thoughts. Zavoi:But your mind can perfectly well use mental substitution to form thoughts by combining concepts, as you do when you feed the animals. Agreed, as long as by mental substitution you mean replacing the variable with an actual value. Zavoi:Gödel’s goal is to construct a statement asserting “This statement is not provable in Peano Arithmetic.” It’s fairly easy to construct a formula template ¬Provable(#φ) that asserts that the formula φ is not provable. But if you naïvely try to construct the Gödel sentence by replacing φ with “¬Provable(#φ)”, then you end up in an infinite regress, and you can never finish writing the formula. Thank you for writing out the math so carefully, Zavoi. You have saved me some trouble, because I was going off some apparently poorly-worded Wikipedia entries and other sources of a popular understanding of Goedel's theorem. Essentially you are saying that there is another way of conceptualizing Goedel sentences besides the way I diagrammed, and that that way is in fact spelled out in the proof using the diagonalization lemma. While I am still skeptical that a thought can refer to itself no matter what tricks are used (although I do agree that a thought can refer to another thought), I must now turn to consider this new candidate as a possible way of forming a thinkable thought in my mind, and it could take some time for me to assess it in light of my views. This is analogous to my position on objective value and objective ethics: I await a conception that is coherent. Every theory presented must be evaluated anew. Meanwhile, I have another set of questions to throw into the mix for anyone who is interested: Zavoi:Substitute(#φ,w,x) ≡ x is the Gödel number of the formula gotten by substituting “w” in for every blank in the formula φ. The Goedel number #φ is a composite of prime numbers that is effectively a set of "instructions" for getting φ. This might work if we are also assuming as in the Peano axioms that there are an infinite set of natural numbers, then I would turn to the question of conceptualizing infinity. Can we really do it? Is it enough to imagine something increasing without bound? I realize going down that road leads to some inconvenient places like ultrafinitism... You might say, "Why bother (and what does this have to do with political theory)?" To me, these questions relate directly to praxeology. If everything I do is an action in the praxeological sense, that means every thought I choose to think is an attempt to experience more happiness. Or put another way, if I consciously think something (for example if I picture something), it is implicit in that thought that it is potentially useful for me to "act as if" [useful for me to act as if it is the case]. So "X" means "I deem that X is useful to act as if." Of course, anything that follows from a set of thoughts that I deem are useful to act as if (i.e., premises), is itself useful to act as if, at least provisionally [but provisional usefulness is usefulness]. If this seems to contradict what I wrote in previous posts, please suspend judgment as I clarify this more below. Now the human endeavor of doing mathematics (assuming axioms and deducing from them) is also action, as is every individual act of stating/thinking axioms deducing the results that follow. Hence, rather than some ill-defined "truth," it is implicit in each statement I make in doing mathematics that I deem such statement useful to me (useful for me to "act as if"). There can be nothing relevant to any single human action (including a single thought) except its deemed usefulness to the actor (in the broadest sense). This leaves no room for thoughts to be "true," or even "accurate" or "correct" without these words merely being considered as terms referring in one way or other directly and exclusively to the notion of usefulness. The different words merely attempt to indicate the type and scope of such usefulness. Strict mathematical formalism, as I understand it - where symbols are manipulated without checking to make sure they actually correspond to thoughts that are possible to think - seems to turn Mises's conception on its head. I see you are saying that this is not the case with Goedel's theorem, and I can no longer say I await an explanation - now I have to stop and evaluate the explanation you have given. However, there are other concepts in math that some have contested as being more or less "impossible to think" or "pure formalisms with no content." Existence proofs, infinity, proofs by contradiction, etc. This is a minority viewpoint, but it seems to me relevant to consider it in terms of praxeology, as Mises's conception seems to include mathematics as a branch of praxeology. This is not to say we must correct any mistakes in math to further political theory or praxeology as a whole. But rather that if praxeology advances, any mathematical results that seem to contradict it could indeed become relevant to the larger debate over methodology, politics, economics, etc. In other words, I think there is some sense in which praxeologists must eventually take on all these fields at once, lest errors in one established field be held up as evidence against the new praxeological conceptions. Likewise, if errors can be identified in other branches of praxeology that are already established, and these errors can be shown to have arisen from a misunderstanding of the larger field of human action as Mises conceived of it, that would provide a powerful "in" for praxeological ideas. Zavoi:The resulting thought is something like: “The result of self-substituting the ‘The result of self-substituting the _-concept does not follow from the premises’-concept does not follow from the premises.” Does this make sense? Well I'm much more optimistic now that I know you, at least, see things in this way. I have not heard anyone acknowledge the thought aspect of this before. As far as I can tell, most thinkers seem to equate thinking with words, usually with their own pet exceptions, but no one seems to go the full way. Regarding your passage here, as I alluded to above, I am not fully convinced that we can just assume it's possible to conceptualize "_-concept". Perhaps Goedel's result contains an obvious way to do so (I'll need to study it more), or perhaps it relies on another concept - like infinity in the Peano axioms - that is itself unthinkable. Or perhaps I am barking up the wrong tree =) Summing up, I now have something to chew on as a possible conceptualization of Goedel sentences. Re: Loeb's theorem, I don't think it will apply in a non-trivial way to what I have stated above, because I am not creating an axiomatic system; I am experiencing sensations and starting my analysis with an "empirical" model based solely on data. The data are not axiomatic, because they are not assumed but rather directly felt hence it wouldn't even mean anything to "deny" them (so "data" is a slight misnomer). The models I create based on the data (the "real world," "other people," etc.) would be premises of a sort, but to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if." So why all the talk about "X follows from the premises" and now I say something a little different? Am I contradicting myself? No, I am speaking at different levels of analysis, with different pieces of my views included or not included based on where I believe the reader is at. In most cases and at most levels of analysis, I deem X "useful to act as if" because X follows from my premises (or my models, or if you like, my "useful to act as if"s - note how this can include mathematical axioms). One thing I did fail to touch on in previous posts are cases where I speak directly of my private sensations (example: X = "I am in pain."). These can be viewed as exceptions, depending on how things are phrased, or not: it's still the case that, if I am in pain it's useful for me to act as if I'm in pain, albeit perhaps trivially so - it seems to me purely a matter of word usage. Why anarchy fails | Post Points: 20
That was my impression of your view, or if you don't like "follows from the premises" then just use the word "true" and substitute it above to dissolve the Liar Paradox. In any case, I see that this section may be moot with respect to Goedel sentences because they are not strictly equivalent to the Liar Paradox (although I still think it applies to the philosophical controversy over such paradoxes - Curry's Paradox, etc.).
Note: I am not using the "a thought can't refer to itself" argument to prove that my "follows from the premises" argument is correct. I am saying that it is only from the perspective of the former that I can coherently say the latter, so I would like to discuss the former claim irrespective of the latter, as we are doing presently.
Zavoi:The thought that “The result of substituting ‘Cats’ for ‘X’ in ‘X are carnivores’ follows from my premises” is a thought about a thought, not a thought about a sentence.
I can picture "Unknown thing is a carnivore" (possible mental image: fuzzy/shadowy creature-thing eating meat on a regular basis), but is that really the same as picturing "X is a carnivore"? I ask not because I know the answer off hand - I'll have to think more about this - but to stimulate discussion or other reader's thoughts.
Zavoi:But your mind can perfectly well use mental substitution to form thoughts by combining concepts, as you do when you feed the animals.
Agreed, as long as by mental substitution you mean replacing the variable with an actual value.
Zavoi:Gödel’s goal is to construct a statement asserting “This statement is not provable in Peano Arithmetic.” It’s fairly easy to construct a formula template ¬Provable(#φ) that asserts that the formula φ is not provable. But if you naïvely try to construct the Gödel sentence by replacing φ with “¬Provable(#φ)”, then you end up in an infinite regress, and you can never finish writing the formula.
Thank you for writing out the math so carefully, Zavoi. You have saved me some trouble, because I was going off some apparently poorly-worded Wikipedia entries and other sources of a popular understanding of Goedel's theorem. Essentially you are saying that there is another way of conceptualizing Goedel sentences besides the way I diagrammed, and that that way is in fact spelled out in the proof using the diagonalization lemma.
While I am still skeptical that a thought can refer to itself no matter what tricks are used (although I do agree that a thought can refer to another thought), I must now turn to consider this new candidate as a possible way of forming a thinkable thought in my mind, and it could take some time for me to assess it in light of my views. This is analogous to my position on objective value and objective ethics: I await a conception that is coherent. Every theory presented must be evaluated anew.
Meanwhile, I have another set of questions to throw into the mix for anyone who is interested:
Zavoi:Substitute(#φ,w,x) ≡ x is the Gödel number of the formula gotten by substituting “w” in for every blank in the formula φ.
