“To you it has been given to know the secrets of the kingdom of heaven, but to them it has not been given.”
Matthew 13:11, NRSV-CI
Phenomenon of Phenomenology
What in the world is a philonoemist (pronounced fill-o-noy’-mist)? You won’t find it in the dictionary, because I made it up (as a good patent lawyer, I know that one should not shy away from being one’s own lexicographer).
Like ‘philosopher’ means lover of wisdom (derived from the Greek words for love (philo) and wisdom (sophia)), ‘philonoemist’ means lover of meaning. The word ‘noema’ derives from the Greek word νόημα meaning ‘mental object.’ The philosopher Edmund Husserl used ‘noema’ as a technical term in phenomenology to stand for the content of a thought, judgement, or perception.
Since it is my contention that the lessons of Gödel’s incompleteness theorems (discussed here and here) are lessons about how to discover true meaning, and since Gödel deeply appreciated and drew from Husserl’s work, the term philonoemist seemed apt, if playfully so.
Husserl introduced phenomenology in the late 1800’s, just in time to influence Gödel as he came into his intellectual prime. Phenomenology is the philosophical study of how we, as conscious beings, experience things. Many now include it as a fifth main field of philosophy, among the big four of ontology, epistemology, logic, and ethics.
The Stanford Encyclopedia of Philosophy[1] offers a pithy comparison of each field in its entry on phenomenology, to wit:
- Ontology is the study of being (what is);
- Epistemology is the study of knowing (how we know);
- Logic is the study of valid reasoning (how we reason);
- Ethics is the study of right and wrong (how we act);
- Phenomenology is the study of our experience (how we experience).
For Gödel, phenomenology was more of a methodology than a new area of philosophical study; a methodology anchored in his belief that concepts and their meanings are first experienced and best known through intuition. He studied philosophy as a hobby rather than a profession, and to help him see reality more distinctly.
Gödel’s view directly opposes Wittgenstein’s view that any truth beyond the formality of logic systems should not be considered much less discussed, for the very reason that it was beyond the formality of logic system. In other words, philosophy should only be used to explore the world we already know objectively.
This objectivist view artificially reduces the universe of thought to only those truths that are provable, which is ultimately a trivial universe of ideas that become self-evident, one-by-one, as they emerge. It is predicated on the illusion that observation can be decoupled from intuition.
Every bit of information collected, whether it be from our senses or from scientific instruments, when it is interpreted, it is interpreted intuitively first. The operation of logic, reason, and rationality occurs only after filtering through intuition. This happens whether we are aware of it or not.
“Mechanical rules cannot obviate the need for meaning, and what gives us access to meaning, namely, intuition, cannot be dispensed with even in mathematics, indeed, even in arithmetic.”
Palle Yourgrau, A World Without Time, pp. 57-58
Abstraction is Extraction of Explication
Intuition requires the ability to generalize, analogize, and ‘abstract up.’ Intuition makes use of the concept that the form of a thing is the essence of the thing, following the language of Plato, Aristotle, and Aquinas.
In this vein, phenomenology is a labor of self-reflection. It attempts to reconstruct our original use of basic ideas in an attempt to recover what we meant in the first place by our most fundamental acts of thought.
This is fully consistent with how our capacity for reason works in the first place, and so to a certain extent is a natural exercise. But to fully harness its power requires discipline.
The method of phenomenology that Gödel cultivated involved “probing the limits of formal methods in capturing intuitive concepts.”[2] This recognizes that formal methods are necessarily limited, and being limited implies that there is truth and meaning beyond them, and if there is truth and meaning beyond them, perhaps we can use intuition to explore the beyond.
“The intuitions Gödel had in mind, however, were the result of the highly rational exercise of turning one’s attention to the nature of the concepts themselves, not of turning one’s ear to the marketplace of ‘ordinary language’ and everyday conversation. Language as such, for Gödel, had nothing to do with it.”
Palle Yourgrau, A World Without Time, p. 179
Once again, this is contrary with the Wittgenstein approach of focusing entirely on precise use of language and rules of syntax to avoid metaphysics, which he considered mere rhetorical trickery.
If intuitional delving into meaning was like metaphysical cave exploration, then Gödel’s spelunking tools would include recursion and self-reference to plumb the depths and test the limits, and analogy and generalization of concepts to find a way back up and out.
Gödel believed that generalizability confers (or at least implies) meaning. In other words, when a concept can be generalized, it has aspects that are applicable to a broader range of other concepts, thus forming a sort of meta-concept. This is meaning.
According to Gödel’s second incompleteness theorem (that a system’s consistency cannot be proven from within the system itself), the fact that consistency itself cannot be proven means that there is no way to ensure paradoxical contradictions won’t occur.
Indeed, when taken together with the first incompleteness theorem (that any non-trivial consistent system has true statements that cannot be proven), this ensures that paradoxes will occur by necessity.
