5,112 words
All previous parts can be found here.
Being, Nothing, Becoming
In our last installment, we introduced Hegel’s Logic and made a few small steps in trying to comprehend the standpoint from which it begins. We have spent a good deal of time discussing Hegel’s concept of “absolute knowing,” and it is in the Logic that we see it in operation. We have found that the Phenomenology of Spirit is an account of how all forms of human consciousness aim at absolute knowing, which is called such because it knows the Absolute. The Absolute is the whole, it is all of existence understood as a One, as an organic totality, rather than as a heap of indifferently related elements (as Bertrand Russell and other bad philosophers have conceived of it).
To know the whole is, ultimately, to know that it has its being in and by means of its crowning achievement, the development within it of human subjectivity. To know the whole is to know that it is a reflection of ourselves: the world exists so that self-conscious spirit can come to be in it. Thus, in a fundamental sense, the distinction between subject (us) and object (the world we know) is overcome in absolute knowing. Famously, Hegel says in the Phenomenology that the Absolute must be conceived in “the whole wealth of [its] developed form.”[1] It is the task of Hegel’s philosophical system to give an account of that wealth. The Phenomenology, which constitutes a kind of introduction to Hegel’s system, merely prepares us for it.
What I have said so far gives a general sketch of the relation of the Phenomenology to the Logic, of absolute knowing, and of the Absolute. It is time now, however, to speak more in terms of specifics, and a bit more technically. What the Phenomenology really does, if we have followed the argument of the text, is to place us in the standpoint from which absolute knowing can begin. As we saw in our discussion of the Phenomenology, all consciousness is “oppositional”: it involves a subject confronting an object in its otherness; it involves an object standing opposed to a subject. However, in seeking knowledge, what subjectivity aims at, in one way or another, is to cancel this opposition between self and an opposing other.
We arrive at the threshold of absolute knowing when we realize that true knowledge – or, more accurately, the wisdom sought by the philosopher – is possible only by leaving the “opposition of consciousness” behind; only through a standpoint that transcends the subject-object distinction. Hegel’s philosophical system will demonstrate in concrete terms how this distinction is overcome through our recognition of ourselves in nature. At the end of the Phenomenology, however, all we are left with is the abstract and contentless idea of the transcendence of subject and object. And this is exactly where the Logic begins.
Hegel’s entire philosophical system will display the Absolute in “the whole wealth of its developed form.” But we cannot expect to find that wealth right at the beginning. Instead, it unfolds in the development of the three divisions of Hegel’s system: Logic, Philosophy of Nature, and Philosophy of Spirit. The beginning must be, of necessity, undeveloped. It must be simple, not complex. And it must be abstract, not concrete. The Absolute itself, for Hegel, is the one, true concrete. Therefore, the beginning point from which we develop our account of the Absolute cannot be concrete. It must, in essence, be empty.
Further, in The Science of Logic, in a chapter titled “With What Must the Science Begin?,” Hegel argues that the Logic cannot commence with any determinate presuppositions at all. In other words, it cannot assume anything specific. As we discussed in our last installment, the ideas or “categories” that will unfold in the Logic are both the fundamental categories of thought and of being. If we want to know what those categories truly are, then we cannot begin with any presuppositions about them. We cannot, in other words, begin with any ideas about what is “obviously” true and can therefore be taken for granted.
If we did so, we would run the risk of repeating the mistakes of the past. And the Logic would be valueless, because it would simply be based on unchallenged presuppositions. Such an approach would be fundamentally unphilosophical. In insisting that his philosophy must be presuppositionless, Hegel is being true to the age-old spirit of philosophy itself, which has always held that everything must be called into question, including our most cherished assumptions, and that nothing may be taken for granted.
As I pointed out last time, the requirement of presuppositionlessness is also fundamentally Cartesian. Descartes opens his Meditations by doubting literally everything. He suspends belief in all that he had ever believed to be true. Hegel does not begin with doubt as such, but with an approach that winds up being the equivalent. In the language of Husserlian phenomenology, he “brackets” everything that exists, including his own beliefs. It is also important to see that Hegel’s conception of how the Logic must begin is entirely in keeping with his conviction (discussed in part five) that we are living in the age in which mankind comes to awareness of itself as radically free and self-determining.
The Logic itself is radically free: entirely free of all presuppositions, and thus free of all the baggage of history. As such, Hegel’s philosophy is quintessentially modern, setting the past aside and beginning entirely anew. As we will see, the Logic is also radically self-determining, in a quite literal sense: the Logic will develop itself before our eyes, just as it developed itself before Hegel’s eyes. It develops itself entirely from itself, without any input from something that exists outside the science. Since the Logic is an attempt to give an account of the whole, there can be nothing outside it!
Hegel is insistent that in beginning the Logic we cannot even presuppose any of the forms of thought, or logical concepts that we believe we already know. We cannot presuppose what philosophy is, or what logic is. We cannot presuppose the validity of specific forms of logical inference, or of the so-called “laws of logic” (such as the “law of non-contradiction”). Thus, all that we can begin with is just pure thought as such; a thought that thinks nothing concrete. It is thought that prescinds from the subject-object distinction; a thought, in other words, that is neither of the subject nor of the object (though this thought will transform in the Logic into a thought that is both of the subject and of the object). Again, this is where the Phenomenology left us: with absolute knowing in its most abstract, germinal form.
But how can we do anything with this most empty and abstract of beginnings? What can we say about it? All we can truly say about it is that it is. Literally nothing else can be said at this point, because this abstract, empty thought with which we begin is a thought of nothing determinate whatsoever. It simply is. And so the first category of the Logic becomes being (Sein).
This is a highly significant beginning. Though the Logic will run through myriad categories or ideas, being is never really left behind. Every subsequent category can be understood to be an elaboration of being. Thus, the Logic is an ontology, an account of the nature of being (though it can also be argued that the entire system is an ontology). As it “moves forward,” the Logic expands upon its beginning in being. What it is moving toward, of course, is the Absolute (or what the Logic calls “Absolute Idea”). And each category of the Logic is understood by Hegel to be a “provisional definition” of the Absolute.
What can we say about this “being” we have arrived at? As we have already established, it is absolutely contentless. In The Science of Logic, Hegel describes it as “Being, pure being, without any further determination.”[2] But if this is the case, how exactly are we to differentiate being from nothing? This is an excellent question. We normally take being and nothing to be absolute opposites. Yet pure being really is nothing at all; nothing determinate. Hegel makes precisely this point. He states, “Being, the indeterminate immediate, is in fact nothing [Nichts], and neither more nor less than nothing.”[3] Thus, nothing becomes the second category of the Logic.
This is a paradigm example of the dialectical thinking involved in what Hegel calls speculative philosophy. The understanding (Verstand) – Hegel’s term for rigid “common sense” thinking – insists that opposites are opposites; that being cannot be nothing and that the very idea that it could be nothing is nonsensical. However, speculative philosophy, like mysticism, displays the identity of opposites. Hegel’s philosophy will do this over and over again, in different ways. Philosophy, for Hegel, must involve an overcoming of common sense, which is “pre-philosophical.” It is the standpoint of those chained to the floor of Plato’s cave, arguing about shadows. In a sentence so important that Hegel italicizes it, he says, “Let those who insist that being and nothing are different tackle the problem of stating in what the difference consists.”[4]
But there is more. While we can see the logic of identifying being and nothing, the mind nevertheless rebels against this. Pay attention here to how your mind deals with these ideas. You see why Hegel begins with being: what can we say about our empty and abstract thought-beginning other than to say that it is, that it has being? Second, you see why this being, however, is indistinguishable from nothing, because it has no content whatsoever. It is, quite literally, no thing (a helpful point which, unfortunately, can only be made in English). But as soon as you think this you balk at it: “No! Being is not simply nothing! This cannot be! These ideas are polar opposites.” Thus, your mind transits back to being and you think it through again . . . and arrive, again, at nothing. From being to nothing, then back to being, then back to nothing. And so on.
