The Merits and Drawbacks of Intellectualism: A Look at Paradoxes (Column 654)

There are ideas so stupid that only intellectuals could believe in them.

(George Orwell)

In the previous column I discussed intuition. My conclusion was that it is a cognitive faculty that underlies our logical and scientific thinking. From this I argued that doubts about its validity, which implicitly expect an answer in scientific or philosophical terms, are mistaken. Intuition is the basis for our rational thinking and cognition, not the other way around. Does that mean intuition is free of error? Certainly not. But when we ask what could count as a problem for an intuitive claim—namely, what could be more fundamental than an intuitive claim so as to override it—there seems to be nothing of the sort, since intuition stands at the base of everything.

It is possible, however, that an intuitive claim, even a very fundamental one, will be challenged by an observation (which we saw in the previous column is itself based on intuition) or by an opposing logical argument (which is also based on intuition, as we saw there). The upshot is that, at bottom, this is a clash between intuitions; in such a case we must decide which of the two we choose and which we reject. Thus, intuitions are examined mainly through their consistency—that is, by demanding coherence and fit within our overall intuitive system. This brings us to the issue of paradoxes. As we shall see shortly, a paradox is usually a case in which an intuition is challenged by an argument or by a counter-intuition, and this situation forces us to re-examine our intuitions. To understand this better, let us first look at a few examples.

Examples of Paradoxes

The first example is the Surprise Test Paradox, discussed in columns 601 – 603, where I dealt directly and explicitly with the meaning of paradoxes and, consequently, with ways to resolve them. A teacher enters the classroom and announces that during the coming week there will be a surprise quiz. The argument shows that it is impossible to give such a quiz. It cannot be given on Shabbat, because if it has not been given by then—then on Friday night the students already know it will be given the next day, and therefore it will not be a surprise. But if so, it also cannot be given on Friday, for on Thursday night the students know it cannot happen on Shabbat and therefore it must happen on Friday, and again they will not be surprised. And so on back to Sunday. This argument proves that there is no such thing as a surprise quiz.

Well, what’s the problem? Then there isn’t. The problem is that we clearly know there is. Our experience (observations) or our intuition tells us that surprise quizzes exist. Thus, a contradiction is created between the claim from experience/intuition that tells us there are surprise quizzes and the logical argument that proves there are not. This is what happens in most paradoxes: on one side stands a logical argument that seems valid and proves X, and on the other an intuition or observation that tells us X is not true.

The same occurs in Zeno’s paradoxes. The paradoxes of Achilles and the tortoise or the arrow in flight prove that there is no motion in the world, since the concept leads us into contradictions and paradoxes. On the other hand, our experience and our intuitions clearly tell us there is motion in the world. Again we have a clash between argument and observation or intuition. In this sense, a paradox is a logical argument that proves that some claim we believe is not true.

Another example is Russell’s set paradox. We speak of two kinds of relations between a set and something (a smaller set) contained within it. The first is the relation of inclusion: every subset is included in the set itself. For example, the set of numbers {2,3} is included in the set of all natural numbers less than 10. The second is the relation of being a member (element): the set {2,3} is not an element of the set of natural numbers less than 10. But 1 is an element of it. That is, {1} is a subset of the larger set, and 1 is an element of that set.[1]

Let us continue and say that every set is also a subset of itself, and in this sense it is included in itself. But can a set also be an element of itself? Think, for example, of the set of all sets, or the set of mathematical “objects.” Each of these is an element of itself. The set of all sets is itself a set, and therefore it is an element of itself. The set of all mathematical objects is itself a mathematical object and therefore it is an element of itself. The set of things described by collections of words is itself something described by a collection of words and therefore it is an element of itself. But of course most sets do not contain themselves as an element. For example, the set of all natural numbers is not itself a natural number; that is, it is not an element of itself.

Next we construct the set of all sets that do not contain themselves as an element. As noted, these are most sets. Now we ask: is this set itself an element of itself? If the answer is yes, then it belongs to the sets that contain themselves as an element, and therefore it is not an element of itself. But if it is not an element of itself, then it is one of the sets that are not elements of themselves, and as such it is indeed included in the set of all sets that do not contain themselves as an element.