Gödel proved these theorems by applying recursive formulae and self-referencing statements to a rigorous and formalistic framework to thereby expose the fact that paradox is not due to imprecision language or rhetorical tricks, but rather is something unavoidable.
Moreover, Gödel demonstrated that the truth and meaning in these unavoidable things called paradoxes cannot be examined from within the system, but required abstracting up. Or, playfully, extraction of elucidation from inextrication requires abstraction.
Smokescreen
What we are discovering is that the very presence of paradox indicates that a concept is real and meaningful. As Gödel said, “[t]he argument that concepts are unreal because of the unresolved logical (intentional) paradoxes is like the argument that the outer world does not exist because there are sense deceptions.”[3]
In other words, where there’s smoke, there’s fire.
When Gödel mentioned sense deceptions, he was referring to optical illusions, which provide interesting insights about the interplay between sensing (observing) and perceiving (intuiting). There is a real world out there that our eyes and brains are designed to make sense out of, and optical illusions demonstrate how the interpretation provided by our brains completely dominates the information received from our senses.
Take, for example, the famous shadow illusion shown in Figure 6. Squares A and B on the checkerboard are the same shade of gray, but we perceive them to be very different because our brains take into account, and compensate for, the shadow seemingly being cast by the cylinder.
Once you see the isolated squares side by side, there is no denying that they are the same shade, and yet we cannot keep our brains from overriding the signals received by our eyes and ascribing contextual meaning that unavoidably alters what we see.
Figure 6: Square A appearing darker than square B is only an illusion
Freely distributable image, credit to Edward H. Adelson
So which is real, the actual shades of the squares or the interpreted meaning of the image as a whole created in our minds? Certainly, the interpreted meaning is more useful, which seems to indicate that we humans don’t actually deal in observed facts (as the objectivists insist), but rather deal in concepts that are already filtered by interpretation and intuition.
Whereas Figure 6 presents the shading of squares A and B as the ‘illusion,’ it is ultimately the concept of the perceived shadow and the adjustments made automatically in our brains that is our reality. Similarly, while we may be tempted to treat concepts leading to paradox as mere illusions, the existence of the paradox is both an indication that a real concept exists and that a deeper (or higher) meaning is lurking.
As Gödel’s methodology recognized, effective thinking must appropriately blend the formal and the intuitive, the observed facts and the abstracted concepts.
Bootstrapping
What Wittgenstein got wrong was that language doesn’t confer meaning by virtue of its precision, but by its ability to generalize.
Just as mathematics is the language, and in some sense an abstraction, of science, so logic is the language and abstraction of philosophy. But if mathematics and logic are confined to their ability to precisely state rules of science and philosophy, their true magic is lost.
Mathematics contains truth and beauty independent from whether it describes the physical world (while admitting that the physical world is wondrous and beautiful as well, in part because it can be described by mathematics), and the same holds for logic independent from philosophy.
Objectivists contend that mathematics is only useful to the extent that it describes what science observes, much like logic is only useful to the extent that it describes what philosophy observes in objective reality.
Relegating mathematics and logic to usefulness only in understanding the objective reality of science and formalist systems while ignoring their direct and independent application to more abstract concepts is like thinking with half your brain tied behind your back.
For Gödel, the ability to generalize was the key to unlocking meaning.
“He then seemed to apply implicitly his principle of uninhibited generalization (and analogy) to infer that, if we think hard enough and see things in the right way, metaphysical concepts become as sharp and can be seen as clearly as the concepts of mathematics.”
Hao Wang, A Logical Journey, p. 9
This inclination to generalize even led Gödel to do what many philosophers have done since he published his incompleteness theorems, and indeed what I am doing throughout this blog, namely to generalize Gödel’s work in logic and mathematics and apply it to philosophy and religion.
While it is important to note that Gödel considered it appropriate to generalize his work, he also realized that metaphysical arguments are often only useful to those who already believe them. To those disinclined to assign truth to intuitions, the metaphysical arguments are easily disregarded and left unexplored.
However, a person who is willing to accept that a higher meaning exists may very well apply those intuited meanings toward looking for more convincing arguments, thus cobbling together a rationale that would not otherwise emerge, and in bootstrap fashion.
This echoes what we learned about the function of reason according to Mercier and Sperber, not to mention Aristotle and Aquinas, as well as how mathematicians think, as discussed here.
Ultimately, Gödel’s phenomenological methods were a rejection of materialism and an affirmation of the primacy of mind and its powers separate from matter. This was very much contrary to many of his contemporaries in science and mathematics who more and more were putting their faith in the impending discovery of physical mechanisms to explain the mind.
Ninety years after Gödel exploded onto the scene, we are still no closer to that panacea. In solving the deep mysteries, Gödel advocated a balanced approach stating:
“The possible worldviews [can be divided] into two groups [conceptions]: skepticism, materialism and positivism stand on one [the left] side; spiritualism, idealism and theology on the other [the right]. The truth lies in the middle, or consists in a combination of these two conceptions.”
