This is a working draft, last updated on 6 September 1998.
The term 'objectivism' has been made famous (or infamous) by Ayn Rand, who chose it as the name for her philosophy of reason, individualism, and freedom. However, there have been objectivists other than those of the Randian variety, even in contemporary philosophy. The most prominent of these, though his work in philosophy is less well known than his spectacular results in logic and the foundations of mathematics, is Kurt Gödel (1906 – 1978).
Gödel is certainly no Randian, for his objectivism is bound up with a thorough-going Platonism. Yet his ideas are eminently worth investigating. Among those privileged to have interacted with Gödel, none has reflected on his ideas more fruitfully than Hao Wang, who presented his reflections on Gödel in several books prepared from the early 1970s until his death in 1995 (footnote #1). Wang's ideas are interesting in their own right and provide a useful counterpoint to Gödel's Platonism. In the end, however, I find neither thinker fully satisfying, which is why section three of this paper is entitled "Beyond Gödel and Wang".
Kurt Gödel is most famous for Gödel's Theorem, also known as the Incompleteness Theorem. Wang presents the following statements of the theorem (Wang 1996, 3):
Many have been the interpretations of this theorem (Nagel and Newman 1958, Hofstadter 1979, Penrose 1990, etc.), ranging from those who celebrate it as an indication of the limits of reason to those who use it to prove the superiority of mind over machine. A particularly fascinating generalization of the theorem was presented by Gödel himself:
A completely unfree society (i.e., one proceeding in everything by strict rules of "conformity") will, in its behavior, be either inconsistent or incomplete, i.e., unable to solve certain problems, perhaps of vital importance. Both, of course, may jeopardize its survival in a difficult situation. A similar remark would also apply to individual human beings (Wang 1996, 4). (footnote #2)
Gödel developed the Incompleteness Theorem in 1930 at the age of 24. He credited his ability to discover this theorem, and find similarly fundamental results on other topics, to his philosophical objectivism (ref). Indeed, it is little known that for most of his life Gödel did not continue to work in logic and the foundations of mathematics, but instead pursued philosophical studies that were intended to elucidate the meaning and value of objectivism. It is on these philosophical studies that this essay concentrates, even here focusing not on Gödel's philosophy of mathematics but on his more general philosophical insights.
As noted, Gödel combined objectivism with Platonism. However, as both Wang and Gödel recognize, objectivism is the broader term. By 'Platonism', Gödel means the view that there exists "a non-sensual reality, which exists independently both of the acts and the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind" (1951, 323). By objectivism he means that "objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions" (1951, 311). Wang further defines objectivism in a domain as 'the belief that every proposition in it is either true or false' (LJ 243), although it is not clear if this is Gödel's view as well. (footnote #3) Objectivity is important for Gödel because of his rationalism, which 'puts universals at the center and views them as stable and knowable by us' (LJ 9); thus Gödel's emphasis on "one of the basic problems of philosophy, namely the question of the objective reality of concepts and their relations" (LJ 7), i.e., the problem of universals.
In his conversations with Wang, Gödel usually defended objectivism rather than Platonism (perhaps because he knew that Wang was not sympathetic to Platonism). Wang reports that Gödel emphasized 'the epistemological priority of objectivity over objects' (LJ 210) and claimed that 'objectivity is better defined for us than objects' (LJ 243). Gödel holds that "out of objectivity we define objects in different ways" (LJ 243), and thus his Platonism does not lead to the view that immediate or infallible knowledge is possible. On the contrary, Gödel's is a 'liberal position of objectivism' (LJ 218) that admits 'the fallibility of our knowledge' (LJ 210).
Gödel's objectivism may have been 'liberal' in this sense, yet it is uncompromising, strongly held, and, Gödel believed, firmly grounded (for example, Gödel thought that objectivity is not seriously challenged by perceptual or set theoretical paradoxes (footnote #4), although as we shall see he did hold that certain conceptual paradoxes are problematic). For Gödel, "there is nothing concealed" about existence and it is "the clearest concept we have", which "helps us form a good picture of reality" and "is important for supporting a strong philosophical viewpoint" (LJ 150). Further, Gödel holds that thought is in no way metaphysically creative, i.e., thought is not creative in the original, precise sense of making something from nothing (LJ 224). Thus "the purpose of philosophy is not to prove everything from nothing but to assume as given what we see as clearly as shapes and colors" (LJ 305) (footnote #5). For Gödel, the power of thought is limited to understanding, analyzing, combining, or reproducing elements given in objective reality; these elements include both particular elements and elements that are, in some sense, abstract. It is these abstract elements (about which more presently) that give our conceptual understanding (and practical problem-solving) an objective quality. One indication of this objective quality is that Gödel speaks of concepts as standing fast over time, with only the perception or understanding of a concept changing and growing as human beings gain a greater or clearer grasp of reality. As Gödel says, "'Trying to see (i.e., understand) a concept more clearly' is the correct way of expressing the phenomenon vaguely described as 'examining what we mean by a word'" (LJ 233).