Russell used this paradox to attack the intuitive notion of “set” (=a collection of elements), and indeed, following this paradox mathematicians decided that this notion is contradictory and not a well-defined mathematical concept. Instead they defined the concept of a set in a more precise mathematical way, thus eliminating the contradiction/paradox. Thus we see that this paradox functioned as a proof by contradiction against the intuitive concept “set.” It proved to us that this notion is contradictory and should not be used in its intuitive sense.

What Is a Paradox?

From these examples we can begin to understand what a paradox is. At first glance it is a proof by contradiction, namely an argument showing that what we thought to be true is not so. The paradox takes an accepted notion or claim and shows, by a logical argument, that it leads to a contradiction. In this way it proves by contradiction that the claim is false.

Thus, for example, Zeno proved there is no motion; Russell proved there is no (intuitive) set; and the Swedish Army paradox proved there is no surprise drill. Why are all these considered paradoxes? Is the proof of a theorem in geometry (by contradiction or otherwise) a paradox? Paradoxes are not merely proofs by contradiction. They are indeed arguments that prove that claim X is false, or that concept X is empty—but that is not enough. For something to count as a paradox, even after the proof we must still suspect that there is a flaw in the proof/argument, and that in fact claim X is true after all, or that the concept in question is not empty. This was the motivation of mathematicians to resolve Russell’s paradox, i.e., to define the concept of set better so that it fits our intuitions but is free of contradictions. It is also why, throughout the generations, people have sought and proposed various solutions to Zeno’s paradoxes, since it is obvious to all of us that there is motion in the world even if we have not found a flaw in Zeno’s arguments (most people cannot point to the flaws in his arguments). Likewise, no one is convinced by the Swedish Army paradox that there are no surprise quizzes or drills. We look for the flaw in the argument, since it is clear to us that the concept “surprise quiz” is not empty and the claim that surprise quizzes exist is true.

An important note in parentheses. It is now clear why pointing out that the conclusion of the argument is false does not constitute a solution to the paradox. For example, many think that by the very fact that it is clear Achilles overtakes the tortoise, we have shown that Zeno’s argument is incorrect. But this is a mistake. In fact, the opposite is true: pointing out that the conclusion is false is what establishes the paradox. Precisely for this reason we have a paradox before us. A solution to a paradox is a reconciliation between the claim and the argument, not pointing out that there is no reconciliation. In the first dialogue of my book The First Being I pointed out that the main arguments against Anselm’s ontological argument rely on the claim that if we adopt the argument we arrive at absurdities (we will prove the existence of the perfect island, and so on). But that is of course not a solution; it is merely pointing out that we have a paradox. This argument, in which we have not found a defect, leads to a conclusion that seems absurd to us. A solution is either pointing out a flaw in the argument, or accepting the conclusion.

What follows from all the above is that a paradox is a clash between logic and intuition. Sometimes the clash is with another logical argument or with an observation, but as noted, these too are based on intuition (the premises of the logical argument, and the trust we place in our observational instruments). Therefore, in general I will assume here that a paradox is a clash between intuition and a logical/mathematical argument.

Another note. From here on I will not distinguish between a question about the content of a concept and a question about the truth of a claim. For my purposes, a paradox proves that claim X is false, where X can also be the claim that some concept has content and meaning. From here on, a paradox is a clash between an intuitive claim and a logical argument.

Two Kinds of Solutions and Three Ways to Relate to Paradoxes

In reality there is no room to accept paradoxes as such. That is, we must decide whether the argument is correct (i.e., the claim X we have held until now is indeed false, or concept X is empty), or whether our prior conclusion is correct (X is true, or X has content and meaning; it is not empty). The Law of Non-Contradiction, the logical principle that denies the possibility of contradictions, is not only a component of our thinking but also a property of reality itself. In reality there are no contradictions. If X is true, then “not-X” is not true, and vice versa. If concept X has content, one cannot say it is empty. I will allow myself not to dwell on this trivial point, even though here on the site (and elsewhere) questions repeatedly arise as to whether logic is a property of our thought or of reality. In Quine’s terms, “the square-round dome of Berkeley College” is not only an empty concept; I can also assert with certainty that in reality Berkeley College has no such dome (not that it goes about without a head covering, Heaven forbid, but that its dome is not contradictory).

