In the Phaedrus, Socrates outlines two essential procedures of classification. One is to bring a plurality under the appropriate form.

The aim of our science, it is often said, is to “cut nature at its joints” (Campbell, O’Rourke, and Slater 2011). But does nature have joints? And what determines where those joints are?

Particular categories exfoliate from others via principles of division which both separate (from taxonomic predecessors) and unite (among their taxonomic successors). So in the end we have a unified grouping united by specification commonalities. And just herein lies the difficulty. For in committing ourselves to a claim of the format “All items of category C are united in possessing feature F” we undertake risky factual commitments when definitions are not at issue. “Swans are white” – but unfortunately there are those antipodean exceptions. “Species breed true by type” – but if they did so invariably then there could be no evolution. The history of science is a litany of unraveled taxonomies as new phenomena fail to conform to established patterns (Oppenheim 1936, Hempel and Oppenheim 1936).

Scientific Categorization

Categorization is crucial to all cognition: without categories, we have the possibility of neither generalization nor individuation: an appreciation for kinds of things and for identification of things, essentially as of a kind. Navigation in our world, or any world, requires its conceptualization in terms of categories. The task of science is of a piece: an extended cultural tradition of attempting to navigate our world in terms of understanding and prediction. The goal is to represent a world well enough to accommodate ourselves to it and, where possible, to accommodate the world to us. As in all conceptual navigation, as in all cognition, categorization is crucial.

It is possible to write the history of science as the history of an attempt to get the right representation of the structure and mechanisms of the world around us. That representation is inevitably in terms of categories – categories of both things and mechanisms. In that sense, the history of science is the history of an attempt to conceptualize the world in terms of the right categories.

What makes them the right categories? Here there are two answers.

Reality is a manifold of variation; the variety, diversity, and complexity the world affords is virtually endless: individually and collectively, its constituents exhibit an ever-proliferating array of features and characteristics. But the human mind, though also vastly complex, cannot keep in step. Its focus has to be not in the endless detail of real individuality, but in its patterns and regularities. The world deals in the variant detail of specifics, while the mind has little choice but to address this via types and in kinds. Reality proliferates, the mind categorizes, in effect endeavoring to deal with an analog Reality by digital means.

Categorization as a Conceptual Necessity

To be intelligibly and usefully managed, information must be systematized. To this end categories are an indispensable resource. Categories provide for the conceptual basis of generalization and identification. Categorization is an essential and unavoidable instrumentality for conceptually navigating a world – indeed for being able to conceptualize a world that needs to be navigated.

Categories map out the conceptual structure by whose means we map out the domain of our knowledge. They provide for the table of contents and the index by whose means our knowledge is organized and made accessible. They render our otherwise incoherent mass of information cognitively manageable and provide an informatively convenient framework for our cognition about the realms of the actual and the possible.

All cognition beyond a dull mental hum demands categorization. All experience beyond passive reception of a perceptual blast demands categorization. All action beyond flailing – and perhaps even that – requires the guidance of categorization. Categories are essential in any cognition, any experience, and any action as we know them, and indeed in any cognition, experience, or action for beings to any degree like us. Successful action in the world demands information about the world, and information as we know it is conceptualized in terms of categories. As creatures that must operate by means of information, we must categorize.

There is a long philosophical tradition in which the emphasis is on a search for the highest or most general categories. Suppose that someone says, with sublime vagueness, “I am thinking of something.” What would this be? And suppose that we can get answers to just ten questions. What should we ask? We would be well advised to adopt something like the following list:

Is it concrete or abstract?

Is it unique or does it have many instances?

It is not merely in terms of logic, metaphysics, and the natural world around us that we use categories in an attempt to predict, explain, manipulate, and understand. We employ and apply categories in ethical and social cognition as well. Categorical misapplications with regard to the natural world can result in failed attempts to predict, explain, understand, or intervene in the course of nature. Categorical misapplications in the social realm can carry an additional burden of ethical mistake and negative social consequences.

