Friday, September 30, 2016

Absolute Idealism, Mathematics, and the Problem of the One and the Many

In previous posts on this blog I developed the rough outlines of what I like to call “Absolute Idealism 2.0”: a contemporary form of Absolute Idealism, informed by modern mathematics and digital physics rather than, say, Hegelian dialectics. In this post I want to investigate how Absolute Idealism 2.0 deals with a well-known conundrum in the history of philosophy: the Problem of the One and the Many. This problem arises for any strongly monist metaphysics, i.e. any theory that recognizes just a single “substance” or entity as the explanatory ground of reality as a whole. How does this single entity, this ‘One’, produce ‘the Many’, the unsurveyable multitude of finite physical objects, coming and going in space and time? Absolute Idealism is, of course, a form of strong monism in that it reduces all of reality to a single ‘Absolute Mind’. Thus the Problem of the One and the Many is especially acute for Absolute Idealism: how does this Absolute Mind produce the multifarious reality in which we find ourselves? In this post I want to argue for a decidedly un-Hegelian solution to this problem, a solution that shows the intimate link between mathematics and the Absolute Mind. I want to show, moreover, that this link is a two-way street: not only can we found mathematics on the self-generating structure of the Absolute Mind, we can also use mathematics to elucidate the nature of the Absolute Mind. First, however, a word of warning to the reader: what follows are mostly philosophical analyses and arguments concerning self-causation and self-awareness; mathematics enters the discussion only in the second half of this post, and then mostly in an informal manner. Nevertheless, I trust that when the suitable formalization is provided, the mathematical claims I make will check out.

Absolute Idealism and Leibniz’s question
First, however, let’s take a closer a look at Absolute Idealism in general and why it can still be seen as an attractive metaphysical position (despite its near universal rejection by contemporary philosophers). I define Absolute Idealism, roughly, as the theory according to which reality as a whole exists because it is thought and/or experienced by an Absolute Mind, which in turn exists because It thinks/experiences itself. Thus, on this definition, the Absolute Mind bootstraps itself into existence through its own awareness of itself
which is why I refer to this Absolute Mind as “Absolute Self-Awareness” (ASA). This ASA, then, exists only as the object of its own awareness. To paraphrase Berkeley: the esse of ASA is its percipi per se – that is: its being is its self-perception. In earlier posts I explained why this amounts to the claim that ASA is self-causing, in the sense of being the cause of its own existence (I will review this argument below). This is indeed a central claim in all Absolute-Idealist thought, from Plotinus ("The One [...] made itself by an act of looking at itself") to Fichte (“The I exists only insofar as it is conscious of itself”) all the way to physicist John Wheeler ('the universe exists because it observes itself').

In my view, this self-causing capacity of ASA makes it a very strong candidate for being the correct answer to Leibniz’s famous question: why does reality exist? and why is it the way it is? Since there is nothing outside reality as a whole, the reason why that whole exists, and why it is the way it is, can only lie within itself – that is to say: only a form of self-causation can answer Leibniz’s question. Now, we know (with Cartesian self-evidence) that self-awareness exists. Hence, given the arguments for the self-causing capacity of self-awareness, what better candidate is there than self-awareness for being the self-causing cause of reality as a whole? Of course, in order to vindicate that proposal, it is not sufficient to merely point out the self-causing capacity of self-awareness; we also have to show how self-awareness produces reality as we experience it. That is: how does ASA explain this physical universe we inhabit, consisting of myriad physical objects which come and go in space and time, governed by laws of nature? The latter question, of course, is the really hard question that any version of Absolute Idealism faces.

The Absolute as ‘cosmic computer’
This is where Absolute Idealism 2.0 comes in. Inspired by ideas from the American Idealist Josiah Royce, I have argued that ASA – due to its internal recursivity, i.e. the fact that it is its own object of awareness – generates an infinite sequence (namely: self-awareness, awareness of self-awareness, awareness of awareness of self-awareness, …) isomorphic to the sequence of the natural numbers N={0, 1, 2, 3, …}.
Thus, presupposing a structuralist account of mathematics ("mathematical objects are identical insofar as they are isomorphic"), we can conclude that ASA is aware of N through the recursivity of its self-awareness. Indeed, since ASA is the self-causing cause of reality as such, we can say that the natural numbers exist only because ASA generates them. We can then construe ASA as aware of all possible computations, i.e. all computable functions from N to N (I explain this more fully here). Next, by appealing to the claim from digital physics that physical processes are thoroughly computational, we can describe the physical universe as a complex computation existing in the structure of ASA’s self-awareness. To be a bit more precise: the physical universe is that complex computation in which ASA ‘sees’ its own essence (namely, self-causation through self-awareness) best reflected. Thus ASA completes its self-awareness by mirroring itself in the computational structure of the physical universe. In this way I explain the Wheeler universe, i.e. the universe that creates itself by evolving the very observers whose acts of observation bring the universe into existence. This self-observing and self-creating universe is, in my view, the computational mirror image of ASA. Of course, Wheeler’s hypothesis is by no means yet an established scientific theory, although it is a distinct scientific possibility. Were Wheeler’s hypothesis to be borne out by experimental data, then that would also be indirect evidence for my account of ASA (I explain this more fully here).