The Goedel number #φ is a composite of prime numbers that is effectively a set of "instructions" for getting φ. This might work if we are also assuming as in the Peano axioms that there are an infinite set of natural numbers, then I would turn to the question of conceptualizing infinity. Can we really do it? Is it enough to imagine something increasing without bound? I realize going down that road leads to some inconvenient places like ultrafinitism...
You might say, "Why bother (and what does this have to do with political theory)?"
To me, these questions relate directly to praxeology. If everything I do is an action in the praxeological sense, that means every thought I choose to think is an attempt to experience more happiness. Or put another way, if I consciously think something (for example if I picture something), it is implicit in that thought that it is potentially useful for me to "act as if" [useful for me to act as if it is the case].
So "X" means "I deem that X is useful to act as if." Of course, anything that follows from a set of thoughts that I deem are useful to act as if (i.e., premises), is itself useful to act as if, at least provisionally [but provisional usefulness is usefulness]. If this seems to contradict what I wrote in previous posts, please suspend judgment as I clarify this more below.
Now the human endeavor of doing mathematics (assuming axioms and deducing from them) is also action, as is every individual act of stating/thinking axioms deducing the results that follow. Hence, rather than some ill-defined "truth," it is implicit in each statement I make in doing mathematics that I deem such statement useful to me (useful for me to "act as if"). There can be nothing relevant to any single human action (including a single thought) except its deemed usefulness to the actor (in the broadest sense). This leaves no room for thoughts to be "true," or even "accurate" or "correct" without these words merely being considered as terms referring in one way or other directly and exclusively to the notion of usefulness. The different words merely attempt to indicate the type and scope of such usefulness.
Strict mathematical formalism, as I understand it - where symbols are manipulated without checking to make sure they actually correspond to thoughts that are possible to think - seems to turn Mises's conception on its head. I see you are saying that this is not the case with Goedel's theorem, and I can no longer say I await an explanation - now I have to stop and evaluate the explanation you have given. However, there are other concepts in math that some have contested as being more or less "impossible to think" or "pure formalisms with no content." Existence proofs, infinity, proofs by contradiction, etc. This is a minority viewpoint, but it seems to me relevant to consider it in terms of praxeology, as Mises's conception seems to include mathematics as a branch of praxeology.
This is not to say we must correct any mistakes in math to further political theory or praxeology as a whole. But rather that if praxeology advances, any mathematical results that seem to contradict it could indeed become relevant to the larger debate over methodology, politics, economics, etc. In other words, I think there is some sense in which praxeologists must eventually take on all these fields at once, lest errors in one established field be held up as evidence against the new praxeological conceptions. Likewise, if errors can be identified in other branches of praxeology that are already established, and these errors can be shown to have arisen from a misunderstanding of the larger field of human action as Mises conceived of it, that would provide a powerful "in" for praxeological ideas.
Zavoi:The resulting thought is something like: “The result of self-substituting the ‘The result of self-substituting the _-concept does not follow from the premises’-concept does not follow from the premises.” Does this make sense?
Well I'm much more optimistic now that I know you, at least, see things in this way. I have not heard anyone acknowledge the thought aspect of this before. As far as I can tell, most thinkers seem to equate thinking with words, usually with their own pet exceptions, but no one seems to go the full way.
Regarding your passage here, as I alluded to above, I am not fully convinced that we can just assume it's possible to conceptualize "_-concept". Perhaps Goedel's result contains an obvious way to do so (I'll need to study it more), or perhaps it relies on another concept - like infinity in the Peano axioms - that is itself unthinkable. Or perhaps I am barking up the wrong tree =)
Summing up, I now have something to chew on as a possible conceptualization of Goedel sentences. Re: Loeb's theorem, I don't think it will apply in a non-trivial way to what I have stated above, because I am not creating an axiomatic system; I am experiencing sensations and starting my analysis with an "empirical" model based solely on data. The data are not axiomatic, because they are not assumed but rather directly felt hence it wouldn't even mean anything to "deny" them (so "data" is a slight misnomer). The models I create based on the data (the "real world," "other people," etc.) would be premises of a sort, but to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if."
So why all the talk about "X follows from the premises" and now I say something a little different? Am I contradicting myself? No, I am speaking at different levels of analysis, with different pieces of my views included or not included based on where I believe the reader is at. In most cases and at most levels of analysis, I deem X "useful to act as if" because X follows from my premises (or my models, or if you like, my "useful to act as if"s - note how this can include mathematical axioms). One thing I did fail to touch on in previous posts are cases where I speak directly of my private sensations (example: X = "I am in pain."). These can be viewed as exceptions, depending on how things are phrased, or not: it's still the case that, if I am in pain it's useful for me to act as if I'm in pain, albeit perhaps trivially so - it seems to me purely a matter of word usage.
Philosophers; we've been babbling incessantly for over 5000years, but have we really said anything? I guess it depends on what you mean by "say." Or what I mean by "mean."
^
That I have "preferences" in some foggy mentalese sense is not at all "self-evident" to me.
So you really are just a robot spamming articles you've not really processed?
Freedom of markets is positively correlated with the degree of evolution in any society...
AJ:That was my impression of your view,
This is not my view. Is it yours?
AJ:or if you don't like "follows from the premises" then just use the word "true" and substitute it above to dissolve the Liar Paradox.
This is closer to my view.
AJ:I can picture "Unknown thing is a carnivore" (possible mental image: fuzzy/shadowy creature-thing eating meat on a regular basis), but is that really the same as picturing "X is a carnivore"?
This is fine, because “X is a carnivore” is not a thought in and of itself; it’s an incomplete concept waiting for something to be dropped into it.
AJ:Agreed, as long as by mental substitution you mean replacing the variable with an actual value.
Yes.
AJ:This might work if we are also assuming as in the Peano axioms that there are an infinite set of natural numbers, then I would turn to the question of conceptualizing infinity. Can we really do it? Is it enough to imagine something increasing without bound? I realize going down that road leads to some inconvenient places like ultrafinitism... … However, there are other concepts in math that some have contested as being more or less "impossible to think" or "pure formalisms with no content." Existence proofs, infinity, proofs by contradiction, etc.
…
However, there are other concepts in math that some have contested as being more or less "impossible to think" or "pure formalisms with no content." Existence proofs, infinity, proofs by contradiction, etc.
Here’s the thing about that: The notion of “follows-from-the-premises” is not finitistically expressible for any system capable of expressing it.
For example, take simple Propositional Calculus (PC), where we go around making statements such as (p∨q)∧¬p → q. There are an infinite number of possible statements in PC, but the matter of their provability is completely finitistic – given any statement in PC, it is possible to algorithmically check whether or not the statement is provable, and we know that this algorithm is guaranteed to terminate after a finite number of steps (e.g., we could test all 2N combinations of values for {p, q, r…} and see if they all work out).
However, PC is far too simple to be able to express within itself the formula “is provable in PC.” At the very least, it would have to be able to talk about an infinite number of objects (e.g., natural numbers), so that all of the (infinite number of) statements in PC can be represented. But now the system is too complicated to express its provability finitistically.
Take the following statement in PA:
∃x(¬∃a∃b(¬∃m∃n(¬(m=S0)∧¬(n=S0)∧m•n=a)∧¬∃m∃n(¬(m=S0)∧¬(n=S0)∧m•n=b)∧a+b=x+x)∧∃y(x=SSy))
In other words, “Goldbach’s conjecture is false. (¬GC)” From a finitistic/constructivist standpoint, it’s tempting to say that ¬GC is not even meaningful: “There exists x? Well what is it?”
But consider the question of whether ¬GC is provable in PA. We could start with the premises of PA, start applying inference rules, and fumble around in the dark until we find a way to derive ¬GC. However, there is no guarantee that this algorithm will terminate: if ¬GC is indeed unprovable, then we will be searching for a proof forever; and what’s worse, we would never know this – we would always think that the proof of ¬GC might be right around the corner. In fact, Gödel’s theorem establishes that no finitistic proof-finding algorithm can exist*: if it did, we could feed Ultimate() into it and thereby prove both Ultimate() and ¬Ultimate().
*(Assuming the system is consistent.)
Therefore, when you think “‘Cats are carnivores’ follows from my premises,” you are thinking a thought that is not finitistically expressible. If you deny your ability to think such thoughts, then you must discard the notion of “follows from my premises,” and this entire discussion becomes moot.
AJ:If everything I do is an action in the praxeological sense, that means every thought I choose to think is an attempt to experience more happiness.
This arguably leads to an infinite regress – if I choose to think X because I think that thinking X will benefit me, why did I choose to think that thinking X will benefit me? Etc… But let’s accept this for now.
AJ:So "X" means "I deem that X is useful to act as if."
That depends on what you mean by “means.”* Do you mean “→” or “↔”? The former is no problem: “If Santa Claus exists, then it’s useful for me to act as if Santa Claus exists.” The latter is, at the very least, strange: “Santa Claus exists if and only if it’s useful for me to act that way.”
*(Yes, it’s come to that. It makes sense in context.)