So what are concepts? According to Gödel, concepts exist in (or compose) a "non-sensual reality". Gödel was reasonable enough to admit that "concepts are there, but not in any definite place. They are related each in the other and form the 'conceptual space'. Concepts are not the moving force of the world but may act on the mind in some way" (LJ 149). For Gödel, "concepts are not objects: we perceive objects but understand concepts" (LJ 235).
Although in his conversations with Wang he held that "concepts are not objects", Gödel certainly did not shy away from saying things that are very close to this. Consider the following:
[Russell's] vicious circle principle in its first form applies only if one takes the constructivistic (or nominalistic) standpoint toward the objects of logic and mathematics, in particular towards propositions, classes, and notions, e.g., if one understands by a notion a symbol together with a rule for translating sentences containing the symbol into such sentences as do not contain it, so that a separate object denoted by the symbol appears as a mere fiction.Classes and concepts may, however, also be conceived as real objects, namely classes as "pluralities of things" or as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.
It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. They are in the same sense necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions and in both cases it is impossible to interpret the propositions one wants to assert about these entities as propositions about the "data", i.e., in the latter case the actually occurring sense perceptions. Russell himself concludes in the last chapter of his book on Meaning and Truth, though "with hesitation", that there exist "universals", but apparently he wants to confine this statement to concepts of sense perceptions, which does not help the logician.... One formal difference between the two conceptions of notions would be that any two different definitions of the form alpha(x) = phi(x) can be assumed to define two different notions alpha in the constructivistic sense. (In particular this would be the case for the nominalistic interpretation of the term "notion" suggested above, since two such definitions give different rules of translation for propositions containing alpha.) For concepts, on the contrary, this is by no means the case, since the same thing may be described in different ways.... The difference may be illustrated by the following definition of the number two: "Two is the notion under which fall all pairs and nothing else." There is certainly more than one notion in the constructivistic sense satisfying this condition, but there might be one common "form" or "nature" of all pairs. (Gödel 1944, 128-129)
Yet there are times when Godel seems to think that one does not need objects for objectivity. Consider the following:
The alternative under which "there exist absolutely undecidable mathematical propositions" ... "seems to imply that mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions, that is to say, some form or other of Platonism or "realism" as the mathematical objects. [There exists no term of sufficient generality to express exactly the conclusion drawn here, which only says that the objects and theorems of mathematics are as objective and independent of our free choice and our creative acts as in the physical world. It determines, however, in no way what these objective entities are -- in particular, whether they are located in nature or in the human mind or in neither of the two. These three view about the nature of mathematics correspond exactly to the three views about the nature of concepts, which traditionally go by the names of psychologism, Aristotelian conceptualism, and Platonism.] For, the empirical interpretation of mathematics, that is, the view that mathematical facts are a special kind of physical or psychological facts, is too absurd to be seriously maintained. (1951, 311-312)If mathematics describes an objective world just like physics, there is no reason why inductive methods should not be applied in mathematics just the same as in physics. The fact is that in mathematics we still have the same attitude today that in former times one had toward all science, namely, we try to derive everything by cogent proofs from the definitions (that is, in ontological terminology, from the essences of things). Perhaps this method, if it claims monopoly, is as wrong in mathematics as it was in physics. (1951, 313)
One ingredient of [nomiminalism] is perfectly correct and really discloses the true nature of mathematics. Namely, it is correct that a mathematical proposition says nothing about the physical or psychical reality existing in space and time, because it is true already owing to the meaning of the terms occurring in it, irrespectively of the world of real things. What is wrong, however, is that the meaning of the terms (that is, the concepts they denote) is asserted to be something man-made and consisting merely in semantical conventions. The truth, I believe, is that these concepts form an objective reality of their own, which we cannot create or change, but only perceive and describe.
Therefore a mathematical proposition, although it does not say anything about space-time reality, still may have a very sound objective content, insofar as it says something about relations of concepts. (1951, 320)
Here it is relations of concepts — and not objects — that provide objectivity.
Apparently Gödel even conjectured that humans have a physical organ that handles abstract impressions (LJ 233, 235) and explicitly rejected the possibility that "an inner perception — our own experience" (LJ 235) is sufficient for this; on the other hand, he usually expressed his thoughts on the acquisition of conceptual knowledge by focusing more soberly on the role of intuition.
Gödel placed great confidence in intuition, but on his view intuition is more akin to trained insight than to any kind of feeling; indeed "appealing to intuition calls for more caution and more experience than the use of proofs" (LJ 301). Although Gödel saw great value in and personally longed for a sudden illumination such as Descartes, Schelling, and (he believed) Husserl had experienced, he insisted that "intuition need not be conceived of as a faculty giving an immediate knowledge of the objects concerned" (LJ 226). He thought that in mathematics "as in the case of physical experience, we form our ideas also of those objects on the basis of something else which is immediately given. Only this something else is not, or not primarily, the sensations" (LJ 226 = CW 2:268). For Gödel, the sensations are "data of the first kind"; but there exist also "data of the second kind", which are "the abstract elements contained in our empirical ideas" (LJ 228). Here is how Gödel explicates this notion:
That something besides the sensations actually is immediately given follows ... from the fact that even our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, for example, the idea of object itself.... It by no means follows, however, that the data of this second kind, because they cannot be associated with actions of certain things upon our sense organs, are something purely subjective, as Kant asserted. Rather they, too, may represent an aspect of objective reality..." (LJ 225 = CW 2:268).