But now we have arrived at a third thing that displays itself to thought: the transit we make, in thought, between being and nothing. From being to nothing and from nothing to being. But what is this except becoming (Werden)? Isn’t becoming precisely the movement from being to nothing (or “non-being”) and vice versa? Consider a simple example. Georg is not a piano player, then he learns to play the piano and becomes one. He moves from not being a piano player to being one. Georg then has a stroke and loses the ability to play the piano (i.e., he goes from being to not being). The third category of the Logic is thus becoming.
And now we have our first dialectical triad: being-nothing-becoming. And it is a paradigm case of the dialectic – perhaps the clearest case in all of Hegel. In a previous installment, we discussed how the oft-used formula “thesis-antithesis-synthesis” is problematic when applied to Hegel, though useful up to a point. Thesis: being. Antithesis: nothing. Is becoming a “synthesis”? Not really, at least not in the sense of a literal fusion of the two. But somehow it seems to reconcile their opposition. And in doing so, it elaborates the initial concept of being. In other words, “becoming” is a more adequate account of being. It can easily be argued, indeed, that for Hegel being is becoming. Earlier, I spoke of how the Logic is radically self-determining: it develops itself, entirely from itself. Bear in mind that this is Hegel’s account of the whole, of all of reality. All of reality is developmental; all of reality is becoming; to be is to become.
Hegel’s Automatic Writing
Some commentators would say that in this first triad of the Logic we have an illustration of Hegel’s dialectical “method.” But this is an inaccurate and misleading way to speak. Look carefully at how I have explained the transition from being to nothing to becoming and you will see that at no point is a “method” applied to a subject matter. This is entirely in keeping with Hegel’s presuppositionless approach: nothing may be presupposed by the Logic, not even “Hegelian dialectic.”
If you think carefully about the foregoing, you will see that the ideas simply develop themselves before our gaze. Thought about being leads to nothing – indeed, it must lead to nothing, and could lead nowhere else (for our abstract and empty beginning has no content that would allow us to say anything else about it). Thought about nothing then leads us back to being. And this “leading back,” the movement between being and nothing, leads us to think becoming. Hegel does not see himself as the author of the Logic. Instead, he is engaged in a kind of automatic writing. The Logic writes itself, and Hegel regards the transitions between its categories as necessary.
This is a singularly strange conception, which most Hegel scholars don’t seem to fully appreciate. How exactly can Hegel trust his own process of writing the Logic? To begin with, in the dialectic of being-nothing-becoming, Hegel certainly seems to be making good on his claim to have done away with all presuppositions. And the dialectic of these ideas seems to unfold naturally and with real necessity. Of course, this does not mean that all the dialectical transitions in the Logic (or elsewhere in Hegel) unfold so neatly. Indeed, even sympathetic critics have charged that there is an arbitrariness to some of Hegel’s dialectical inferences.
The more interesting reason that Hegel trusts his dialectical process in the Logic is purely metaphysical. Hegel is comfortable with the idea that the Logic simply unfolds before him, as a process of discovery rather than personal creation, because he regards the dialectic of his entire system as an expression of being itself. Hegel believes that philosophy (preeminently, his own philosophy) comes into being as, in effect, an outgrowth of nature. He believes that philosophy, as an account of the whole, is itself the highest expression of the whole. Nature, in other words, gives rise to its own self-understanding. This self-understanding is, in fact, nature’s perfection and completion; it is the telos of nature.
Through this sublime vision, Hegel is overcoming the traditional philosophical conception (very strongly represented in the modern period, up to and including Kant) that the world and our understanding of it are two quite separate things. This distinction is simply yet another expression of the rigid distinction that is usually made between subject and object. As one commentator puts things:
Knowledge is not simply a representation – a sort of mirror image – of the natural world, though in some sense and to some extent . . . it is representative. It is the actual product and actualization of the potencies inherent in natural processes. As such, it is a more complete and adequate revelation of the true character of the natural world. For the dialectical series is essentially teleological, and the outcome is the explication and the truth of the preceding process. Knowledge, therefore, is the proper essence of its subject matter, not just the image of it, but the actual presentation in conscious form of its true reality.[5]
Philosophy, for Hegel, is an organic outgrowth of the universe itself. Thus, when Hegel set pen to paper and composed the Logic, he believed that, in effect, the universe was speaking through him. Though Hegel never says this explicitly, it is a clear implication of his position that he believed himself to be the vehicle for the universe’s self-understanding. As fascinating as this theory is, we cannot actually offer it as a justification of Hegel’s procedure in the Logic without begging the question: it is the task of the Logic, and of the entire system, to demonstrate that it is the self-understanding of the universe.
Having now discussed the first dialectical triad of the Logic, let us take a step back and understand its context within the science as a whole. The Logic has three major divisions: the Doctrine of Being, Doctrine of Essence, and Doctrine of the Concept. The Doctrine of Being in turn has three subdivisions: Quality, Quantity, and Measure. “Quality” also has three subdivisions: Being, Being Determinate (Dasein), and Being-for-self (Fürsichsein). (Are you beginning to notice a pattern?) In turn, the three subdivisions of “Being” are being, nothing, becoming. The triad we have just covered thus makes up the first part of Hegel’s account of “Quality.”
The reader can be forgiven for finding all of this rather complex and baffling – for it is both of these. Clarity might possibly be achieved if we were to go through each of these subdivisions within divisions, and all of Hegel’s dialectical transitions. For this, however, a book-length commentary would be necessary. So we will try to approach clarity through a second-best method: an account of only the major divisions, and major transitions, in the Logic. We will approach the Hegelian terrain, in other words, from a very high altitude.
Let us first try to understand what is going on the Logic’s first division, the Doctrine of Being, in such a way that we can understand, among other things, how being-nothing-becoming figures within it. If, as I have said, the Logic is an ontology then obviously the Doctrine of Being must be very important. Hegel’s choice of titles, however, is misleading. The Doctrine of Being does not, in fact, present Hegel’s final word on the nature of being. Instead, we must emphasize that the entirety of the Logic is an ontology, and “what being is” does not emerge fully until the very end of Hegel’s account. In reality, the Doctrine of Being only deals with one particular interpretation of being – one which is only part of Hegel’s ontology, a part which, if taken in isolation, would be an inadequate account of being.
Heidegger is usually careful to point out that Hegel’s use of “being” (in the Logic) is idiosyncratic, and that the Doctrine of Being is not a full statement of Hegel’s ontology. Heidegger, of course, expresses his interpretation of Hegel in his own idiosyncratic language – but what he says can be useful to us. In one essay, he states that being in the Logic expresses “the first, simple objectivity of the object.”[6] In one of the lecture courses, Heidegger states “that which Hegel designates with ‘being’ we call ‘objectness,’ which is a designation that indeed captures what Hegel himself also means.”[7] In the same course, Heidegger states, “‘Being’ for Hegel: Beingness in the sense of the immediate representation of the object in its objectness as re-presentedness.”[8]
What all these statements mean is that the categories of the Doctrine of Being are the categories of the giveness of objects in direct perception: of the “objectness” of the perceptual object. As we have already said, all the categories of the Logic have both an objective and subjective dimension: they are simultaneously the categories of being and of thought. The Doctrine of Being presents us with the “being” of objects as encountered at the perceptual level, and its categories are the fundamental categories involved in our experience of objects. Objects, as perceived by the senses, always exhibit Quality, Quantity, and Measure.