The problem is even worse when you’re dealing with “follows from my premises”:
(1) is not problematic: “If Santa Claus existed, I would have found out about it by now.” (2) is the sin of map-territory conflation, of whose dire consequences Löb warns us.
Does either (1) or (2) represent your view?
AJ:Hence, rather than some ill-defined "truth," it is implicit in each statement I make in doing mathematics that I deem such statement useful to me (useful for me to "act as if").
Rather, it is implicit in the act of making/believing the statements… Identifying “usefulness” with the statement itself is the same kind of extreme subjectivism that leads to (2) above.
AJ:Re: Loeb's theorem, I don't think it will apply in a non-trivial way to what I have stated above, because I am not creating an axiomatic system; I am experiencing sensations and starting my analysis with an "empirical" model based solely on data. The data are not axiomatic, because they are not assumed but rather directly felt hence it wouldn't even mean anything to "deny" them (so "data" is a slight misnomer).
Earlier you stated:
AJ:But when we really get down to it, both statements are "analytic" for most people, because most people have a set of assumptions that already entails that there have been black dogs in the world. For example, most people have seen black dogs, and one of their assumptions is that what they see (under certain conditions) is "in the world."
I took this to mean that people begin with a set of premises (assumptions, axioms…) and then employ rules of inference (entailment) to derive other beliefs. This is the very essence of an axiomatic system. And you also seem to have claimed that this axiomatic system is powerful enough to express “follows-from-the-premises.” If so, then Gödel, Löb, & friends apply.
AJ:…but to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if."
Again, could you clarify which direction the implication goes in?
Zavoi:This is fine, because “X is a carnivore” is not a thought in and of itself; it’s an incomplete concept waiting for something to be dropped into it.
Looks like we're in agreement here.
Zavoi: Therefore, when you think “‘Cats are carnivores’ follows from my premises,” you are thinking a thought that is not finitistically expressible. If you deny your ability to think such thoughts, then you must discard the notion of “follows from my premises,” and this entire discussion becomes moot.
Not expressible in that "system," which in this case probably means that thought or line of logic itself. But what I mean is something a little different.
Zavoi: This arguably leads to an infinite regress – if I choose to think X because I think that thinking X will benefit me, why did I choose to think that thinking X will benefit me? Etc… But let’s accept this for now.
The word "think" is too ambiguous, so let me clarify, first by way of metaphor:
"I" (my conscious mind) = Computer user
My unconscious mind = Computer
Action (in the praxeological sense) = All input from user to computer
Experience = sensations = All output from computer to user*
*Since the user is human, all output must be perceivable in the five senses (via monitor, speakers, etc.), and I am saying thought is the same
Now the words "think" and "thought" are used in both senses: (1) the action ("input") of giving an instruction to the subconscious to, say, move one's finger or call up a certain memory, and (2) the experience/sensation ("output") of, say, "seeing how a concept works." To keep the senses of the words "think" and "thought" clear, I will color all instances of those words red if they refer to action (input), and blue if they refer to experience/sensations (output).
This seems to clear up the infinite regress: "if I choose to think X [leading to an output thought I call "X"] because I think that thinking X will benefit me, why did I choose to think that thinking X will benefit me?" The answer to the underlined being, "You didn't." In other words, one doesn't choose to think, one chooses to think. Thinking is just seeing (and since it's not [praxeological] action, it'd be clearer not use verbs...but I don't want to make this any weirder language-wise than it already is).
Zavoi:That depends on what you mean by “means.”
That's a valid question at this level of analysis, but I really just mean that it is a norm/convention* of thinking the thought X that "I deem that X is useful to act as if." In other words, the thought (meaning thought) only presents to me as a thought if I deem** that it is useful for me to act as if.
*habit? not sure what word to use here
**or perhaps better to say "my subconscious mind deems" - who knows? I'm just saying that that appears to be my internal convention: thoughts/conclusions are only present to me as thoughts/conclusions because they are supposed to be useful. Who or what "deems" them that way, I do not know, perhaps a combination of instinct and built-up subconscious belief structure.
So I believe I am making a point in this section about definitions, rather than anything contentual. That doesn't mean I'm saying something pointless; when definitions have led to confused conclusions about content, clarifying the definitions can lead to new conclusions all by itself. (This is not to say that there are necessarily any confused conclusions here or in math or politics or wherever, but that what I am doing is looking out for any.)
Zavoi: AJ:Hence, rather than some ill-defined "truth," it is implicit in each statement I make in doing mathematics that I deem such statement useful to me (useful for me to "act as if"). Rather, it is implicit in the act of making/believing the statements…
Rather, it is implicit in the act of making/believing the statements…
Now I can state this more clearly: It is an internal norm/convention of each sensual presentation in my mind of a thought/statement (e.g., an axiom; I'll call it a "verbal statement" if stated) in doing mathematics that I deem* such thought/statement useful for me to act as if.
*See double-asterisked note above
Zavoi:Earlier you stated: ...
At the start of the thread I was speaking at one level of analysis, but now we are at a deeper one. I hope the above has clarified what I mean.
Zavoi: AJ:…but to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if." Again, could you clarify which direction the implication goes in?
So coming back to this, it's not a matter of implication. That would be saying something about content; I have basically been trying to make a point about definitions or internal norms/conventions or the innate structure or character of belief...I don't know what to call it exactly. In any case, I think much of what I'm aiming to say, aside from my main claim that a thought cannot refer to itself, is not so much a matter subject to proof or disproof, but rather a perspective that can be deemed elucidating or not.
AJ:Not expressible in that "system," which in this case probably means that thought or line of logic itself. But what I mean is something a little different.
Now I’m starting to get confused about what you’re actually trying to say.
When you say “‘Cats are carnivores’ follows from my premises,” what set of premises are you referring to? Well, it’s the same set of premises that supposedly entail the belief “‘Cats are carnivores’ follows from my premises,” is it not?
AJ:Now I can state this more clearly: It is an internal norm/convention of each sensual presentation in my mind of a thought/statement (e.g., an axiom; I'll call it a "verbal statement" if stated) in doing mathematics that I deem* such thought/statement useful for me to act as if.
So you’ve got this conscious mind, sitting at the display of the unconscious, viewing the output of thoughts. When it sees something, it concludes that the unconscious must have deemed it useful, or else it would not have outputted it.
While I see some other problems with that view, at least it does not literally identify a statement with the usefulness of that statement. This seemed to be what your position was, as suggested by your casual replacement of one with the other:
AJ:to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if."
…and saying that this equivalence is valid simply by convention or by definition:
AJ:So coming back to this, it's not a matter of implication. That would be saying something about content; I have basically been trying to make a point about definitions or internal norms/conventions or the innate structure or character of belief...I don't know what to call it exactly.
Now, what I’m saying basically comes down to this: “X” and “X follows from my premises” and “X is useful to believe” are three very different things. It does not follow from my premises that it’s raining outside, although it may in fact be raining. If someone orders me at gunpoint to believe in Santa Claus, it is useful then for me to believe in Santa Claus, and I may even delude myself into believing it, but that does not cause Santa Claus to come into existence.
Confusing these things is, as I mentioned, conflation of the map and the territory, which causes your mind to become tied up in all sorts of knots.
Perhaps you don’t even disagree with this; I’d like to know.
Hi Zavoi. Since others may also be following this (there are so many lurkers on this board), I will be going off on some tangents that may not be directly relevant to the discussion, but that I think may flesh out some of what I've been saying. Feel free to respond to whatever or not, as you please.
Zavoi: When you say “‘Cats are carnivores’ follows from my premises,” what set of premises are you referring to?
My beliefs, including any tentative "beliefs" (models, temporary assumptions, believed sensory data, etc.) I may be working from at that moment.
Zavoi: Well, it’s the same set of premises that supposedly entail the belief “‘Cats are carnivores’ follows from my premises,” is it not?
What is entailed is the belief "Cats are carnivores." The "follows from my premises" part is just a reference to the fact that the premise beliefs logically entail the conclusion belief, and is only needed in certain cases in discussion with others, or in internal contemplation about how the mind works.
Zavoi:So you’ve got this conscious mind, sitting at the display of the unconscious, viewing the output of thoughts. When it sees something, it concludes that the unconscious must have deemed it useful, or else it would not have outputted it.
Or rather, thoughts/conclusions are only outputted as thoughts/conclusions relevant to the task at hand because they are supposed to be useful for that task (in this case, doing math).
What do I mean by "outputted as thoughts/conclusions"? I posit there is a special marking system in the "user interface" for distinguishing concluded thoughts/beliefs from the other various sensation patterns. In my case, I can sometimes notice a very slight and extremely brief sensation of "certitude" in my eyes, temples, and/or chest when such a thought image appears as a relevant belief or conclusion, rather than as something else. For someone else, it could be that such sensations are tinted orange, or for someone who thinks in sounds it could be that those auditory sensations have an distinctive edgy sound as if generated by a computer, or are accompanied by a lemony scent, or who knows. I suppose everyone is different.