In opposition to Kant, Gödel claims that conceptual categories such as object and entity are not purely subjective impositions of the mind, but contain something objective. A prominent example is the phenomenon of, in Gödel's words, "thinking together" multitudes as unities, i.e., regarding an individual thing as a unit of a set or concept. About this phenomenon, Gödel says:
The distinction between many and one cannot be further reduced. It is a basic feature of reality that it has many things. It is a primitive idea of our thinking to think of many objects as one object....In the general matter of universals and particulars, we do not have a merger of the two things, many and one, to the extent that multitudes are themselves unities. Thinking together may seem like a triviality. Yet some pluralities can be thought together as unities, some cannot. Hence, there must be something objective in the forming of unities. (LJ 254)
For Gödel, thinking together pluralities as unities seems on the face of it to be very much an instance of intuition, for it is based not on "sensations or mere combinations of sensations" (data of the first kind), but on abstract qualities or elements of the sensations (data of the second kind). This indicates the power of intuition.
Yet this is not the whole truth, for Gödel holds that intuition contributes only so much to knowledge. For example, in the realm of mathematics, he thinks that "our real intuition is limited to small sets and numbers" (LJ 216) and that there is a "big jump" from seven plus or minus two (Miller 1956) to the entire set of finite numbers (LJ 213), which is achieved through the abstraction of adding one continually to yield larger and larger numbers (LJ 213). Abstractive actions or procedures of this kind are instances not of intuition itself, but rather of idealization performed on the materials provided by our intuitions (and provided, I would add, by perception, experiment, experience, etc.). Thus Gödel thinks that in the case of "thinking together" or abstraction, "idealization is decisive" (LJ 260).
In general Wang found that he 'detected from [Gödel's] observations' (LJ 216) a 'dialectic of the criterion of intuition as a way of characterizing our strong and stable beliefs and the procedure of idealization to purify and extend our conceptions' (LJ 230). Together, intuition and idealization serve to create an ever-expanding spiral of knowledge, and such a dialectical approach 'is thought to be instructive, because the interaction of intuition and idealization, vague though they are as concepts, has worked so well so far, and we have every reason to believe that it will continue to work well' (LJ 230).
Is knowledge gained through intuition and idealization on solid ground? Gödel thinks so. Yet 'he judges the familiar strategy of taking intersubjective agreement as the ultimate criterion of truth to be an inconsistent half-measure. In particular, his belief in our capacity to know objective reality as it is, is at the center of his dissatisfaction with Kant's philosophy' (LJ 221) (footnote #6). If intersubjective agreement is not enough, whence knowledge that can be truly termed objective?
Part of the key seems to lie in Gödel's notion of "absolute knowledge". Wang writes of this idea that it is 'knowledge that is feasible and applies to central and stable conceptual achievements. He sees this kind of absolute knowledge as the highest ideal of intellectual pursuit. His favorite example is Newtonian physics' (LJ 302). The significance for Gödel of absolute knowledge and of the Newtonian example derive from 'the central importance of the axiomatic method for philosophy' (LJ 244):
The Newtonian frame is a kind of absolute knowledge. It is a psychological backbone. In this sense absolute knowledge is the frame or backbone or axiom system of a good theory. The backbone of physics remains in Newtonism. Experience fills in the gaps after absolute knowledge is obtained. (LJ 302 – 303)
see Gödel 1961, pp. 381-385 for discussion of the importance of an improved conceptual method, which G associates with phenomenology -- connection here to Rand's formula "nature, to be apprehended, must be obeyed"? long quote follows.... (381-385)
As for the rightness or wrongness, or, respectively, truth and falsity, of these two directions [toward or away from metaphysics] is concerned, the correct attitude appears to me to be that the truth lies in the middle or consists of a combination of these two conceptions.
Now, in the case of mathematics, Hilbert had of course attempted just such a combination, but one obviously too primitive and tending too strongly in one direction. In any case there is no reason to trust blindly in the spirit of the time, and it is therefore undoubtedly worth the effort [[at least][ once to try the other of the alternatives mentioned above -- which the results cited leave open -- in the hope of obtaining in this way a workable combination. Obviously, this means that the certainty of mathematics is the be secured not by proving certain properties by a projection onto material systems -- namely, the manipulation of physical symbols -- but rather by cultivating (deepening) knowledge of the abstract concepts themselves which lead to the setting up of these mechanical systems, and further by seeking, according to the same procedures, to gain insights into the solvability, and the actual methods for the solution, of all meaningful mathematical problems.
In what manner, however, is it possible to extend our knowledge of these abstract concepts, i.e., to make these concepts themselves precise and to gain comprehensive and secure insight into the fundamental relations that subsist among them, i.e., [into] the axioms that hold for them? Obviously not, or in any case not exclusively, by trying to give explicit definitions for concepts and proofs for axioms, since for that obviously needs other undefinable abstract concepts and axioms holding for them. Otherwise one would have nothing from which one could define or prove. The procedure must thus consist, at least to a large extent, in a clarification of meaning that does not consist in giving definitions.