The Problem of the One and the Many
Of course, Absolute Idealism 2.0 is as yet no more than a vague proposal. Numerous problems have to be solved and explanatory gaps have to be filled before it can be called a properly scientific theory. One of those problems, however, is of a distinctly philosophical nature, namely, the ancient Problem of the One and the Many. As already noted, this problem arises for any strongly monist metaphysics, i.e. any theory that recognizes just one single “substance” or entity as the explanatory ground of reality as a whole. For then the following question arises: How does this single entity, this ‘One’, produce ‘the Many’, the multitude of finite physical objects, coming and going in space and time? The problem follows especially from the fact that this single primordial entity must be ontologically self-sufficient (since there is by definition nothing outside or prior to it), indeed it must be the cause of its own existence. But to say that it is
ontologically self-sufficient is to say that it doesn’t need anything beyond itself. So why then did it produce anything beyond itself? Qua self-causing, it causes just itself, and nothing more. It would seem, then, that just from this primordial 'One' we cannot explain how 'the Many' came into existence. In other words, we appear to have a dilemma: to solve Leibniz's question we need a self-causing being, but precisely its self-causing capacity creates the problem of the One and the Many.

The absolute simplicity of the Absolute
But is the problem of the One and the Many really a problem? Why can't the self-causing cause of reality be Many right from the start, realizing itself in multiple self-causations? If self-causation can happen once, then why not a second time, and a third, and a fourth ...? Indeed, why not infinitely many times? This would give us a monadological ontology, where reality is multiple right from the start, consisting of infinitely many centers of self-causation. However, as attractive as this solution might seem, it won't work, for the following reason. Multiplicity requires differences and therefore a medium in which these differences are realized. And this medium is something. Thus the medium needs to be explained too, ultimately by the self-causing cause of reality, which therefore must be ontologically prior to any such medium. The self-causing cause of reality, therefore, cannot be multiple. It must be essentially one and utterly undifferentiated.

Let’s take a closer look at the above argument. First take the claim that multiplicity requires differences. This is obvious. If things do not differ in any way, how then can they be distinguished? And if they can’t be distinguished, how can they be multiple? Here Leibniz’s Principle of the Identity of Indiscernibles kicks in: if things differ in no respect from each other, they are one and the same (i.e. numerically identical). Now consider the second claim: differences require a medium to be realized in. A simple thought experiment shows that we must take space and time to be the most fundamental media of multiplicity. In other words: without space and time multiplicity would strictly speaking be inconceivable. Imagine all objects as devoid of distinguishing properties. Then our last resort, before having to conceive of these objects as one and the same (because of the Identity of Indiscernibles), is to think of them as separated in time or space. Take, for example, two identical geometrical structures, e.g. two spheres. Since they share all their properties, what keeps us from concluding that they are actually one and the same object? There are only three possibilities: either we locate the two spheres in two different positions in space, or we locate them at the same spatial position but at different points in time, or we locate them at different points in both space and time. So the existence of time or space is the minimal presupposition we need to conceive of multiplicity. But time and space are something, thus they already presuppose the self-causing cause of reality. Therefore, since the self-causing cause is ontologically prior to spacetime, it cannot possibly be conceived as multiple. It simply makes no sense to say that self-causation can 'happen more than once'. Thus the Problem of the One and the Many is genuine.

The self-causation of self-awareness
So how can we solve this problem in terms of Absolute Idealism 2.0? In order to get a foothold on this issue, let's retrace the argument for the self-causing capacity of self-awareness and see what this implies for the nature of the Absolute Self-Awareness (ASA) that supposedly underlies reality as a whole. First notice that
to be truly self-aware, it is not enough that you are aware of your empirical properties, e.g. what your name is, what your body looks like, where you live, what you are doing right now, etc. You must also be aware of the fact that you are self-aware. In other words: self-awareness must itself be one of the objects of which it is aware. This follows from the essence of self-awareness, since "a self-awareness unaware of itself" is clearly a contradiction in terms. Self-awareness must therefore have a circular structure: it must include self-awareness of self-awareness. This circularity of self-awareness fits the circularity of self-causation: just as the self-causing cause is its own effect, so self-awareness is its own object of awareness. Self-awareness, after all, cannot exist without being aware of itself. This circularity is therefore a necessary condition of self-awareness. And, clearly, it is also a sufficient condition, since if there is an awareness that is its own object of awareness, then that awareness ipso facto amounts to self-awareness, however empty it may otherwise be. Thus the essential circularity of self-awareness implies its self-causing capacity, since it is both a necessary and a sufficient condition of its own existence. In a slogan we can say that the self-realization of reality (in the sense of self-creation, self-causation) is its self-realization (in the sense of self-awareness).

Clearly, then, this ASA is not individual human self-awareness, as it differs among individuals. None of us has brought him- or herself (let alone the entire universe) into existence. As empirical individuals we are biologically conditioned, brought into existence by others, subject to time. The experience of our own self-awareness may give us empirical access to the self-causation that can answer Leibniz's question, but to make full sense of this answer we have to generalize beyond ourselves. We have to project self-awareness to something that transcends us, "the Absolute", the unconditioned 'thing' that conditions all of reality.