This might seem obviously wrong, because we can easily have momentary thoughts presented to us that we immediately recognize as not useful. I would just say that's the process of reaching a conclusion. Once I reach a conclusion (that is, a set of sensations bearing the markings of a "certified conclusion" or belief), however temporarily, I deem it to be useful until it is overturned by new evidence, including rethinking and new sensory evidence from the (presumed) outside world. This just means that reasoning is the process of attempting to derive useful conclusions (rather than "truth").
I suppose this only seems weird to us because there is (or at least formerly was) almost never any survival value in realizing this, so we have a tendency to gradually consider all useful models as certainties, just like how scientists sometimes seem to want to rename their theories to laws. It's just not expedient most of the time to keep track of such things. As very young children, we learn that our deliberate imaginings have a mighty competitor (called "reality"*) in terms of which set of sensations we can most reliably base our actions on in order to elicit more pleasant sensations, at least long-term (fantasizing can be very satisfying for a brief while).
*...and characterized by the sensations being more lasting and only being manipulable through this thing called one's "body"
Zavoi: While I see some other problems with that view, at least it does not literally identify a statement with the usefulness of that statement. This seemed to be what your position was, as suggested by your casual replacement of one with the other: AJ:to say "X" or "X follows from my premises" means to me more like, as above, "I (provisionally) deem X useful to act as if." …and saying that this equivalence is valid simply by convention or by definition:
As per above, "X," "X follows from my premises," and "I (provisionally) deem X useful to act as if" are all different ways to explain what a thought or assertion is. Why the need to explain what a thought or assertion is? Just because in some instances an utterance might not be intended as an assertion, even thought it may look like one. As Adam Knott notes, I might utter the words, "It is a fact that Elvis is dead," but not be asserting anything - I could just be practicing for a play or testing a microphone.
So then, if we are to evaluate an assertion, such as those that underly so-called logical paradoxes like Curry's Paradox, we need to understand what we are doing when we evaluate it as an assertion. What does it mean in terms of human action? Most of the time it doesn't matter, because the context makes it clear enough. Self-referential statements are a rare example where this has to be clarified. For thoughts as well, if we are to analyze or evaluate them, in rare cases it becomes necessary to explain what we mean (or just clarify in our own minds what we mean) by the notion of a thought/conclusion/belief/etc., or what conventions are implicit in the act of thinking, concluding, or stating "X."
Zavoi: It does not follow from my premises that it’s raining outside, although it may in fact be raining.
If I get your point here correctly, for instance, I might believe (that is, deem it useful to act as if) it's not raining outside now. Yet 10 seconds later I look outside and realize it has been raining for a long while, proving that my previous belief is (and was) un-useful. That would probably be a good time to reassess whatever premises led to that conclusion, or try refiguring to make sure I didn't make an error in the deduction. I guess the key is that I didn't claim that X was definitely useful for me to act as if, but that I deemed it so based on the beliefs I had at that time.
Bottom line is that I can never make an objectively accurate statement about any real world, or even about what would definitely be useful for my to act as if at any given time, since I cannot access any real world but through the fallible mechanism of (apparently) my sensory organs and pre-conscious filtering and interpretation of the same. On the most basic level, all I can say is I experience patterns of sensations and strive to make them more satisfactory through action, including efforts to develop models and reach conclusions that are more and more useful to act as if.
Zavoi:If someone orders me at gunpoint to believe in Santa Claus, it is useful then for me to believe in Santa Claus, and I may even delude myself into believing it, but that does not cause Santa Claus to come into existence.
Under those circumstances, I may indeed deem it useful for me to act as if Santa Claus exists for the duration of the incident. My beliefs are just my deemed useful-to-act-as-if's, and "existence" is a belief system in itself (albeit apparently an extremely useful one - see below). So to say that Santa does not come into existence just because I deem it useful for me to believe in him, to me simply means that once this gunman leaves me alone I will no longer deem it useful for me to believe in him (unless the experience has so warped my mind that the belief persists until new evidence leads me away from it).
Again, I am not capable of making a purely objective statement about any real world, so I cannot speak (or think) of Santa Claus "existing" in that sense, only in terms of a useful model of the same. Stepping back into common-level analysis, what then accounts for the intense feeling of certitude people have about their beliefs? Evolutionary utility, perhaps. The deep level of analysis we're discussing here is of course usually not needed for survival and procreation.
Zavoi:Confusing these things is, as I mentioned, conflation of the map and the territory, which causes your mind to become tied up in all sorts of knots.
In terms of the map and territory, at this level of analysis I am saying that sensations are the map, and the territory is a useful model. If one acknowledges that the whole reason for presuming the territory "exists" is because such a presumption is useful*, then of course it is only coherent to treat every statement about the territory (such as, "X exists") as a statement about what is useful.
*I don't seem to have gained very much happiness by refusing to differentiate between dreams/fantasies and (what appear to be) the direct testimony of my eyes, ears, etc. To me, believing in a "real world" has been the ultimate useful assumption =)
AJ: What is entailed is the belief "Cats are carnivores." The "follows from my premises" part is just a reference to the fact that the premise beliefs logically entail the conclusion belief, and is only needed in certain cases in discussion with others, or in internal contemplation about how the mind works.
But the belief that such-and-such follows from your premises is itself a belief that you have. You seem to have drawn a distinction between thoughts (“Cats are carnivores”) and meta-thoughts (“‘Cats are carnivores’ follows from my premises”), such that the premises entailing the thoughts are not the ones entailing the meta-thoughts you are expressing now. But you cannot get outside your own mind like this – you cannot position yourself to see a bird’s-eye-view of the logical structure of your own thoughts (as you can do when you analyze PA or PC as an “outside observer”). Any beliefs you have about your own mind (such as “‘Cats are carnivores’ follows from my premises”) are themselves a part of that same mind.
AJ: If I get your point here correctly, for instance, I might believe (that is, deem it useful to act as if) it's not raining outside now. Yet 10 seconds later I look outside and realize it has been raining for a long while, proving that my previous belief is (and was) un-useful.
Or, perhaps you never find out that it was raining, and you go the rest of your life believing that it didn’t rain. That still doesn’t mean it didn’t rain. (I think?)
AJ: In terms of the map and territory, at this level of analysis I am saying that sensations are the map, and the territory is a useful model. If one acknowledges that the whole reason for presuming the territory "exists" is because such a presumption is useful*, then of course it is only coherent to treat every statement about the territory (such as, "X exists") as a statement about what is useful.
So, you perceive a certain regularity in your sensations, and you invent the notions of “things” and “physics” and “universe” in an attempt to model and predict these sensations. “Real things” can be said to exist in the same way that the letters on my computer screen can be said to “exist” (even though it’s just a certain pattern of pixels turning on and off as I scroll, just as the “things” you believe in “are” “really” just a certain pattern of sensations). Is this an accurate summary?
I am almost inclined to agree with this, as a way of answering brain-in-a-vat skepticism (sure, I can only know what I perceive, but what I perceive is my reality). The problem with this is that it’s impossible for me to believe that the world consists of AJ’s sensation-modeling-framework, just as it’s impossible for you to believe that the world consists of Zavoi’s sensation-modeling-framework. So it’s necessary for us to assume a common reality, relative to which our own models might be wrong, before either of us can say anything and expect it to be understood by the other. (This may be what the private language argument is about; I haven’t read up on that.)
In any case, though, firstly, “usefulness” doesn’t come in to this, except in the trivial sense in which anything I do is ipso facto considered useful by me. Secondly, the Löb objection still applies: if my model of reality itself contains the assertion that anything contained in my model obtains in reality, then I will end up believing absolutely everything.
It may in fact be the case that all my beliefs are correct, by virtue of my believing them, but I cannot actually believe that.
Zavoi:But the belief that such-and-such follows from your premises is itself a belief that you have. You seem to have drawn a distinction between thoughts (“Cats are carnivores”) and meta-thoughts (“‘Cats are carnivores’ follows from my premises”), such that the premises entailing the thoughts are not the ones entailing the meta-thoughts you are expressing now. But you cannot get outside your own mind like this – you cannot position yourself to see a bird’s-eye-view of the logical structure of your own thoughts (as you can do when you analyze PA or PC as an “outside observer”). Any beliefs you have about your own mind (such as “‘Cats are carnivores’ follows from my premises”) are themselves a part of that same mind.
If I believe "If A then B" and I also believe "If B then C" then it seems to follow from my premises that "If A then C." Of course I could be in error about it really following from my premises, but that's what my mind serves up for me. The thought "If A then C" is tagged in the "user interface" as a logically necessary conclusion. That I can see, because that is the very thing that I must see in order to recognize the thought as a (supposed) logically necessary conclusion.
So the distinction is not, for instance, between “Cats are carnivores” and “‘Cats are carnivores’ follows from my premises”, but rather between the thought "Cats are carnivores" tagged as a model that seems to fit the data (or just a hypothetical fancy) and the thought "Cats are carnivores" tagged as a logically necessary deduction.