Now in fact, there exists today the beginning of a science which claims to possess a systematic method for such a clarification of meaning, and that is the phenomenology founded by Husserl. Here clarification of meaning consists in focusing more sharply on the concepts concerned by directing our attention in a certain way, namely, onto our own acts in the sure of these concepts, onto our powers in carrying out our acts, etc. But one must keep clearly in mind that his phenomenology is not a science in the same sense as the other sciences. Rather it is [or in any case should be] a procedure or technique that should produce in us a new state of consciousness in which we describe in detail the basic concepts we use in our thought, or grasp other basic concepts hitherto unknown to us. I believe there is no reason at all to reject such a procedure at the outset as hopeless. Empiricists, of course, have the least reason of all to do so, for that would mean that their empiricism is, in truth, apriorism with its sign reversed.
But not only is there no objective reason for the rejection [[of phenomenology]], but on the contrary one can present reasons in its favor. If one considers the development of a child, one notices that it proceeds in two directions: it consists on the one hand in experimenting with the objects of the external world and with its sensory and motor organs, on the other hand in coming to a better and better understanding of language, and that means -- as soon as the child is beyond the most primitive designating -- of the basic concepts on which it rests. With respect to the development in this second direction, one can justifiably say that the child passes through states of consciousness of various heights, e.g., one can say that a higher state of consciousness is attained when the child first learns the user of words, and similarly at the moment when for the first time it understands a logical inference.
Now one may view the whole development of empirical science as a systematic and conscious extension of what the child does when it develops in the first direction. The success of this procedure is indeed astonishing and far greater than one would expect a priori: after all, it leads to the entire technological development of recent times. That makes it thus seem quite possible that a systematic and conscious advance in the second direction will also far exceed the expectations one may have a priori. (381-385)
For Gödel there are two main factors in the acquisition or validation or axiomatic knowledge: intuition and "fruitfulness" (LJ 243). This kind of absolute knowledge does not yet exist in philosophy, but it does exist in many or perhaps all of the sciences (or portions thereof, such as mechanics with Newtonism and number theory with the Peano axioms). Gödel thought that "logic and mathematics (just as physics) are built up on axioms with a real content which cannot be explained away'" (Gödel 1944, 132), and seems to have thought that such axioms are possible also in metaphysics.
One goal of Gödel's is that philosophy should come to absolute knowledge. Thus his claim that "philosophy as exact theory should do for metaphysics as much as Newton did for physics" (Wang 1974, 85). "The beginning of physics was Newton's work of 1687, which needs only very simple primitives: force, mass, law. I look for a similar theory for philosophy or metaphysics" (LJ 167). (As we shall discuss shortly, Wang finds this overly ambitious and would settle for knowledge that is less than absolute in Gödel's sense.)
Note that Gödel does not say that ideally philosophy is only an exact theory, but that philosophy as exact theory will provide absolute knowledge. This leaves open the door for philosophy as something else, for example a philosophy to live by. Indeed, Gödel says that one function of philosophy is to investigate 'the meaning of the world', which consists in "the separation of wish and fact" (LJ 309). Wang finds two meanings in this separation. One meaning is the 'methodological principle' that 'we keep our wishes separate from our investigation of the facts, not allowing our wishes to distort our vision' (ibid.). The other meaning touches on the gap between actuality and what we wish for, with the result that 'we strive to satisfy our wishes because the actual situation does not agree with what we wish for' (ibid.). This provocative suggestion is explored by Wang at some length, so I shall treat of it more fully in the next section. The key point here is that Gödel's objectivistic vision does not exclude the search for personal meaning; indeed, Gödel seems to think that there are two complementary approaches to or functions of philosophy, and that each is valid.
Needless to say the foregoing account merely scratches the surface of Gödel's thought, and there is much more to it than can be explored in a short essay (especially as I have focused on those aspects to which I most sympathetic, and have not spoken of less central aspects of his thought, such as his theological monadology). From my perspective, the essence of Gödel's philosophy consists of an objectivism that is grounded by a form of Platonism. It is this Platonism that Wang would prefer to dispense with, although, as we shall see, not without consequences for an objectivistic first philosophy.