There are a range of lessons to be learned here, both regarding the nature of categories – studied here in a realm independent of “nature's joints” – and with regard to social construction and ethical action.

Categorization Errors in Social Cognition: Oversimplification

A major purpose of categorization is simplification. The world around is a complex place, with no event exactly repeated in all particulars and no process repeatable without some variation. Effective cognition and effective action demand that we work with a simplified representation of such a world, in which items, events, and processes are conceptually grouped in terms of categories.

One clear danger in any such categorization is oversimplification: a categorization that is too simple to do justice to the genuine complexity with which we are attempting to deal. That oversimplification can be particularly troublesome when it is complex social realities which we are trying to address.

Consider for example categories very often used in thinking about groups of people in our society, and the ethical emotions we find appropriate for different groups:

• • The homeless.

• • The mentally ill.

• • The drug-addicted.

• • Petty criminals.

We sympathize with the homeless and want to relieve their plight. A similar sympathy seems appropriate with the mentally ill, though the measures to be taken would seem to be very different. With regard to the drug-addicted we may have mixed emotions, particularly in recognizing lives that have been characterized by repeated bad but voluntary choices. In thinking of petty criminals our sympathy may waver or disappear.

What is wrong with this exercise is the illusion that these categories are genuinely distinct. Many people fall into several or all of them. The most extensive survey of the homeless, taken on a single night in 2014, reached the consensus that 45% of the homeless had some form of mental illness, with serious mental illness in an estimated 25% (Torrey 2021).

The present book is the third in a series of collaborations, its two predecessors being:

• • Beyond Sets: A Venture in Collection-Theoretic Revisionism (Frankfurt: Ontos Verlag, 2010).

• • Reflexivity: From Paradox to Consciousness (Frankfurt: Ontos Verlag, 2012).

In this book, as in the others, a key philosophical concept (collectivity/totality, reflexivity/self-orientation, and in this case categoricity/sortalization) is subject to critical scrutiny and innovative exploration.

A great deal of water has flowed under the philosophical bridge since the tract on The Conception of Types in the Light of the Modern Logic (Der Typusbegriff im Lichte der Neuen Logik) by Carl G. Hempel and Paul Oppenheim (Leiden, 1936). This classic venture at renovating categorization in the heyday of logical positivism was predicated on now-outdated perspectives and requires updating in the wake of the wider horizons that subsequent philosophizing has opened up. The present deliberations represent an attempt to accommodate these broadened perspectives.

The authors are grateful to Estelle Burris for aid in preparing their text for publication.

A number of the lessons regarding categories that we have emphasized throughout also play a role in classical puzzles and paradoxes. We have attempted to counter a range of tempting philosophical conceptions of categories, arguing among other things that:

• • Categories are not set-like entities.

• • Categories typically carry the intentionality of their membership conditions rather than being defined purely extentionally in terms of membership.

• • Both categories and the similarity relations that travel with them can be essentially vague.

• • Categories function pragmatically in terms of salient purposes and interests, and their application can thus vary with different purposes and interests.

• • Categorical classifications need not form tree-like structures of exclusive and exhaustive sub-categories.

Each of these points appears again in a consideration of category mistakes and a range of paradoxes, in either classical or contemporary form:

• • The Sorites and its ancient kin: the Phalakros and the Millet Seed.

• • The Ship of Theseus, the Statue and the Clay, Dion and Theon and the Problem of 1,001 Cats.

• • The Sancho Panza Hanging Paradox and The Contract of Protagoras.

• • The Liar.

Paradox is often presented in terms of a set up followed with a final “gotcha” question:

• • Does a trio of sand grains constitute a heap or not?

• • Once we have replaced each plank, do we have the ship of Theseus or not?

• • Is this lump of clay a statue or not?

• • Is the Liar sentence true or not?

But the appropriate response is sometimes not an attempted “solution” in the terms in which the “gotcha” question is phrased but a rejection of the question and perhaps the entire setup. For the paradoxes we will survey, the proper is often “The question, and perhaps the entire set up, is based on a misunderstanding of the nature of the categories in terms of which it is phrased.” For a range of classical paradoxes, such an approach allows us to characterize fairly precisely what that categorical misunderstanding amounts to, even if it doesn't always offer the kind of solution that the “gotcha” question seems to demand.

