On Concepts and Conceptualizations (Column 662)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

IYCh (=if Ḥamina’i wills) in the coming week and a half I won’t be in the country. So I’m leaving you this column as food for thought. I’ll try, as much as possible, to be reachable on the site, on WhatsApp, and by email.

This past Shabbat I read R. Kook’s words in his essay “Derekh Ha-Teḥiyah,” and they reminded me of a topic I’ve long wanted to write about: the relation between our insights themselves and their linguistic and logical formulation. I addressed this issue from another angle in the series of columns 379 – 381, but here I wish to examine it through the prism of the analytical study (iyyun) customary in yeshivot. I’ll begin with a glossed quotation from R. Kook.

The Initial Distinction: The Thing Itself and Its Representations

At the head of R. Kook’s collection of essays appears the essay ‘Derekh Ha-Teḥiyah’, and at its beginning he writes as follows:

“Spiritual being as it is cannot be recognized through any search or inquiry. Knowledge, rational investigation, philosophy—these indicate only the external signs of life. Even when they deal with the inwardness of life, they see only the shadows of life and not its inner essence, and all the might of rational proofs is only to clear a path for the spirit by which to approach the antechamber of sensing spiritual reality.”

R. Kook here distinguishes between the idea itself and its rational–logical expressions. In our thinking we use definitions and arguments to examine insights and ideas. But the arguments and definitions are merely representations of those insights and ideas, and it is a mistake to identify the insight with its articulated rational representation.

In my series on poetry (107 – 113) I dealt extensively with the question of defining abstract concepts such as poetry, intelligence, democracy, and the like. I argued there that such concepts are difficult to define and that, in general, a definition fails to capture their essence. It serves as a representation or external expression of the idea itself through several of its main features. Thus, for example, it is hard to convey to someone the idea of a democratic state through a conceptual, logical description. One can, of course, propose a definition containing several essential features—separation of powers, civil rights, elections, and so on—but this does not truly transmit the idea itself. One who has not lived it does not truly understand what is at issue. This can also be connected to the question of conventionalism versus essentialism. Conventionalism regards concepts as arbitrary definitions of a community of speakers; on such a picture it is natural to see the concept as the aggregate of its features and nothing more. But on an essentialist conception, the concept is a kind of Platonic Idea (which may or may not exist in a world of Ideas), and the features are descriptions of it but not it itself—just as the properties of a person or of a table are not the person or the table themselves (see my discussion of Leibniz’s “Identity of Indiscernibles” in columns 383, 519, and elsewhere).

It is a fact that there are disputes about such concepts—meaning their definitions are not unequivocal. Everyone sees them a bit differently. One might think that the dispute reflects the fact that these are two different ideas, and therefore there is no point using the same term for them. On that view it would be a pseudo-dispute. But I have often written that the very fact that disputes are conducted indicates that both sides agree they are speaking about the same concept or idea (otherwise there would be no point in arguing about it), but that they disagree about it. The upshot is that the idea is not merely the sum of its features. The features try to capture an abstract idea and to formulate and conceptualize it explicitly. But the idea and the concept are the abstract thing, not the conceptualization that is only its representation.

I have often brought the example of Mary’s Room, which illustrates why a perfect physical understanding of optics does not truly give us an immediate understanding of concepts like light or color (see, for example, in column 452, where I also touched on the distinction I am making here). More generally, the formulas in physics are representations of the facts of reality themselves, and it is incorrect to identify the representation with reality or with the concepts represented. I’ll return to this example below. And more generally still, I have often argued that with respect to both concepts and ideas we can apply Kant’s distinction between the thing-in-itself (noumenon) and the thing as it appears to us or to our intellect (phenomenon). Underlying this claim is a Platonic view of concepts, which regards them as expressions of certain Ideas (essentialism) and not merely social conventions.

It is important to understand that I am speaking not only about inaccuracies in the definitions of concepts, but also about the fact that definitions, by virtue of being formulated and logical, do not succeed in capturing the thing-in-itself (just as we saw that the concept “democracy” precedes the set of features that represent it). True, we have no other alternative for handling such concepts and ideas than by way of definition, analysis, and logical argument—but it is important to realize that these are only tools by which I can touch the thing represented by them. The formulation and the argument are a medium that represents the concept itself, and therefore through them—and only through them—I can attempt to touch it.

I think these are the “shadows” of which R. Kook speaks in the passage quoted above. The articulated rational representations constitute a faint representation of the “light,” which is the idea itself. His terminology is rooted in Kabbalah: in the kabbalistic picture, the vessels give the light its particular form and distinguish it from other entities or states. The light is clothed each time in different vessels and therefore assumes a different form each time. Indeed, later in the essay R. Kook explains that at the foundation of all reality in its varieties stands that very same light, and the differences among parts of reality derive from the fact that this light is clothed each time in different vessels.

Combining Intuition and Conceptualizations: R. Kook’s Cave Parable

He now continues and adds:

“But so long as a person is immersed in his senses and their narrow bounds, he will not recognize and will not know spiritual being, and only faint shadows will appear thereby. And if he relates to the shadows as if to true reality, then those shadows will be a heavy burden upon him and will diminish his material and spiritual strength together, until he seeks to flee from them as from a harmful thing.”

As we have seen, a person must use intellect, definitions, and arguments to deal with concepts and ideas. But one who identifies the idea with its concrete formulation will not truly arrive at knowledge of the idea itself. He grasps the shadows rather than the light. In such a state, the formulations will be a burden that prevents him from grasping the true thing. Because of the clinging to conceptualizations and verbal formulations, a person may err and conclude that something is true or false because there is such-and-such an argument for or against it.