The emptiness of the self-mirroring mirror
That form of self-awareness, where there is only self-awareness of self-awareness and of nothing else, I call "pure self-awareness". As can be seen from the above, ASA is such pure self-awareness, because qua self-causing cause it is nothing but circular awareness, i.e. awareness that only has itself as object of awareness. It is aware of nothing but its own awareness. But note that this seems to make ASA strangely empty, an awareness of nothing in particular. We can compare pure self-awareness to an empty mirror that we somehow bend around so that it mirrors itself. A photon trapped inside this self-mirroring mirror would then bounce back and forth endlessly between its two sides. In this sense the mirror can be said to reflect its own image infinitely many times. But at the same time the reflected image remains the image of an empty mirror. And the mirror image of an empty mirror is also empty... Pure self-awareness is like that: although it is aware of itself, it is aware of nothing in particular. ASA, we can say, is an 'empty Absolute'.

This emptiness of ASA, its awareness of nothing, is reinforced by the fact that in ASA the subject is the object of which it is aware. Thus in ASA there is no subject-object opposition, no difference between the awareness and what it is aware of. And since there is as yet nothing besides ASA (as it constitutes the self-causing cause of reality), there really are no differences marking ASA at all. This also follows from that fact (noted above) that differences require a medium to be expressed in, together with the fact that ASA – qua self-causing cause of all reality – must be ontologically prior to any such medium. Thus, as said, ASA must be utterly undifferentiated, and its awareness of itself must be an awareness of nothing in particular.

The immediacy of Absolute Self-Awareness
This lack of difference between subject and object in ASA is also reinforced by the fact that the self-awareness of ASA must be attained immediately; it cannot in any way be mediated. This follows from reflection on the possibility of self-causation. It is clear that the self-causation, which timelessly ‘kick-starts’ reality, cannot in any way be mediated; it must take place immediately, ‘in one fell swoop’, or it doesn’t take place at all. Suppose, a contrario, that a hypothetical self-causing cause C first has to effectuate a mediating cause C’ which only then produces C itself. In that case self-causation would clearly be impossible. C only has causal power when it exists, but it exists only as soon as it has caused itself. This means that it can’t cause C’ prior to causing itself. Therefore C’ can’t be causally prior to the effectuation of C. Therefore self-causation is only possible at once: the self-causing cause must immediately be its own effect. Thus ASA must be an immediate self-awareness, such that the awareness immediately is its own object of awareness.

Contra Hegel
This immediacy of ASA distinguishes the present approach from the dialectical version of Absolute Idealism as it can be found in Fichte and above all Hegel, where the Absolute’s self-awareness is essentially mediated by otherness. In such dialectical Idealism, the Absolute Self comes to know itself only through contrast with the Not-Self, such that the Absolute Self must first ‘posit’ the Not-Self before it can ‘posit’ itself as a determinate Self (via the negation of the Not-Self, the “negation of the negation”). But, as the above makes clear, this mediation destroys the Absolute’s self-causing capacity. Especially in Hegel, therefore, the existence of the Absolute – and thereby the existence of reality as such – remains an unwarranted assumption, since Hegel cannot explain it by recourse to self-causation. Stressing the essential mediation of the Absolute by otherness, Hegel writes in the Phenomenology: "Of the Absolute it must be said that it is essentially a result, that only in the end is it what it truly is [...]." (Hegel 1977: 11) But the Absolute, as that which explains all of reality, cannot be "essentially result", because result of what? By definition nothing can precede the Absolute. So the cause producing the Absolute as result can only be the Absolute itself, such that the Absolute is both beginning and result at the same time. And, moreover, nothing can mediate that transition from beginning to result, because the Absolute must already be the Absolute right from the start, i.e. it must immediately be its own result. Thus, to repeat, the self-causation of the Absolute must be immediate; it must take place at one fell swoop. In denying this, Hegel takes for granted the existence of the thought process that leads up to the Absolute as its conclusion (and, indeed, for Hegel, the Absolute is this thought process coming to self-understanding). In Hegel, therefore, the existence of thought is presupposed; its existence is not explained by the Absolute, rather the Absolute is explained by it. Thus Hegel fails to solve Leibniz's question why there exists anything at all, simply because Hegel presupposes the existence of thought. As Edward Halper notes vis-á-vis Hegel, "thought itself needs to be accounted for as much as anything else":

"On the one hand, the scope and power of his idealism is truly impressive. On the other, the recognition that his entire philosophy is a determination of thought raises exactly the sort of question that the traditional ultimate cause is supposed to resolve: why is there any thought at all? The comprehensive structure of the Hegelian categories, that is, their self-contained development that fulfills and attains itself – in short, all that makes the Hegelian system so attractive – makes the question of its ultimate origin all the more pressing." (Halper 2011: 184, 183)