In columns 601 – 603 I explained that when we face a paradox, we have two ways to proceed:

• Assume the argument is correct and conclude that claim X, which we had held until now, is indeed false. Here the paradox functions as a proof by contradiction.
• Find a flaw in the argument and retain claim X. A special case of this approach is to show that the conclusion that follows from the paradox does not, in fact, contradict claim X (we erred in understanding the argument and its conclusion, even though there is no formal flaw in the argument itself).

I explained there that there can also be a third option:

• Suspect that there is a flaw in the argument even if we have not found it, and still remain with X. This will happen when our intuition about X is very strong and the argument that purports to prove that X is false is complex (and we suspect it contains a flaw we have not noticed. This is even more true if our logical and philosophical skills are not the best).

Option C would seem to express intellectual dishonesty: they proved to you that you are wrong, and you still insist on holding on to X. But that is not necessarily the case. There can certainly be situations in which a person feels that the argument is complex and harbors a suspicion that it contains a flaw. If his intuition regarding X is strong, there is nothing wrong with his remaining with it—at least for the time being, until matters are clarified otherwise.

I think all of us have felt this way at times about various arguments presented to us. We can see it in debates about the existence of God, where a strong argument is advanced that proves His existence, yet the atheist is unimpressed and does not repent. Likewise with those who believe or do not believe in free will and are unmoved by strong arguments against their position, even if they have not found a flaw in them. The same holds for various ideologies (political, economic, etc.). This phenomenon resembles clinging to our views despite the difficulties raised against them, at the price of adopting flimsy explanations for those difficulties (I discussed this in my article on Occam’s razor, and in columns 30 and 440). Here we go further and remain with position X without any explanation for it. But, as noted, such an approach is not necessarily dishonest.

In parentheses I will note that this is where the philosophical dispute between Karl Popper and Thomas Kuhn lies (see about it in column 647). Karl Popper argued that any counterexample—that is, any empirical observation whose results contradict the theory’s prediction—should falsify the theory. Thomas Kuhn, by contrast, argued that in practice this does not happen. A scientific community does not discard a good theory (a paradigm) with every counterexample that emerges. A minimal quantity of counterexamples is needed before we abandon a good theory and replace the paradigm. What underlies this is that the counterexample is a logical argument proving that our current position (the paradigm) is false. But the scientific community prefers not to abandon the theory and to assume that perhaps there is an error in the experiment, or that the theory can be refined so as to accommodate this case as well. This is the preference for intuition and common sense over logic—and, as noted, this is not necessarily dishonest.

Logic vs. Intuition: Which Should We Prefer?

If we are caught in a conflict between logic and intuition, it is interesting to think, in general, which of the two we trust more. Regardless of what we saw in the previous column (that intuition is a cognitive tool), I think that if I ask people, most would answer that logic is of course more reliable. Intuition is elusive and contentious; your intuition is not like mine. Not for nothing did we see there that people treat intuition as a feeling. Logic, by contrast, is objective and universal. Who will dispute a valid logical argument?! It is the strongest thinking tool we have. It is true, as I explained above, that it may be that the logical argument generating the paradox merely seems valid to us, but in fact contains a flaw (which we have not yet found). Still, so long as we have not found the flaw and the argument seems valid to us, it appears that most people will trust logic and not intuition.

Surprisingly, our actual attitude toward paradoxes is the opposite. The vast majority of people who face a paradox will adopt claim X by force of intuition and suspect some flaw in the logical argument that leads to the opposite conclusion. No one considers saying there is no motion in the world, and everyone is sure that there is some flaw in Zeno’s arguments—although most people will not know how to point to that flaw. The same with the surprise quiz/drill. I described in those columns my frustration with it, for although I could not find a flaw in the logical argument, I was all the while certain that surprise quizzes exist (see the columns there for my description of seesawing between these two possibilities). Needless to say, it is clear to everyone that the concept of a set has a well-defined content, and we continue to use it in everyday language (as opposed to mathematicians, who insist on contradiction-free theories), while Russell’s paradox may honorably wait on the sidelines.