But in the final analysis, intuition must be the supreme judge, with the definitions and arguments as a medium that enables us to reach it (or at least to approach it). We cannot—and therefore it is also not right—to refrain from using these tools; yet at the same time it is very important to remember that these are not the things themselves but only a way to encounter the thing (the concept or the idea) in itself.

This, I believe, is precisely the meaning of Plato’s Allegory of the Cave (the term “shadows” hints that R. Kook may be aware of this context of his words):

“The allegory, presented in his book The Republic (opening of Book VII), tells of a group of prisoners confined from a young age in a dark cave, chained such that their heads always face one direction and cannot turn. Behind the prisoners there is a wall, and behind it burns a fire that never goes out, illuminating the wall at which the prisoners gaze. Between the fire and the wall pass beings carrying statues. The statues, borne above the height of the wall and illuminated on one side by the fire, cast their shadows onto the wall the prisoners watch. When one of the beings utters a sound, the prisoners hear the echo returning from the wall. Consequently the prisoners think that what they see on the wall is in fact real life. Their lives revolve around the shadows and their voices; the prisoners develop a culture and a host of theories about them and build their spiritual world according to the figures on the wall.”

“One of the prisoners manages to free himself from his chains. His muscles are atrophied from sitting chained, and some force drags him up a steep dirt path out of the cave. He is scratched and bruised, and when he reaches the outside he cannot see anything, accustomed as he is to the darkness of the cave and unable to face daylight. After some time he begins to see blurry shadows—as he saw on the cave wall. But as time passes and he gets used to the light, he can look around and see that the beings are not shadows; he can gaze upon nature, the birds, and the heavenly bodies at night. He can also notice his own shadow.”

“The prisoner decides to return to the cave to tell the others about the wonders of the outside world and to free them. But when he returns to the cave he again cannot see anything, for his eyes have adjusted to the sunlight and cannot see in the dark, and the other prisoners mock him and claim he has become blind. He tries to explain to them the world outside, but they dismiss his words with scorn. The prisoner who went out to the light stands his ground and tries to enlighten the eyes of his chained comrades, but they decide that if anyone tries to release them from their bonds and take them out of the cave, they will kill him so as not to be blinded themselves.”

Here we see the relation between things themselves and their shadows, the limitation of those who take the shadows to be the thing itself and do not recognize that these are only representations of the thing, and consequently their inability to accept the full truth even from one who was exposed to it and tries to convey it to them.

R. Kook goes on to explain how to avoid this failure:

“But however much a person flees from the shadow, the shadow pursues him, and there is only one remedy to be rid of it, namely the addition of light. This increase of light can come only through elevation into the innerness of the very essence of spiritual being, and for this a person has no psychic means other than the ‘deep sublimity’ of the sense of divine faith. This is the knowledge of knowledges and the feeling of feelings that connects the spiritual being of a person, existent in actuality, with the higher spiritual being and commingles his life with the lofty, boundless life devoid of any physical weakness…”

Faith, likened here to light, is the instrument by which one can grasp the things themselves. This light penetrates through the definitions and arguments that represent them to them themselves. It dispels the shadows and enables us to see what they represent. In my terms, I think R. Kook identifies intuition with faith (I explained this identification at the end of column 653). His claim is that after formulation and conceptualizations, we must not stop; we must continue and penetrate through them to the understandings of the thing itself. Thus one can leave the cave—or at least try to understand what the shadows moving on its walls represent.

But of course this does not mean we should skip the conceptualizations, definitions, and arguments:

“Indeed scientific culture, founded on proximate intelligibles and clear senses and the natural morality that issues from them and under their influence, prepares humanity to absorb into itself the abundance of light of the general spiritual psychē. Then the cultural orientation of intellect, sense, and material morality will receive great strength and a divinely based, penetrating, and lasting order. Not so if the abundance of light comes without the paving of a path of intellectual culture and ordinary morality; then this light will generate only darkness.”

As noted, the shadows (conceptualizations) are our tool for grasping reality. We cannot truly leave the cave. We have no ability to grasp the thing except through the shadows, for only by using recursive reason can we progress toward grasping the ideas in themselves. Intuition without recursive intellect is not worth much, and likewise recursive intellect without intuition. We need the combination of both. When someone comes who has seen the things themselves, we must not dismiss his words out of attachment to the shadows. We should try to understand that the shadows are representations; then perhaps we can strive toward understanding what they represent.

He now explains the other side of the coin: what happens if we do not make use of the shadows and try to advance directly by faith (intuition)? He describes what happens when one uses intuition without the supervision of recursive intellect:

“From the eruption of spiritual existential light from its sphere onto uncultivated ground comes the murky side of idolatry; thus it led to false imaginings and the very evil impulses of the heart. And from the tendrils of this Sodom-vine humanity has not been purified to this day. Even today culture has not yet reached the measure that divine sympathy of absolute good dwells in the depth of the soul of the collective organs; therefore the divine bond within them is an alien and foreign god that acquires bizarre, caricatured qualities. We now see in them marks of wickedness and tyranny, and the essence of morality is melting away and leaving their hearts.”

Intuition is a wild instrument that can lead us to delusions, fantasies, and deceptions. Anyone can claim anything in the name of his intuition. Intuitions also often mislead us, and using intuition alone, without intellectual oversight, is primitive thinking. The role of the intellect (conceptualization, definition, and argument) is to critique the use of intuition, to guide and direct it. But clinging to these conceptual tools is no less idolatrous: the notion that definition grasps the thing itself and that science yields understanding of things themselves is also a kind of idolatry. In column 655 I described what happens when one uses intellect without common sense—that is, without intuition.