Over the course of many years, Hao Wang had the good fortune of conversing with Kurt Gödel about key issues in philosophy, logic, and the foundations of mathematics. Wang sums up his approach to preserving these conversations as follows:
All along I have thought of my study of Gödel's ideas as a way to arrive at and communicate what I take to be the most reasonable of his views on the issues he studied, rather than an attempt to depict faithfully the body of his philosophical thought. (LJ xi)[Wang 1996] has two purposes: first, to present Gödel's life and work as completely and coherently as I can, and, second, to use his outlook as an illustration of how one might try to arrive at a comprehensive philosophy.... (LJ 20)
Wang had his own unique perspective in philosophy, which I find quite fascinating and full of wisdom. He was brought up in China and came to the United States to complete his graduate studies in philosophy at Harvard. Because of his upbringing and his subsequent formal training, he maintained a keen interest in integrating the best of Chinese and Western philosophy. His background in the humanistic philosophies of China led him to seek a conception of philosophy that would go "beyond analytic philosophy" (the title of Wang 1985) and "do justice to what we know". He was more interested in pursuing or continuing philosophical projects (such as his provocative suggestion for an updated version of Francis Bacon's Instauratio Magna) than in focusing narrowly on the problems of philosophy. However, his focus on projects, and on developing a worthy project for himself at a time when ambitious projects in philosophy were decidedly out of fashion, can make his thought somewhat scattered and unsystematic, although pregnant with possibilities. Wang's ideas deserve an essay of their own; however, in this paper I shall focus on his reflections on Godel and his own views on topics closely related to philosophical objectivism.
Unlike Godel, Wang is not especially interested in establishing or strengthening the philosophical foundations for his views.
For Wang, objectivism is an idealization that is acceptable because of the prevalence of intersubjective agreement (LJ 23); while this is not enough for Gödel (LJ 231), it seems to be for Wang.
Gödel's criticism of "Aristotelian realism" as an alternative to Platonism (Godel 1951):
I have purposely spoken of two separate worlds (the world of things and of concepts), because I do not think that Aristotelian realism (according to which concepts are parts or aspects of things) is tenable. (321)...the Platonistic view is the only one tenable. Thereby I mean the view that mathematics describes a non-sensual reality, which exists independently of the human mind and is only perceived, and probably perceived very incompletely, by the human mind. (322-323)
Quote from Hermite by Gödel:There exists, unless I am mistaken, an entire world consisting of the totality of mathematical truths, which is accessible to us only through our intelligence, just as there exists the world of physical realities; each one is independent of us, both of them divinely created and appear different only of the weakness of our mind; but, for a more powerful intelligence, they are one and the same thing, whose synthesis is partially revealed in that marvelous correspondence between abstract mathematics on the one hand and astronomy and all branches of physics on the other.
Comment by Gödel on this passage:
So here Hermite seems to turn toward Aristotelian realism. However, he does so only figuratively, since Platonism remains the only conception understandable for the human mind. (323)
conceptualism = anti-platonism (concepts do not exist)
Some old material on Aristotle's philosophy of mathematics:
One manifestation of Aristotle's undifferentiated unity of epistemology, philosophy of science, and methodology, is his emphasis on the acquisition of knowledge. Compared to twentieth-century philosophers of science, Aristotle is simply not interested in justification. The acquisition of knowledge is an active process; the rules which govern it must be applied in the midst of scientific investigation itself. And these rules could be ascribed variously, some answering questions of epistemology, some of method, and some of metaphysics. Thus the central aspect of his first philosophy that I would like to examine is Aristotle's account of the formation and development of conceptual knowledge.
It may be slightly misleading to speak of Aristotle's "account" of concept-formation and the growth of understanding, since he never sets it forth in any one text. However, by looking at relevant passages (especially from the Posterior Analytics, the Metaphysics, and the theoretical sections of the biological works), we should be able to see what Aristotle's view consists in.
As always in the study of Aristotle, it will be helpful to examine earlier views on the topics at hand, especially those of Plato. Plato, of course, had held that there were myriad Forms that caused the existence of the physical, sensible world. One puzzle that could be and was raised concerning his theory was the unity of sensible individuals: how, for example, can '(a) man' be one thing if it contains or mirrors multiple Forms such as rational, two-footed, animal, hairless, opposable-thumbed, and so on? Alternatively, if the thing itself is thought to instantiate only the essential Form of the thing, how then can it possess any characteristic other than the essential characteristic? How, then, can we differentiate between things which possess the same essence? These problems will be highlighted if we look at an example of Plato's method of division and definition, an extended case being the attempt to define the statesman's object of concern (i.e., man) at Politicus 263e-267a. In this passage, Plato concludes that the statesman is sovereign over herds of two-footed, walking (wingless), hornless, land-dwelling, tame, gregarious living creatures - that is, men. Yet his method reveals that these divisions cannot be progressive but are independent, since we can just as well observe winged bipeds, hornless water-dwellers, wild gregarious creatures, and so on. In the Topics, Aristotle criticizes Plato's method on this account: two-footedness is not a difference that exists among or is exhibited only by tame animals, but properly speaking is exhibited by animals with feet (and similarly for other differences). Plato's divisions are independent. Further, Plato's divisions are guided by intuition (as indicated by Politicus 264e, where the division in question is applied only after it is affirmed that 'practically everyone, even the most witless, would judge this to be so'); and it is only after the divisions have been made that they are tested against the natural objects (as Phaedrus 265e recommends).
What notion of concepts does this practice reveal? According to Plato, we construct our divisions of things in a manner that keeps them independent and isolated, and then (after we have formulated them) we test them out in order to discover if they are justified. Thus Plato's theory of knowledge seems to be (at least in this respect) a justification theory. Whether or not this holds true of Plato's entire theory, Aristotle's criticisms of Plato's theory reveal his own theory and practice to be quite different from Plato's, and from any sort of justification theory.