Surprisingly, in most cases where there is a paradox that sets intuition against logic, our initial tendency is in favor of intuition and against logic. This is surprising especially in light of what I described above: most people, when asked, would not say this. They would tell you they prefer logic to intuition (you can see on this site how many critiques there are of my “excessive” trust in intuition).

And Now to the Intellectuals

I have often quoted here the aphorism of George Orwell—cited as the motto for this column—that there are ideas so stupid that only intellectuals could believe in them. It sounds like a witty quip, and people feel that sometimes it is even true (at least if they are not intellectuals). Now we can understand why this is even to be expected.

A simple person, untrained in analytical and sophisticated thinking, tends to follow his heart’s sense—that is, his intuitions. Even if there is a logical-philosophical argument that contradicts them, he tends to dismiss it and ignore it. For him, common sense trumps any intellectual move. This is why ordinary people generally do not arrive at revolutionary ideas and are more inclined to conservatism—that is, to cleave to the status quo. Intellectuals, by contrast, are people exposed to new and revolutionary ideas and thought. They engage in abstract philosophical arguments and tend to assign them significant weight. Naturally, when a counter-intuitive argument arises, the intellectual will adopt it with a higher probability than the simple person. An intellectual builds a worldview on the basis of arguments and repeatedly challenges current thinking and the status quo. Naturally, in the clash between argument and intuition, he will assign greater weight to the argument than the simple person. There is another reason for this difference: the intellectual is also more skilled in logic and therefore is less inclined to suppose that there is a flaw in the argument that he has not noticed. If the argument seems valid to him, he assumes it is valid. By contrast, a simple person untrained in logic finds it easier to suppose that the argument contains some logical flaw he did not discover.

This, in my view, is why there are more atheists among intellectuals, and also more who believe in determinism—even though both of these views contradict common sense and accepted beliefs (intuitions). Intellectuals here follow arguments that seem convincing to them and are willing to abandon common sense and be more open to changing their existing intuitions. Incidentally, I think this is also why intellectuals tend to lean left. The right is conservative and believes in accumulated wisdom; it casts doubt on logical arguments, however reasonable and persuasive. It prefers to wait and see whether there is a flaw in the argument, or whether in reality it may not be applicable. The intellectual is persuaded by an argument and quickly adopts new ideas and thought. After all, he has a logical proof that the status quo is wrong or that there is a better, more successful ideology.

Orwell’s words are, of course, a critique of intellectuals. But note that in light of what I have said here, this is not precise. The picture I have drawn contains a critique of both sides. In columns 601–603 I pointed out the problems with clinging too tightly to intuition and closing oneself off to new arguments that challenge it. A conservative must also understand that accepted positions may contain flaws, and the way to criticize and examine them is by means of arguments. And of course Orwell is right that the intellectual, too, must understand that sometimes logical arguments contain flaws, and that one ought not scorn common sense and intuition.

The Third Way

Above I noted that when facing a paradox where we have not found a flaw in the logical argument, there are two possible responses: remain with intuition and assume there is a flaw in the argument, or adopt the argument’s conclusion and abandon intuition. What is the criterion? When is it right to do this, and when that? I have no criterion, and I think there cannot be one. It seems to me that in such situations the decision is made by a second-order intuition (one that determines how much credibility we should assign to our intuitions), and therefore there is no way to mechanize or formalize it. It is a matter of gauging how strong the intuition is and how persuasive the argument is (and also how complex it is, such that there may be a flaw hidden from me. This is also a function of my logical and philosophical abilities, of course).