Self-intuition in Plotinus and Schelling
That ultimate origin, therefore, must be an immediate self-awareness, where the subject of awareness immediately is its own object. To stress our difference from Hegel, we should note that such immediate self-awareness cannot be mediated by concepts, let alone by inferences. A concept, after all, has its meaning only in relation to other concepts (as Hegel stressed with his Spinozist dictum “omnis determinatio est negatio”). Thus if the Absolute were a conceptual self-awareness, it would presuppose (i.e. be mediated by) a system of concepts, which would destroy its self-causing capacity. Using the traditional distinction between concept and intuition (i.e. sensation), we can put this by saying that ASA, rather than being conceptual self-awareness, must be a form of self-intuition, such that the intuition and the intuited are immediately one and the same. Rather than to Hegel, therefore, we should turn to Plotinus and Schelling (at the time of his Identity System), given the emphatic stress they both put on the immediacy of ASA. For both Plotinus and Schelling, the Absolute (what Plotinus called “the One”) is essentially an immediate self-intuition.

Plotinus states the immediate unity of subject and object in the One quite explicitly, calling the One " a kind of immediate self-intuition" (Enneads, V.4.2.18): "It will have only a kind of simple intuition directed to itself. But since It is in no way distant or different from Itself, what can this intuitive regard of Itself be other than Itself?" (Enneads, VI.7.38-39)
To stress this immediacy Plotinus compares the One to a "self-touching" rather than explicit self-knowledge (cf. Enneads, V.3.10.40-43). It is in part from this utter lack of difference between subject and object in the One that Plotinus concludes the absolute simplicity of the One, its utter lack of internal differentiation. For Plotinus, therefore, the One is in a sense completely indeterminate, an 'empty Absolute' as we said earlier. More or less the same thought can be found in Schelling's Identity System, where the Absolute is thought as "Absolute Identity" or the "Indifference Point" where subject and object coincide. This agreement between Schelling and Plotinus is of course no coincidence, given the influence Plotinus exerted on the development of Schelling’s thought.

Again: The Problem of the One and the Many – Hegel’s revenge?
But it might now seem that we have gotten ourselves in a highly problematic situation. If the Absolute is an immediate unity of subject and object, without any internal differentiation, an awareness therefore of nothing in particular, how then can we possibly hope to solve the problem of the One and the Many? How could such an 'empty Absolute' generate differences and complexity? Isn't this precisely the situation Hegel (1977: 9) warns us against when he derides Schelling's Absolute Identity as "the night in which [...] all cows are black"? For Hegel, such an indeterminate Absolute can only lead to "acosmism", i.e. a denial of the full reality of the physical universe as compared to the thoroughgoing unity and simplicity of the Absolute. It is precisely to avoid such acosmism that Hegel denies the immediacy of ASA, seeing the essential mediation of the Absolute by otherness as the solution to the One-Many Problem. The fact that Hegel thereby loses the self-causing capacity of the Absolute is something that he takes in stride: it's the price he pays for having that solution. But is Hegel right in this? Is he correct in thinking that we face a dilemma here: either the immediacy of the Absolute required for self-causation but then no multiplicity, or the mediation of the One by otherness required for multiplicity but then no self-causation? In my view, this is a false dilemma: the immediacy of ASA, which implies its utter simplicity, does not exclude the fact that ASA is at the same time infinitely complex. Indeed, as I will argue in the following, utter simplicity and infinite complexity coincide in the Absolute qua pure self-awareness.

The coincidence of simplicity and complexity in the Absolute
In fact, the coincidence of utter simplicity and infinite complexity in ASA was already suggested by the metaphor we used to clarify the nature of ASA: the self-mirroring mirror. Although an initially empty mirror mirroring just itself remains empty, it can also be said to reflect its own image infinitely many times, as a photon trapped inside it would bounce back and forth endlessly between its two sides. Thus the self-mirroring mirror contains both nothing and infinity. ASA is like that: both utterly undifferentiated and infinitely complex. Setting aside this metaphor, we can see how this coincidence of emptiness and infinite complexity in ASA follows from the recursivity inherent in its nature as pure self-awareness. The latter, to repeat, is an awareness that is its own object of awareness, i.e. an awareness of awareness. But the awareness of which it is aware already is pure self-awareness. Thus ASA is not just awareness of awareness, but also awareness of awareness of awareness, and awareness of awareness of awareness of awareness, and so on ad infinitum. Thus, although in one sense ASA is aware of nothing in particular, in another sense it is aware of an infinity multiplicity. We can understand this recursivity of pure self-awareness more formally as follows: if we describe awareness-of-something as a function f(x)=y where f given input x produces awareness-of-x as output y, then pure self-awareness (since it is its own object of awareness) becomes the recursive function f(x)=x which generates the infinite sequence x=f(x)=f(f(x))=f(f(f(x)))=f(f(f(f(x))))= … So, although in a sense empty, ASA at the same time contains a recursively generated infinity. This solves the problem of the One and the Many.