Thus, the initial account of being in the Logic is also what common sense would identify as being. Ask the man on the street what he thinks it means “to be” and he will usually, in one way or another, say that it means to be there, in front of him, as an object of perception. Thus, Heidegger says elsewhere that “What Hegel calls beings and being we [meaning “I, Martin Heidegger”] designate with the terms extant [Vorhandene] and extantness [Vorhandenheit].”[9] Vorhanden and Vorhandenheit are terms often used by Heidegger, and they are usually translated as “the present-to-hand”: what we experience as given to us, in merely gazing upon what presents itself perceptually.
Heidegger correctly notes that Hegel’s fully developed conception of being, which emerges only at the end of the Logic, corresponds to what is called in the tradition “actuality.” Heidegger writes, “What we call ‘being,’ in accordance with the beginning of Western philosophy, Hegel calls ‘actuality’ [Wirklichkeit]; and this designation is not a coincidence but is pre-determined in Aristotle as the first end of this beginning [of Western philosophy]: energeia – entelecheia.”[10] We need not worry about what these Greek terms mean right now. Suffice it to say that Hegel’s “actuality” – his mature conception of being – is a reworking of Aristotle’s “actuality” (which can translate both energeia and entelecheia). This emerges at the end of the Logic in a very striking manner.
From Being to Essence
Now we must look more closely at Hegel’s categories of “being.” As we have seen, the first triad of the Doctrine of Being (within the subdivision Quality) is being-nothing-becoming. And becoming is an elaboration of being (which is so empty and abstract it is indistinguishable from nothing). Being is becoming – being is not static, it is dynamic and developmental. The category of becoming is then succeeded in the dialectic by “determinate being” (Dasein [11]). A determinate being is one which becomes or maintains itself actively in being. But what does it become? What does it maintains itself as? We can only answer by reference to the concept of quality. Things are determinate through possessing definite qualities of various kinds – and through not possessing others, or not possessing the same combination of qualities as other things.
Again, it would be impossible to go through all of Hegel’s categories and dialectical transitions. So, now that the idea of quality has cropped up, let us confine ourselves just to understanding the major dialectical transition from quality, to quantity, to measure. As we have seen, the categories of the Doctrine of Being are those of perceptual “objectness.” But what can this have to do with quality, quantity, and measure? Very simply, whatever is given to us present-at-hand as object exhibits these three moments. Hegel is thinking about perceptual experience in the most fundamental terms possible. Every object we encounter is experienced by us as possessing some quality or qualities. And it always possesses those qualities in some quantity – i.e., to some degree. For example, two plates are hot, but one is hotter than the other.
Hegel argues that the mind is able to consider quantity only by becoming deliberately indifferent to quality. For example, there are eleven items on the desk in front of me. One is the laptop which I am using to write this article. Then there are seven books (all by or about Hegel), one coffee cup, one saucer, and a napkin. Normally when I deal with these objects I am focused on their qualities. For example, if I wish to take a sip of coffee, I do not pick up my copy of The Science of Logic, for it has none of the qualities of a coffee cup. And when I wish to consult a point in the Logic I do not reach for the saucer. When I wish to count all the objects on the table, however, I deliberately discount what each object is. I become indifferent to the qualities of the objects and treat each merely as a “unit,” and then I begin: “one, two, three, four . . .” etc.
Similarly, if I just wish to count how many books are before me, I do not consider whether they are hardcover or paperback, primary or secondary literature, colorful or dull, I merely identify the books as books and begin counting. What Hegel has succeeded in demonstrating is that quality and quantity are conceptually bound up with each other and that they are, in a sense, opposites, which is not normally how we think about them. This is the brilliance of the Logic: Hegel teaches us to see familiar concepts in entirely new ways.
Thesis: quality. Antithesis: quantity. Measure is the third term that “reconciles” the two. Consider this example: all men exhibit the quality of height, but some men are taller than others; i.e., they each possess the quality in a different degree. But they cannot possess height in just any degree. If a man were two inches tall, or twenty stories tall, he wouldn’t really be a “man” anymore, would he? This is where measure comes in. There must be something that determines a range within which qualities can exist in some quantity or degree. Measure is a limit, in effect, to the quantity of a quality. This sounds reasonable. But exactly what is it that determines the measure? For one thing, measure would have to differ from case to case. There is a minimum and a maximum height for men (though defining that is difficult), but it is obvious that the same measure won’t apply to giraffes. What do we appeal to in order to know the measure?
It is at this point that the transition is made from the Doctrine of Being to the Doctrine of Essence. The essence of something is its inner being. In German, the word for essence is Wesen, and it is apparently derived from gewesen, the past participle of the verb “to be” (sein). Linguistically, Wesen thus is intimately related to being. In certain contexts, Wesen can even be translated as “being,” usually when what is being referred to is a being, rather than being-as-such (in German this distinction is much less confusing than it is in English).
As we have seen, the categories of the Doctrine of Being are those of objectness or perceptual being. When we arrive at measure, however, we realize that whatever determines the range within which qualities can exist in some quantity is not an aspect of objects that is apparent in simple perception. What is it that determines the range within which a man may possess height, without ceasing to be a man? Mustn’t it be the nature or essence of manhood? This is, at least, where thought goes in response to the issue. Thought turns to the inner being of the thing, and we realize that there is a dimension to the being of things that does not appear to the senses.
Thus, when the dialectic moves from the Doctrine of Being to the Doctrine of Essence, an appearance-reality distinction is discovered. From this new standpoint, the categories of the Doctrine of Being seem fundamentally naïve. The transition from Being to Essence is like the escape from Plato’s cave: we realize that surface appearance is mere shadow, and we seek the truth not in “objectness” but in an essence which does not appear. Also, in the Doctrine of Essence, the dialectic seems to change. Whereas in Being one category is superseded by the next, in Essence we encounter pairs of terms. Sometimes these have the character of “opposites,” but at other times they are merely correlative. Each member of these pairs is shown to be what it is only in relation to its other.
The categories of the Doctrine of Essence are those of what Hegel calls “reflection” (Reflexion). This concept is closely related to one we discussed earlier, “understanding” (Verstand), which refers to rigid, “common sense” either-or thinking. Reflection cannot transcend the conceptual either-ors discussed in the Doctrine of Essence. It cannot think speculatively: in other words, it cannot perceive that these conceptual oppositions are not as opposed as common sense thinks that they are. As an example, take the appearance-essence opposition itself, which is fundamental to this entire section of the Logic. It is customary (especially in Platonism, or any form of thought influenced by it), to take essence as primary and to treat appearance as “mere” appearance; merely as shadows on the wall of the cave, and perhaps as deceptive. But in response to this, Hegel makes some very sensible (and very Aristotelian) points.
First of all, what appears in appearance? Isn’t it, in fact, the essence, the nature of the thing, that is making an appearance? When a man manifests himself to us in various ways – running a mile, reading a book, solving an equation, changing a tire – aren’t all these an expression of his manhood or manness? If so, then it is a mistake to rigidly dichotomize appearance and essence, and to think that appearances somehow “cut us off” from essence. No, appearances are in fact the means by which essence displays itself or, we could say, realizes itself.
