Henry Flynt

(c) 1995 Henry A. Flynt, Jr.

The following is an experiment in specializing the "Is there language?" trap to enumeration (to an arithmetical posture generic to enculturation). The objective is to provide a refutation of arithmetic beyond any inconsistency proof attainable by logistic.

In order to be reading this, you the reader supposedly have faculties and knowledge. Alternately: there is a supposed shared culture which this exercise engages.

Specifically: The exercise is cast in English for the reader whose native language is English. The knowledge or shared culture includes the supposition that the natural language English (E) is achieved . Namely:

- Utterances are explicitly compound, and thus subject to partition into components and enumeration of components.

- The language can characterize itself in respect to enumerating the components in its utterances.

- The language can assert the existence of its utterances.

- Utterances are concretely realized: actual communication occurs.

- The language is your medium of thought.

The exercise entertains the illusoriness of natural language and arithmetic. Then isn't the careful exposition--especially the defined distinctions--a pedantic absurdity? The answer is that the exposition engages the knowledge you must have to be a reader: the presumed shared culture. English must be supposed to be shared, otherwise it is not a natural language. The shared culture purports to be a community of meaningful discourse (even though there are conceded to be isolated difficulties). As culturally competent, you are compelled to respect the competence which the exposition displays. In the shared culture, certain simple or basic distinctions or theses, even though simple or basic, should be solidly operative and should be accountable for. If the culture goes crazy trying to live with these distinctions, then the exposition is a controlled reduction to absurdity.

The specimen comes from concrete natural language which purports to be able to characterize its own utterances quantitatively. (E) Consider the supposedly non-uniquely self-applicable and self-validating sentence

(a) There is a concrete sentence containing eight words.

and the correlative question

(b) Is there a concrete sentence containing eight words?

The exposition is cast in the language which is supposedly your medium of thought. It is not necessary or desirable to dissociate the concrete sentence from the thought which the sentence expresses. For convenience, give the word "ideated" the definition "entertained mentally by myself." Then in addition to (a) and (b) above, consider

(e) There is an ideated sentence containing eight words.

and

(f) Is there an ideated sentence containing eight words?

The specimen, then, is a juncture of enumeration and your possession of a medium of thought. What is at stake is whether you have to endorse arithmetic in order to socially have a mind.

A refutation does not address a confession of ignorance. If you confess ignorance, you are dismissed. If you claim possession of an English nobody else has ever known, an English not written here or anywhere, or claim a cultural competence which denies the conditions presented above, then you cannot credibly uphold (a) and dispute the promised result. The refutation eventuates by default.

Marshalling the initial examples,

(a) There is a concrete sentence containing eight words.

(b) Is there a concrete sentence containing eight words?

Supplementarily,

(n) There is no concrete sentence containing eight words.

The item of knowledge (a) should be a contingent fact; its truth should be contingent. It should be doubtable and deniable without self-refutation. And yet, (a) is true a priori. It is true if it exists. (b) compels an affirmative answer; and (n) is self-refuting. In other words, (a) is (pro forma) self-validating; it is true in every case, including the case in which it is false or meaningless.

So (a) should be contingent. (Equally germane, it should be nontrivial.) But it cannot be. (Pedantically) (a) compels a contradiction in how you know it.

To elaborate on why (a)'s self-validation is outrageous.

Is it possible to expunge arithmetic from the consciousness which entertains 'There is an ideated sentence containing eight words'? Marshall the pertinent examples:

(e) There is an ideated sentence containing eight words.

(f) Is there an ideated sentence containing eight words?

(p) There is no ideated sentence containing eight words.

The existence of your medium of thought is underneath all specialized pedantic questions. You have to have the instinct for the mirage, for awaking. It requires self-detachment to realize that your apparitional discursive mentation, which ostensibly establishes itself, does not manifestly establish itself at all. What is at issue is not the apparition that you think in the linguistic medium--but the presumption that this mentation is not-gibberish. If you think the detachment is a pose, purely pro forma, you remain in the mirage.