In the final analysis, our intuition gives us the premises upon which a logical argument can be built to reach conclusions. Moreover, conceptualization and argument are not a substitute for intuition; they only attempt to articulate our intuitions more explicitly and precisely. Still, we must not give up formulation and conceptualization, for they constitute our only way to arrive at understanding the thing itself—and, alas, they can also in turn reshape the intuitions themselves. Through conceptualizations we can discover problems in our intuitions (see columns 654 – 655 on the dilemma of how to decide contradictions between recursive thinking and intuition, and the relation to paradoxes).

I now wish to enter this process in greater detail, to demonstrate and sharpen what is at issue. I’ll do so through the sugiyot that attempt to define the giving of a get (bill of divorce), and afterward broaden the discussion to lamdanut in general. The source is a short essay I wrote about thirty years ago in Bar-Ilan’s weekly sheet, and from a slightly different angle in my article in Iyyun on Zeno’s paradox of the arrow in flight.

Talmudic Discussions of Giving a Get

It seems that the verse in the Torah that received the most discussions and Talmudic sugiyot per word is the verse “and he shall write for her a bill of severance and give it into her hand” (Deut. 24:1–3). Every word here is expounded into mountains of halakhot occupying dozens of sugiyot, mainly in tractate Gittin. Here I wish to focus on the last two words: “and give into her hand” (and a bit on the word “keritut” [severance]). The Gemara derives from here that the act of divorce involves giving a get into the woman’s hand. Dozens of sugiyot try to clarify what that “giving” is and also what the “hand” is into which the get is to be given. In each sugiyah various cases are brought that exemplify valid and invalid givings, with the assumption that from this huge set of examples the learner can distill general definitions of the act of giving and of the receiving “hand.” These definitions will enable us to decide regarding cases not appearing in the Gemara—and apparently there are very many of them—whether the giving is valid or invalid.

The Aharonim indeed try to use these examples to extract a general definition. Because there are so many examples, the task is not easy. Typically they propose some formulation and test it against various examples. Seeing that it fails the test, they tweak it and put a revamped formulation on the table. That, too, fails when faced with other examples and requires further tweaking. Thus they slowly approach a definition that tries to fit all the examples in the Talmud. This description raises the question: why doesn’t the Gemara give us the general definition? Why does it prefer to give a collection of many examples from which we ourselves should try, by trial and error, to approach the definition—if we can succeed at all? It seems the Sages had such a definition, since they themselves had to decide for each example whether it is a valid giving or not. It is reasonable they did so in light of some definition. Why, then, didn’t they shorten our path and give us the definition itself rather than countless examples? In column 482 I discussed the casuistic structure of the Talmud and explained the advantage of a way of thinking and debate through examples versus a positivism that works through rules and conceptualizations. I briefly mentioned there also the sugiyot of giving a get; here I wish to enter these sugiyot in greater detail to sharpen the matter.

Defining the Giving of a Get: Discussions of the Aharonim

At first blush, giving a get is the husband taking the get and handing it to the wife. That is a physical definition of the act of transfer. But from the Talmudic sugiyot it emerges that Ḥazal understood that there is no necessity for a physical transfer. Thus, for example, it is clear from the sugiyot that if the husband places the get in her courtyard, or placed it in his courtyard and transferred ownership of the courtyard to her, the woman is divorced. In other words, here too the giving is valid even though no physical hand-off occurred. The Rishonim and Aharonim discuss the exact definition of valid giving of a get. Thus, for instance, the author of Ketzot Ha-Ḥoshen (sec. 202, s.k. 5) proves that an act of acquisition (hakna’ah) is not necessary, since it is accepted that if it was written on items prohibited for benefit the get is valid (see Gittin 20a), and according to some Rishonim there is no acquisition in items prohibited for benefit. It seems the basis is that the Torah’s language requires giving and not acquisition. Yet we saw that acquisition can effect the transfer of the get, and apparently that does not operate by virtue of acquisition but because the acquisition constitutes a juridical transfer. That is, the transfer need not be physical.

There are lengthy discussions in the Talmud and its commentaries about how the giving (physical) must be performed: what is the law when the woman takes it from the husband’s hand or pocket; what if the husband brings his pocket close to the woman so she will take it; “take your get from atop the ground,” and many more. Each such discussion requires many examples throughout Shas, and it is no wonder that the resulting definitions are quite complex and vague. The first three simanim in Kehillot Ya‘akov (on Gittin) discuss the definition of “giving.”

In siman 14 he deals with givings that leave something with the husband. There too he of course needs many examples (this is your get and the paper is mine; writing a get on her paper; leaving with him the paper between lines or between words; leaving a small margin outside the get; writing on the hand of a slave or the horn of a cow and giving her the cow or the slave, or only the hand or the horn; “they were presumed to own the tablet”; or a case where the woman pays the scribe’s fee. He discusses whether, when the paper belongs to the husband, he must transfer it to the woman; he also treats the derashah “one book,” not two or three, and more). In the end he arrives at a very convoluted definition.

Siman 15 deals with acquisitions that entail transfer and with the various modes of transfer (the woman’s taking with or without the husband’s assistance; “take your get from atop the ground”; acquisition by courtyard with the get inside; placing the get in her courtyard with her slave present or not, and many more). Siman 16 deals with the issue of meḥusar ketzitzah (lack of final severance). To illustrate, I will detail a bit of what happens in his final siman (17).