Furthermore, if essence realizes itself (i.e., makes itself real) precisely in its appearing, then it is difficult to see what reality essence has when considered apart from appearance. Just as Aristotle maintains, contra Plato, that the forms have no reality separate from their instantiation in the world, so Hegel maintains, contra Kant, that appearances do not cut us off from “what things really are” (i.e., “things in themselves”) but are instead the means by which the being of things displays or reveals itself. And Heidegger, we may note, is in complete agreement with Hegel on this point.
Thus, the standpoint of the Doctrine of Essence and its appearance-essence (or appearance-reality) dichotomy is fundamentally flawed. And so the dialectic passes on to the Doctrine of the Concept (Lehre vom Begriff; where Begriff is sometimes rendered in older translations as “notion”). This is the third and final division of the Logic and, as a dialectical third, we must expect that it will “reconcile” the standpoint of the first two divisions. In other words, the Concept will be the “reconciliation” of Being and Essence.
It is to the Doctrine of the Concept that we will turn in our next installment, which will complete our discussion of the Logic. The Doctrine of the Concept culminates in what is without question the key idea in all of Hegel: Absolute Idea. It is Absolute Idea that is true being for Hegel, and thus understanding this idea is key to understanding Hegel’s ontology, and his entire system.
Notes
[1] G.W.F. Hegel, The Phenomenology of Spirit (trans. A.V. Miller, 1977), 11.
[2] G.W.F. Hegel, The Science of Logic, trans. A.V. Miller (Atlantic Highlands, NJ: Humanities Press, 1989), 82. (Henceforth, SL.) Italics in original.
[3] Hegel, SL, 82.
[4] Hegel, SL, 92.
[5] Errol E. Harris, Formal, Transcendental and Dialectical Thinking (Albany: State University of New York Press, 1987), 171.
[6] Martin Heidegger, Pathmarks, ed. William McNeill (Cambridge: Cambridge University Press, 1998), 329. Essay translated by Robert Metcalf.
[7] Martin Heidegger, Hegel, trans. Joseph Arel and Niels Feuerhahn (Bloomington, IN: Indian University Press, 2015), 8.
[8] Heidegger, Hegel, 41.
[9] Martin Heidegger, Hegel’s Phenomenology of Spirit, trans. Parvis Emad and Kenneth Maly (Bloomington, IN: Indiana University Press, 1988), 41.
[10] Heidegger, Hegel, 40. I have transliterated the two Greek terms into Roman letters.
[11] Please note that Hegel’s use of Dasein has no relationship to Heidegger’s use of the same term.
16 comments
Hello, it’s Observer on my members account. This article alone was well-worth the subscriber’s fee, and I don’t think there’s anything like it elsewhere. It’s about time I chipped in, given how much personal value this series alone has given me over the years.
Personally, I think this is where I realize that Hegel is not going to have all the answers due to some fundamental issues (though I suppose that’s why we’re here to listen to what Heidegger has to say). But I think you’ve granted me the gift to realize that just because those fundamental issues are there, does not mean that Hegel is not worth putting in the work. The rest of the progression after the introduction of Being and Nothing demonstrates that fact. That realization alone is invaluable perspective, and I thank you for providing it.
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If you’ll entertain me briefly, I’ll hazard an explanation to Hegel as to why Being and Nothing must be different in my book. It is because indeterminacy is not Nothing (though it may not be any particular being), nor can we say that pure indeterminacy and pure nothingness are alike in any way except an insufficient analogy. The difference between Being—whether we think of beings, pure being, and even indeterminacy that somehow manages to be—and Nothing is so vast that it cannot be quantified. Even the tiniest shred of Being’s presence vitiates the basis of the analogy. I’ve been listening to a lot of Houlgate’s famed lectures on Hegel, and every time he exhorts the audience to avoid thinking “Being is Being is Being, Nothing is Nothing is Nothing” and learn to love the vanishing, I cannot help but think “But the so-called Nothing being considered is not yet even Nothing. It’s merely an indeterminacy, and indeterminacy is! All we ever began with was Being and Being, Being prior to determinate being and difference, and Being never gets to vanish in the first place.”
I am not sure what can disabuse me of this prejudice at this point. “Nothing” seems to not be pure nothingness but is rather like some kind of Scholastic prime matter. Perhaps I will slide further into my Parmenidean madness and insist that we can’t even say that Nothing is, for Nothing “Isn’t”. What are your thoughts, Mr. Cleary? I’ll grant you the final word on the matter.
Mr. Cleary would give a different answer, but here’s how I evaluate it.
While Mr. Cleary is right to criticise Bertrand Russell, the English philosopher is on to something here. The set theory states that a “set of all sets” can’t exist. Likewise, a “generalised Being” and a “generalised Nothing” are incorrect terms. There isn’t a single Being, only infinite number of different Beings. At the very least, we should say that Being and Nothing are beyond the reach of any rigid logical system. Therefore, any attempt to build a philosophic system based on logical inference from those Hegelian “observations” concerning Being and Nothing is self-contadictory and bound to fail.
Next, even if we accept the argument, there is a huge number of logical errors and inconsistencies in the Hegelian model. For example, he infers that “Being is Nothing” from his statement that both Being and Nothing can’t be described properly. Sorry, that’s logically incorrect. One example to the contrary: the Truth cannot be defined (Tarsky’s Theorem), and the Falsity cannot be defined either. This doesn’t mean that “true is false”. It just means that True and False are diametrically opposite, but escape any attempt of rigid definition.
Another example comes from topology. For a non-empty set T, you can introduce a trivial topology {∅, T}. Although there is a duality between ∅ and T, and both are singular in the sense of being open and closed simultaneously, you cannot claim that ∅=T. Otherwise you run into a contradiction.
So, you can’t say that “Being is Nothing”, or you crash the entire system based on this premise.
There is also a huge problem with the Hegelian “dialectical method”. Popper justly criticised him on the logical grounds, although Popper was incorrect generally. Note that the “thesis-antithesis” system roughly corresponds to “A∧~A” in mathematical logic. And it’s correct that “A∧~A ⇒ B“, where B is an arbitrary statement. I.e. you can infer anything in such a system. Although there is no direct relationship to Hegel’s dialectic here, it illustrates that Hegel’s dialectical constructions are suspiciously convenient to reach any conclusion Hegel wants.
Finally, note how certain people like the idea that “Being is Nothing”. I.R. Shafarevich discussed it in the “Socialism Phenomenon”. It’s true of Hegel, Meslier and a lot of others. It looks like “Being is Nothing” is either an indicator of a broken mind or even its cause. You might think of a mental “gimbal lock” here. My opinion is that it might be a source of significant amount of destructive ideologies out there.
(I am RR). I appreciate the extensive reply! I will try to summon a response.
First, I am generally not a fan of ZFC set theory foundations for mathematics. I think that mathematics, properly speaking, is the science of quantity, which is what the Greeks thought and what mathematicians from Descartes to Gauss believed prior to developments in the 19th century. I am skeptical of the “ontological underpinnings” of sets, elements, etc., and whether they form a proper basis for quantity. I am also skeptical about the scope of Godel’s incompleteness theorems (and Tarski’s undefinability theorem for that matter) because 1) the fact that it relies on what I find to be problematic axioms; and 2) because most examples of supposed incompleteness seem to be problems of self-reference, and self-reference problems tend to be unforced errors of language. All of these contribute, in my view, into a confusing of what mathematics is and a failure of mathematics to be a coherent, unified field of inquiry. I offer that as a disclaimer, in case there is a point you want to contest later on that actually boils down to a clash in axioms, so neither of us waste our time!