Our aim is to point out the crucial role that categories play – and that our assumptions regarding the nature of the categories play – in creating these puzzles in the first place. Although our analysis may at times implicitly suggest a, what we have to offer need not stand or fall with so ambitious a goal.

An Arc of History

Our concern is with categories in general, including chemical categories, biological categories, color categories, social and cultural categories, even mystery categories and categories of fictional entities. But much of the philosophical history regarding the nature of categories in general is buried beneath a history focused on a very specific set of categories: the categories, envisaged in a universal classification as the highest kinds or genera. A range of alternative and conf licting proposals as to which are the categories in this sense stretches from Aristotle through Kant and Husserl to the present.

Which are the highest categories is a topic of controversy, both historical and contemporary. We have also noted the ancient debate (as old at least as Porphyry) as to whether the categories are (mere) human contrivances or reflect objective differences in the nature of things via natural kinds. The philosophical history reveals a range of different approaches to the nature of categories – envisaged alternatively as metaphysical, epistemic, linguistic, or pragmatic. But it also reveals an intriguing trail of development through these approaches. In broad strokes, the history of categorytheorizing exhibits an increasing naturalization. We will trace that trail of changing interpretations of the categories through exemplars in Aristotle, Avicenna, the Ramist Revolt, Locke, Kant, Peirce, Frege, Husserl, and Ryle to contemporary debates.

The two central areas of debate, both historically and in contemporary guise, are debates as to which the categories are and to what they are. The lesson we draw from history is that both areas of controversy reflect deep philosophical mistakes. One mistake is the presupposition that there is some unique and ultimate set that are the categories. Another is the mistaken assumption that categories must fall to one side or the other of a false metaphysical/epistemic divide.

Aristotle

Aristotle was equivocal as to whether categories are about language or about reality. But this is understandable, given that our informative talk about things is an endeavor to make manifest what things actually are, adequatio ad rem: “approximation to reality” usually taken as “correspondence to fact.” All of the Aristotelian categories indicate different aspects of what something is or does.

Aristotle's interpreters are not unanimous with regard to the nature and function of his Categories. But his main Anglo-Saxon expositors have approached the matter from an epistemic point of view.

Preliminaries

Categories, categorization, and the organization of information are crucial topics throughout the broad realm of cognitive studies, not merely in philosophy but in both theoretical and empirical work in linguistics, psychology, and the brain sciences. Each of these disciplines offers intriguing findings regarding the ways we categorize—findings of importance for the understanding of human knowledge.

The central philosophical point that categories are crucial for cognition, in general, is fully recognized in the empirical work of other disciplines. As the linguist George Lakoff emphasizes,

A point that is clear philosophically, and essentially a priori, is that categorization and discrimination are linked. To recognize things as belonging to different categories is to be able to discriminate, at least conceptually, between items in those different categories. The “at least conceptually” allows us to say that we can discriminate between prime numbers greater than a googol and non-primes greater than a googol, though actually being able to name any of the former may be beyond our abilities. With that proviso, indeed, to recognize things as belonging to different categories is to discriminate between them.

This in no way forces us to say that all discrimination entails categorization: that if we recognize that this thing is different than that, it must be because we have assigned the two things to different categories. Categories are kinds, with the kinship of kinds determined by the pragmatic context of our purposes. Two things x and y will always belong to different sets or different arbitrary collections: we need to merely think of (1) the set to which the coins in my pocket and x belong as members and (2) the set to which the coins in my pocket and y belong as members.

What Is It to Be Rational?

Man, it has been said, is the rational animal. Our doings need not be determined by automaticity or instinct, but can result from deliberative thought. Deliberative thought based on beliefs and evaluations regarding ends can be the determining of actions. Accordingly the full exercise of our human rationality calls for trying to do the best we can manage under the circumstances to meet our genuine needs and our appropriate wants—that is, to realize our best interests.