The basic question raised by the Rishonim and Aharonim is whether acquisition can also be considered a valid giving of a get. On the face of it, yes—for in the examples above the Gemara itself says he can transfer the courtyard to her when the get is inside. The Rishonim explain that transferring the courtyard when the get is inside does not necessarily operate by virtue of acquisition but is a form of transfer (mesirah). As noted, this is a juridical transfer of the courtyard that counts like a physical transfer. That is with respect to transferring the get that is in his courtyard. Placing the get in the woman’s courtyard is surely a transfer, for the courtyard is the woman’s domain like her hand (or her agent), and placing the get in it is literally physical giving. There you have the first complication: there are acquisitions that operate as valid giving, but not by virtue of the kinyan; rather, the kinyan serves as a mode of juridical transfer. What shall we say about a kinyan that has no element of transfer?

Rashi’s view at the start of Perek Ha-Zorek (Gittin 77b) is that kinyan agav (by means of real estate) is effective for a get. The Rishonim wonder how there is giving here, for we do not require tzeburin (the item acquired by agav need not be placed on the land through which it is acquired), and therefore this is a mere acquisition with no act of giving. Indeed the Gemara itself requires that the husband transfer to the woman the place upon which the get lies (and then there is a juridical transfer, as we saw). Without such a transfer, kinyan agav does not work according to many Rishonim (see, for example, Ran, Ramban, Ritva, and Rashba ad loc., who explain that there is no act of giving). The Ran and his colleagues hold, as we saw, that the kinyanim effective for a get are not so by virtue of the acquisition but because they involve juridical transfer. Therefore, in their view, a bare acquisition, such as agav without tzeburin or by sudar (handkerchief), which are acquisitions without an element of transfer, do not work. But according to Rashi even such acquisitions are effective. If so, in his view the requirement that the get be “gathered” (tzeburin) is puzzling. They also challenge Rashi from the Gemara (BB 151b), which writes that transferring the get via kinyan sudar is ineffective—again indicating that a kinyan lacking an act of transfer is ineffective.

In Ketzot Ha-Ḥoshen sec. 242 s.k. 4 he discusses Rashi’s position and attempts to reconcile it. He explains that although a regular kinyan agav does not need tzeburin, with a get—if he transfers it without it being gathered—the get may indeed be acquired and there is a valid act of transfer, but there is a deficiency in keritut (as in a get in her hand with a string in his hand—i.e., one who hands his wife a get while holding a string tied to the get; see BM 7a). [Admittedly this is difficult, for if he actually transfers the place to her, why does Rashi require that it be acquired via agav and not by kinyan ḥatzer, as the other Rishonim understood?] He then continues and argues that even Rashi would concede that kinyan sudar would not work as giving a get, because that is a mode of sale and purchase (exchange) and not a mode of transfer, just as priestly gifts cannot be given via ḥalipin (see BM 12a).

In Kehillot Ya‘akov siman 17 he brings these words of the Ketzot and shows that this is a dispute among the Rishonim. Then, in subsection (gimel), he cites an amoraic dispute in the Yerushalmi regarding a get written on the horn of a cow, which he transfers to her by mesirah, and discusses what mesirah is at issue (handing over the halter—then it is like partial transfer—or seizing the cow itself). He then passes to a case where he sells her the get (for money), which again is a mode of sale and not transfer. After that he moves to kinyan hagbahah (lifting), kinyan yad (hand), ma‘amad sheloshtan, and many more.

In the end, even from the little we have seen here (a small part of siman 17), one can see how convoluted the theory is and how many assumptions it contains. We saw that kinyan is effective but not by virtue of its acquisitive character; rather, it is a kind of juridical transfer—and all this only if there is no problem of keritut—and that too depends on disputes; and the application to each kinyan depends on considerations and disagreements. See how many details and nuances there are in the general definition, and how many Talmudic examples he addresses (we have seen only a small fraction) merely to clarify the question of kinyan in a get. And that is without the definition of giving itself in the first three simanim there, and even after all this we have not yet handled all the examples brought in the Talmud and in the Rishonim. This description naturally returns us to the question: why didn’t the Gemara state directly whether kinyanim are effective for a get, and which ones? Why does it present us with dozens and hundreds of examples instead of stating straight out the result/conclusion?

Questions About the Aim and Meaning of the Lamdanic Enterprise: Lamdanut and Scientific Research

The lamdanic process I’ve described here looks like scientific research. In scientific research we collect various facts from observations we’ve conducted and try to build a theory that explains them. We test the proposed theory against additional cases and correct it if needed. In the end we should have a theory that explains all the cases we’ve encountered, and hopefully such a theory will give us predictions for what will happen in future experiments and cases not yet examined. The advantage of a scientific theory is its simplicity: it has a small number of principles, concepts, and theoretical entities with which one can explain very many examples and predict the results of future experiments.

How should we relate to a situation where the theory we ultimately obtain contains a number of components similar to the number of examples it purports to explain (each example prompted us to add to the theory another component—like an additional entity, a caveat, a sub-principle, etc.)? Clearly such a theory is not worth much. And if, because of its complexity and vagueness, this theory also does not allow us to predict any future result, then it seems to have no scientific value at all.

But notice that this is exactly what happens in our case. The theory that emerges regarding the giving of a get—if we want it to fit all the dozens and hundreds of examples in the Talmud and its commentaries—turns out so convoluted and vague that its number of components is roughly the number of examples. In fact, we don’t really have a formulation that succeeds in encompassing all of them. Hence it is not reasonable to rely on this vague and complex theory to rule on cases not appearing in the Talmud and its commentaries (this is the analogue of predictions of a scientific theory, except that here we have no source to tell us what the law truly is in those cases).