Second, I think that paradoxes involving “set of sets, sets which contain themselves, etc.”, are a specific type of ontological problem which has emerged over and over again since Ancient Greece. The “third man argument” against Plato’s “naive” (initial) theory of forms is similar in structure, and I would argue that Aristotle’s argument in Metaphysics that Being is not a genus was constructed specifically to avoid such a paradox. Everything that is, genera, species, and individuals, belong to Being. But if Being were a genus, a “super-genus” if you well, then it would be a genus which contains itself. So, in order for Being to be an arche, it cannot be a genus. Instead, it has to be something more open-ended and indefinite. From this angle, you can find the traces of Heidegger’s own attempt to flesh out the ontological difference, the difference between Being and beings. And with Aristotle and Heidegger alike, you no longer need to worry about a self-containing paradox, since Being merely is and only has that relationship to itself.
Third, I think Hegel’s issue is a problem that is more complicated than a simple contradiction of Being and Nothing. I think that what Hegel takes to be Being, even “Pure Being”, and what Hegel takes to be Nothing, even “Pure Nothing”, are simply not radical enough to serve the role of what these either term seems to be referring to and ought to refer to. “Pure Being” turns out to still “be” in some way, and “Pure Nothing” turns out to be more than mere nothing (the radicality of nothingness escaping Hegel’s grasp). So, when Hegel sketches the initial dialectical scheme in the beginning of the SoL, he is actually mapping out a dynamic that is different from what he portrays them to be, and that is the reciprocal reliance of form and matter, ideality and reality, etc., and the indeterminacy of taking either primordial concept to their radical extreme. The vanishing part is interesting. But the BLUF is that indeterminacy is not nothing, and Being vs. Nothing is not the engine of vanishing. That’s all that needs to be said. It is not a successful presuppositionless inquiry, albeit the work as a whole has been given new value to me thanks to Cleary’s exposition.
I do think that there is something going on that seems to make Hegel’s dialectical movements seem meandering and arbitrary. You are keen to point that out! Perhaps it is the explosion principle somewhere. Or perhaps it is the sheer freedom of the spirit in motion as it contemplates itself. In any case, caution must be taken in understanding it.
Thank you for your thoughtful answer, RR! I enjoyed your mode of argument.
Well, yes, we have an axiomatic difference. I don’t think there is a single Being. I also accept set theory foundations, and I don’t think mathematics is about quantity. Even numbers are not about quantification, and there is infinitely more than numbers in mathematics. While I don’t think it is applicable everywhere, such things as Gödel’s Incompleteness Theorem provide answers to some philosophical questions in a way. Let me explain: while the Truth isn’t limited to mathematical domain, it incorporates mathematics as well. So, given that Tarsky’s Theorem prohibits strict definition of the Truth, you may say that you can’t define “generic Truth” also, because mathematical Truth is a part of it.
Then, it’s neither a problem of language nor of self-reference. The proof for the “set of all sets” is actually trickier: it involves a contradiction with a set of all sets which don’t contain themselves. Well, you can say that there is a self-reference, but indirect. For the Gödel (and Tarsky) theorem, a wholly different approach is used: there is an explicit construction of a true statement that cannot be inferred from a given set of axioms and inference rules S. It’s a strict theorem of propositional calculus, and I don’t have problems with it. Like with the Truth, if you can’t have a full set of axioms in propositional calculus, you can’t have it in any larger domain. The theorem just shows you a basic property of our world, and it manifests everywhere: in mathematics, physics, biology, philosophy and everywhere else.
The problem is actually even larger than that. There are axiomatic statements that yield very different worlds depending upon whether you accept them as true or not. You can think of Euclidean and Lobachevsky geometries, for example. Or the Continuum Hypothesis (whether 𝔠=𝔄₁ or not). You can state that 𝔠=𝔄₁ or 𝔠≠𝔄₁ and get different resulting systems that are both coherent. So, I don’t accept a single Being hypothesis.
And “Gödelian” statements are not self-referential at all! They are normal true statements which just cannot be inferred. You can look up some examples. Also, self-reference can be used as a convenient tool during the proof, but self-referential things are only a part of a larger picture. E.g. a lot of proofs of unapplicability of Turing machines use the self-reference problem as a basis, but a) it’s a perfectly valid problem in itself and b) the problems under consideration are absolutely practically-related, like whether a Turing machine M stops or not.
I suppose you get my position by now, more or less. I think that a discussion of the Pure Being is incorrect in itself, because it’s beyond any descriptive powers of language. Whenever it arises, we get quite arbitrary philosophical systems.
And what Hegel does is even more objectionable: if he called his system “Wissenschaft“, then he has an obligation to be rigorous. And he isn’t.
I also don’t accept any attempts of “minimalistic” approach toward philosophical constructions. Everyone has an infinite number of axioms, and you cannot really infer something appropriate just from Being and Nothing, or “cogito ergo sum” etc.
Let’s also try to accept your position. In my opinion, Hegel’s “Being is Nothing” trick is sufficient to invalidate his constructions, whether it is his greatest sin or not. Being is not Nothing, or we can’t have a meaningful discussion at all.
Yes, I think your criticism is also valid. This can also highlight my position: the more “radical” your definition of pure Being is, the more abstract it gets. We can say that Being is, but we can’t imagine it. Whenever we try, we think of particular beings, e.g. sky, clouds, Jupiter or NGC 5194. Eventually, it escapes any description or consideration. With Nothing the situation is even more problematic: if we denote it as N, then, whenever we propose a statement P(N), we already consider it as a being (even if we say that Nothing isn’t). You can’t build a coherent system just from that. (But you can construct whatever you want. 🙂 )
Further, we get to another Hegel’s problem from there. He may have thought that his system is presuppositionless, but it’s also not true. There is an infinite number of axioms that he accepts by default. For example, he thinks that language is sufficient to describe a proper philosophical system. Or that the structure of reality and logic is time-independent and space-independent. Generally, the whole Hegel’s argument also incorporates all his environment, including the German language, society etc. I am particularly sure that if he were born 70 years later, he would reach absolutely different conclusions based on those premises. He might have become something of a Social Darwinist, maybe.
I think he had a predetermined idea of what he wanted to get, and all else served as a disguise.
Thank you for the substantial reply and for keeping me company as we wait for the next installment of the series! I have a few remaining points:
1) I think you’re right that there is no “single” Being, at least not in the usual, definite sense. That is why Aristotle argues that Being is not a genus, why Heidegger treats Being as “transcendent, universal, and indefinite” and makes the ontological difference (beings vs. Being). The point is that Being is not a being. But what is it then? It certainly isn’t nothing, and it seems to be. I think this is probably end-stage level ontology (to the point where Heidegger would argue that it cannot be an ontology, but I digress).
2) I’m confused by your added distinction of “the set of all sets which don’t contain themselves” and what it entails. It was my understanding that this was an additional clarification added to avoid Russell’s paradox.
3) What are some good examples of non-inferrable true statements that do not involve self-reference? I’ve struggled to find any whenever I’ve looked into the subject.
4) On your position of Hegel’s radical abstraction as you mentioned towards the end, do you get a sense of “khora” like from Plato’s Timaeus?
On your points.
1. Yes, I accept the argument that Being is not a genus. However, it is supernatural and supralogical, so our attempts at its philosophical (i.e. logical) discussions are insufficient to reach appropriate conclusions. Categories like genera are good for natural applications only. Maybe it’s better to describe Being as a predicate applied to various objects (B(N)⇔∃N), thus not encountering paradoxical problems we are unable to solve. Anyway, I find all existing descriptions of Being incomplete or just wrong.
2. For the problem of a universal set, the proof is quite clear. The fact that U∈U isn’t a problem in itself, so we define ordinary and extraordinary sets.