Does Rationality Have Different Forms or Versions?

Rationality has three different modes, cognitive, evaluative, and practical, depending on whether the issue at hand relates to matter of beliefs, evaluation, or action. And accordingly rationality has three departments concerned, respectively, with its cognitive (epistemic), evaluative (normative), and procedural (practical) dimensions. In each case, we look to the best estimate we can make of

Gödel's Belief in a Leibnizian Conspiracy

Kurt Gödel's 1930s demonstration of the provability incompleteness of axiomatic arithmetic was a monumental achievement in mathematical logic and marked him as “one of the most significant logicians in history.” In the mid-1940s, Kurt Gödel embarked on a systematic study of Leibniz's logic which continued for at least another decade. During this time, I myself was writing my Princeton doctoral dissertation on Leibniz's Cosmology, and we had something of a tug-of-war over the Leibniz material in Firestone Library—each recalling for his own needs material out on loan to the other. (Unfortunately for me, we never made any direct contact.)

Gödel described himself as, unlike Einstein, “following Leibniz rather than Spinoza.” As Gödel studied Leibniz via Louis Couturat's classic La logique de leibniz, he became convinced that resistance to the logico-mathematical Platonic realism of his own position was prefigured in a conspiracy of suppression and silence that had kept Leibniz's similar insights from being properly understood and appreciated. And the more Gödel studied Leibniz, the more keenly he suspected that Leibniz might have anticipated parts of his own work—and especially his demonstration of arithmetic's provabilityincompleteness. Gödel came to this view because he saw Leibniz as a precursor and a kindred spirit whose problematic reception was a foreshadowing of his own difficulties.

But while there is little doubt that Gödel saw Leibniz as a precursor engaged in a kindred inquiry, there remained in his mind questions about the extent of anticipation about findings. Moreover, the matter of motivation remains somewhat obscure. Was he worried that he might have been fully anticipated? (After all, in mathematics all the credit goes to whoever gets there first.) Or was he hopeful of finding that he had succeeded where the great Leibniz had tried and failed? Perhaps we will never know. But either way, Leibniz's work on matters of provability and demonstrative systematization in mathematics was of deep concern to Gödel.

Curiously, Gödel saw Leibniz as a precursor not only in logic and the foundations and epistemology of mathematics but also in metaphysics, the general theory of reality.

Since the inauguration of philosophy among the ancient Greeks, the definitive mission of the enterprise has been to address the “big questions” regarding mankind, the world, and our place in the human and natural scheme of things. The task has been to elucidate how we should understand these matters and what we can and should do to implement this understanding in action.

These issues set the philosophical agenda as cultivated in Greek antiquity, and give rise to a tripartite division of the discipline into logic (the theory of thought), metaphysics (the theory of existence), and ethics (the theory of human conduct). It equipped the agenda of Greek philosophy with such topics as truth, knowledge, mathematics, justice, novelty, reality, and analogous key issues of enduring human concern.

But these fundamental issues were only the starting point. Questions always have presuppositions and these raise further questions. Over time the range of issues or the question-agenda grows substantially larger and becomes changed in matters of emphasis and orientation. Later questions were given out of attempts to grapple with earlier ones. The problem-agenda of a particular thinker or era is bound to reflect the state of knowledge of the day, the climate of opinion of the culture, and the specific connections of individuals. It is bound to be diverse and variable.

The individual philosopher is of course free in the choice of problems, unfettered in his work, and unhampered by any need for agenda balance. With individuals this does not matter. But at the communal level it does. But the wider community bears an obligation to the discipline as a whole and neglects major areas of thematic orientation at its own peril.

The definitive task of the philosophical enterprise is systematic in its constituting mission to provide a comprehensively integrated account of the salient problems. The situation of medicine is not dissimilar here. The individual physical may coordinate in the eye but the medical professor cannot afford to leave the ear out of it. And essentially the same thing holds with regard to philosophy. If the enterprise is to succeed, then somehow the collective spirit of the discipline must put its hidden hand to the task of overall correlative integration and systematization.

