“Paradoxes have always seemed specifically designed to convince us that we are simply not smart enough to take up whatever topic brought us to them.”
Rebecca Goldstein, Incompleteness, p. 50
Conceit and Exhaustion
It cannot be over-emphasized how fearful we can become when confronted with our own limitations by paradox. This is particularly true for those advancing the cause of human progress through science, mathematics, philosophy, psychology, sociology (and pretty much any other field of study that shuns mysticism).
However, it is conceited to think that the lesson of paradox is that we are not smart enough. It is conceited because it assumes that paradox is a surmountable problem if we only knew a little more or tried a little harder.
The issue is not our intelligence, but rather the inherent limitations of our existence in this world. In other words, we need to recognize there are transcendent truths that cannot be understood except through submission to a higher power and purpose, not begrudgingly admit of our stupidity and try harder next time.
Despite the truth of Gödel’s incompleteness theorems, which demonstrated that paradox is unavoidable, many elite intellects persist in couching paradox as something to be removed, dismissed, and, as a topic of conversation, ostracized. We will call these elites “objectivists.”
As we’ve seen in earlier posts, Wittgenstein and Russell belonged to this elitist club of objectivists. Joining them was twentieth century scientific philosopher Willard Van Orman (WVO) Quine, who turned up his nose particularly high at metaphysics and paradox. Talking specifically about Quine’s contributions to philosophical discourse, the contemporary British philosopher Paul Horwich commented:
“[I]t is now widely believed that the sciences exhaust what can be known and that the promise of metaphysics was an intellectually dangerous illusion.”
Palle Yourgrau, A World Without Time, p. 168
The objectivists believe that the job of science is to know everything that can be known, and that there is nothing left after everything is known. A consequence of this belief is that concepts within logical systems have no meaning beyond their usefulness in discovering or explaining other concepts.
This is intellectual nihilism.
As Gödel demonstrated, the only logic systems that can be exhausted of all true statements are ones that are so simple that they are trivial, and therefore already have no meaning. In contrast, all non-trivial systems are inexhaustible of their truths, and thus cannot be rendered meaningless.
Canaries in the Objectivist Coalmine
How fascinating that the objectivists were more troubled by the existence of paradox than they were about concluding that the search for meaning is pointless. The inability to talk about anything worth talking about concerned them less than having to admit that there was anything unknowable.
For Wittgenstein in particular, “[u]nknowability is regarded as a sign that a mistake in the use of language has been made.”[1] He along with Russell and their ilk dismissively lumped paradox into the myriad of philosophical problems that arise from syntactical confusion and ambiguous language. In doing so, they ignored any prospect of discovering meaning from paradox.
For objectivists, meaningfulness derives from verifiability. Paradoxes in logic are like dead canaries in the coalmine – if you find one you better get out quickly.
Interestingly, objectivists also admit that the criteria for what constitutes meaning cannot be verified. As such, by their own logic, objectivists must either concede that (1) not everything meaningful is verifiable or (2) that nothing is meaningful. The objectivists trapped themselves in a logical box of their own making, and happily locked it from the inside.[2]
In their effort to remain consistent and avoid all paradox, the objectivists doomed themselves to transacting in a worthless intellectual currency, like playing with Monopoly™ money.
“For Wittgenstein, nothing that was of genuine value – such as the beautiful, the good or the meaning of life – could actually be stated (as opposed to ‘shown’), and everything that could be said, which amounted to the substance of physical science, was absent of value.”
Palle Yourgrau, A World Without Time, p. 166
This is the consequence of refusing to engage with paradox for fear of admitting the system has limitations, a system that the objectivists insisted could be designed in such a way to remove all contradiction and exhaust all truth.
A Whirlwind Tour of the Tractatus
Let us dig a bit more at the objectivist philosophy by spending a few paragraphs exploring Wittgenstein’s most cited work, the Tractatus Logico Philosophicus.
The introduction of the Tractatus, written by none other than Bertrand Russell, begins with the innocuous-sounding premise, “The whole function of language is to have meaning, and it only fulfills this function in proportion as it approaches to the ideal language which we postulate.”[3]
Russell is saying that it is possible to devise a perfect language where everything has a definite and unambiguous meaning, and that through such a language a complete and consistent system of logic can be built.