Hyperset theory and the structure of Absolute Self-Awareness
Earlier I noted that the relation between (absolute) self-awareness and mathematics is a two-way street: not only can we base mathematics – at least the existence of N and of computation in terms of functions from N to N – on ASA, but we can also use mathematics to clarify the nature of ASA. The latter holds in particular for the coincidence of utter simplicity and infinite complexity in ASA, which we can clarify by means of the theory of hypersets (i.e. nonwellfounded sets). In contrast to standard Zermelo-Fraenkel set theory, hyperset theory allows self-membered sets, i.e. sets that contain themselves as elements. Thus in hyperset theory, as developed by Aczel (1988), we can have a set Ω={Ω} (i.e. a set Ω which has only one member, Ω itself). As Aczel (1988: 6-7) points out, this set can be represented in different ways (in fact, infinitely many different ways): on the one hand, we can represent Ω with utter simplicity as Ω={Ω} (as we did above), but we can also represent it by means of an infinite recursive unfolding as Ω={{{…}}}. This infinite unfolding, which of course follows from the fact that Ω is self-membered, is clearly the reason why Aczel calls this set "Ω" ("big omega"), which is a standard symbol for infinity in mathematics. But, as noted, Ω can be represented in infinitely many other (finite) ways as well, namely: Ω={{Ω}}, Ω={{{Ω}}}, Ω={{{{Ω}}}}, etc. Clearly, in all these cases we are not talking about different sets; they are merely different ways of representing (describing) one and the same object, the set Ω. When we represent Ω as Ω={{{...}}}, Ω still has only one member, despite its infinite description; it is just that this one member in turn contains one member (namely, itself), which in turn contains one member (namely, itself), and so on. The descriptions "Ω={Ω}" and "Ω={{{...}}}" are therefore co-extensive, despite their striking difference in complexity: whereas "Ω={Ω}" is utterly simple, "Ω={{{...}}}" contains an infinite multiplicity. Since both descriptions are just as true of Ω, we can say that Ω is both utterly simple and infinitely complex. These diametrically opposed properties therefore coincide in Ω.

The relevance of all this to our discussion of pure self-awareness follows from the fact that we can use set theory to describe the logical structure of consciousness-of-something. Just as a set contains its members, so a state of consciousness can be said to contain the object(s) – or the representation(s) thereof – of which it is conscious. Thus, for example, when I see a dog, my perceptual consciousness at that moment can be described as a set S such that S={dog}. The philosopher of mind / cognitive scientist Kenneth Williford (2006) uses the logical resources of set theory to great effect in order to analyze the logical structure of complex states of consciousness-of-something as these develop over time. The details of his account of consciousness are not relevant now, apart from the fact that Williford pays special attention to the self-referentiality of consciousness-of-something. This self-referentiality refers to the fact if I am aware of something (e.g. I see a dog), then I am also aware that I have this awareness (i.e. I am aware that I see a dog). Thus the self-referentiality of consciousness-of-something means that the latter always involves self-awareness as well. Therefore, as Williford (idem: 127-128) points out, if we use set theory to analyze the structure of consciousness-of-something, we have to use hyperset theory to capture this self-referentiality. That is: if we describe my perceptual awareness of a dog as the set S={dog}, then to capture the self-referentiality of this awareness we have to turn S into a hyperset, namely, S={dog, S}. The object of the awareness described as S={dog, S} is not just the dog but also this awareness itself, such that my awareness of the dog is also aware of itself. As Williford points out, the hyperset description S={dog, S} can be seen to imply infinite complexity, since it can be rewritten as S={dog, {dog, S}}, S={dog, {dog, {dog, S}}}, S={dog, {dog, {dog, {dog, S}}}}, and so on.

In this way, the claim that every state of consciousness-of-something involves self-referentiality can be seen to imply an infinite regress, and this has been taken as an argument against that claim, since the regress seems to imply that even such a simple state of consciousness as seeing a dog would be infinitely complex. Given the fact that human brains are finite objects, capable of only finitely many different brain states, the infinite complexity implied by self-referentiality would seem to be ruled out by neural physiology (a classic statement of this type of argument against self-referentiality can be found in Ryle 1990 [1949]: 156). Williford, however, removes the sting from this argument by repeating Aczel's point that although a hyperset like S={dog, S} can be seen as having an infinite recursive unfolding, the set itself remains essentially a finite object with only two elements, namely the dog and S itself. As such, according to Williford (idem: 130-1331), the logical structure depicted by a hyperset like S={dog, S} is perfectly capable of being represented by a finite object such as the human brain. Be that as it may. For us the infinite complexity seemingly implied by self-awareness poses no problem at all, because – in our construction – self-awareness belongs ultimately not to finite empirical objects, such as human brains, but to ASA, which is infinite. Surely ASA, the self-causing cause of reality as a whole, can enter into infinitely many different states if anything can! But at the same time, as hyperset theory shows, despite this infinite complexity, ASA remains utterly simple. For as we have seen, ASA is a form of pure self-awareness, i.e. an awareness which has no other object than itself. Thus, using hyperset theory, we have to describe ASA as the hyperset S={S} – which, indeed, is just Aczel's set Ω. So what we have said above about Ω, that it is both utterly simple and infinitely complex at once, holds for ASA as well. These diametrically opposed properties coincide in the Absolute.