Again, it's a convention of the interface. Just like there is a certain bleep your computer makes when you try to do something that it doesn't permit. It's not that you can see how the computer works. Maybe there is no manual or you've never read it, but you know from experience with that interface that this bleep means "The action you just tried to perform is not permitted." You can't really know absolutely that, say, the bleep doesn't actually mean something like, "Click 1 centimeter to the left 10 times and I'll let you do that." But it seems to have proven useful for you to interpret that bleep as indicating that an action is not permitted.
In the same way, it seems to have proven useful to me to interpret sensations tagged in a certain way as logically necessary conclusions, tagged in another way as models, tagged in another way as daydreams, tagged in another way as raw sensory data about the "real world", etc.
Zavoi:Or, perhaps you never find out that it was raining, and you go the rest of your life believing that it didn’t rain. That still doesn’t mean it didn’t rain. (I think?)
I'm saying that to believe is just to deem it useful to act as if. I'm not (and I assume you also are not) in a position to talk about whether "it actually rained" in an objective sense. Strictly speaking, I'm claiming that all I can usefully mean by "it actually rained" is that I deem it useful to act as if it actually rained.
Consider Moore's paradox: "It's raining, but I don't believe it is."
As some have pointed out, if the assertion is taken to be an assertion of belief, then it becomes "I believe it's raining but I don't believe it's raining," which is just a contradiction. If the assertion is taken to be a statement about the "real world" (not a belief about it), then it seems paradoxical, but I am questioning this supposed ability to comment directly on any real world, given that one cannot perceive any such thing as an objective observer. In other words, any assertion I make that is supposed to be about a real world cannot be anything more than a belief. "It's raining" [as an assertion] = "I believe it's raining" = "I deem it useful to act as if it's raining."
I find it useful to say "It actually did rain" particularly when I had, at some time T in the past, deemed it useful to act as if it weren't raining at that time T but I now deem it useful to act as if it had been raining at T. Speaking loosely I would say, "It was raining, but I did not believe it was." Strictly, though, the statement seems to purport to make an objective assertion about some reality, so it is overstating things - it is making a statement that is impossible for me to make since I can't have such knowledge. If it is taken as a subjective statement about my beliefs then it is fine, or not only fine but completely undeniable insofar as it does correspond with my beliefs at the moment it is uttered.
Zavoi:So, you perceive a certain regularity in your sensations, and you invent the notions of “things” and “physics” and “universe” in an attempt to model and predict these sensations. “Real things” can be said to exist in the same way that the letters on my computer screen can be said to “exist” (even though it’s just a certain pattern of pixels turning on and off as I scroll, just as the “things” you believe in “are” “really” just a certain pattern of sensations). Is this an accurate summary?
If I'm interpreting you correctly here, this is indeed what I'm saying.
Zavoi:The problem with this is that it’s impossible for me to believe that the world consists of AJ’s sensation-modeling-framework, just as it’s impossible for you to believe that the world consists of Zavoi’s sensation-modeling-framework. So it’s necessary for us to assume a common reality, relative to which our own models might be wrong, before either of us can say anything and expect it to be understood by the other.
I'd call it a problem of communication. In order to communicate I have to presume the listener conceptualizes the world and the current context in a similar enough way for the inferential gap to be crossed. If not, communication becomes impossible.
Imagine you're away on vacation and an alien, who is fluent in English*, is taking care of your seven dogs. While on vacation you remember that one of your dogs needs to be given his medication. You phone the alien to tell him to give the dog the medicine. Luckily, the dog that needs the meds is white, and all the other dogs are black. So your conversation might go like this:
You: Give the medicine to the white dog.
Alien: I see no white dog.
You: It's the one that isn't black.
Alien: I see no black or white dogs. Color-wise, all seven of them are mostly red by volume. [Alien has some kind of "x-ray" vision]
You: No no, we humans refer to the color of a dog by the color of its fur, not its innards.
Alien: The dog is a living being, is it not? I see no living white fur component to any of them.
You: Well, I mean give the medicine to the dog that has white fur attached to its skin.
Alien: Understood, there is only one dog like that, so I know which one you mean.
You return from vacation to find the medicine sitting where it was and the dog very ill. You ask the alien, but he says, "I transfered ownership title over the medicine from you to the dog. What more did you want?"
*As might be clear from the illustration, the very idea that an alien could become fluent in English is perhaps not a fair assumption!
When I say to someone "It's raining," I might be doing one of many possible things:
1. Declaring that I believe it's raining.
2. Declaring that it "objectively" is raining.
3. Stating a hypothetical situation for the person to consider.
4. Effectively commanding the person to believe it's raining.
5. Practicing for a play.
6. Trying to impress them with how deep and rich my voice sounds when those particular syllables are uttered.
...etc...
And of course, to expect these feats of communication to cross the inferential gap, I have to presuppose that there is enough shared context. I can only expect to communicate most things under the assumption that other humans are first of all not automatons and actually do act in the Misesian sense and furthermore feel emotions and desires of at least a similar structure to mine. Given those assumptions, the more similar the circumstances and culture in which we grew up, and the more similar our circumstances now, the more faithfully my communications will cross the gap.
Communications even to our own selves often don't cross the inferential gap, as anyone who often writes personal notes can attest.
So instead of saying I have to assume others share a common reality, I would say I have to assume others are sufficiently like me and face sufficiently similar types of circumstances to those that I face.
Zavoi:In any case, though, firstly, “usefulness” doesn’t come in to this, except in the trivial sense in which anything I do is ipso facto considered useful by me.
In terms of communicating or interacting with others, I would say I try to guess others' motivations (that is, what is useful to them) based on whatever similarities it appears that they share with me. I can do that through pure behaviorism where I am really just trying to see how they're "wired" (though this is technically no longer communication or interaction) or otherwise treat other humans like black boxes, or through praxeology where I start from the assumption that they are subjective actors like myself (I think most people do praxeology every day...).
I think most people also take the former approach on a daily basis, like when they think, "If I tell people I am a janitor, the result is generally X. If I tell people I am a sanitation technician, the result is generally Y. I like result Y more than result X, so I tell people I am a sanitation technician instead of telling them I'm a janitor. " That line of thinking treats people as black boxes, with certain inputs yielding certain outputs. It doesn't analyze the notion of my image as a janitor versus my image as a sanitation technician in others' minds in terms of how those two images of me might be useful to them; it just looks at the output results statistically and chooses the input that gives the preferred output.
Zavoi:Secondly, the Löb objection still applies: if my model of reality itself contains the assertion that anything contained in my model obtains in reality, then I will end up believing absolutely everything. It may in fact be the case that all my beliefs are correct, by virtue of my believing them, but I cannot actually believe that.
This still assumes there is some "actual reality." I think this goes back to what I said at the beginning of this post: there is no assertion about any reality contained in my model, I simply mean that what I refer to casually as "reality" just is a model*. Regarding "correct," if you mean this in terms of correspondence to reality, the same applies.
*Not a model of something, but a model for predicting which actions are most likely to lead to more satisfactory sensations.
AJ:If I believe "If A then B" and I also believe "If B then C" then it seems to follow from my premises that "If A then C." Of course I could be in error about it really following from my premises, but that's what my mind serves up for me. The thought "If A then C" is tagged in the "user interface" as a logically necessary conclusion. That I can see, because that is the very thing that I must see in order to recognize the thought as a (supposed) logically necessary conclusion.
If the premises are:
then “If A then C” follows from the premises. So does “If not C, then A.” But one thing that does not follow from the premises is “‘If A then C’ follows from the premises.” That is a much more complicated statement, asserting the existence of a valid chain of reasoning starting with the premises and ending with “If A then C.” As it so happens, such a thing does exist, but the two-premise system we have here does not “know” this (i.e., it doesn’t follow from the premises).
If I understand you correctly, you’re saying that when you_red see a belief pop up on your “internal computer”_blue, tagged as a “true” statement, you_red know that that statement must follow from your_blue premises, because otherwise it would not have been produced.
But, in your view, you_red do not know that the statement follows from your_red premises. You_red have your_red own set of premises (e.g., “Anything produced by me_blue follows from my_blue premises”) that produce beliefs such as “‘Cats are carnivores’ follows from my premises.” And then we must ask: are you then going to tell us that you (on a third, higher level)_green notice that you_red produce the belief “‘Cats are carnivores’ follows from my_blue premises,” and thereby conclude that “‘Cats are carnivores’ follows from my_blue premises” follows from your_red premises? And then where does that belief come from? Etc…
The solution to this problem is to stop it where it starts: to recognize that your_black beliefs are what is produced by your_black premises, and this belief follows from those very same premises. (“Your” is written in black for a reason.) This is different from saying that believing X is equivalent to believing that I believe X:
AJ: Strictly speaking, I'm claiming that all I can usefully mean by "it actually rained" is that I deem it useful to act as if it actually rained.