A set A is ordinary if it doesn’t contain itself, and extraordinary otherwise.
Theorem 1. A set E of all ordinary sets cannot exist.
Sketch of the proof. Let’s assume otherwise: E exists and contains all ordinary sets.
Is this set E extraordinary or not?
If we assume that it’s extraordinary, then E∈E by the definition of an extraordinary set, so it’s ordinary – contradiction.
Then, if we assume that it’s ordinary, E∉E by its own definition, so it’s extraordinary – another contradiction.
So, E cannot exist. ∎
Yes, it’s connected to the Russell paradox. You can claim that the problem stems from self-reference, but actually it appears at a later point. That’s important!
You can also explore the field of Turing machines to see how this is adjacent to computation theory. For example, let’s look at the self-applicability problem.
Theorem 2. Self-applicability problem is unsolvable via Turing machines.
Sketch of the proof. We assume that there exists a Turing machine M which yields 1 for another machine T if T is self-applicable and 0 if T hangs on its signature.
Let’s construct another machine M’ based on M this way: it outputs 1 if T is not self-applicable (hangs), and M’ hangs itself when T is self-applicable.
If M’ is self-applicable, than M'(sig M’) hangs because of its building scheme ⇒ M’ is not self-applicable ⇒ contradiction.
If M’ is not self-applicable, then M’ hangs on M’. But it must output 1 and stop by its definition! So, it must be self-applicable ⇒ another contradiction.
So, M’ cannot exist ⇒ M cannot exist, and the problem is unsolvable. ∎
(Note that a Turing machine is not applied to itself, but its signature, which is absolutely valid input data. In practice, it’s possible to run a computer programme on its own code, e.g. readelf -s `which readelf`, and it’s perfectly normal.)
Here we have an obvious practical takeaway: there can’t exist a definite method to check whether a specific Turing machine stops on any given input or hangs indefinitely. You can find many more appropriate examples there. E.g. unsolvability of the inclusion problem for algebraic manifolds over a field with zero characteristic (Kanel-Belov, Chilikov 2019; in Russian). Do you see how that works?
1) So, if Being is just a predicate, then in what sense is the predicate referring to when you apply it to a subject? It can’t be referring to nothing. That is an issue. Predicates either refer to essential qualities or accidental qualities, and these are things. e.g. Socrates is a rational animal, versus Socrates is old. And in the former sense, it is very easy to think of rational animal as the species human being, belonging to the genus animal (in which Socrates is an animal is also a true statement). So, is Being a genus? It starts to act like one in many ways. The arguments against it are mostly technicalities that don’t seem to completely eliminate the insight that led us to consider it as something genus-like.
Perhaps Being is forever doomed to be equivocal. But I think there is a lot of work to be thought about. I like to think about the nature of the copula, the debate of whether existence is a predicate, and ontological modalism (when we say that something is, do we mean that it is intelligible? possible? actual? etc.). There is much more to be said on this topic.
3) To begin, what do you mean by inference? Do you mean a straightforward process of deduction? Because if that’s what you mean, then yes it is impossible because there are too many “unknowns”, so you cannot triangulate to an answer. But suppose I happened to have a savant-like memory of numbers, and I recognized many that filled in the pattern of operations present, and that gave me enough to solve the rest of the equation (or perhaps, even less believably, knew it all in one shot). That would be a use of abduction, and abduction is an important part of reasoning altogether. Does inference exclude abduction?
Perhaps what is not inferrable to one person is inferrable to another person. That is to say, it’s a “skill issue”. lol. It’s certainly a skill issue for me! Jokes aside.
Second, I am unfamiliar with the moving parts of Matiyasevich’s theorem, but if it is predicated on some kind of recursion, then wouldn’t that be a kind of self-reference? In computer science, recursive functions are functions that call themselves. And if I’m correct in this sense, then I think it demonstrates how pervasive problems of self-reference are and how key they are to Godel’s incompleteness proofs, given how difficult it is to summon a decisively non self-referential example.
“Predicates either refer to essential qualities or accidental qualities, and these are things. e.g. Socrates is a rational animal, versus Socrates is old.”
I think that the bold part of the quotation is not necessarily correct. Let’s say that a predicate P(x) has a characteristic (or underlying) set U(P)={x: P(x)=1}, if such a set can exist. We may postulate that some predicates have corresponding underlying sets, while some don’t (e.g. Being or Truth). Those that do are regular predicates, which correspond to ordinary genera. Those that don’t are, therefore, irregular, and we can’t operate with them like with the first group. What to do with them is another consideration.
You might object that it’s a mathematical argumentation, but mathematics is a part of reality, so, if you cannot specify the appropriate genera for mathematical predicates, you also cannot define them for any larger domains.
Anyway, I don’t intend to get into the weeds here. I already made my position clear. One great problem of a huge number of philosophers like Hegel is that they try to use some abstract logic to build their theories, which is wrong in itself. Another one is that they make a lot of logical mistakes in the process. 🙂 Hegel’s example is especially egregious: it’s not “Wissenschaft” what he proposes.
It’s all worth pondering, but I prefer to go in a different direction.
“To begin, what do you mean by inference? Do you mean a straightforward process of deduction? Because if that’s what you mean, then yes it is impossible because there are too many “unknowns”, so you cannot triangulate to an answer.”
There’s a lot to answer here!
First, it seems like you don’t understand the subject under consideration. I recommend you to read the proofs – Gödel’s texts are much better than Hegel’s! I don’t know about publications in English, but I’ve read a Russian book with a proof of the Gödel’s Theorem for high school students. It’s a long proof, but quite accessible.
The scope of the Gödel’s Theorem is formal axiomatic systems. By inference, I mean the process of proving a statement in the system via any possible rules in such a system. You can use deduction, induction, abduction or even seduction if they are formally defined in the theory. However, I am curious to know how abduction or seduction axioms would look like! 🙂
Second, you’re incorrect in your assessment of the Matiyasevich equation above. “Too many unknowns” is never a problem. Of course, when such an equation has a solution, it would be likely quite long even to write. For example, from the second large square you obviously get q=b^(5^60). Given that 5⁶⁰ is slightly (several percent) greater than 5∙10⁴¹, with e.g. b=2 we get that q>10^(1.5∙10⁴¹), so q>10^(10¹⁵⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰). To write such a number in ⅕”∙⅕” decimal digits, you would need about 100000 Sun masses of paper, and that’s quite a lot. 🙂 Anyway, mathematics doesn’t operate with vague concepts like “practical unsolvability” – if an equation requires 10¹⁵⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰⁰ years to solve, it’s solvable. If it’s not solvable, it cannot be solved at all.
Here you ought to prove that a solution to the equation does not exist, which is very different from finding one. It’s proven that for any formal theory there are values of K that provide an unsolvable Diophantine equation, and that cannot be proven in that particular theory.
“Second, I am unfamiliar with the moving parts of Matiyasevich’s theorem, but if it is predicated on some kind of recursion, then wouldn’t that be a kind of self-reference?”
You need to read the proofs, or we won’t get to the substance. If you’re uncomfortable with “recursively enumerable sets”, it’s equivalent to calling them “listable sets”. By definition, a set A is recursively enumerable if A = {<a₁, . . . , aₙ> | ∃x₁, . . . , xₘ: F(a₁, . . . , aₙ, x₁, . . . , xₘ) = 0}, where F is a computable function. There is no self-reference.
“In computer science, recursive functions are functions that call themselves.”
That’s not correct. Look up a proper definition.
“And if I’m correct in this sense, then I think it demonstrates how pervasive problems of self-reference are and how key they are to Godel’s incompleteness proofs, given how difficult it is to summon a decisively non self-referential example.”