The mental nature of sets
In order to properly understand the relation between ASA and mathematics, it is important to note that the usefulness of set theory for understanding the logical structure of consciousness-of-something is not just a happy coincidence. For where does our understanding of sets derive from if not from the structure of consciousness? A set, after all, is commonly defined as collecting its elements into one whole. But where do we find this collection of elements into a unity if not in consciousness? After all, as Kant stressed, consciousness is essentially unifying (“synthesizing”): no matter how many and diverse the contents of a state of consciousness are, these contents are experienced as one whole by the experiencing subject, as making up a single state of its consciousness. No wonder that we can use set theory to describe the structure of consciousness-of-something!

Indeed, it seems clear, at least to me, that there is no other source of set-formation, i.e. of the collecting of elements into a unity, than this synthesizing capacity of consciousness. Remember that we are not just talking about the unity that characterizes physical objects, say, a molecule which in a sense unifies diverse atoms. Such unification can, perhaps, be accounted for in terms of physical properties and natural forces alone. But, obviously, the process of set-formation is much, much more encompassing: a set can collect into a unity any number of the most diverse elements, including elements that do not form a single object. Thus e.g. we can have the set {2, redness, my dentist}. The elements in this set obviously have nothing to do with each other and do not naturally form a unity (let alone a physical unity). So where does their unification into one set come from if not from consciousness, respectively from my thinking about 2, redness and my dentist in one single act of consciousness?

Contemporary set theory seeks to downplay the problem of what sets are by taking an extensional view of sets, where sets are defined by their elements. But even given such extensionalism, the fact remains that a set is something over and above its elements (cf. Potter 2004: 22). If a set were nothing but its elements, then we would have to say that the empty set, {}, is nothing because it contains no elements. But clearly the empty set is not nothing, it is a set (cf. Gardner 1977: 15). Enderton (1977: 3) makes the same point when he remarks that “a man with an empty container is better off than a man with nothing – at least he has the container”. Indeed, the assumption that the empty set is nothing (because it contains nothing) can be seen to lead to a contradiction, for if the empty set were nothing, then the set containing the empty set, {{}}, would contain nothing and would itself be the empty set, such that {{}}={}, which is of course absurd. So, even if the identity criteria of a set lie wholly in its elements (= extensionalism), the fact remains that the set is not identical to its elements: it is something over and above them, indeed it is what collects them into a single whole. Clearly, as we have already noted, the elements do not collect themselves into a whole. So what then is a set in contradistinction to its elements?

It is a well-known fact that modern set theory does not have a clear answer to this question. Thus Hallett (1984: 37-38), who speaks of "the mystery of the 'oneness' of sets", points out that in modern set theory "the concept of set is taken as primitive and is left unexplained". Are we to suppose, if we are mathematical platonists, that sets are simply primitive entities residing up there in ‘Plato’s heaven’? But aren’t we then simply acquiescing with a brute fact, leaving the nature of sets unexplained? This is a very unsatisfactory state of affairs, especially given the foundational role of set theory in mathematics. Set theory is often seen as the foundation of mathematics, in the sense that the bulk of known mathematical objects can be understood as set-theoretic constructions. Thus the inscrutability of sets has wide-ranging consequences. If we don’t really know what sets are, then – given set-theoretic reductionism – we really have no idea what mathematical objects in general are. Indeed, as Russell (2004: 58) said: “mathematics may be defined as the subject in which we never know what we are talking about”.

But doesn’t the intimate connection between sets and consciousness offer a way out of this problem? For why can’t we say that a set simply is the unity created by a consciousness-of-something, i.e. the synthetic unity that comprises the contents of that consciousness into one whole? Of course, such a “psychologistic” or “intuitionist” approach to set theory has been rejected because it would undermine the objectivity and infinite complexity of mathematical objects (after all, which human subject can really collect an infinite multiplicity into one state of consciousness?). But the threat of psychologism falls away when we take an Absolute-Idealist standpoint and define sets as the unities inherent in ASA. Since ASA is the ultimate cause of reality as a whole, sets (as the unities inherent in ASA’s unfolding consciousness) would have an objective and transsubjective status. And, as noted above, infinitely complex conscious states (and therefore sets with infinitely many elements) would surely be no problem for the Absolute.

Note, by the way, that this strategy of seeing sets (and mathematical objects in general) as constructions of an absolute mind, is not completely unheard of in mathematics. As Potter (2004: 38) notes, one strategy that seems to be implicit in the thinking of many mathematicians is to regard platonism as a limiting case of intuitionism: it is, roughly, what intuitionism would become if we removed all the constraints on the creating subject. An account of mathematics along these lines can be found, for example, in Wang (1974: 81-90). In that light, an Absolute-Idealist philosophy of mathematics certainly fits in with pre-existing tendencies in the philosophy of mathematics. I venture that such an Absolute-Idealist approach is the only one we have to make sense of what sets are. Moreover, given set-theoretic foundationalism, the nature of mathematics as such would be greatly elucidated by such an Absolute-Idealist approach to sets. Mathematics would then become a structure inside ASA. I will say more about this in the concluding remarks below.