According to Aristotle, human beings differ from the other animals in that we possess 'reasoning and thinking' (logikon kai dianoia), whereas the other animals have only perception (De Anima 415a4ff.). While Aristotle readily admits that certain animals do possess the faculty of imagination (phantasia) and even (for those animals who can perceive the passage of time and distinguish between past, present, and future) the faculty of memory, it is only human beings that possess thought, reason, speech, and the other conceptual abilities. This is the "difference" of man. Yet what is the nature of this ability to form concepts, which ability underlies the most distinctive human characteristics? The most famous passage in which Aristotle attempts to explain this process is the final chapter of the Posterior Analytics:
From perception there comes memory, just as we have said, and from memory (when it occurs often in connection with the same thing) comes experience; for memories that are many in number form a single experience. And from experience, or from the whole universal that has come to rest in the soul (the one apart from the many, whatever is one and the same all together in each of those things), there comes a principle of skill or of understanding - of skill if it deals with how things come about, of understanding if it deals with what is actually the case. Thus these states neither belong in us in a determinate form, nor come about from other states that are more cognitive; but they come about from perception (An. Po. II.19, 100a3-11).
Now the fact that Aristotle speaks here (and in a similar vein elsewhere) of 'the one apart from the many, whatever is one and the same all together in each of those things', has been seized upon by those who would impute to his metaphysics the "real specific forms" of neo-Platonism - that is, the doctrine that there are real forms (both universal and abstract) which somehow exist in each member of a species, with some manner of "determining element" in each particular providing for individuation. While this interpretation falsifies even this passage (the preposition para in line 7 effectively rules it out, since it implies the separation of the universal from the particular), it also goes against the argument of Metaphysics Book VII. As Aristotle argues in VII.13, the universal cannot be the ousia ("substance" or "being") of a thing, since if that were the case then Socrates or any individual man would contain not only the universal 'man' but also the universal 'animal', which would set up a regress of substances inside of substances inside a primary substance, which is obviously non-sensical. Consider 1038b35-1039a2: 'It is plain that no universal attribute is an ousia, and this is plain also from the fact that no common predicate indicates a "this", but rather a "such"' (that is, a natural kind). Aristotle, as David Balme has argued ('Aristotle's biology was not essentialist'), was no neo-Platonist in his metaphysics.
If this is true, then what was Aristotle in his epistemology?
Aristotle takes as fundamental that what exists in a primary sense are particular, determinate individuals. These individual things are each unique, and each possesses certain characteristics which make it unique. The purpose of Aristotle's epistemology is to make these things known--to 'hunt out' (thereuein) the nature of the thing, just as Plato said, and to 'discover exactly what it is' (heurein hoti pot' estin, Sophist 221c). For Aristotle, the only way to understand "what exactly a thing is" is through understanding the way that it is similar to and differs from other things that exist. Thus in Posterior Analytics II.14, Aristotle recommends that in order to best proceed in the task of definition, one should first of all 'pick out the anatomies and the differences'; and thus also he devotes the entire long introduction to the Historia Animalium (his "Inquiry into Animals") to the differences and similarities between animal kinds, which can differ variously in their parts, in their modes of living, in their habits, and in their actions (487a11). Similarly, his treatise on the Parts of Animals seeks to explain the fundamental causes of the differences exhibited among these parts.
What is the nature of these differences? In the introduction to the Historia Animalium, Aristotle writes as follows:
Some animals are the same in all their parts, while others are not the same. Sometimes the parts are identical in form [in eidos ], as, for instance, one man's nose or eye resembles another man's nose or eye, and in like manner with horses, and with all other animals which we reckon to be of the same form [eidos ]. . . In other cases the parts are identical, save only for a difference in the way of "the more and the less", as is the case in animals which are of the same kind [genos - HA 486a14ff.].
Now the technical terms Aristotle that uses for "the more and the less" are to mallon kai hetton and, more generally, huperoche kai elleipsis (cf. Metaphysics VIII.2 1042b32-35). In two entries to his "philosophical dictionary" (Book V of the Metaphysics) Aristotle sets forth some grounds for his use of these terms, when he speaks in Chapters 13 and 14 of quantity and quality (to poson kai to poion ):
We call a quality primarily a difference in the being (ousia ) of things, for example, man is an animal of a certain quality because he is two-footed, and the horse is so because it is four-footed. . . . Also all the attributes of substances in motion (e.g. heat and cold, whiteness and blackness, heaviness and lightness, and others of this sort) in virtue of which, when they change, bodies are said to change [ - we call these qualities]. (1020a33-b12).
Of the things that are quantities by their own nature some. . . are modifications and states of this sort of substance, e.g. much and little, long and short, broad and narrow, deep and shallow, heavy and light, and the other terms of this sort - and also great and small, and greater and smaller. . . (1020a17-24).