AJ: "It's raining" [as an assertion] = "I believe it's raining" = "I deem it useful to act as if it's raining."
This I think I’m willing to agree with:
AJ: I'd call it a problem of communication. In order to communicate I have to presume the listener conceptualizes the world and the current context in a similar enough way for the inferential gap to be crossed. If not, communication becomes impossible. […] So instead of saying I have to assume others share a common reality, I would say I have to assume others are sufficiently like me and face sufficiently similar types of circumstances to those that I face.
[…]
I think it comes down to a difference in terminology: “common reality” vs. “sufficiently similar types of circumstances.”
AJ: If the assertion is taken to be a statement about the "real world" (not a belief about it), then it seems paradoxical, but I am questioning this supposed ability to comment directly on any real world, given that one cannot perceive any such thing as an objective observer.
AJ: This still assumes there is some "actual reality." I think this goes back to what I said at the beginning of this post: there is no assertion about any reality contained in my model, I simply mean that what I refer to casually as "reality" just is a model*. Regarding "correct," if you mean this in terms of correspondence to reality, the same applies.
I agree with this: The model does not contain “assertions” about reality*. In other words, the model’s assertions in fact derive from nothing more than the premises of the model (including sensory input), and there is no mystical connection between the “real world” and the model, such that any two models will always reach the same conclusions.
*(“Reality” here refers to not only the outside world, but statements about mathematics, logic, etc.; in short, everything that is claimed to be true. It’s one thing to say that your model of the external world is only a provisionally useful model of sensations, but it’s another thing to believe that everything you believe (including everything you have stated in this thread) is also like this.)
I also might agree that our model of the “external world” is a model of sensation-patterns, and that we may even be consciously aware of this.**
**(And perhaps this is the substance of your argument, in which case maybe we don’t actually disagree on anything.)
However, I don’t agree that this applies to all beliefs whatsoever. There’s an important distinction to be made: The model does not contain “assertions” about reality, but it does contain “assertions about reality.” That is, the assertions produced by the model purport themselves to be about something called “reality.” An outside observer sees this happen because the modeler is “receiving” certain generated thoughts, but from the perspective of the modeler, these statements simply are. The modeler’s premises do not entail that anything entailed by the modeler’s premises can thereby be tagged with “reality.”
AJ: I think most people also take the former approach on a daily basis,
I actually think that most people use a theory-of-mind (putting yourself in the other’s shoes) to predict their responses, since there isn’t a lot of statistical data available for the black-box approach. But I’m not a psychologist, so I don’t know.
Zavoi:But, in your view, you_red do not know that the statement follows from your_red premises. You_red have your_red own set of premises (e.g., “Anything produced by me_blue follows from my_blue premises”) that produce beliefs such as “‘Cats are carnivores’ follows from my premises.” And then we must ask: are you then going to tell us that you (on a third, higher level)_green notice that you_red produce the belief “‘Cats are carnivores’ follows from my_blue premises,” and thereby conclude that “‘Cats are carnivores’ follows from my_blue premises” follows fromyour_red premises? And then where does that belief come from? Etc…
As I say, it's a convention of the interface. How do I know it's a convention of the interface? That is a model that seems to fit the data. It's like if I say, "That bleep means the mouseclick won't be processed." How do I know? From experience. I could be wrong, but that interpretation has seemed to serve me well my entire life, or my entire time of working with the Windows OS.
Note that this reflects the level of certainty I have about logical conclusions (what are tagged "logically necessary conclusions" in my mind): it's to the level where if I were to doubt (the usefulness of) it, I would have to doubt (the usefulness of) everything I think I know. So people say, "If A implies B and B implies C, but A doesn't imply C, then everything I know is in question! It is meaningless [read: useless] to question that!"
In the end, I can't know what the computer "really means," I can simply choose interpretations that seem more useful to me, based on experience. I am here saying that the fundamental conception of these matters in terms of "true and false" is not as useful to me as a conception in terms of "more useful and less useful to me." And I am suggesting the same may be true for everyone who is looking at this level of analysis.*
*...and possibly even at everyday levels of analysis - I've often shot myself in the foot by saying something just because it was "true" yet very un-useful to say at that time And I think that was a direct result of my superstition about "truth," that is, thinking there was some objective truth knowable and statable by me, rather that just useful or less useful notions on which to base the actions I take.
Zavoi:I think it comes down to a difference in terminology: “common reality” vs. “sufficiently similar types of circumstances.”
Right, the reason I prefer not using the word "reality" is that it carries extra baggage about there being some objective outside world, even though I cannot speak of such a thing other than as a model for my subjective pursuit of happiness.
Zavoi:In other words, the model’s assertions in fact derive from nothing more than the premises of the model (including sensory input)
Quick clarification: Not all sensory data are premises (thoughts); only interpretations can be. I may have been sloppy about that at times. In other words, this following collection of dots (analogous to sensory data) may not be a premise in my mind:
Zavoi:*(“Reality” here refers to not only the outside world, but statements about mathematics, logic, etc.; in short, everything that is claimed to be true. It’s one thing to say that your model of the external world is only a provisionally useful model of sensations, but it’s another thing to believe that everything you believe (including everything you have stated in this thread) is also like this.)
Zavoi:That is, the assertions produced by the model purport themselves to be about something called “reality.”
Zavoi:I actually think that most people use a theory-of-mind (putting yourself in the other’s shoes) to predict their responses, since there isn’t a lot of statistical data available for the black-box approach. But I’m not a psychologist, so I don’t know.
AJ:1. I experience sensations. 2. I connect these sensory "dots" to form concepts/thoughts/premises (which also present to me as sensations) that I deem more or less useful to me to "act as if" in order to experience more satisfactory sensations than I am now. 3. I also logically deduce things from these premises, or maybe my subconscious does (probably some kind of interplay between conscious and unconscious). I can of course also deduce from previous deductions.
2. I connect these sensory "dots" to form concepts/thoughts/premises (which also present to me as sensations) that I deem more or less useful to me to "act as if" in order to experience more satisfactory sensations than I am now.
3. I also logically deduce things from these premises, or maybe my subconscious does (probably some kind of interplay between conscious and unconscious). I can of course also deduce from previous deductions.
I think this terminology is kind of idiosyncratic, but we can stick with it. I’ll use “deduce from the model” to refer to the entire process by which assertions (be they premises or more indirect deductions) are produced from sensory dots.
AJ:As I say, it's a convention of the interface. How do I know it's a convention of the interface? That is a model that seems to fit the data.
So the fact that your “inner computer” is taking in sensory data and outputting a model of that data – this “fact” is itself just part of model that fits the data? What is this data? The experience of the inner computer’s assertions being useful? And what data produces the model asserting that the fact that your “inner computer” is taking in sensory data and outputting a model of that data is itself part of a model fitting the experience of the inner computer’s assertions being useful? Can’t you go arbitrarily deep with this?
AJ:In my conception this would be, "the conclusions (tagged as being) deduced from the model are tagged as being deduced from the model." So it wouldn't really be saying anything.
This is the point I’m trying to make: for a model to assert X is not the same thing as for the model to assert that the model asserts X.
Is “(tagged as being)” supposed to be there? It makes a big difference to leave it out:
The conclusions deduced from the model are tagged as being deduced from the model.
This is not at all obvious, if this “tagging” is something that the model is supposed to be aware of (which I assume it is, since you are talking about it now). If the model asserts “A bee is approaching a flower,” it is just that – we cannot use a “convention” to finagle this assertion into “The model asserts that a bee is approaching a flower” (and vice-versa) without causing a disastrous Löb explosion.
From Wittgenstein’s Tractatus:
5.633:Where in the world is a metaphysical subject to be found? You will say that this is exactly like the case of the eye and the visual field. But really you do not see the eye. And nothing in the visual field allows you to infer that it is seen by an eye.
When you see something, you may think to yourself “Okay, all I really know is that my eye is seeing this” (although you cannot see this fact). You can do this because your mind is not totally visual, and your eye is not your only source of knowledge. But this changes when we move from the visual field to the totality of your mind itself. If you try this, you would end up thinking: “A bee is approaching a flower. Okay, all I really know is that my model deduces that a bee is approaching a flower. Okay, all I really know is that my model deduces that my model deduces that a bee is approaching a flower. Okay…” Either you must stop this regress at the source, by saying that “A bee is approaching a flower, and that’s that”; or at the next step, by saying “My model deduces that a bee is approaching a flower, and that’s that.” Either way, you cannot think that every thought of yours is “just” the fact that it is deduced from your model.
AJ:Indeed, this seems a necessary trick for my mind to pull on me, as it would not be very useful for me to consider these matters of epistemology at such a deep level at all times.
This isn’t just a trick used to avoid overthinking ordinary things; it’s necessary to prevent the regresses from going on forever. Without referring to “reality,” your belief system becomes a castle in the air, as useless as a calculator which when input with “2+2=” answers the question “What does this calculator output when input with ‘2+2=’?”
