Starting at least as long ago as Euclidean Geometry ~300 BC and lasting until the 1930s, there was a generally held assumption amongst scholars that it should be possible, in principle, to build mathematical/logical systems within which any statement constructed within the rules spelled out by that system could be unambiguously determined to be either true or false. The epitome of this academic tradition was the publication of a three-volume work, Principia Mathematica in 1910, 1912, and 1913 by Alfred North Whitehead and Bertrand Russell. They attempted to build, from the ground up starting with only a small number of axioms and inference rules, a formal system within which all properly formulated mathematical statements could in principle be determined to be either true or false. Examples of statements that could in principle be proven to be either true or false within this system would include the relatively simple such as, “2 + 2 = 4”, to the more complex such as Fermat’s Last Theorem, “no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two.”
No scholars are currently continuing to work on this project. Why not? Because it is now recognized that this project is doomed to failure. Just as no intelligent person would continue working on trying to build a perpetual motion machine now that we understand the implications of the Laws of Thermodynamics; Similarly no one who understands the implications of a paper published by Gödel in 1931, “On formally undecidable propositions in Principia Mathematica and related systems I”, would continue trying to build a formal system within which, in principle, all mathematical truths could be established. Gödel’s paper proved that any formal system that grows complicated enough that it can allow statements having the property of Self-Reference will not be capable of deciding whether or not some of those self-referencing statements are true or false.
And the implications of Gödel’s incompleteness theorem go far beyond mathematics. Epistemological questions having to do with how certain we can be about what things mean and how we establish whether or not assertions can be categorized as being either ‘true’ or ‘false’ were knocked off their presumed foundations.
Here is an example that illustrates the effects of self-reference within the domain of the meaning of language. Consider the following sentence,
“This sentence is false.”
Now try to decide if this sentence is ‘true’ or ‘false’. Thinking about this leads to what Hofstadter  characterizes as a strange-loop. If one decides that the statement is ‘true’, then reflection immediately leads to a contradiction, since the statement itself asserts that it is ‘false’. However, as soon as one then switches to the opposite conclusion, that the statement must be ‘false’, reflection reveals that what the sentence asserts, that it is ‘false’, is ‘true’, so it makes no sense to characterize it as ‘false’. The reason for this back-and-forth inability to decide whether the sentence is ‘true’ or ‘false’ is that the sentence involves self-reference, it is making an assertion about itself. And the formal proof about why self-reference makes this sentence undefinable in terms of being ‘true’ or ‘false’ is to be found in Gödel’s incompleteness theorem.
But Gödel’s theorem was not published until 1931, and was not appreciated by anyone except a small number of mathematicians and logicians for decades after that. That historical fact is the reason I found it remarkable to discover repeated uses of various forms of self-reference in passages dealing with epistemological issues in Moby Dick, a book published in 1852.
I have not done exhaustive research on Melville, but based on a perusal of a few anthologies that contain some of his personal letters and published short works, and having now read Moby Dick, I would characterize his epistemological position as being something along the lines: Experience can not be reduced to a single meaning, regardless of how exhaustively it is written about, depicted in the visual arts, systematized scientifically, dogmatized by theologians, subjected to critical analysis, reflected upon, or lived. It always remains in the end, mysterious, and subject to ongoing multiple interpretations. That applies to “Whales”, it applies to his book about a whale, “Moby Dick”, and it probably applies to most everything else in life.
Obviously, Melville did not have explicit knowledge of the reasoning employed in Gödel’s incompleteness theorem. However, the fact that he embedded many of his epistemological discussions within language that involves self-reference suggests that he perhaps had some kind of an intuitive understanding of the utility of self-reference in advancing his position. Many aspects of Melville’s writings have convinced me that he was an intellectual genius, so the conjecture that he might have had an intuitive sense of some of the implications of Gödel’s incompleteness theorem does not strike me as completely implausible. In the remainder of this essay, I will explicate three examples from Moby Dick where epistemological questions about ‘meaning’ and ‘truth’ are discussed within self-referential passages.
One early example occurs in the first three paragraphs of Chapter 3 where Ishmael describes a painting he is looking at on the wall of the Spouter-Inn:
Entering that gable-ended Spouter-Inn, you found yourself in a wide, low, straggling entry with old-fashioned wainscots, reminding one of the bulwarks of some condemned old craft. On one side hung a very large oilpainting so thoroughly besmoked, and every way defaced, that in the unequal crosslights by which you viewed it, it was only by diligent study and a series of systematic visits to it, and careful inquiry of the neighbors, that you could any way arrive at an understanding of its purpose. Such unaccountable masses of shades and shadows, that at first you almost thought some ambitious young artist, in the time of the New England hags, had endeavored to delineate chaos bewitched. But by dint of much and earnest contemplation, and oft repeated ponderings, and especially by throwing open the little window towards the back of the entry, you at last come to the conclusion that such an idea, however wild, might not be altogether unwarranted.
