The Ultimate Insight: On the Explanatory Power of Absolute Self-Awareness

A recurrent theme on this blog is the idea of Absolute Idealism 2.0, i.e. a contemporary, ‘mathematized’ version of the age-old philosophy of Absolute Idealism, which runs from the ancient Upanishads to the Neoplatonist philosopher Plotinus to German and Anglo-American Idealists such as Schelling, Hegel, Green and Royce. Absolute Idealism 2.0 takes over their central insights but develops them in a novel way consistent with modern science, in particular with the central role of mathematics in physics. The following post gives a broad overview of the central principles of Absolute Idealism 2.0. and how these principles enable us to make sense of reality-as-a-whole.
The self-creating power of Absolute Self-Awareness We can provisionally define Absolute Idealism as the philosophical theory that everything exists because it is thought and/or experienced by an Absolute Mind, which in turn exists because It thinks/experiences itself. Thus, on the Absolute-Idealist view, the Absolute Mind constitutes its own existence by thinking/experiencing itself, that is, by being self-aware. The Absolute Mind, then, should first and foremost be defined as Absolute Self-Awareness. The idea that (pre-reflective) self-awareness has a self-creating aspect is most often associated with the German Idealist Fichte and his case for the “self-positing of the I”, but roughly the same idea can be found with other Absolute Idealists as well:

• The Vedantic sages of the Upanishads: “In the beginning this world was only Brahman, and it knew only itself (Atman), thinking: ‘I am Brahman.’ As a result, it became the Whole.” (Brihadaranyaka Upanishad 1.4.10) 

• The Neoplatonic philosopher Plotinus: “The One [...] made itself by an act of looking at itself. This act of looking at itself is [...] its being.” (Ennead VI, 8, 16, 19-21)

• The German Idealist Schelling: “it is through the self's own knowledge of itself that that very self first comes into being” (Schelling 1800: 27).

• The American Idealist Royce: “if whatever exists, exists only as known, then the existence of knowledge itself must be a known existence, and can finally be known only to the final knower himself, who, like Aristotle's God, is so far defined in terms of absolute self-knowledge” (Royce 1899: 400).
   

By highlighting this self-creating aspect of Absolute Self-Awareness, Absolute Idealism is very attractive in that it offers a clear-cut and intuitively plausible answer to Leibniz’ famous question: “Why is there something rather than nothing?” According to Absolute Idealism, there is something rather than nothing because Absolute Self-Awareness is self-creating. I call this answer to Leibniz’s question intuitively plausible mainly because of two reasons: (1) we are all self-aware and so we know – with Cartesian self-evidence – that self-awareness exists, indeed it is the only existence we are absolutely certain of, and (2) we have a glimpse of the self-creating power of Absolute Self-Awareness in the (self-)awareness we have of our own free will. This last point needs some elucidation.

Kant and the “unconditioned causality of freedom
Of course, when I talk of the self-creating power of self-awareness, I am not talking of individual human self-awareness. None of us has brought him- or herself and the universe into existence. As empirical individuals we are biologically conditioned, brought into existence by others, subject to time. So how then can I say that our own self-awareness gives a glimpse into the self-creating power of Absolute Self-Awareness?

In a way we do experience a degree of self-causation in ourselves, namely, insofar as we exercise positive freedom, i.e. autonomy. Positive freedom requires a capacity for what Kant called “spontaneity”, the “unconditioned causality of freedom” (Critique of Pure Reason, B561/A533), i.e. a capacity to initiate an entirely new course of action and/or thought ‘out of nothing’, unmotivated and/or uncaused by prior givens. But isn’t such a capacity for radical spontaneity – for initiating something out of nothing – precisely what is needed in order to answer Leibniz’s question, i.e. to explain how reality has lifted itself into existence preceded by nothing?

Insofar as our self-awareness reveals in us this ‘unconditioned causality of freedom’, then, we have all the more reason to take our self-awareness as the key to answering Leibniz's question. It is, moreover, the self-positing nature of self-awareness that explains this unconditioned causality of freedom in the first place. Obviously we aren't self-causing in any absolute sense (since, to repeat, we have not created ourselves), but we are relatively self-causing in that we can at least intervene in the causal order of reality by spontaneously initiating a completely new causal chain of events.