Your (IMO false) claim that “A calculator can’t ‘tell us what 2+2 really equals’; it can only ‘tell us what it thinks 2+2 equals’” is much different from the (IMO true) observation that “A calculator can’t tell us ‘what 2+2 really equals’; it can only tell us ‘what it thinks 2+2 equals.’”
The English language is extremely bad at making this distinction. The distinction is the one by which “Alex admires Samuel Clemens’s writings” can be read as either true (the extensional reading) or false (the intensional reading) in the case where Alex says “I admire Mark Twain’s writings” but does not know that Samuel Clemens is Mark Twain. Your claim is intensional; the (IMO true) observation above is extensional.
"Hume differentiated "matters of fact" and "relations of ideas." Are you suggesting that Hume conceived of "relations of ideas" as Mises conceived of his praxeology?"
I believe that is exactly how Mises characterized Praxeology. At its most basic, it is merely a relation of ideas, while the fundamental axiom is undeniable, that is, by denying action, one is acting. In a lecture given by David Gordon at Mises U, he essentially agreed that Praxeology was simply tautological. Of course, some of the subsidiary axioms are deniable, though they are very reasonable assumptions.
Zavoi: AJ:As I say, it's a convention of the interface. How do I know it's a convention of the interface? That is a model that seems to fit the data. So the fact that your “inner computer” is taking in sensory data and outputting a model of that data – this “fact” is itself just part of model that fits the data? What is this data? The experience of the inner computer’s assertions being useful? And what data produces the model asserting that the fact that your “inner computer” is taking in sensory data and outputting a model of that data is itself part of a model fitting the experience of the inner computer’s assertions being useful? Can’t you go arbitrarily deep with this?
All I'm saying above is that it has apparently served me well to treat certain sensations (data) in certain ways. If you're going to ask how I know it has served me well, I cannot answer but that it seems to be working to my satisfaction, better than anything else I've tried. I don't know if certainty can get any higher than that.
Why I think you might be seeing infinite regresses is that you seem to be thinking in terms of truth and falsity, whereas I am conceiving things purely in terms of utility.
Zavoi:This is the point I’m trying to make: for a model to assert X is not the same thing as for the model to assert that the model asserts X.
First, a model is a thought/concept/notion. Can a thought/concept/notion refer to itself? It doesn't seem so. Second, the model needn't assert that it asserts something for me to decide it is useful to act as if. Like I've been saying, it is from experience that I "know" it is useful to act as if [the model]. It is not that the model tells me it is useful to act as if; just like it's not that a screwdriver tells me it's useful for screwing in screws, or that my computer tells me what a certain bleep means. It's just that it's apparently served me well to treat sensations tagged in a particular fashion in certain ways.
Zavoi:...if this “tagging” is something that the model is supposed to be aware of (which I assume it is, since you are talking about it now).
You seem to be treating "reality" as "everything," including all private sensations. This would result in a misinterpretation of what I'm saying. By "reality" I specifically mean those sensations that appear to be more lasting and that cannot apparently be manipulated or influenced through pure thought alone, but only through the means of what I term my physical body. (If you want to include things like "logic" and "mathematics" I will explain why there is ambiguity in those terms, and that they are conceived variously as including both actions and imagined sensations. Better to include a specific mathematical statement or something, rather than "mathematics.")
Zavoi:If the model asserts “A bee is approaching a flower,” it is just that – we cannot use a “convention” to finagle this assertion into “The model asserts that a bee is approaching a flower” (and vice-versa) without causing a disastrous Löb explosion.
I think this pinpoints the issue. Strictly speaking, "A bee is approaching a flower" is not an assertion. It's a "deemed useful to act as if."* At a given moment, it may be that all that is perceptually present for me is that one image. It may have a gold lining. Perhaps in all my experience the only thoughts I have that have a gold lining are those that I am fairly certain of, among all the thoughts I ever had. The gold-lined thoughts have proven to be fairly reliable for me to act as if. So it seems useful for me to treat this thought as also fairly reliable to act as if. That is all I mean. From this perspective, there is no assertion saying "It is useful for you to act as if a bee is approaching a flower," there is just - for example - a long history of precedents (assuming my memory serves me).
*Bearing in mind that by seeming to assert this very sentence, I am being inconsistent. This is simply because the English would get too unwieldy if I were to always say, "I deem it useful to act as if X and I am suggesting that you will do, too" instead of just "X." Or in this case, "I deem it useful to act as if 'A bee is approaching a flower' is not an assertion but rather a 'useful to act as if', and I am suggesting that you will deem it useful as well."
Zavoi:When you see something, you may think to yourself “Okay, all I really know is that my eye is seeing this” (although you cannot see this fact). You can do this because your mind is not totally visual, and your eye is not your only source of knowledge.
Rather, "All I really know is that I am sensing this," where sensing has none of the normal English connotations of actually being connected with something "out there" that is being sensed, but rather is just a private subjective sensation.
Zavoi:But this changes when we move from the visual field to the totality of your mind itself. If you try this, you would end up thinking: “A bee is approaching a flower. Okay, all I really know is that my model deduces that a bee is approaching a flower. Okay, all I really know is that my model deduces that my model deduces that a bee is approaching a flower. Okay…”
See above paragraph about assertions.
Regarding intensional vs. extensional, note that this distinction is not present in thought, only in words. If my claim is taken as intensional (and therefore "false," er, not useful) it can only be a problem of the words I have used to express it, not of the notion itself.
Summing up, I think we are running up against several elements of differing conception or just ambiguity: 1. What "reality" encompasses. 2. Whether it's more useful to analyze fundamentally in terms of truth or in terms of utility (human action). 3. The interface metaphor and how one's conscious mind can "know" things. 4. What it really means to "deem" something (in my use of the phrase "deem something useful to act as if"), who's doing the deeming, etc. (and I am not sure of the answer to that one).
AJ:Second, the model needn't assert that it asserts something for me to decide it is useful to act as if. Like I've been saying, it is from experience that I "know" it is useful to act as if [the model]. It is not that the model tells me it is useful to act as if;
So not all knowledge is part of a useful-to-act-as-if model, but there is also “direct” knowledge, which proceeds directly from experience/sensation without any intervening model. Here you seem to be saying that your knowledge of the usefulness of “acting-as-if-[the model]” is of this type. This is the “simply is” knowledge I’ve mentioned, which stops the infinite regress caused by the subsumption of all knowledge under a useful-to-act-as-if model.
AJ:By "reality" I specifically mean those sensations that appear to be more lasting and that cannot apparently be manipulated or influenced through pure thought alone, but only through the means of what I term my physical body.
By this are you referring just to the “external world”? I will not dispute (not here, at least) that “knowledge” of the external world (“There is an object called the sun, and it will rise tomorrow”) is just the most useful model of one’s sensations. But by “reality” I was referring to all knowledge, irrespective of its topic. Taken this way, my claim that “Knowledge is about reality” can be seen as trivially true, but it seems that you’re saying that there is no real knowledge, only “pseudo-knowledge” such as “A bee is approaching a flower.” (By pseudo-knowledge I refer to statements that appear to be direct at a shallow level of analysis where we ignore the clichéd outer brackets, but, seen at a deeper level, are actually statements about some model.) But we cannot even imagine what it would be like to have nothing but pseudo-knowledge, since for a thought to be pseudo-knowledge is for it to be really about something else (e.g., “A bee is approaching a flower” is really saying “It is useful for me to act as if a bee is approaching a flower”), and whatever this real topic is, it is actual knowledge and not pseudo-knowledge.
AJ:This is simply because the English would get too unwieldy if I were to always say, "I deem it useful to act as if X and I am suggesting that you will do, too" instead of just "X."
If for some reason we can’t go from
X ⇔ I deem it useful to act as if X and I am suggesting that you will too
to
I deem it useful to act as if X and I am suggesting that you will too ⇔ I deem it useful to act as if I deem it useful to act as if X and I am suggesting that you will too, and I am suggesting that you will too
then your assertion-reduction template is not universally applicable to all X. If so, then we have direct knowledge, the infinite regress stops, and Löb’s theorem is no longer a problem.
AJ:Regarding intensional vs. extensional, note that this distinction is not present in thought, only in words. If my claim is taken as intensional (and therefore "false," er, not useful) it can only be a problem of the words I have used to express it, not of the notion itself.
The intensional/extensional thing was just a way of clarifying an ambiguity in English, without which it would have been completely unnecessary to mention this at all. All knowledge is in fact deduced from a model (in the same way that Alex in fact admires Samuel Clemens), but it is not the case that all knowledge is of its being deduced from a model* (in the same way that Alex does not intensionally admire Samuel Clemens).
*(Although some of it might be, but then it’s really pseudo-knowledge.)
