© 1995 Henry A. Flynt, Jr.
My 1961 text "Concept Art" outlined the genre which I called concept art. This genre depended crucially on my vitiation of the knowledge-claims of mathematics--correlative to Philosophy Proper. Actually, a negative universal result enters here. A rigorous presentation would have to be positioned to avoid being self-defeating. The present explanation is only heuristic.
I rejected the content claims made for traditional mathematics.--And I rejected the claims for proof-discovery, in a syntactical sense, made both by traditional mathematics and by the new logistic or positivist formalism. Let me expand.
For knowledge-claims to be warranted, the relation of a word to its intension would have to be an objective (metaphysically real) relationship. (The tenet of the real, invisible link of token or type to meaning.)
For the knowledge-claim that "a proof is discovered" to be warranted, the relation of rule to instance would have to be an objective (metaphysically real) relationship. (It's a strict analogy to the case of the word.)
Philosophy Proper nullified such beliefs. (Along with their negations. That is why it is no better than heuristic to announce that "a word does not have an objective relation to its intension.")
Translating into professional jargon, one might say that I attacked the semantics of metamathematics. (On the other hand, any attempt to borrow the jargon is dangerous, because the professional usage is subtilely narrower than the meanings which semiotic terms have acquired colloquially.) I saw the pretensions of metamathematics as a displacement of the indefensible pretensions of mathematical Platonism.
"Concept Art" does not make a syntactical inconsistency proof the crux of the matter. Anticipating what I will say below, a syntactical inconsistency proof would be an astute hypocracy result to show that consistency is not unassailable.
Let me dwell for a bit longer on vitiation of the knowledge-claims of mathematics. With his famous Second Problem, Hilbert had popularized the notion that the entire question of the validity of mathematics came down to showing that a game whose rudiments were mechanically specifiable would not yield an inconsistent outcome.
Replying, my appraisal is as follows. To present the consistency question as the ultimate cognitive issue concerning mathematics is an outrageous piece of misdirection. When you don't take the semantics of the metatheory seriously, then the game is a lie--its consistency is no more than a pseudo-problem. In none of my 1961 or 1962 work did I call explicitly for an inconsistency-derivation in a conventionally accepted mathematical theory.
It is not my topic here, but Hilbert's stance concealed another deception. Even though the test of consistency may be used to sieve out junk from mathematics, the major ideas are not discarded when they are found inconsistent. The proclamation of consistency as the ultimate test of the worthiness of mathematics was insincere. Hilbert's metaphor of dragging theories before a judicial tribunal and banishing inconsistent theories forever was utter propaganda.
1961 concept art does not begin until logic and mathematics have been recognized to be "false." Concept art renounces truth-claims as a purpose for mathematical systems. A non-cognitive use of language is permissible in which the reader treats the text like a Rorschach blot. The notion of a syntactical system is borrowed, but the system becomes "tokenetics-rich."
The preliminaries explain why there was an almost exclusive emphasis on syntax in the original concept art. Concept art needed the flattening of mathematics to syntax, so that it could disrupt the enterprise metasyntactically. I utilized the projection of mathematics onto its logical tree-structure because I wished to scramble this tree-structure. If it helps you understand to think of this approach as cruel, you are free to do so.
I thought that I had been driven to concept art by a series of intellectual hammer blows: positivism, formalism, Cage's indeterminacy. All mathematics would be replaced by an endeavor which involved a Rorschach-blot semantics, and which did not claim to be cognitive, at least not in the inherited sense. Its value was a beauty which was non-sentimental; later I would say "the invention of new mental abilities."
Many years later, I was told that "the public doesn't know that mathematicians claim beauty as a value of mathematics." I was told that I had done something illicit in appealing to such an arcane idea. That indeed was one cause of my inaccessibility--that I ranged freely across the walls which separated science people from poetry people, etc.
Whatever involvement I had at that time in bizarre models (again, to risk the professional jargon) would be more nearly shown by "Energy Cube Organism." But note: even when I bring this in this enterprise, I still have not asked for an inconsistency derivation in a seemingly consistent (serviceable) theory. "Energy Cube Organism" asks for an overtly inconsistent system of propositions to be given a realization "in the world."
One who calls concept art elitist doesn't understand to what branch of endeavor it belongs. Concept art is not my entry in a bric-a-brac contest. It needed to be done, I would say it was inevitable--for reasons independent of popularity.
My turn to the problematic of inconsistency proofs in 1977, with "Proving that (the metatheory of) arithmetic is inconsistent," harbored a degree of confusion; at the same time, it radically altered the problematic of the "inconsistency derivation."
One reason for me to address syntactical consistency was that if consistency of the system was an unassailable absolute, that would belie my cognitive nihilism. But I didn't need to compete with Wette, or to pull a Russell or a Kleene-Rosser or a Rosser. I could kill the system at a pure metalinguistic level--as above.
The endeavor did not clarify until 1988. It was about whether you have to invest in arithmetic in order to culturally participate in the "game" called knowledge and to socially have a mind. Is arithmetic inescapable in the purported containing shared medium of thought? The upshot was selection of a question, oriented to culture and consciousness, which identifies the truth of arithmetic with common sense--or more to the point, with the existence of language. So a correlative of the "Is there language?" trap emerged.
I capped the venture off by proposing that my new "paradox" be rotated to become a horizontal inconsistency derivation (which indulges the conformist game-rules, or metalogic). This proposal was a rash speculation, a conceit. I imagined a universal inconsistency derivation in any "consistent system" strong enough to code a certain portion of its metalogic. Such a result would be astute hypocracy. If it needed machinery reminiscent of the Diagonalization Lemma, I wouldn't consider it anything more than sophistry. Its only real value would be to rattle the profession.