The centrality of these terms 'quality' and 'quantity'* in Aristotle's scientific practice and theory of knowledge cannot be underestimated, though they may at first seem obscure. Aristotle's basic idea appears to be this: all objects differ from one another, and they must differ in certain specifiable ways. These differences (in parts, habits, modes of living, etc.) can be of two types - they can be differences in nature or kind, or differences in degree. Now Aristotle's insight here is that things which are identical in sort or kind (the same poios - for example, a certain part of animals, say, eyes) differ only in their quantitative measurements (they may be more rounded, lighter or darker, larger or smaller, etc.). And Aristotle's contention is that, to the human mind, the fundamentally cognizable, conceptually knowable differences between things are not the quantitative, measurable differences, but differences of sort or kind: the primary difference of being or substance is one of quality.
Yet (as a first approach) it is not correct to say that this statement, which is true of the activity of the mind, is also true of the things themselves, since the actual differences between things exist only on a quantitative continuum. However, Aristotle holds that the human mind works only through an understanding of the universal, as Plato first set forth in the Phaedrus (249b-c, 265d, 273d-e). Aristotle has explicated this important principle of Plato's by maintaining that we "understand things universally" only when we (temporarily) disregard or omit or subsume the specific measurements of the differences between certain things, and consider these things to be identical (in quality or kind), even though in reality they do exhibit these manifest quantitative differences. This process of omission or subsumption occurs when, as Aristotle says, we grasp things 'in abstraction' (ex aphaireseos), that is, when we grasp things on the principle that the relevant quantitative specifications of its qualities must exist in some degree but may exist in any degree (within a certain range).
This is what I mean by the "algebraic" nature of concepts in Aristotle's epistemology: concepts or universals refer "one over many", with the variable nature of the measurements being considered (temporarily) irrelevant for the purpose of universalized understanding ("whatever is one and the same all together in each of those things"), much as is done in mathematical algebra. [As Mandelbrot notes (p. 5a), the English word "algebra" derives from the Arabic al-jebr ("the re-integration or reunion of broken parts"), which is related to the verb jabara ("to re-unite, re-integrate, restore, consolidate"). The Arabic phrase for algebraic computation is silm al-jebr wa'l-muqabalah, "the science of comparison and re-integration", which is a perfect characterization of Aristotle's theory of the formation of concepts.]
Of course, this is only true within certain bounds. That is, the level of abstraction desired or needed determines what the bounds of that abstraction should to be, in other words how wide the range should be in which the specifications of the differences between things are disregarded. [There seems to me good reason to believe that this process of "bounding" is the genesis for the Greek term for definition, which is horismos, cognate with the words for boundary (horos) and separating or delimiting (horidzein).] Thus the qualitative differences between things are indices along which quantitative measurements can be made--which measurements are, within a certain range, regarded as inessential, and which objects (in this range or genos ) are considered to be qualitatively and quantitatively identical, even though in actuality differing - but only, as Aristotle puts it, quantitatively by "the more and the less".
For Aristotle, one is not a number; the numbers (what we would call the whole numbers or natural numbers) begin at two. One is a metron, a measure of things (Metaphysics 1016b24-6, 1052b1ff., 1087b33ff., etc.), which is indivisible in all respects of magnitude (De Anima 430b20-21, Metaphysics 1052b20-7 and b31-53a18, and Heath pp. 217-219). Aristotle holds that "number" (arithmos) is always a plural term; thus it has been noted that in Aristotle and in Greek number theory generally the numbers do not conform to our natural numbers but are conceived to be set of units (cf. Burnyeat 1987 pp. 226, 235 and Euclid Elements Book VII Def.1: 'A number is a plurality composed of units'). This fact may be related to its etymology - LSJ connect arithmos to the verb *aro, whose reduplicative form ararisko means in its basic sense "to join together". Thus Greek arithmos is similar in meaning to Arabic al-jebr - both refer to a "joining or binding together" of one over many.
Furthermore, Aristotle's definitions of 'one' and of 'point' are very similar: 'one is that which is indivisible in respect of all magnitude and which has no position, while that which is similarly indivisible but which has a position is a point' (Metaphysics 1016a24-26). This brings up the issue of dimension and of the definitions of the most fundamental things in geometry. Euclid begins Book I of the Elements with this topic, and Simplicius commented that Euclid's negative definition of point ("a point is that which has no part", Definition I - cf. Heath 1908 ad loc.) arises from the fact that point "is arrived at by first detaching surface from body, then line from surface, and finally point from line". Poincare (1905, Ch.III Sec.3 and 1913, Part IX) elaborated on Euclid's starting points in a manner similar to that of Simplicius: surfaces are the boundaries of solids, curves are the boundaries of surfaces, and points are the boundaries of curves (the fundamentality of points arising from the fact that points are non-continuous while surfaces and curves are continuous - points have no boundaries).
The connection to the "algebraic" function of concepts or definitions should be clear. While the Greeks had no algebra in our sense (that is, in mathematics), for Aristotle words or concepts unify over more than one instance and stand for more than one "thing within certain bounds". In the case of numbers, however, this is doubly so: every numeral "3" unifies over any single instance of three things (regardless of material makeup, etc.) and is in itself one symbol for these "magnitudes of three" or "three things"*. This is why unity is so special for Aristotle: the single numerical symbol stands for any instance (of, in this case, a single thing), but it does not and cannot unify over plural things; it refers to an existent unity - and in Aristotle's metaphysics every existent thing or ousia is a unity [though animals have parts, for example, these parts are not really living parts unless they are functioning in and for the whole animal].