This underscores the difference between empirical, individual self-awareness and Absolute Self-Awareness: what the former has relatively and finitely, the latter has absolutely and infinitely. That is to say: Absolute Self-Awareness has (or is) absolute freedom. The self-evident experience of our own self-awareness gives 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 prereflective self-awareness to something that transcends us, the Absolute, the unconditioned 'thing' that conditions all of reality.

The mathematical unfolding of Absolute Self-Awareness
What then is the precise relation between Absolute Self-Awareness and individual self-awareness as it is found in you and me? To answer this question we have to move from Leibniz’ question to the next question: why is the universe the way it is? We do not just want to know why something exists, we also want to know why this something is the way it is, i.e. why reality has developed into this infinitely complex universe in which we find ourselves. It is especially here that I take the age-old philosophy of Absolute Idealism into a new direction, making it fit for the future by drawing on ideas from modern physics and mathematics. It is here that Absolute Idealism becomes Absolute Idealism 2.0.

Making creative use of some seminal ideas from the American Idealist Josiah Royce, I argue that the recursivity inherent in Absolute Self-Awareness – in short: its awareness of itself, its awareness of that awareness, its awareness of the awareness of that awareness, and so on – establishes an intrinsic connection between self-awareness and the recursively generated natural numbers and even the recursively generated set-theoretical universe of pure sets, which in a way contains the whole of mathematics (more about this here and here). Thus, from this perspective, the Absolute Mind comes out as a deeply mathematical being, generating – through the recursivity of its self-awareness – all of mathematics, and subsequently mirroring itself in those mathematical structures that best reflect its transcendent splendor.

The universe as the mathematical self-image of the Absolute
In my view, the resulting mathematical mirror image of the Absolute Mind is our physical universe (which, as modern physics shows, is indeed thoroughly mathematical in nature). Through mirroring and recognizing itself in this mathematical universe, and particularly in those mathematical structures that emulate intelligence (such as the algorithmic structure of the human brain), the Absolute Mind increases its own self-awareness and thus teleologically realizes its essence. In this way I explain the apparent fine-tuning of the universe, i.e. the fact that surprisingly many of nature’s fundamental constants – such as the ratio of the masses of electrons and protons, the energy density of the vacuum, even the three-dimensionality of space – are “just right” for the evolution of life. This bio-friendliness of the universe follows from the fact that the universe is the mathematical mirror image of the Absolute Mind.

It is, moreover, the self-recognition of the Absolute Mind in mathematical structures (such as the algorithmic structure of the human brain) that infuses these structures with phenomenal consciousness: it explains why the mathematical structure of the brain is “accompanied by an experienced inner life” (Chalmers 1996: xii). In this way I aim to solve the notorious “Hard Problem of Consciousness”. Moreover, as it is the mathematical structure of the universe as a whole in which the Absolute Mind mirrors itself, we must see the entire universe as infused with phenomenal consciousness, thus arriving at a panpsychist view of the cosmos.

This, then, answers the question we raised above about the relation between individual human self-awareness and Absolute Self-Awareness. Individual self-awareness, as experienced by individual organisms, is nothing but the self-reflection of the Absolute in specific mathematical structures, notably in those algorithms that “simulate” intelligent volitional agency, algorithms such as the ones that underlie the functioning of our brains. In this sense, Absolute Self-Awareness is the pre-reflective core of every finite individual form of self-awareness. One could say that each empirical instance of individual self-awareness (human or otherwise) is, as it were, a navel in the physical universe, connecting the latter through a transcendental umbilical cord with the Absolute Self-Awareness that grounds reality as a whole.

Explaining mind-body dualism
This explanation of what individual consciousness is – namely, the self-reflection of Absolute Self-Awareness in the complex algorithm that simulates brain functioning – also allows us to make sense of the apparent duality of mind and matter and, notably, the apparent supervenience of the former on the latter. This is one of the major difficulties faced by any kind of Idealism: if matter is just an appearance in consciousness, why and how then can it seem that matter exists apart from consciousness and, indeed, that (individual) consciousness appears to depend on matter?

From the perspective of Absolute Idealism 2.0, this duality between mind and body comes down to the distinction between, on the one hand, the mathematical structure of the recursive self-unfolding of Absolute Self-Awareness, and the latter’s self-recognition in certain privileged parts of that structure on the other. The mathematical structure in which the Absolute reflects itself is the structure of matter, i.e. the structure discovered by physics. But it is the self-reflection of the Absolute in this structure, the fact that it recognizes itself in it, that – so to speak – infuses the structure with phenomenal awareness. It is this act of self-recognition that explains why the mathematical structure of matter is – as Chalmers put it – “accompanied by an experienced inner life”. This holds in particular for the structure of the brain (human or otherwise), which is the kind of mathematical structure in which the Absolute recognizes most of itself (intelligent and volitional agency); hence the infusion of this structure with individual consciousness.