The ontological priority of simplicity
Let’s continue with our discussion of the coincidence of utter simplicity and infinite complexity in ASA. Here it is important to note that these two aspects of its being are not on a par: its simplicity is ontologically primary, its complexity secondary. This follows from the fact, already noted, that self-causation must be immediate and that ASA must therefore be immediate self-awareness. Thus the self-awareness of ASA cannot in any way depend on the infinite sequence generated by its internal recursivity (which we formalized as x=f(x)=f(f(x))=f(f(f(x)))=f(f(f(f(x))))= … ). For then its self-awareness would be mediated (indeed, infinitely mediated) by prior conditions, such that before being fully self-aware it would first have to be aware of its self-awareness, and aware of that awareness of self-awareness, and aware of that awareness of awareness of self-awareness, and so on. In other ways: if that infinite sequence belonged to its essence, ASA would have to embark on an infinite regress in order to attain self-awareness – which simply means that it would never attain self-awareness at all.

Even worse, such mediation (indeed, any mediation) would make its self-causation impossible right from the start. Therefore, qua self-causing, ASA must first of all be immediate self-awareness (and as such utterly simple and awareness of nothing) and only secondly will it be infinitely complex. That is: only as a result of the recursivity inherent in its immediate and pure self-awareness (i.e. awareness that immediately is its own object of awareness) is it also aware of its self-awareness, aware of that awareness of self-awareness, and so on. Thus ASA's infinite complexity follows from its immediate self-awareness. This infinite complexity is therefore ASA's first product beyond itself, i.e. the first entity apart from its essence. This means that, although in a sense part of its being, this infinite complexity is experienced by ASA as something external to it, i.e. as something apart from the self-experience that constitutes its immediate self-awareness. In other words: this infinite complexity is ASA's first object of experience as different from itself qua subject of experience. Herein lies the ontological origin of the subject-object opposition.

Is the recursive structure of ASA well-ordered?
This asymmetry between the simplicity and complexity of ASA is also important for another reason, namely, for a proper understanding of how ASA produces the set of natural numbers, N. We have already seen the basic idea: the sequence x, f(x), f(f(x)), f(f(f(x))), …, generated by the recursivity of ASA, is isomorphic to the sequence of the natural numbers, i.e. the sequence 0, S(0), S(S(0)), S(S(S(0))), …, generated by the recursive successor function S(n)=n+1. Thus, given a structuralist view of mathematics, we can conclude that ASA is aware of N through the recursivity of its self-awareness. If we take ASA to be the self-causing cause of reality, we can then say that the natural numbers exist because ASA generates them. However, this is not yet the full story. The natural numbers form what is called a well-ordered set in that it contains a first or “least” element (usually taken to be 0) from which all other elements can be constructed through iterated application of the successor function (which establishes a well-ordering relation among the elements of N). So if the sequence generated by ASA is to be truly isomorphic to N, we must show this sequence to be well-ordered as well, which means that it must have a least element.

Here the hyperset approach to ASA cannot help us: hypersets cannot be well-ordered because they have no least element (more precisely: hypersets are non-wellfounded and wellfoundedness is a precondition for any well-ordering). Thus a hyperset like Ω={Ω}, which we considered above, clearly has no least element, since its recursive unfolding never ‘bottoms out’ – which was precisely the reason why we could rewrite “Ω={Ω}” as “Ω={{{…}}}”. Since we used Ω to elucidate the structure of ASA, this might seem to pose a problem for our claim that ASA generates N. However, this is where the noted asymmetry between the simplicity and complexity of ASA comes to our rescue. We saw that ASA, qua self-causing cause of reality, must first of all be immediate self-awareness; only secondarily does its infinite complexity arise. That is, only as a consequence of the prior presence of immediate self-awareness is the recursively generated structure of ASA generated (i.e. the sequence: awareness of self-awareness, awareness of awareness of self-awareness, awareness of awareness of awareness of self-awareness, and so on). ASA’s immediate self-awareness, then, is the least element we need for its recursive structure to be well-ordered. Hence, that structure is isomorphic to N: the immediate self-awareness, which is the essence of ASA, fulfills the same role that 0 plays in N.

This structural similarity between 0 and immediate self-awareness is all the greater given the fact, noted above, that immediate self-awareness is in a sense empty, an awareness of nothing in particular. If we take into account the mental nature of sets as the unities effectuated by consciousness-of-something, we can then quite literally reconstruct 0 as the set {} resulting from the unification process inherent in immediate self-awareness. Since all consciousness-of-something unifies the contents of consciousness into a whole, immediate self-awareness must do the same, only this time an empty whole is formed, given the fact that immediate self-awareness is awareness of nothing in particular. In that sense, immediate self-awareness unifies nothing – which doesn’t mean that it doesn’t unify but that it unifies an empty state of awareness. From an Absolute-Idealist standpoint, therefore, where sets are understood as the unifications effectuated by ASA, the empty set is simply ASA’s most primitive state, its immediate self-awareness qua awareness of nothing. At the same time, however, this empty set is also a hyperset in that it recursively includes itself as its sole element, so that we get the sequence {}, {{}}, {{{}}}, … , which is Zermelo’s set-theoretic reconstruction of the natural numbers. In a sense, then, we can say that the empty set, understood as ASA in its most simple state (where it is awareness of nothing in particular), is also a self-bracketing set in that it recursively places itself inside its own brackets.