This raises the question: what is the aim of the lamdanic enterprise at all? Why is it important to go through all these examples and build a theory if, in the end, we get something clumsy and ill-formulated, very vague and open to many interpretations, and mainly ad hoc (i.e., the formulation was constructed after the fact from the examples to fit them all, sometimes by force and with difficulty)? Will such a theory really enable us to determine the law in other cases that come before us (the analogue of predictions of a scientific theory)? Can we rely on it for halakhah? Does this not portray lamdanut as an empty tool? On the other hand, what is the alternative? Even if we memorize all the Talmudic examples, as well as those appearing in the commentaries and the poskim, how can we draw conclusions from them for future cases? To do that, we must understand the principles underlying them. If so, it seems we have no other path but the “scientific investigation,” that is, the lamdanic analysis as I described it here.

The Meaning of the Lamdanic Clarification

As we saw at the beginning of the column around R. Kook’s essay, arguments and conceptualizations are a medium whose purpose is to create an immediate, intuitive understanding of the idea or concept in question. So too with the giving of a get: the various cases come to illustrate the concept and the process in question from different angles. Through them we are meant to generate an intuitive understanding of the valid process of delivering a get. How is this understanding formed? We saw that the way to reach direct understanding is through conceptualizations and arguments—and this is exactly what we do in the lamdanic process. Contrary to what many of us might think, its aim is not to reach a comprehensive scientific formulation that will cover all cases. My claim is that this analytical process is itself the aim.

It is a process of putting forward different general formulations, rejecting them via counter-examples, proposing improved formulations, rejecting them against further examples, further improvement, more rejections, and so on. In the end we indeed have no economical and effective formulation of a general theory, but something very complex and vague—yet that does not really matter. In the course of this very process there forms within us an immediate understanding of the process of giving a get. This is an understanding that cannot be articulated in words as a concise, focused, effective theory—but that does not matter. It is an understanding formed in our intellect during the process of debate, conceptualizations, and their rejection. We now have good intuitions about giving a get, and when other cases come before us we can draw on them to decide the law (whether we are dealing with a valid or invalid giving). We will not always be able to justify it explicitly, and even if we do, it will usually be a fairly vague argument, not sharp and decisive. But none of that matters, because what determines is our intuitive, immediate grasp of the concept—not the formulations, definitions, and arguments, which are only its representations.

This is apparently also why the Gemara did not give us the final result—i.e., the definition itself. It used dozens and hundreds of examples precisely because it had no way to articulate it as a general theoretical definition that would be accurate for all cases. Reliance on examples will bring us to a better understanding than providing a general definition that is vague and complex and does not truly capture the essence of all the examples—and therefore also cannot underpin an accurate treatment of all the cases we might encounter. This of course brings us back to column 482 on the Talmud’s casuistry mentioned above.

One can say that the lamdanic process is actually a kind of via negativa (negative theology) of attributes. In this process we learn everything that cannot be an exhaustive definition of giving a get, since we negate formulation after formulation. In the end we are left with a very vague and highly complex formulation that is not effective at all—but that is only on the logical plane. In the very process there forms within us a good intuitive understanding, even if not articulated, which resides within us and not in the words with which we summarize that ongoing effort. This resembles what Maimonides describes regarding understanding divinity via the negation of attributes. He also notes that after all the negations we do not arrive at a positive articulated understanding, but are left with a positive intuitive understanding of the Holy One—one we cannot articulate in words. See more on this at the end of the column.

Processes and States

In my two articles mentioned above (the short essay in Bar-Ilan’s weekly sheet and my article in Iyyun on Zeno’s arrow), I noted the uniqueness of giving a get. I explained there that we are dealing with an action—that is, a process—and that any attempt to define a process through boundary states is doomed to failure (see there for mathematical and physical explanations and implications). If our aim is to define the transfer of the get from husband to wife, we are dealing with an action, i.e., something dynamic. Yet all attempts to define it use boundary states that by their nature are static: we seek a definition via the before-state and after-state (e.g., previously the get belonged to the husband and in the end it is the wife’s property—defining acquisition as transfer; or previously in the husband’s hand and now in the woman’s—defining physical transfer).

In those articles I showed that one cannot define dynamic concepts via the boundary states through which they pass. This is why giving a get demands so many examples and why the Talmud cannot offer us an immediate definition in their stead. There is no reason to assume we will succeed more than the Sages themselves; hence it is not right to expect that at the end of the process we will have a general theoretical definition that is effective and exhaustive. Had there been such a definition—they would have given it to us directly.

Note that the lamdanic process itself as I’ve described it is also a dynamic process and should not be defined via its boundary states. Its importance lies in the process itself and not in the result at which we arrive. The analysis of concepts and ideas through comparisons to examples, forming various generalizations and rejecting them again and again—this is a process that generates within me, as a side effect or residue of the process, the intuitive insights regarding the concepts and processes at issue. On the logical plane, the result at the end, i.e., the formulation of the general and final theory, does not truly capture the process; therefore one cannot define it on its basis. What matters is the process itself. This is precisely the meaning of R. Kook’s distinction in the passage quoted at the beginning.

By way of analogy: developing a spacecraft that will reach the moon or Mars can fail, and even if it doesn’t, it may appear very wasteful and superfluous. Yet this process yields various significant by-products created along the way—scientific and technological inventions that serve us in many contexts unrelated to astronomical research and space exploration. These by-products remain with us, accompany us, and give us great value long after the mission itself failed or was forgotten.

Neural Networks

Modern science has afforded us an opportunity to understand better all that has been said so far. I mean what in computer science and neuroscience is called “neural networks.” I won’t enter here into details (I’m no expert), but I’ll describe what we need. One can find abundant material online (see, for example, on Wikipedia here).