No, you’re not correct. I don’t know why you are so stubborn about that. The Gödel’s Theorem has been proven more than 80 years ago. Gödel himself wasn’t happy with his result, as far as I know. A lot of people have tried to bypass the theorem, to absolutely no avail.
You can look at it this way: for any statement Gödel builds a corresponding arithmetic formula. Then he shows that some arithmetic formulas cannot be proven in a given theory. An arithmetic formula is not self-referential. With Matiyasevich Theorem, you can provide examples in the form of Diophantine equations, like the one I used. That equation is not self-referential, either. There are no references to the Russell Paradox or something like that anywhere in the theorem. If you don’t agree, please point out where self-reference might be, in your opinion.
1) I do not think that predicate logic as it has been constructed is a suitable method for thinking about the problem. For starters, it presumes certain metaphysical understandings of predication. If you build a syntax in which existence cannot be a predicate, or in which its mappings are unclear, then what are you going to prove with such a syntax? The same conclusion you began with. Obviously, not a great method of inquiry.
2) On the topic of defining inference. I think at this point there is going to be an impasse. It seems ridiculous to me to consider acts of reasoning to be “axioms”, especially in the sense to offer the ability for any act to be considered an inference as long as it is formalized. It seems to cheapen the apodictic force of reason. Furthermore, what would then govern what it means for something to be well-defined? Would we need even further axioms, axioms for the axioms or axioms for axiomatization? How long would this process go on for? Finally, this is an unsatisfying and roundabout way of dealing with the question of inference. Do we not use inferential reasoning all the time? It shouldn’t be difficult to pin down its meaning(s) without transforming it into something alien.
3) The self-reference is clearly in the recursive structures built by the sets mentioned. Enumerative recursion of sets is a structural self-reference. That is another problem with using set theory. What is the relationship to ontology and sets? What is the being of a set? Structure has always been associated with ontology, so why should we ignore structural reference as self-reference, a thing-referencing-itself?
This is not a matter of comfort, by the way. Recursion has many useful applications. I am simply calling it what it is, which is a type of self-reference.
4) “That’s not correct.” https://en.wikipedia.org/wiki/Recursion_(computer_science) “Recursion solves such recursive problems by using functions that call themselves from within their own code.” Curt dismissals which run contrary to virtually every treatment of the problem in the field are not helpful.
5) Godel proves that some kinds of arithmetic formulas cannot be proven. Which kinds of arithmetic formulas? Self-referential ones! https://en.wikipedia.org/wiki/On_Formally_Undecidable_Propositions_of_Principia_Mathematica_and_Related_Systems#Outline_and_key_results Godel’s system is just an arithmetic mapping tool, one which mapped mathematical statements on to numbers and more primitive arithmetic operations, and what he did was essentially demonstrate that certain kinds of “formulas” will never generate a well-defined number.
1. This was just an example out of the blue. I don’t think myself that it’s correct. However, it illustrates my point: introducing Being inevitably leads to contradictions in the system. So, we either forbid it or exclude from discussion. (I love that formulation: “introduction of the axiom of Being to any logical system leads to a contradiction”. I’m sure, however, that you would object to it.)
2. Let me explain. I’m not a formalist myself (Gödel’s Theorem destroys any formalism). But any positivist system essentially “asymptotically converges” toward some formalism. Hegel’s system and intentions seem like a positivist approach, and its logical conclusion is logical positivism. Therefore, it’s enough to address the latter to highlight holes in the Hegelian constructions. Obviously, we can’t use Hegel’s language, because it’s (likely deliberately) introduced to obfuscate the matter. It would take years of debates about vaguely formulated notions, and eventually no side would be sure what they meant. 🙂
Axioms by definition are taken for granted. That also illustrates another point: any logical system of philosophy (e.g. Cartesian, Kantian, Hegelian) that attempts to build something upon a reduced axiomatic basis is doomed for failure. Actually, I think that all of them serve as poor diguises of nihilism and tend to produce only justifications for it (whether the authors intended so or not). And any finite axiomatic basis must be considered “reduced”, so, “logical philosophy” can be put aside. Of course, there are some points to take from it, but overall, it’s not correct.
And I don’t agree with this: “It shouldn’t be difficult to pin down its meaning(s) without transforming it into something alien”.
3,4. Fine, I turn pedantic mode on at 1%. 🙂
Concerning recursive sets: they are those with a way to list elements. For example, prime numbers. You don’t get it: they are built constructively.
Then, you should have noticed that we were discussing Matiyasevich Theorem (mathematics), not programming. In computer science (i.e. computation theory, if it’s named properly), recursion is rigidly defined. (By the way, Gödel himself uses recursive functions in his paper, and it shouldn’t have raised those objections if you have read the original. If you haven’t, you should, or what are we talking about?)
A mathematical recursive function can be defined this way.
The basis functions there are o(x)=0; s(x)=x+1; Iₘₙ(x₁, …,xₘ)=xₙ, if n≤m. Then we add superposition operation, e.g. h(x₁, …,xₙ)=f(g₁(x₁, …,xₙ), …, gₖ(x₁, …,xₙ)), where f, g₁, …, gₖ are already defined. Further, we tell that function h(x₁, …,xₙ₊₁) is a result of recursion operation upon f(x₁, …,xₙ) and g(x₁, …,xₙ₊₂), if:
h(x₁, …,xₙ, 0)=f(x₁, …,xₙ);
h(x₁, …,xₙ, y+1)=g(x₁, …,xₙ, y, h(x₁, …,xₙ, y)).
The functions obtained from basis ones through a finite number of those operations are called primitive recursive. If we add minimisation operation, we get partially recursive functions: p(x₁, …,xₙ)=μₜf(x₁, …,xₙ, t)={min t: f(x₁, …,xₙ, t)=0}. (That’s not a proper full definition, of course.)
As you can see, recursive functions are clearly defined. If φ(n) is recursive, it may or may not be expressed via φ(0), …, φ(n-1), which are already defined. Forward references are strictly forbidden here, as well as self-references!
Example: with φ(n)=n! we get φ(0)=1; φ(n)=nφ(n-1). There are no self-references there.
If the name misleads you, call them “inductive functions”, because they are inductive, obviously. However, you cannot exclude them, or you would be left with a system that is insufficient even for Peano arithmetic. Think of Presburger theory, for example.
Considering recursion in programming: it’s not self-referential either. Let’s look at the factorial example. I can write e.g. a function in C.
uint64_t fact(uint8_t n) {
if(!n) return 1;
return n*fact(n-1);
}
(Of course, for some n result won’t fit in uint64_t, but that’s an illustration.) We can take a different approach without recursion:
uint64_t fact(uint8_t n) {
uint64_t acc=1;
for(unsigned i=1; i<=n; i++) acc*=i;
return acc;
}
Actually, we can perform recursion unwinding with the use of a stack. In hardware, function calls are implemented through stacks, and early computers didn’t have call instructions at all. In e.g. Strela, you cannot call anything, only perform jumps. So, it’s not what you think of it.
By the way, these theories have very intricate relationships between each other. For example, Presburger arithmetic lacks multiplication and is insufficient. A computer that lacks any tools for recursion implementation is not Turing-full. You can see that multiplication and recursion are connected.
Or, maybe, you are of the opinion that we should forbid multiplication at all? 🙂
5. No, you don’t understand.
In reality, Gödel eventually constructs an arithmetic formula. It is irrelevant what is used as a basis; this arithmetic formula can’t “know” anything about Gödel numbers or any part of encompassing theory. It is a statement about some properties of natural numbers. Arithmetic has no tools to explore something beyond it. (How can a formula like ∀y∀z (x=y·z→y=1∨z=1) reference things like statements? It can’t.)