The Self-Bracketing Set: An Absolute-Idealist philosophy of mathematics as a whole?
I want to finish this post with some general considerations and open questions concerning this notion: the self-bracketing set. This notion, as we have seen, results when we view ASA under the aspect of unification inherent in any form of consciousness-of-something. Thus, if we remember that ASA is a form of pure self-awareness (i.e. awareness that is its own object of awareness), it then becomes clear that we can describe ASA as a self-unifying unity, in other words: a self-bracketing set. We have already seen how this gives us N, the set of the natural numbers, and also the concept of computation qua functions from N to N. But the question now becomes: can we do more with this conception of ASA as the self-bracketing set? Can we, in fact, use this idea to develop a philosophical foundation for mathematics as a whole, instead of just for N and the concept of computation? After all, sets are foundational for mathematics; according to many, the whole of mathematics can be derived from the axioms of Zermelo-Fraenkel (ZF) set theory. So if sets are ontologically grounded in the unifying aspect of ASA (as I have argued above), then it would seem logical to say that mathematics as a whole is ontologically grounded in ASA. That is to say: ASA, by being the original source of set formation, would also be the origin of mathematics as a whole. This, of course, would require that we show how the ZF axioms follow from the nature of ASA – an admittedly very tall order. I hope to be able to do this somewhere in the future. For now I will limit myself to the power set axiom, which is of central importance in ZF because it is the main operation through which the universe V of pure sets is built up from the empty set onwards (and the whole of mathematics supposedly fits in V). So does what we have said so far about ASA allow us to derive the power set operation (which collects into one set all the subsets from a previously given set)? Interestingly enough, yes! For the power set operation is implicit in the move from N to all functions from N to N: the cardinality of the set of all these functions is precisely the cardinality of R (the set of real numbers), which is equinumerous with the power set of N (i.e. Pow(N)). If, as I have argued earlier, ASA is through its inner recursivity aware of N, then ASA is also aware of all functions from N to N, which means that ASA performs the power set operation on N. Since this is such an important result, I will rehearse my argument here:

As we have seen, each consecutive level in the recursively generated sequence of ASA’s self-reflection, generated by the function f(x)=x, corresponds to a natural number, such that x=0, f(x)=1, f(f(x))=2, f(f(f(x)))=3... etc. Since ASA knows itself as identical with itself on each such level – because x=f(x)=f(f(x))=f(f(f(x)))= ... – this self-knowledge amounts to a knowledge of equality relations between the natural numbers. For example, ASA knows that its identity on reflection levels 4, 7 and 15 is the same as its identity on level 2 – and this amounts to the equality relation (4, 7, 15)=(2). But such a relation is just a mapping from N to N. Hence, by being aware of its self-identity on all the levels of its self-reflection, ASA is aware of all functions from N to N, including all computable functions. Therefore ASA is not only aware of N but also of Pow(N) since Pow(N) is the cardinality of the set of all functions from N to N.

It would seem, then, that we already have one centrally important ingredient required for an Absolute-Idealist account of mathematics as a whole, namely, the power set operation. However, in order to get at V, we must iterate the power set operation from {} onwards. But as we have just seen, the power set operation implicit in ASA applies directly to N instead of beginning with {}. Does this mean that we can't build up V in terms of ASA? Or is it the case that when we analyze the functioning of the power set operation in ASA more closely, we will find that that operation can be seen to start with {} after all? For now, however, I don't have any answers to these questions. I hope to be able to figure this out – and with more formal rigor than was hitherto possible for me – in the future.  

-Aczel, Peter (1988), Non-well-founded sets. CSLI: Stanford.
-Enderton, Herbert B. (1977), Elements of Set Theory. New York: Academic Press.
-Gardner, Martin (1977), Mathematical Magic Show. London: Penguin.
-Hallett, M. (1984), Cantorian Set Theory and Limitation of Size. Clarendon Press: Oxford.

-Halper, E.C. (2011), "The Ultimate Why Question: The Hegelian Option", in: J.F. Wippel (ed.), The Ultimate Why Question: Why Is There Anything at All Rather than Nothing Whatsoever? The Catholic University of America Press: Washington, D.C.
-Hegel, G.W.F. (1969), Science of Logic, trans. A.V. Miller. George Allen and Unwin: New York.
-Hegel, G.W.F. (1977), Phenomenology of Spirit, trans. A.V. Miller. Oxford University Press: Oxford.  
-Potter, M. (2004), Set Theory and its Philosophy: A Critical Introduction. Oxford University Press: Oxford.
-Russell, B. (2004), Mysticism and Logic. Dover: Mineola, New York.
-Ryle, Gilbert (1990 [1949]), The Concept of Mind. Penguin Books: London.
-Wang, H. (1974), From Mathematics to Philosophy. Routledge & Kegan Paul: London.
-Williford, Kenneth (2006), "The Self-Representational Structure of Consciousness," in: U. Kriegel & K. Williford (eds.), Self-Representational Approaches to Consciousness. MIT Press: Cambridge Mass.

No comments:

Post a Comment