This explains also the importance for Aristotle of the "indivisibility in all respects of magnitude" exhibited by the point and by unity. For a curve, we can put forth as our definition "the boundary of a surface" or 'breadthless length' (Topics 143b11ff.), which must exist in some length but may exist in any length (the bounds here are, at least in mathematical abstraction, infinite, i.e. non-existent). But there is no such "algebraic" characterization of a point, for it cannot vary in itself (it can only vary in its position, but this applies to any figure). Thus, the point is defined negatively by Euclid and his predecessors, since there is nothing prior to it in thought by which one could define it positively. Aristotle objects to this, but he cannot improve upon it. Simplicius and Poincare seem correct in their explications of Euclid's definitions, but Aristotle would presumably still object that curve, surface, and solid are prior only 'to us', but not 'absolutely'. The problem here seems to exemplify a phenomenon noted by Russell: 'The most obvious and easy things in mathematics are not those that come at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle' (Introduction to Mathematical Philosophy, p.2).
The point, then, is indivisible; but it is not a measure (metron) of length. The measure of length is a unit that we establish, e.g. the foot (Metaphysics 1052b32, 1087b35). In the case of numbers, however, the unit of measure is simply unity, 'that which is one' (to hen). Unity, for Aristotle, is not a number but the arche (cause or starting-point) of numbers (Metaphysics 1052b32). Once again, language may be a guide. The Greek verb derived from arithmos is the verb for counting (arithmeo). Thus the phenomenon that Aristotle has in mind seems to be that we do not "count to one" or "count one" (in Greek, counting, by its very relation to the word arithmos, implies plurality); yet the process of counting does begin at one. This appears to be in harmony with Dedekind's explication of the Peano axioms for the natural numbers (see his letter to Keferstein) - aside, of course, from a difference of nomenclature: for Peano arithmetic, one is a number, but is not (as is every other number) a successor number; for Aristotle, however, this special quality of unity means that we ought not call it an arithmos, but "the measure and starting-point of numbers". [This "being a starting-point" is even more strongly implied by subsequent explications of Dedekind's work; see, for example, Hao Wang's 1987 re-stating of Dedekind's analysis "more explicitly", in which it is pointedly the first axiom that "1 is a number" (Wang 1987, p.200).]
1. Wang 1974, 1986, 1987, 1996.
2. This interpretation is remarkably similar to F.A. Hayek's idea of (the futility of) what he called the "synoptic perspective"; see Hayek 1962, 62 and Sciabarra 1995, 46. [Within the text, quotations from Gödel are contained within double quotes and quotations from Wang are contained within single quotes; indented quotations are labeled in the preceding paragraph, though in general such quotations in section one are from Gödel and in section two are from Wang. It is important to bear in mind that many of the quotations from Gödel are actually reconstructions by Wang of his conversations with Gödel (see LJ 132-137 regarding Wang's contacts with Gödel and his approach to preserving the substance of their conversations).]
3. Although see Gö 1961, 379: "...one clung to the belief ... that every precisely formulated yes-or-no question in mathematics must have a clear-cut answer. I.e., one thus aims to prove, for [what are according to the nominalistic conception of mathematics] inherently unfounded rules of the game with symbols, as a property that attaches to them so to speak by accident, that of two sentences A and ~A, exactly one can always be derived. That not both can be derived constitutes consistency, and that one can always actually be derived means that the mathematical question expressed by A can be unambiguously answered."
4. LJ 238. See also Godel 1951, 321: "I wish to repeat that 'analytic' here does not mean 'true owing to our definitions', but rather 'true owing to the nature of the concepts occurring [therein]', in contradistinction to "true owing to the properties and the behaviour of things' [[see also 1951, 309: the true is objective, the merely demonstrable is subjective]]. This concept of analytic is so far from meaning 'void of content' that it is perfectly possible that an analytic proposition might be undecidable (or decidable only with [a certain] probability). For, our knowledge of the world of concepts may be as limited and incomplete as that of [the] world of things. It is certainly undeniable that this knowledge, in certain cases, not only is incomplete, but even indistinct. This occurs in the paradoxes of set theory, which are frequently alleged as a disproof of Platonism, but, I think, quite unjustly. Our visual perceptions sometimes contradict our tactile perceptions, for example, in the case of a rod immersed in water, but nobody in his right mind will conclude from this fact that the outer world does not exist."
5. This may be the reason why, for Gödel, everyday knowledge is more important for philosophy than scientific knowledge (LJ 130 etc.). As Gödel says, "Philosophy consists of pointing things out rather than arguments."
6. Wang indicates that Gödel was sympathetic to phenomenology as 'a precise formulation of the core of Kantian thought' (LJ 157, see also LJ 171), yet he also discusses extensively that Gödel admired Leibniz and Husserl (as well as Plato) while Wang himself admired Marx and Kant (and to some extent Aristotle).
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©1998 by Peter Saint-André