Let me reproduce some of what I wrote in 1988 about whether arithmetic is culturally inevitable.
Let us consider 2+3=5, or,
(1) Two and three makes five.
(Or for that matter, consider 2+3=4, or,
(2) Two and three makes four.
Are "two', `three', `four', and `five' nouns, or are they quantifying modifiers?
A whole number could be conceived as a quantifying modifier in such a sentence as
(3) Three pens lay on the table.
Ontologically, the subject of the sentence is nonmathematical beings (pens); any implicit reality-claim pertains to them. "Three" does not appear as a being in its own right, but only as a quantifying modifier of the pens.
But if this is the way [experiential] that number is understood, then the eternal laws of arithmetic can fail at any time. Suppose you have a stream of soap bubbles. You count two of them, then another three of them. Then you recount the totality to ascertain the sum. It is perfectly possible that you will get four, not five. (And, given a plurality of absolutely indistinguishable entities, you are also at risk of counting one entity twice.)
It may be objected that (i) the circumstance that the bubbles can vanish, and (ii) the recount to establish the total, are irrelevant. It may be said that once you have counted two, and again three, then you have "five counted"; recounting doesn't enter into it. But this objection treats 2+3=5 like a definition, and then the numbers are not quantifying modifiers. I will treat the "definitional" conception shortly; here let me finish with quantifying modifiers. If you test 2+3=5 on the basis of authentically temporal counts of apparitions as psychological acts, it can turn out false if the apparitions are such as to appear and vanish. ["It's no fair because the entities are evanescent." Counting itself is always evanescent.] Curiously, the counting itself is an enunciation of counting-tokens successively, so that the counting-tokens themselves normally arise and vanish. Also 2+3=5 will be false if one is adding volumes of water and alcohol. Also it will be false if one adopts the standpoint of perception, and considers material bodies to have vanished if they disappear when one's perceptual field shifts.
To fall back on the arithmetical laws as definitions, here, exposes something about arithmetic as ideology and psychic deformation. By imposing definitional arithmetic laws on apparitions, one makes apparitions more "stable" than they really are. Note that if you replaced the stream of soap bubbles by a photograph of them, you would have replaced the bubbles with a plurality whose reality-characteristics more nearly corresponded to pure addition's laws. Arithmetic doesn't apply to the world of things; it contributes to constituting the world of things.
Impositions of laws of arithmetic as definitions makes the apparitions more fixed, cut-and-dried than they really are. Extra-experiential patterns are imposed on apparitions.
We have already noted the other conception of the numbers in e.g.
(1) Two and three makes five.
--namely that the number-names are nouns, subjects and objects of sentences. Now reality-claims about [the reality-character of] nonmathematical beings are [is] irrelevant to arithmetic. So the number-names are nouns; what are the beings which they denote? The question has no answer which is a common-sense answer. The nature of pure whole numbers, abstract whole numbers, is a question which only receives esoteric, sectarian answers. (Similarly for the "definitional" interpretation of arithmetical laws--on which I will comment more below.) In this case, arithmetic behaves like a crackpot discipline. It is not cultural enough to provide a target for my type of attack.
I am placed in the embarrassing position of having to do mathematics a favor before I can annihilate it. The burden is on me to give life to the straw man before I can kill him. The question is whether there is any concept or law of mathematics which can be entirely assimilated to the broadest purported shared medium of thought--which is indispensable to natural language or common sense. It appears that 2+3=5 does not have any single meaning or function. It is used as a quantifying modifier relative to simultaneously present material bodies. Some consider it to be a truth about operations in an authentically temporal mental sequence. It is presented as an absolute abstract truth. Etc. Either it is experientially falsifiable; or else it is an esoteric matter on whose nature no two experts can agree. I am put in the position of having to do mathematics a favor--of trying to show that it is genuinely cultural--running through all possible roles of 2+3=5 until I find a role which you have to invest in in order to have cultural participation in knowledge at all and to socially have a mind.
The point is that one can participate in language without espousing absolute abstract arithmetic at all. The sentence
(4) The dog bit the man.
has a noun phrase of two words and a verb phrase of three words, adding up to a sentence containing five words, but you don't have to know that. You don't even have to have writing. It is claimed that the deepest involvement of arithmetic in language is in the grammatical category of number. (Nouns, pronouns. Dog/dogs; it/they.) (This implies that if there is a facet of arithmetic which one has to invest in, it is enumeration.) But if number-endings were lost, that would not stop one from talking (even if it were thought to result in a loss of information). And in any case, the matter of number-endings concerns what I have called numbers as quantifying modifiers. Only reality-claims for the nonmathematical beings which are the subjects of the sentences matter; and 2+3=5 can be experientially falsified at any time.
When number-names are nouns and are expected to have denotations, it is an embarrassment to the experts to be asked to produce these denotations. In the twentieth century, experts began to claim that arithmetic is a game. (Perhaps this is what is implied when arithmetical laws are conceived as "definitions.") But consider:
"Everybody knows that this move is illegal." But what is its illegality? Are mathematicians going to translate what a game is and what rules are into the containing shared medium of thought--into natural language and common sense? Certainly there is no standard way of doing this. In fact, mathematicians have strenuously avoided the translation of "game"-truths into culture. The conception of mathematics as a game is a sectarian conceit--and cannot even be a target for my negative universal approaches.
Advanced mathematics is deeply unclear and duplicitous about where the numbers it invokes are located on the spectrum of conceptions of number portrayed here.