He continues, saying, “Most questions and propositions of the philosopher result from the fact that we do not understand the logic of our language.”[4] In this one statement, Russell neatly dispatches the bulk of philosophy going back millennia, before Socrates and ancient Greece, as a mere misunderstanding between logic and language.
So much for the whole ‘standing on the shoulders of giants’ thing.
On this foundation of dismissiveness, we can explore the Tractatus proper. The book is presented as a series of statements and passages that follow one from the other without justification or argumentation, as if they are self-evident. We will concern ourselves only with a few passages that reveal Wittgenstein’s attitudes toward paradox, logic systems, and the role of philosophy.
As an initial matter, Wittgenstein asserts, “What can be said at all can be said clearly; and whereof one cannot speak thereof one must be silent.”[5] This sounds like today’s cancel culture applied to paradox. In fairness to Wittgenstein, the assertion is not so much a dictate to others as a self-imposed rule he is applying to his own work.
He states a similar concept later in the body of the Tractatus, saying, “Everything that can be thought at all can be thought clearly. Everything that can be said can be said clearly.”[6] Fair enough. We can all agree that clear thinking is worthwhile.
In proposition 3.13, Wittgenstein sets forth the basis for why clear and unambiguous language is so important. He says that words are the form that represent a meaning, but they are not that meaning themselves. This implies that paradox is a rhetorical trick in which substance is mistakenly ascribed to something that is only form.
For example, consider the self-referencing statement: ‘This sentence is false.’ Wittgenstein would dismiss the paradox as sleight of hand because the phrase ‘this sentence’ is merely form, and form cannot represent itself without violating propriety. Easy peasy, Viennese-y.
Next, Wittgenstein builds on his view that inconsistencies in logic can be avoided by properly devising a language that has no ambiguity. In proposition 3.325, he announces the need to employ a rigorous symbolism that obeys a set of logical syntax rules so that all ambiguity can be excluded.
But this begs the question of Wittgenstein’s original assumption, namely ‘do all inconsistencies indeed arise from ambiguity?’ It seems that Gödel’s results suggest otherwise – that complete, consistent systems of symbolism and syntax are not possible even if perfectly constructed.
For Wittgenstein, however, the theoretical possibility of a perfectly constructed system of logic is tantalizingly close. All that remains is removing the paradoxes due to self-reference, like Russell’s Paradox. And so he stipulates, “No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the ‘whole theory of types’).”[7]
This is the equivalent of disallowing in set theory ‘let S be the set of all sets….’ For Wittgenstein, this is not an arbitrary rule, but follows directly from words being form that represents meaning, and not the meaning itself (and as such cannot properly refer to themselves as concepts, but only as a form that stands for a concept).
In proposition 3.333, Wittgenstein demonstrates this point by showing that a function cannot be its own argument because, if it was, then the symbol representing the function would mean more than one thing. And in a perfect logic system nothing can mean more than one thing.[8]
This is how Wittgenstein attempts to get rid of paradox–by simply not allowing self-reference or recursion to exist as a matter of syntactical rule. Interestingly, this seems to assume that self-reference is the source of all problematic ambiguities and paradox.
But we know something that Wittgenstein apparently did not. Per Gödel’s theorems, adding rules in this way (that is, by fiat or by axiom), even if justifiable, will not lead to completeness and consistency.
At this point in the Tractatus, Wittgenstein has satisfied himself that a perfect system of logic is constructable, and would result in removing all sources of ambiguity and paradox. He thus proceeds to make some statements about the properties of the logic system and the propositions it contains.
One of the most stunning and philosophically self-revealing statements is set out in proposition 3.42, which stands for the idea that all ‘logical space,’ that is, all possible true statements in a system of formal logic, can be traversed by other true propositions, and indeed starting from any one of such propositions.
3.42 “Although a proposition may only determine one place in logical space, the whole logical space must already be given by it.
(Otherwise denial, the logical sum, the logical product, etc., would always introduce new elements – in co-ordination.)
(The logical scaffolding round the picture determines the logical space. The proposition reaches through the whole logical space.)”