A principal part of our brain consists of a network of nerve cells (neurons). Each such cell can be in one of two states (firing or not, 1 or 0). Between nerve cells there are connections that link them and transmit information from one to another and vice versa. But not every pair of neurons in the network is connected. In each neural network the structure differs. What determines the structure are the connections between the various neurons: the network is defined by which neurons are connected and which are not. In addition, each connection between a pair of neurons has a different weight, and these weights can change according to the data the network receives. In sum, many connections—originating in other cells—enter each neuron. The connections entering a neuron from all directions sum with one another (each multiplied by the weight of that connection), and according to the result (determined by the activation function) the cell decides whether to be off or on. Changing its state causes it to send a new signal to all the cells connected to it, thereby changing their state, and so every cell influences and is influenced by all the other cells connected to it.

From the mid-twentieth century, physicists and mathematicians began to notice that such networks have tremendous computational power. They can perform functions similar to the human brain (whether the brain is nothing but a neural network or contains something beyond that is disputed; in neuroscience it is customary to assume so). For example, they can perform mathematical computations, recognize images or letters (handwriting), understand financial markets, speech recognition, text analysis, and more.

To cause such a network to perform a certain task we must “train” it. Training can be done in various ways, but here I will refer only to what is called supervised learning. Suppose we want to train our network to recognize horses. In supervised learning we feed the network very many images, among them images of different horses and of other objects, with each image accompanied by the correct answer (1 if it is indeed a horse, 0 if not). The network receives the examples one after another and is supposed to output some value for each image it receives as input. We organize the internal weights of the network’s connections such that the result obtained for an input that is a horse will be 1 and for a non-horse input will be 0. Note that if one assigns weight 0 to a certain connection, that cancels that connection between the pair of neurons. That is, training determines the network’s structure and sets the weights on its connections. After enough diverse examples, including creatures very similar to horses that are not horses, a network with certain weights will be formed that should give correct answers for all the training examples (1 for horse images, 0 for others). The assumption is that a network trained and constructed thus will be able to recognize other horses—i.e., images it has not seen—reliably. Of course, reliability depends on the network’s quality and complexity and on the quantity and variety of training examples.

A Comparison with Classical Software

To understand the concept better, try to think what we would have to do to train a classical program to recognize horses. We would have to analyze images of different horses ourselves and understand what distinguishes those images from other images. After arriving at conceptualizations and clear definitions of a horse image, we would have to feed them into the program as instructions: if you see four legs of such-and-such kind and such-and-such a tail, eyes at such-and-such a distance and size, a belly of such-and-such a shape, and so on—this is a horse (i.e., print 1). Otherwise—print 0. This is a very hard task, and generally practically impossible. It is extremely difficult to give a sharp conceptualization that will describe unequivocally all horse images and distinguish them from other images (e.g., donkeys or mules). Note that this is a task every small child performs easily, but a classical program—even a very sophisticated one—probably won’t succeed. In a classical program, the task of recognizing a horse is nearly impossible. The conclusion is that the child’s brain probably doesn’t operate like classical software; he does not really extract the features of horse images but does so in another way.

Indeed, the logical structure of an artificial neural network (on a computer), which is very similar to how the brain functions, performs such a task with great ease. A fairly simple network will succeed in recognizing horses at a good level of reliability after training that is not especially complex or intensive. As I described above, training a neural network involves no analysis of the images and extraction of features that characterize horses—i.e., all that we would have to do before writing our classical program. In the neural-network approach, the program itself does all of that during training. It does so by organizing the weights of the various connections in the network.

But here lies the important point: the network does not really do that. Nowhere in the network will you find what those distinguishing features of horse images are. That information does not exist in the network in any sense. It is not correct to say the network performs the conceptualization that people would have to do in the classical program. Rather, it skips that and blindly constructs the connection weights so that the output will be correct for each example. It does not “know” what a horse is or what its features are. This is not the right language to think about how a neural network operates. It is a blind mathematical process that ties input to output. If you regard the input image as a series of numbers (there is a way to represent an image numerically) and the desired output as a number (1 for input representing a horse image, 0 for another), all the network does is find a function that maps the correct output to each input. At no point is there analysis of horse images and understanding of their features. No one is speaking about horses at all. We are merely playing with numbers. And astonishingly, it works wonderfully. AI practitioners themselves stand amazed at the performance of programs they themselves wrote (such as chattgpt, for example). When they built the program, they did not imagine it would succeed in doing all it does in practice.

The Chinese Room

To understand the meaning better, I return to the example of the Chinese Room mentioned above. The renowned British mathematician Alan Turing proposed a test to give us a criterion for when we can treat a computer as if it were a human being. The test proceeds thus: Reuven sits in a room before two terminals, each connected to a different external room. In one sits a person (Shimon) and in the other an artificial intelligence. Reuven converses via the two terminals with Shimon and with the program, and solely from the conversation must decide which terminal represents Shimon (the person) and which represents the computer. If and when there will be a program that competes on equal footing with Shimon—i.e., if and when there will be a program so sophisticated that Reuven cannot tell which is the human and which is the program—then that program merits treatment as a person (leave aside the moral question and questions of emotions and mental states; we focus here on the ability to think and communicate).