To make it clear: imagine we look at different arithmetic formulas. They are either true or not. Then we start exploring which formulas can be proven in some specific axiomatic theory Θ₁. It turns out that a particular formula G is Gödelian. But we can investigate it in a different theory Θ₂, and it would be provable in this (extended) theory. Let that sink in.
What Gödel shows is that if his formula is provable, than the theory under examination is contradictory; the meaningful part is the arithmetic statement, and its counterpart is auxiliary.
Then, there are various ways to prove the theorem. V.A. Uspenskiy has a different proof through universal function (By the way, he wrote the book I recommended.) The phenomenon is infinitely larger than particular examples built by Gödel.
Behold, I have investigated a little, and next time I’ll provide particular examples.
So, here is a list of bulletproof examples.
1. The pair of (17 Gen r) and (Neg (17 Gen r)) in Gödel’s original paper (page 188; Satz VI and Satz VIII). (By the way, Gödel explicitly states that the unresolvable problem is arithmetic.)
2. The question of whether the theory under consideration is contradictory or not (Gödel’s work, page 196, Satz XI). Before any further objections I need to clarify: by definition a theory T is non-contradictory, if for any statement F either T⊬F or T⊬(∼F). It means that for any statement F you cannot prove both F and its negation in the theory.
3. The question about the existence of roots for the Matiyasevich equation that I have provided.
Particular examples:
4. The Continuum Hypothesis cannot be proven in ZFC (Gödel, Cohen), but can be in ZF+AD (e.g. Pakhomov). I should have mentioned above that it qualifies as an unprovable statement also. Then, there are various further examples which are connected to that, for example, aleph-hypotheses.
This hypothesis also has geometric interpretations, for example, “the Ox axis can be painted with countable colours so that for any monochromatic quartet (e₁, e₂, e₃, e₄) the following is true: e₁+e₂≠e₃+e₄“.
5. Paris-Harrington Theorem (Paris, Harrington, “A Mathematical Incompleteness in Peano Arithmetic”).
6. Kanamori-McAloon Theorem (Kanamori, McAloon, “On Gödel incompleteness and finite combinatorics”).
You need to look especially at examples 5 and 6, because they directly refute your position.
To explain a little, both theorems are connected to the Ramsey theory and partition calculus. The original Infinite Ramsey Theorem concerns partitions of an infinite set. Further, there are the Finite Ramsey Theorem and various extensions of it.
They have the following properties.
A. The Infinite Ramsey Theorem belongs to the field of ZFC applications; it cannot be even formulated in PA, yet alone proven.
B. The Finite Ramsey Theorem can be both formulated and proven in Peano arithmetic.
C. The extensions dealt with in 5 and 6 can be formulated entirely in PA, but cannot be proven in it. (They require some extension, e.g. ZFC.)
D. Furthermore, the results posited via the extended theorems can be checked for particular M, e, r, k directly.
E. All those theorems provide meaningful statements about some properties of natural numbers.
Of course, it’s only the tip of the iceberg.
Hi Oleg,
I greatly appreciate the robust list of examples. I’ll take note and study them over time. In the meantime, I’ll leave the thread to rest, since I think we’ve done enough pollution of philosophy with mathematics haha. Otherwise, it’s been a pleasure, and I wish you the best!
3. If you want an example of a non-inferrable statement, it requires to specify an axiomatic theory Θ to build. Here are some notes.
Generally, examples tend to be difficult to understand for a non-specialist. As far as I know, there are proofs for specific systems and statements that they are equivalent to the induction axiom. (Of course, the initial theory cannot have an induction axiom, or this statement would be inferrable, obviously.) I need to consult my student notes on mathematical logic for that.
There is a very beautiful theorem proven by our great mathematician Yu. V. Matiyasevich: D = R, where D is the class of Diophantine sets (roughly: those that can be defined by a Diophantine equation) and R is the class of recursively enumerable sets. It has some mind-blowing ramifications, and one of them is the Gödel-Matiyasevich theorem of incompleteness. It allows to build non-inferrable statements for a given theory in the form of solvability of a Diophantine equation.
For example, we can state that for a theory Θ there exists a number K for which the Diophantine equation
(elg² + α – (b – xy)q²)² + (q – b^(5^60))² + (λ + q⁴ – 1 – λb⁵)² + (θ + 2z – b⁵)² + (u + tθ – l)² + (y + mθ – e)² + (n – q¹⁶)² + ((g + eq³ + lq⁵ + (2(e – zλ)(1 + xb⁵ + g⁴) + λb⁵ + λb⁵q⁴)q⁴)(n² – n) + (q³ – bl + l + θλq³ + (b⁵-2)q⁵)(n² – 1) – r)² + (p – 2ws²r²n²)² + (p²k² – k² + 1 – τ²)² + (4(c – ksn²)² + η – k²)² + (r + 1 + hp – h – k)² + (a – (wn² + 1)rsn²)² + (2r + 1 + φ – c)² + (bw + ca – 2c + 4αγ – 5γ – d)² + ((a² – 1)c² + 1 – d²)² + ((a² – 1)i²c⁴ + 1 – f²)² + (((a + f²(d² – a))² – 1) (2r + 1 + jc)² + 1 – (d + fμ)²)² + (((z+u+y)² + u)² + y – K)² = 0
is unsolvable, but it cannot be proven in Θ. You’re welcome to find K for your theory yourself. 🙂
You can build examples in other form, of course. For the one above, consult e.g. J.P. Jones’ article in the Bulletin of the AMS.
Another point. I haven’t thought it well, but I have a hypothesis.
Proposition. For any “sufficiently good” theory Θ and non-inferrable in Θ but true statement P there exists another statement R(Θ, P) that is equivalent to P and corresponds to the original Gödel scheme of building a non-inferrable (in Θ) statement.
I’m not sure now whether it’s true (or even provable). Maybe it’s not hard to prove. Maybe impossible. I need to meditate on it for some time, not now. If that’s true, then you can say that any axiom independent of other parts of the system is “self-referential” in the sense of Gödel’s proof.
4. Yes, there is a sense like “khora” in “Timaeus” when you explore Hegel’s system. I’m sure Hegel also drew upon Plato’s ideas. But I get a very different impression from those two philosophers. For Plato, “khora” is a medium that helps to project ideal entities upon this world, so it looks more auxiliary. It also spoils things to some degree. And Plato generally avoids discussing khora in “Timaeus” – he loves triangles so much! 🙂 I think he deliberately shies away from it, because the first rule of khora is not to speak about khora. On the contrary, Hegel operates with his “prime matter” as if it’s bricks. It would have been better for him if he also approached it more indirectly (although only slightly better).
Oh, and here’s another question:
“Thus, the initial account of being in the Logic is also what common sense would identify as being. Ask the man on the street what he thinks it means “to be” and he will usually, in one way or another, say that it means to be there, in front of him, as an object of perception. Thus, Heidegger says elsewhere that “What Hegel calls beings and being we [meaning “I, Martin Heidegger”] designate with the terms extant [Vorhandene] and extantness [Vorhandenheit].”[9] Vorhanden and Vorhandenheit are terms often used by Heidegger, and they are usually translated as “the present-to-hand”: what we experience as given to us, in merely gazing upon what presents itself perceptually.”
If the Doctrine of Being roughly corresponds with present-at-hand (vorhanden), then what aspect of Hegel’s system best corresponds with ready-to-hand (zuhanden)?
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