Simple people tend to cling too tightly to their intuitions. In the terminology above, we can say this is “extreme right” in the philosophical sense of the term—absolute conservatism. “The extreme left,” by contrast, is the stance that, upon encountering a brilliant and persuasive argument, immediately changes its mind, ignores common sense and intuition, and sometimes even tramples them. These are the nonsense-ideas that only intellectuals can believe.

I think here, too, one should take the middle path. It recognizes that a paradox has two sides and there is no simple, general way to choose between them. Those who take this approach give due regard to intuition and common sense, but not absolute trust. Arguments are an important tool for challenging our intuitions, and it is important to understand that sometimes a paradox is a proof by contradiction that my intuition is wrong. In column 602 I pointed out that paradoxes are an important tool against our natural conservatism. Here I add that one must also beware of the opposite mistake: disparaging our intuitions and over-trusting logic (and also facts and science). Sometimes there is a flaw in the logical argument, even if we have not yet found it. You will find countless examples of this on the site. If the two sides are right and left (in the intellectual sense), then here I propose the Third Way.

A Look at the Halakhic-Analytic Plane

More than once I have discussed the common division between lamdanim (analytic scholars) and decisors (poskim). The lamdanim know how to ground every position on logical foundations, analyze its premises, and demonstrate its consistency. Decisors, by contrast, are usually endowed with a lower level of analytic rigor; what distinguishes them is the ability to decide between the different positions. One of the founders of modern lamdanut, R. Chaim of Brisk, served as the rabbi of the city of Brisk. He was known for trying to avoid decisions as much as possible. When a halakhic question came before him, he would refer the questioner to R. Simcha Zelig the Dayan. In another case it is told that he sent a question to R. Yitzchak Elchanan, the greatest decisor in Europe at the time, and asked him to answer yes or no—without reasons. For every reason, R. Chaim knew he could raise counter-arguments. He asked for a decision, and that was all.

Where does this come from? Why did R. Chaim not fulfill his role as the city’s rabbi? I think it stemmed from his lack of ability to decide. A person with strong analytical abilities can ground every position on reasonable and consistent foundations, and in such a situation he loses the ability to determine which position is correct. The decisor who decides usually does so on the basis of intuition or common sense, but analytical abilities usually interfere with this. R. Chaim was an intellectual who gave preferential weight to logic and disparaged common sense and intuition (even though he understood that without them he could not decide halakhah), and so it is with many lamdanim. This is also why in Brisk they tend to be stringent in order to satisfy all the opinions that exist in halakhah. People think this is the result of great fear of Heaven—but regardless of how much fear of Heaven there was (and is) there, I think the explanation is simpler. Once you can ground every view on solid reasoning and display its inner logic, you have lost the ability to determine who is right. For you, logic is the yardstick, and the only way to decide is to find a contradiction or a logical proof favoring one position or against the other. But in an analytical world like Brisk, you will not find such contradictions. Every contradiction can be resolved by sophisticated logical structures, and thus all positions remain standing. Therefore, in practice, one must be stringent to satisfy them all. The only way to decide is to activate intuition and common sense. But these tools are not legitimate in Brisk (as is known, they do not ask “why?” there, only “what?”).

This distinction is very similar to what we saw above. In Brisk they are “leftists”—that is, halakhic and Talmudic intellectuals. As such, they follow logic and arguments and have no trust in intuition and common sense (I have often explained that they live under the illusion that they do not use these tools at all—as if they really ask only “what?” and not “why?”). The decisors are the “right”—that is, they rely more on common sense and intuition. Anyone familiar with these genres knows that halakhic responsa usually contain analysis at a lower level than books of analytic lamdanut.

In the next column I will continue to examine my thesis in the religious sphere. There I will address the question of what constitutes Torah wisdom and greatness in Torah, and the relationship between them.

[1] Is 1 a set or not? Seemingly yes, and then I should have written {1}. But intentionally, when dealing with a subset I used curly braces, and when dealing with an element it appears without the braces. In the new (axiomatic) set theory, which was born out of this paradox, we distinguish between 1 and {1}. In this notation, {1}—that is, the set containing 1—is a subset of the naturals up to 10, but 1 itself is an element of the larger set and not a set in its own right.