But what most puzzled and confounded you was a long, limber, portentous, black mass of something hovering in the centre of the picture over three blue, dim, perpendicular lines floating in a nameless yeast. A boggy, soggy, squitchy picture truly, enough to drive a nervous man distracted. Yet was there a sort of indefinite, half-attained, unimaginable sublimity about it that fairly froze you to it, till you involuntarily took an oath with yourself to find out what that marvellous painting meant. Ever and anon a bright, but, alas, deceptive idea would dart you through.—It’s the Black Sea in a midnight gale.—It’s the unnatural combat of the four primal elements.—It’s a blasted heath.—It’s a Hyperborean winter scene.—It’s the breaking-up of the icebound stream of Time. But at last all these fancies yielded to that one portentous something in the picture’s midst. That once found out, and all the rest were plain. But stop; does it not bear a faint resemblance to a gigantic fish? even the great leviathan himself?
In fact, the artist’s design seemed this: a final theory of my own, partly based upon the aggregated opinions of many aged persons with whom I conversed upon the subject. The picture represents a Cape-Horner in a great hurricane; the half-foundered ship weltering there with its three dismantled masts alone visible; and an exasperated whale, purposing to spring clean over the craft, is in the enormous act of impaling himself upon the three mast-heads.
It is readily apparent that these three paragraphs can be read not only as a literal description of a painting, but also metaphorically as a description of the book Moby Dick. Thus, this three paragraph passage contained within the book Moby Dick is making a reference to the book Moby Dick, or to put this in other words, Moby Dick is making a self-reference. And what is this passage about? It is a speculative discourse about, What are some of the ways one might go about trying to discern the meaning of this painting/book!
There are several other places in the book where the self-referencing is done explicitly rather than indirectly through metaphor. I will provide two examples here. First is the opening paragraph of Chapter 45:
So far as what there may be of a narrative in this book; and, indeed, as indirectly touching one or two very interesting and curious particulars in the habits of Sperm Whales, the foregoing chapter, in its earliest part, is as important a one as will be found in this volume; but the leading matter of it requires to be still further and more familiarly enlarged upon, in order to be adequately understood, and moreover to take away any incredulity which a profound ignorance of the entire subject may induce in some minds, as to the natural verity of the main points of this affair.
And what is the epistemological argument advanced in this self-referencing passage? A question is being raised about whether or not the reader should accept what the narrative of Moby Dick asserts about whales as being the truth!
One of the most complex uses of self-reference is in the Town-Ho’s story recounted in Chapter 54. In a self-referential passage Ishmael informs us that he is going to relate some events that are happening on The Pequod during an encounter with another ship, The Town-Ho, but he is not going to relate these events as they are happening “right now” (in the narrative). Instead, he is going to relate the events to us as he had later related them to some of his friends in Lima Peru (date not specified, but the impression gained while reading the remainder of the chapter is that this must have taken place months or years later than “right now” in the narrative):
For my humor’s sake, I shall preserve the style in which I once narrated it at Lima, to a lounging circle of my Spanish friends, one saint’s eve, smoking upon the thick-gilt tiled piazza of the Golden Inn.
The story(s) narrated in the remainder of the chapter combine self-reference with several levels of recursion, making it difficult for the reader to keep track of where we are in (narrative) time. This is reminiscent of the strange loops Hofstadter describes, well over 100 years later, that are sometimes created when self-reference is combined with recursion. And at the end of this chapter, what epistemological issue is addressed? Whether or not what the narrator has described is ‘true’!
1. The mathematician Roger Penrose’s book, The Emporer’s New Mind (1989) provides a detailed step by step description of how to construct self-referencing statements whose truth or falsity can not be determined within the system in which they are formulated.
2. Douglas Hofstadter in his book, Gödel, Escher, Bach: An Eternal Golden Braid (1979) provides extensive examples of how the implications of Gödel’s incompleteness theorem apply to other domains in addition to mathematics.
3. In fact I don’t think the implications of Gödel’s incompleteness theorem have yet percolated through much of even college educated society, even now in the second decade of the 21st Century. Personally, I would probably rank it among the top 4 most important intellectual discoveries of human history:
(Copernicus/Galileo: The earth is not the center of the universe;
Darwin: Humans are not qualitatively distinct from other animals;
Freud: Human thinking and motivation is often not rational;
Gödel: There are things that are true that we will never be able to prove algorithmically.)
However, whenever I quizzed students in my college classes about this issue (prior to my retirement from teaching a few years ago), few of them had even heard of Gödel, and of those few who had, less than a handful appear to have understood the implications of his theorem.
4. The Melville Log, Volumes I and II, edited by Jay Leda, Harcourt, Brace & Co, 1951; The Portable Melville, edited by Jay Leda, Viking Press, 1952;
Great Short Works of Melville, edited by Warner Berthoff, Harper and Row, 1969.