Against this test John Searle raised the Chinese Room example. Imagine Levi, an Israeli native who does not know a word of Chinese, sitting in a room with two windows. Inside the room there is a vast container filled with Chinese characters in infinite quantity. Through one window he receives questions written in Chinese; his task is to collect characters from the container and assemble from them answers to the question—in Chinese—and send them through the other window. Every time he answers irrelevantly he receives an electric shock. Suppose we have infinite time, and Levi slowly learns to answer in a way that avoids shocks. At the end of the process he conducts a written dialogue in fluent Chinese, receives questions, and answers them appropriately. Has Levi learned Chinese? Does he understand Chinese? Clearly not. He indeed converses in Chinese, and the relation between input and output is identical to that of a person fluent in Chinese; yet there is a crucial difference: the Chinese speaker has understanding behind the answers he sends, while Levi has no aspect of understanding. His cognition is not involved in the process. He does not understand the questions or the answers, but he knows how to match an appropriate answer to each question. He functions like a program containing a function that maps an output to each input. Levi’s level of understanding in that state is no different from the program’s. Both converse, but neither has understanding. They move their lips, but no cognitive aspect accompanies it. Searle sought to demonstrate thereby that even if a computer passes the Turing Test, it does not mean it is a person. What it lacks is the dimension of understanding. The computer correctly pairs input with output (it has constructed a function via its neural network), but it does not “understand.”

One might wonder whether such a thing is possible: can a person conduct a conversation in Chinese without understanding anything? Neural networks show us that yes. If we train a network long enough on enough examples, it will speak Chinese—or any other language you like. This already exists (anyone who has used chattgpt knows this). Is chattgpt a person? Today, when we know such creatures, it is perfectly clear that Turing erred—i.e., even a program that passes this test is not a person. What it lacks is the component of understanding. It seems to me that AI practitioners do not sufficiently attend to this conclusion. Everyone is very enthusiastic about these programs, but few note that so far the conclusion from their amazing performance is the opposite of what was expected: today almost no sane person imagines treating them as human, despite their passing the Turing Test with eyes closed. It seems that the Turing Test has been dealt a severe blow by these programs and their impressive achievements, and it turns out Searle was right.[1]

Back to Lamdanut

We can now return to the question of lamdanut. Note that the process I have described here is very similar to the process I described above regarding the lamdanic analysis of the sugiyot of giving a get. We saw that the learner receives dozens and hundreds of varied examples that “train” him. True, as a human he does try to build theories and conceptualizations and tests them against subsequent examples that refute them—and then he builds more sophisticated theories and those too are rejected. We saw that in the end he is left without a general theory that truly explains all the examples. The reason is that such a general theory would correspond to classical software. Were it possible to arrive at a general, unequivocal theory about giving a get, this would mean that one could construct classical software to decide what counts as valid giving according to a finite set of rigid criteria. But as we saw, even for relatively simple tasks (like recognizing horses) this is practically impossible. By contrast, a neural network succeeds in coping well with such tasks precisely because it does not perform conceptualization. It pairs output to input and generates for each input the correct output. The encounter with the examples builds the internal network and its weights so that it will give the correct answer for future cases.

This is exactly what happens to us during lamdanic analysis. The confrontation with examples—via building theories and conceptualizations and negating them—creates within us the right weights, so that when we encounter other examples we will reach the correct answer (1 = valid giving; 0 = invalid giving). The difference between us and an artificial neural network is that at every stage we do this through conceptualizations and general theories and their rejection, whereas the network simply adapts its weights to the examples. It does not “think” and does not “understand.” At the end of the lamdanic process, the hope is that there has formed within us a correct intuitive understanding of giving a get (a network with the right parameters has been built within us), and when we encounter a new case this intuition will yield the correct halakhic answer. We achieved this via conceptualizations and their rejection—i.e., via definitions and general claims and analysis of them—but only via a negation of attributes: all the conceptualizations fall away along the path, and this process builds within us a network that improves with each additional example, in the hope that we will reach a network that gives us the correct answer. In effect, “da‘at Torah” forms within us—but not in any mystical or spiritual sense. It is a process of intuitive understanding of giving a get. We tend to see in this something mystical because we lack an explicit conceptualization of the final theory (i.e., the conclusion of the lamdanic analysis). But there is nothing mystical here (though certainly something not fully understood by us).

Back to R. Kook and Maimonides’ Via Negativa

In column 653 I sketched artificial intelligence and linked it to faith and intuition. This is the light of R. Kook that helps us eliminate the shadows and penetrate through them to the understanding and knowledge of the thing itself. The description I have given here indeed raises the question whether there is anything in us beyond such a neural network. After all, I described the human lamdanic process, and it seems entirely parallel to what happens in a neural network. In columns 590, 591, and 592, I discussed the definition of AI and its limitations and to what extent we can suppose that our brain is nothing but an advanced neural network (i.e., that there is nothing in us beyond that computational tool). I will not reopen that question here. In any case, the analogy between our lamdanut and the way a neural network is trained, in my view, sheds light and clarifies the meaning of lamdanic analysis. We can now more easily understand the via negativa, at least in the lamdanic context and perhaps also in its original theological context. Therefore I will comment briefly here on Maimonides’ doctrine of the negation of attributes.

I believe I have already noted here the problematic nature of Maimonides’ via negativa. On the face of it, a logical analysis presents it as an empty tool. Maimonides says that the claim that God is merciful does not truly state that He is merciful but only that it is not correct to say He is not merciful. The reason is that, because of our limitations as corporeal and finite creatures, we have no way to know the attributes of the Holy One, and all we can do is negate from Him any human (and indeed any) attribute. But according to this, two difficulties arise regarding claims about God’s attributes, and I will use as an example the claim that God is merciful:

1. Do these attributes teach us anything about God? We have learned that it is not correct that He is not merciful, but equally we do not know that He is merciful. We negate various things from Him, but what remains to us at the end of the process?