Once again, this contradicts Gödel, who proved that there are nooks and crannies in logical space that cannot be reached from the logical scaffolding, no matter how intricately that scaffolding is constructed.
But Wittgenstein’s proposition 3.42 also produces a startling admission–that all true statements in the system are in essence already known from any other true statement. This describes a tautological system, one of mere truism and apparent meaningless triviality.
And if you said as much to his face, Wittgenstein would not disagree.
Wittgenstein considered himself a philosopher, and given his own conclusions about the logical system he envisioned, it is not surprising that he considered the purview of philosophy to be somewhat limited. He stated that, “The object of philosophy is to clarify thought and not to develop new propositions.”[9]
Philosophy was not about intuiting transcendent truths that can neither be proven nor disproven, nor was its business getting in touch with the infinite. Rather, “Philosophy should make clear and delimit sharply the thoughts which otherwise are, as it were, opaque and blurred.”[10]
Wittgenstein accused much of what passed for philosophy with muddying the waters rather than clarifying.
To his credit, Wittgenstein recognized that the mystery of life cannot be solved inside of the universe. But his reaction was not to look outside, but to devise a system in which we dare not even ask the question.
He lamented that, “The solution of the riddle of life in space and time lies outside space and time.”[11] And for treatment he prescribed a placebo: “For an answer which cannot be expressed the question too cannot be expressed.”[12]
In other words, if a question can be articulated, then it can be answered, and if it can’t be answered, don’t bother asking.
For Wittgenstein, it’s not as if mystery doesn’t exist. In fact, he not only believed that mystery exists, he claimed we observe that mystery exists: “There is indeed the inexpressible. This shows itself. It is the mystical.”[13]
While Wittgenstein said we know the inexpressible exists because it shows itself, he believed talking about it is futile because only what can be said is worth saying, and we can’t say anything about the inexpressible.
Compare this Wittgenstein worldview to the Chesterton worldview that even though we cannot explain or express the mystical, we can still use it to understand our world better. Both Wittgenstein and Chesterton agree that the mystery of life cannot be solved inside of the universe, but they differ in how to respond.
Chesterton asserted that while we cannot express paradox we can still discern its truth and use it to better understand what can be understood. Wittgenstein concluded that it is folly to even ask the question.
In fact, it is with this very sentiment that Wittgenstein chooses to conclude the Tractatus: “Whereof one cannot speak, thereof one must be silent.”[14] But not before informing the reader who just slogged through myriad propositions that his effort may very well have been useless:
“He who understands me recognizes my propositions as senseless in that once he climbs atop them all he must throw away the ladder.”
Ludwig Wittgenstein, Tractatus Logico Philosophicus, 6.54
Again, so much for the ‘standing on the shoulders of giants’ thing.
Wittgenstein paints a picture of a bleak world where all true propositions, once understood, have no more use, because if everything true is provable, then we live in a self-evident tautology.
And yet he relegates himself to confinement inside it, like a prisoner who is able to glimpse freedom through the bars but dares not look for fear of experiencing hope. What he doesn’t realize is that the bars only confine him because he has made himself too big.
If he was willing to shrink himself, to humble himself, to admit that there is something worth exploring even though it is beyond the ability for his intellect to comprehend and his language to describe, then he could slip through the bars easily, like an unburdened camel through a needle’s eye.[15]
False Veneer of Freedom
For all their talk about the virtues of objectivity, it turns out that objectivists actually believe in and rely on a wholly subjective world. What do I mean by that? It can’t be said better than Kant can.
Immanuel Kant, one of the central intellects of the Enlightenment and champion of reason and objectivity, believed that there is nothing knowable outside of the knower’s ability to know. If something is knowable, then it only is knowable because there is an observer capable of knowing it.
This means that knowledge does not stand on its own merit but only exists in reference to an observer. That makes Kant’s philosophy subjective because knowledge of the universe is limited to what is obtainable through observation, and thus only in relation to an observer.
There is no objective higher truth or ideal other than the method for making observations. However, objectivism pretends that the results of observation are objective, without appreciating that they have already been filtered through the subjectivity and intuition of the observer.