The Expressive Sufficiency of Language in the Tractatus

The unifying framework of Ludwig Wittgenstein's intellectual life was provided by a project of vast and monumental ambition. Its ground plan was laid out in his earliest work, the Tractus Logico-Philosophicus, with its grandiose vision of a perfect language, a symbolic manifold able to provide the means for accurate and complete characterization of reality.

The crux of the Tractatus lies in its commitment to the problematic idea that language—not exactly as we have it but as we can get it with some logical repairs—is capable of a complete and adequate descriptive presentation of the world's facts. But there is good reason to think that this idea of the ontological adequacy of available language is not only problematic but also illusory—that linguistic completeness and accuracy are an unrealizable mirage, not because they are so difficult to achieve, but because the very idea of their realization is ultimately as incoherent as it the idea of a complete and detailedly accurate mapping of terrain. All in all, the Tractarian position betrays a yearning for an impracticable absolute.

The Tractatus gets off to a peculiar and problematic start by riding roughshod over the crucial distinction between fact and fiction, stating true and actual facts and unrealized possibilities. As it represents the situation: “Logic deals with every possibility and all possibilities are its facts” (2.0131). For logic all possibilities exist, and the distinction between actually and merely possible states of affairs is abolished. In logical rigor, nothing whatsoever is accidental or contingent: “In der Logic ist nichts zufallig” (2.012), and the world (1.2) is throughout a web of necessitation. The Tractatus thus becomes a logical revision to Spinoza's Ethics and on its principles, language and its inherent logic become means for the exposition of whatever can meaningfully be said.

Building Blocks of the Tractarian Position

On this basis, the position of Ludwig Wittgenstein's Tractatus Logico- Philosophicus regarding the relation of language to reality is predicated on his pivotal contentions regarding five crucial concepts. They stand as follows:

• I. World. The world (reality as he also calls it) is the totality of facts about what is actual and possible. (1-1.21)

• II. Facts. Facts are possible states of affairs. They comprise the world. (1.21- 2; 2.04)

• III. Thought.

The outbreak of war in the late 1930s created turmoil throughout the realm of European scholarship. Even in the distant USA, there were massive reverberations, and after America's entry into the conflict, the demands of war had a massive impact as a whole generation of young people was diverted from normal life into military activities. Academic philosophy could not but be affected. The generation born in the 1910–1920 decade was especially hard hit, with its ranks depleted by the demands of war. The present discussion will examine one instructive instance of this phenomenon, the diversion of one particular and a particularly significant individual from philosophical scholarship into a very different pathway of endeavor.

The person at the focus of my deliberations here is something of a paradox. He is the most important twentieth-century philosopher no living philosopher ever heard of. A scholarly official near the top of the Pentagon hierarchy (Alan C. Enthoven) said he “was the most important strategic analyst and thinker of his own time.” A high official in the State Department (Richard Rosencrance) wrote that “Probably no civilian strategist has had more influence in the nuclear age that [he].” I think it is fair to call him “the Clausewitz of nuclear age.” His name is Albert C. Wohlstetter. And he was a trained philosopher of science.

Albert James Wohlstetter was born in 1913 to an affluent Jewish family living in the then-fashionable Washington Heights section of Manhattan. (His father, an attorney, was the chief counsel to the Metropolitan Opera and a major investor in early filmography production.) Wohlstetter attended CCNY (with a scholarship in modern dance!) graduating with a B.A. in 1934. (There was always a touch of gracefulness to his movements.) Like many of his New York City contemporaries, he was enthralled by Trotskyite communism in his early years.

After college, he turned to philosophy—specifically to philosophy of science and logic. From 1935 to 1940 he was a graduate student at Columbia, studying logic and philosophy under Ernst Nagel and Morris Cohen, and statistics under Abraham Wald. He took his M.A. in 1938 with a thesis in symbolic logic with Nagel, and had his first publication in 1940 with a review of J. H. Woodger's book on The Technique of Theory Construction in the Journal of Symbolic Logic.