To this Maimonides claims that nevertheless, by virtue of negation, there remains within us a residual understanding regarding God’s attributes. Negation also teaches us something, even if via negation. After we have negated everything from Him we remain with a certain hylic (rudimentary) understanding of Him.

2. Do such claims say anything at all? Just as one must not say He is not merciful, equally one must not say He is merciful, for that too is a human attribute. There is full symmetry between mercy and cruelty. So what is the meaning of the statement that He is merciful? Equally I could say He is cruel, intending that it is not correct that He is not cruel. If so, we truly remain with nothing. Apparently these claims are contentless. According to this it is unclear how Maimonides can speak of that residual hylic understanding that remains to us at the end of this process.

I will preface by saying that, personally, I do not understand why a doctrine of negative attributes is needed at all. In my eyes there is no problem using positive attributes regarding God (as the kabbalists hold). It seems this matter penetrated to us from Islamic philosophy and has no logical or traditional basis. Still, here I will propose a possible interpretation for those who, like Maimonides, maintain that the attributes of the Holy One must be negative—and I will do so in light of all we have seen so far.

In the terms of column 660 one can say that the negation at issue here is not a contrary negation but an abnegating negation. That is, when we negate His being cruel, we have not reached the conclusion that He is merciful; rather, we have reached the conclusion that there is in Him some hylic something related to mercy. True, we have no way to articulate and conceptualize it, but by philosophical and theological reflection and via the negation of attributes we succeed in creating within ourselves an intuitive understanding of God’s qualities as merciful in a positive yet hylic manner. The negation is not a statement that He is not merciful or not-not merciful; the negation is a peeling away (a hylicization) of the attributes while trying to reach via intuition the abstract concept they describe or represent. In this sense it is indeed correct to say that God is merciful (i.e., it is incorrect that He is not merciful), but it is not correct to say that He is cruel. There is no symmetry between mercy and cruelty. When we negate His being cruel this builds in us some understanding of Him, but it is not the understanding that He is merciful; rather, it is something more abstract for which we lack an explicit conceptualization. The opposite of the attribute “merciful” is not the opposed attribute “cruel,” but the hylic concept that is conceptualized and represented by mercy.

One can say this is a Buddhist negation of attributes and not a negation in the Western logical sense. The question whether God is merciful or cruel is answered with “mu” (see the motto to column 660 and the end of that column), but it is not a neutral mu equally distant from cruelty and mercy. It is a mu of passing to a more abstract plane, which nevertheless is better described by mercy than by cruelty. God’s attribute is the thing-in-itself whose phenomenological manifestations are the mercies as we usually know them in people.

We saw above that all the conceptualizations we negated in the lamdanic analysis of giving a get built something positive within us. They all turned out incorrect, but it is not correct that they now have the same status as their opposites. The correct answer (which cannot be articulated) is a Buddhist negation of them, not their inversion. The rejection of the theories we negated on the basis of examples does not lead to a contrary negation but to an abnegating negation—or better, to their peeling away (hylicization) from their attributes.

A Further Look at Torah Study and Its Meaning

Finally, I wish to return and consider the meaning of Torah study as it emerges from the picture I have described here. I preface that the main points of this section touch on the two remarks about Torah study that appear in column 479.

The Aharonim (see, for example, Birkat Shmuel Kiddushin, sec. 27, and many more) discussed the parameters of the mitzvah of Talmud Torah and wondered whether its essence is knowledge (with study only a means to acquire it) or whether the main value is the study itself (the very engagement with Torah being the aim). From our discussion here it emerges that the question nearly empties itself of content. The essence of study is the very engagement and contact with Torah—but this is precisely also the way to acquire the knowledge. True, the knowledge at issue here is not the usual product of study, i.e., the collection of halakhic details we gathered along the way. That is the domain of beki’ut (breadth learning), which in my view is precisely a means and not an end. The goal is lamdanut, whose main concern is the collection of insights formed within us during that engagement—in the process described here. The aim of lamdanic engagement is not the collection of halakhic–Torah data points but the formation of intuitive patterns within us (da‘at Torah) that enable us to understand the essence of sugiyot and, from that, to make halakhic decisions for additional cases.

There I also explained that this does not mean that study is only a means to halakhic ruling. Study is a value unto itself, but that value merges with the ability to rule halakhah. Acquiring that ability is the study. Moreover, I explained there that the core of study is lamdanic analysis because the aim of study is cleaving to the Holy One, and this very engagement creates that devekut. This is not ḥasidic devekut, whose essence is experience and emotion; rather, the study itself is cleaving to God—“the Holy One, Israel, and the Torah are one” (this is elaborated in Nefesh Ha-Ḥayyim, Gate 4, and in Tanya, chs. 4–5. See my essay here, in column 379, and elsewhere). We can now understand this better. Devekut forms of itself in the process of study: more and more hylic insights that constitute the Torah—that is, the light clothed in the concrete halakhic vessels (R. Kook’s “shadows”)—enter into us and become embedded in us during our pounding on the particulars of halakhah and the examples. As in the kabbalistic picture, the pounding on the vessels/shadows uncovers the light within and behind them.

If the Holy One and the Torah are one, this is precisely the description of cleaving to Him, by which the holy triad is formed that includes Israel as well—bringing Israel into that unity (with the Holy One and the Torah). I elaborated on this idea from almost the same vantage point in my essay “What Is ‘Efficacy’?” where I described devekut as a connection to the metaphysical–ontic essences underlying halakhah (ḥaluyot).

[1] In the past I was asked for my view, and I said that we will be able to decide whether it is a person or not when we meet such a creature before our eyes. Then we will have an immediate perception of it and will be able to form a position. Theoretical claims on such a matter have limited meaning.