Philosophy often appears to be an exercise in searching for truth by assigning objectivity to subjective things. The unexpected result of such activity is the creation of a moral relativism that depends on the frame of reference of the so-called objective observer.
Objective observations as an end to themselves, without being tethered to a fixed ideal or immutable higher truth, result in free-floating, ever-changing moral and ethical standards according to the whim of the individual, so long as they are connected somehow to an objective observation about the world.
Gödel thought that perhaps this tendency toward moral relativism ran deeper than mere misplacement of trust in objectivism. Speaking about strict versus relative codes of ethics, he said:
“Actually, it would be easy to get a strict ethics – at least no harder than other basic scientific problems. Only the result would be unpleasant, and one does not want to see it and avoids facing it – to some extent even consciously.”
Palle Yourgrau, A World Without Time, p. 165, quoting Gödel
What Gödel is saying is that we avoid talking about higher truths beyond rationality and logic, not because we are committed to objectivity, but because we are afraid of being held to a higher standard. This calls to mind Chesterton’s famous quip that, “The Christian ideal has not been tried and found wanting. It has been found difficult; and left untried.”[16]
Most people today look to religion for the comfort it offers and not as a source of truth (or for that matter as a director of action or defender of reason). Instead, they consider science to be the proper sphere for truth and reason, trusting that science follows an objective and rigorous methodology.
This leaves a disjointed philosophy of life where morality and ethics shift with the prevailing societal views so long as they fit the latest scientific orthodoxy, and religion is merely a tube of soothing salve stored away in a first aid kit in case comfort is needed.
The term Moralistic Therapeutic Deism (MTD), introduced by Christian Smith and Melinda Lundquist Denton in their book Soul Searching: The Religious and Spiritual Lives of American Teenagers, captures perfectly this approach to religious faith.
In MTD, the focus of religion is feeling good and being at peace. God created the universe and humanity and a general moral order, but beyond that God is not actively or personally involved in our lives. God doesn’t make mistakes, and therefore all humans are basically good, as are our hopes and dreams for ourselves.
According to MTD, God just wants us to be happy, which we will accomplish by ‘living authentically.’ MTD has no defined historic or religious truth, no eternal standard or ideal, and certainly no coherent doctrine, thus fitting perfectly within modern culture’s version of objectivism that we call ‘believe science.’
Whereas MTD presents a false veneer of freedom, true freedom can be found only in a state of morality accountable to a fixed ideal determined by God and not Man. Only by striving to live life in this manner are we pursuing our own true good, to the benefit of our eternal happiness.
Note that true freedom is not the freedom to do whatever we please or deem to be right in our own eyes. Because to do that is to be enslaved by one’s own appetites and the hubris of one’s own judgement.
Everybody serves a master, and by the grace of God we get to choose what master we serve, and to that choice we are eternally bound. The only true freedom is the freedom from sin, which we experience in those fleeting moments when we choose to do God’s will and live God’s law.
God’s moral law and prescripts of worship were made for the happiness of humankind; so that we could know our purpose and reach our potential through glorifying God, which is our highest good.
[1] Rebecca Goldstein, Incompleteness, p. 91
[2] C.S. Lewis was credited with saying that, “The doors of hell are locked from the inside.
[3] Tractatus, introduction
[4] Tractatus, introduction
[5] Tractatus, preface
[6] Tractatus 4.116
[7] Tractatus 3.332
[8] For the mathematically inclined, proposition 3.333 goes like this. If you have a function refer to itself, that is F(F(x)), then the inner and outer F’s, while represented by the same symbol, actually mean different things. Thus, a function cannot be its own argument because it violates the syntax rule of one symbol corresponding to one and only one meaning.
[9] Tractatus 4.112
[10] Tractatus 4.112
[11] Tractatus 6.4312
[12] Tractatus 6.5
[13] Tractatus 6.533
[14] Tractatus 7
[15] Recalling the words of Jesus, “Again I tell you, it is easier for a camel to go through the eye of a needle than for someone who is rich to enter the kingdom of God.” Matthew 19:24 NRSV-CI
[16] Chesterton essay, “What’s Wrong with the World”