You can't think 'There is no ideated sentence containing eight words' without validating 'There is an ideated sentence containing eight words'. But have you thought 'There is no ideated sentence containing eight words'?

(e) should be a contingent fact. It should be doubtable and deniable without self-refutation. And yet (e) is true a priori. (e) is true in every case, including the case in which it is false or meaningless. (e) compels a contradiction in how you know it.

(c) There is at least one utterance.

(d) Is there at least one utterance?

(q) There is not even one utterance.

Here the outcome no longer depends on the possibility of partitioning of sentences into components.

(c) should be a contingent fact. It should be doubtable and deniable without self-refutation. And yet (c) is true a priori. The reality of "one-or-more" quantification is compelled. (c) is true in every case, including the case in which it is false or meaningless.

So (c) should be contingent; but it cannot be. (c) compels a contradiction in how you know it.

One could question whether arithmetic is involved here--whether this case says anything about whether it is possible to expunge arithmetic from the consciousness which entertains 'There is at least one utterance'. All of arithmetic has been contracted to the opposition of "not even one" and "one or more."

The answer is that an inconsistency is exposed in the epistemology of the opposition of "not even one" and "one or more." So, roughly, the distinctiveness of zero and one is absurd.

Extraneous afterward for the academic fool

In the academic discipline called foundations of mathematics, there are rebuttals of various "radicalisms." Those radicalisms were so insincere that they should never have been announced in the first place. Instead of quoting those rebuttals, certain digressions are included here to deprogram the academically regimented reader.

The reader may have been indoctrinated to the effect that: structure--as of a presumed row of symbols--is a palpable property, ascertainable by visual inspection. Consider the token-series

Substituting periods to indicate which of the elements are the parenthesized ones, the structure of the token-series in question can be construed as ((.))((.)), or as (((.)(.))), etc. One could "empirically observe" the token-series for twenty years--or even peer at it through a very scientific magnifying glass--without discovering which structure is meant or which one is correct.

Concerning the notion that arithmetic can be founded on concrete tokens, all proponents of this notion quickly abandon the concrete token for the abstract type. If tokens really were the issue, then arithmetic would become inseparable from perception and hermeneutics. Given the inscription

one must impute which side is up, as that affects meaning and truth. To use words to say which side is up would be a confession that mathematics is not autonomous from verbal language. (And so much for the notion that stroke-arithmetic needs no hermeneutic.)

Given the contextless inscription

one must impute whether it is a number or a word. To use words to cue this would be a confession that mathematics is not autonomous from verbal language. How then is a mathematical inscription to be distinguished from a verbal inscription?

Consider

(g)

and

(d)

You may have been indoctrinated to overlook the distinction between objective linguistic properties and properties apparent to subjective inspection. You are likely not to register the distinction except when confronted with unknown foreign languages. Referring to (g) and (d), if either is true, then it is true by self-validation. But the attitude that "of course" (g) is true and (d) is false is too easy, mistaking opinion for the establishment of certainty. Assuming, for the sake of making the point--and as a hypothesized proponent of arithmetic would--that languages are objectively real, then the facile judgment overlooks the possibility of a distorted mental state in which you misidentify the language you understand, for example. The inference from seeming identity to objective identity cannot attain certainty so easily. The common-sense judgment that (g) is true and (d) is false does not any more penetrate opinion than proving the existence of God by asking a priest. If this challenge to common and witless inferences seems bizarre, that is because inferences of this sort are almost never exposed and challenged.

In dreams, you may mis-identify the language used.

The illustrations here were framed with the directly self-referential locution 'This sentence', which is available in E. That locution is suspect. But that does not matter, because once the distinction between objective linguistic properties and properties apparent to subjective inspection has been raised, it applies everywhere in a supposed natural language.

- the contradiction is an incrimination no matter how much you wish to swallow it;

- lowering your standards cannot positively establish anything.

Why is "This sentence is in French" a self-validating falsehood? If you happen to be "in the illusion" that the sentence is in French, then mere exhibition of the sentence is manifest proof that it is French. This is a level of skepticism or detachment which the common-sense mind does not experience.