Hao Want, A Logical Journey, p. 153
Science, by definition, can only speak from the materialism side, and thus science alone cannot represent truth. Likewise, pure mysticism can speak only to the spiritual side. So how to combine them?
Chesterton would argue that only Christianity has successfully combined these two conceptions in a non-trivial and sensical manner; in the same way that a key fits a lock.[4]
Fairy Tales are Real
This is not the only parallel between Gödel and Chesterton. The two shared an interest in fairy tales and a belief in their important function. It is said that Gödel’s preoccupation with meaning and precision stemmed from a yearning for security and certainty in his formative years, which perhaps also explains why he was drawn to fairy tales.
In a private letter, Gödel wrote, “Only fables present the world as it should be and as if it had meaning.”[5] For him, fables distill our relationship to the world, to mystery, and to each other, down to their essential meanings, and to that extent they are just as real as reality.
Chesterton lauded fairy tales in their ability to acknowledge both the darkness in the world and that the darkness can be defeated, as captured in the statement, “Fairy tales do not tell children that dragons exist. Children already know that dragons exist. Fairy tales tell children that dragons can be killed.” As delightful and believably Chestertonian as this quote is, it appears to be a summary of a longer passage from Chesterton’s essay, “The Red Angel,” and not a direct quote.[6]
Gödel’s quest for meaning was bolstered by the order he saw in the universe. For him, order implied meaning because without order there is disorder, and making sense out of disorder is not possible.
Gödel also believed in the principle that everything had a reason, and he never stopped trying to understand those reasons. In his efforts, Gödel demanded precision, perhaps looking for a sense of security in the certainty that rigor provides.
It seems that this belief in the existence of meaning and purpose coupled with the need for rigor and certainty were major factors contributing to the mental health issues that afflicted Gödel for much of his life.
In the Bible, Job found comfort in the security of not understanding, trusting that the God who created order had a plan, even though the design of that plan was incomprehensible. Job, like Gödel, had faith that everything had a reason. Unlike Gödel, Job took comfort in knowing that a reason existed even if that reason could not be understood.
Encoding Meaning to Unlock It
The methodology we’ve discussed here may help us to unlock true meaning. It starts simply enough by acknowledging the power of intuition, and quickly becomes complex as it looks to exploit the properties of the formal system being examined so that a higher-level meaning can be extracted.
In the proof of his first incompleteness theorem, Gödel exploited the system of formal arithmetic (or FA), utilizing self-reference and recursion to create a strange loop so that FA could be used to simultaneously talk about itself and talk about the syntactical language used to talk about FA, thus abstracting to a meta level.
“Gödel had succeeded in proving, then, that FA, though in itself a system of formal, meaningless signs, could be ‘double stuffed’ with meaning, i.e., assigned meanings that ensured that it could be used to represent, simultaneously, number theory and the syntax or proof theory of FA itself.”
Palle Yourgrau, A World Without Time, p. 65
In essence, what Gödel did was encode the instruction manual for the ‘machine’ inside the machine itself. In order to demonstrate how meaning could be unlocked through abstraction, he implanted the meaning in an encoded form that was designed to be unlocked in the same way that organically-existing meaning would have to be unlocked.
As I think about how Aquinas advised we go about interpreting scripture with all its embedded layers of meaning, prefigurements of Christ, and cross-references inviting the abstraction of higher truths, I can’t help but wonder whether what Gödel did to encode meaning in the system of formal arithmetic, God already did with Scripture.
[1] Found online at https://plato.stanford.edu/index.html
[2] Palle Yourgrau, A World Without Time, p. 138
[3] Hao Wang, A Logical Journey, p. 238
[4] “But in answer to the historical query of why [Christianity] was accepted and is accepted, I answer for millions of others in my reply; because it fits the lock, because it is like life. It is one among many stories; only it happens to be a true story. It is one among many philosophies; only it happens to be the truth.” G.K. Chesterton, The Everlasting Man, p. 158
[5] Palle Yourgrau, A World Without Time, p. 5
[6] The longer passage reads as follows:
“Fairy Tales then, are not responsible for producing in children fear, or any of the shapes of fear; fairy tales do not give the child the idea of the evil or the ugly; that is in the child already, because it is in the world already. Fairy tales do not give the child his first idea of bogey. What fairy tales give the child is his first clear idea of the possible defeat of bogey. The baby has known the dragon intimately ever since he had an imagination. What the fairy tale provides for him is a St. George to kill the dragon. Exactly what the fairy tale does is this: it accustoms him for a series of clear pictures to the idea that these limitless terrors had a limit, that these shapeless enemies have enemies in the knights of God, that there is something in the universe more mystical than darkness, and stronger than strong fear.”