AJ:1. What "reality" encompasses. 2. Whether it's more useful to analyze fundamentally in terms of truth or in terms of utility (human action). 3. The interface metaphor and how one's conscious mind can "know" things. 4. What it really means to "deem" something (in my use of the phrase "deem something useful to act as if"), who's doing the deeming, etc. (and I am not sure of the answer to that one).
I think #1 is the key point of contention, which is really only a terminology dispute. But the matter of “who is doing the deeming” is important in that we should recognize that the person for whom X is useful is the same as the person who believes “I deem X useful,” and it’s not as though we can believe things about ourselves without those beliefs becoming aspects of ourselves.
Long has written an excellent piece.
“Socialism is a fraud, a comedy, a phantom, a blackmail.” - Benito Mussolini"Toute nation a le gouvernemente qu'il mérite." - Joseph de Maistre
Thanks for the paper, Vichy. It's too long for me to read now, but I read a fifth of the way.
So far the ambiguity of the word "logic" and the conception of things in terms of truth as the fundamental unit of analysis seem misguided. I've been arguing that truth is merely a useful heuristic, but that at a deep enough level of analysis* there is only utility. Perhaps inadvertently, Long illustrates this here:
Zavoi:So not all knowledge is part of a useful-to-act-as-if model, but there is also “direct” knowledge, which proceeds directly from experience/sensation without any intervening model. Here you seem to be saying that your knowledge of the usefulness of “acting-as-if-[the model]” is of this type. This is the “simply is” knowledge I’ve mentioned, which stops the infinite regress caused by the subsumption of all knowledge under a useful-to-act-as-if model.
Well is it really knowledge? Rather than deal with the ambiguity of that word, let me speculate in this way: Perhaps I don't "know" it is useful to act as if, I just get pleasure (some tiny twinge of satisfaction perhaps) from treating it that way. In other words, my subconscious has me Pavlovian conditioned to do so That might be the final connection between conscious and unconscious. I don't know; this is as deep as I've considered this so far.
Zavoi: AJ:By "reality" I specifically mean those sensations that appear to be more lasting and that cannot apparently be manipulated or influenced through pure thought alone, but only through the means of what I term my physical body. By this are you referring just to the “external world”?
By this are you referring just to the “external world”?
I am specifically avoiding reference to anything like that, or rather I am saying that if that term means anything to me, it means [the underlined text above].
Hopefully the above speculation about Pavlovian conditioning addresses your observation about knowledge and pseudo-knowledge. I still like the example of the bleep sometimes emitted by my computer when I click with the mouse. How do I know it means "The click will not be processed?" I don't really know, perhaps, but in the end maybe I just feel good acting as if the clicks that elicit a bleep will not be processed. So it is useful to me, but I don't make that statement at the deep level of analysis in terms of methodological individualism, just at the casual level as third party might as they are observing me and hearing my reports.
So perhaps that was the problem all along. I was (or should have been) making the final statement that "all beliefs are just useful-to-act-as-ifs" at the casual level, not at the deep subjective-individualist level. At the deep level, it may just be that it gives me a certain type of pleasure to act as if certain things, and I call those my beliefs.
I should probably add, for anyone who is just skimming this thread, that this would not necessarily mean anything like "I just believe what feels good" or "It's true if it feels right." But rather that the concepts of truth and belief are just useful heuristics. In the end, I as a consciousness strive to attain satisfaction and avoid pain, and of course my ability to think of anything at all is extremely limited without access to my unconscious memory and processing (not a scientific statement - although it may indeed be valid as one - but just a subjective observation). Since most of the processing apparently happens below conscious awareness, it is not too strange to imagine that pleasure and pain might figure into the interface between conscious and unconscious. You could think of it loosely as your genes motivating you to action, as pain and pleasure are ultimately the only real motivators.
Zavoi:But the matter of “who is doing the deeming” is important in that we should recognize that the person for whom X is useful is the same as the person who believes “I deem X useful,” and it’s not as though we can believe things about ourselves without those beliefs becoming aspects of ourselves.
I think I see where my explanation was misleading. Does the above clear that up?
AJ: I'd like to respond to Long's paper at length,
I'd like to respond to Long's paper at length,
Do it.
AJ: I can probably respond to specific points if anyone cares.
I can probably respond to specific points if anyone cares.
I care.
Could you pick some particular point? The paper raises so many subtle issues and points that I agree with, disagree with, or find relevant/irrelevant that it would take me weeks to cover it all (and I've barely even read half of it yet).
Plus Long is writing with an entirely different goal, and the points I find interesting are sometimes perhaps only incidental to the overall arc of his argument. I think he is writing to show how Wittgenstein backs up Mises, but I am writing here to espouse a more general perspective that happens to touch on some of the issues therein. So even where I take issue with the points raised I wouldn't be refuting the paper but rather talking about tangential issues of only eventual relevance (if any) to the paper itself.
AJ: Could you pick some particular point? The paper raises so many issues and points that I agree with, disagree with, or find relevant/irrelevant that it would take me weeks to cover it all (and I've barely even read half of it yet).
Could you pick some particular point? The paper raises so many issues and points that I agree with, disagree with, or find relevant/irrelevant that it would take me weeks to cover it all (and I've barely even read half of it yet).
I will try to get back to you on that. I just started reading it this morning, so I only just started the fourth chapter.
AJ:Perhaps I don't "know" it is useful to act as if, I just get pleasure (some tiny twinge of satisfaction perhaps) from treating it that way.
But by telling me this, you are indicating that you do know that it is useful to act as if (assuming “pleasure” and “usefulness” are interchangeable). Which brings us to…
AJ:So it is useful to me, but I don't make that statement at the deep level of analysis in terms of methodological individualism, just at the casual level as third party might as they are observing me and hearing my reports. So perhaps that was the problem all along. I was (or should have been) making the final statement that "all beliefs are just useful-to-act-as-ifs" at the casual level, not at the deep subjective-individualist level. At the deep level, it may just be that it gives me a certain type of pleasure to act as if certain things, and I call those my beliefs.
“It is useful, but I don’t know that it is.”
I think this is more or less getting to the heart of the matter, and shows why it is important to distinguish extension from intension. Yes, it may be the case (as seen by a third party, or, as you say, “at the deep level”) that my beliefs are just what it is useful* for me to act as if – just as the “beliefs” of PA are just what is deduced from the Peano axioms. This is a purely extensional statement. It is then tempting to say casually “All my beliefs are just useful-to-act-as-ifs,” discreetly moving to an intensional statement (as if to say P⊢(P⊢S→S)). I think our whole disagreement can be resolved by recognizing this distinction.
*(Where this sense of “useful” is not necessarily something I am aware of; e.g., in the same way that the food I eat may be healthy or unhealthy for me without my knowing this.)
AJ:pain and pleasure are ultimately the only real motivators.
This isn’t really relevant, but it’s something that’s been bugging me about praxeology for a while: Are we defining pain as “that which one is motivated to avoid” (and pleasure “to seek”)? If so, isn’t this an entirely uninformative statement? If, on the other hand, we are using the everyday definition of pain as “the sensation produced when my bodily tissues are damaged,” then the statement is by no means necessarily true (someone could be a masochist, suffering for the sake of truth, etc.).
AJ:I think I see where my explanation was misleading. Does the above clear that up?
I think so. This debate seems close to over.
Zavoi: AJ:Perhaps I don't "know" it is useful to act as if, I just get pleasure (some tiny twinge of satisfaction perhaps) from treating it that way. But by telling me this, you are indicating that you do know that it is useful to act as if (assuming “pleasure” and “usefulness” are interchangeable). Which brings us to… ... “It is useful, but I don’t know that it is.” Not exactly: the very question of what it means to "know" hinges on the perspective one chooses, and that we are apparently taking different side of. Zavoi, bold added:Yes, it may be the case (as seen by a third party, or, as you say, “at the deep level”) that my beliefs are just what it is useful* for me to act as if By the deep level I mean from an individualist-subjective perspective, where I assume nothing about any "reality" or "third parties." At the deep level, perhaps all that is happening is I get pleasure from certain actions, and by now I have long forgotten or pushed into the subconscious any awareness of this process. At a casual level - that of a third-party observing someone, for example - it might be said that "what she calls her beliefs just turn out to be her deemed useful-to-act-as-ifs; in other words, that is all she really means by belief." But like I said, intension and extension are matters of language, not of thought. Whether I call it "the morning star" or "the evening star," in my own mind I am simply, for example, seeing an image of Venus. Even if I believe that the morning star and the evening star refer to two separate things (say, two distinct images in my mind), I am simply then referring to those respective images. In other words, the whole issue of intension and extension is also part and parcel of the issues we've been discussing. Why anarchy fails | Post Points: 5
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Not exactly: the very question of what it means to "know" hinges on the perspective one chooses, and that we are apparently taking different side of.
Zavoi, bold added:Yes, it may be the case (as seen by a third party, or, as you say, “at the deep level”) that my beliefs are just what it is useful* for me to act as if