Hence the duality of brain and consciousness, and the apparent dependence of the latter on the former. The brain as a physical object is simply the underlying mathematical structure as experienced from the outside by another conscious organism (i.e. another brain / individual consciousness), whereas individual consciousness is that very same mathematical structure as experienced ‘from within’, i.e. as an object for the Absolute’s self-recognition. We can call these, respectively, the first-person and the third-person experiences of the mathematical structure of the brain. The fact that individual consciousness appears to be causally dependent on the brain is due to the fact that individual consciousness is ontologically dependent on the mathematical structure in which the Absolute recognizes itself.

The funny thing here is that we, human beings, are in principle capable of both a first-person and a third-person perspective on the mathematical structures of our own brains; for example – to take a rather drastic example – when we open up our skull and use a mirror to look at our own brain. We then experience its underlying mathematical structure in two ways simultaneously: from the inside as the object of the Absolute’s self-recognition, which gives us our individual consciousness, and from the outside, i.e. from a third-person perspective, which gives us this strange lump of grey matter that is supposed to be us. Something similar happens, though less drastically, when we look at a CT scan of our own brain. 

Morality as self-recognition in the other
To repeat: Absolute Self-Awareness is the pre-reflective core of every finite individual consciousness, insofar as the latter is nothing but the self-recognition of the Absolute in the mathematical structure of the brain. Thus, as the Vedanta philosophy based on the Upanishads puts it, we are all in principle capable of discovering the same Self (Atman) as the innermost core of our individual self-awareness. This Universal Self, this core in each of us, is the Absolute Self-Awareness as it reflects itself empirically in the self-awareness of finite organisms in the universe. Thus, the “unconditioned causality of freedom” we detect in our self-awareness really is the unconditioned causality of the self-causing Absolute as the ground of all reality. In that sense the (self-)awareness we have of our own free will does give us a glimpse into the endlessly creative source of the universe.

Insofar as we are capable of this glimpse, i.e. of ‘seeing’ the Absolute as the prereflective core of our own self-awareness, we start to appreciate that the same holds for all living beings. We start to realize that all organisms are essentially nothing but different manifestations of one and the same creative essence, the Absolute, the Universal Self, which senses, thinks and acts through all these organisms. This gives an enormous feeling of connection and love for others. Suddenly you can empathize with others and take their perspective much more easily, because you know they are not fundamentally different from you. You start to experience other beings as different versions of yourself, i.e. of your innermost Self, the creative essence of the universe. It is this empathy with others, through the non-dual sense of cosmic unity, that is the foundation of all sincere morality.

In this way, a kind of self-recognition in others – a seeing of yourself in others – takes place, but the self that is recognized here is not primarily the individual self but rather the Self, the Universal Self, the Atman, the Absolute. One could say that in this way the universe as the mathematical self-image of the Absolute is all the more true to its archetype: just as the Absolute recognizes itself in the otherness of the physical universe, so the universe mirrors this Divine Self-Recognition by evolving organisms that recognize themselves (i.e. their Self) in each other. Through the evolution of this self-recognition among organisms, this “mutual recognition” as Hegel calls it, the universe evolves into an ever improving mirror of the Absolute, thereby contributing to the latter’s essence as self-awareness.       Relation to Enlightenment in Eastern spirituality As the above reference to the Vedanta indicates, this (self-)realization of the Absolute as the prereflective core of our own individual self-awareness has a deep connection to what in Eastern spirituality is known by such terms as "Enlightenment", "Awakening", "(Self-)Realization", and "Liberation". This marks an important difference between Western and Eastern forms of Absolute Idealism. Whereas the Western forms are mostly theoretically oriented, aimed at a purely intellectual understanding of reality, the point of virtually all Eastern spirituality is primarily practical, aimed at a radical existential transformation of human life. Hence the terminological distinction I draw between Western philosophy and Eastern spirituality. It is certainly not the case that philosophical theorizing is entirely absent in the East – quite the contrary, Eastern spirituality contains some of the deepest philosophical thinking ever done. It is just that in Eastern spirituality all theorizing is ultimately subordinated and subservient to the spiritual goal of Liberation: theory for the sake of theory is rejected, because it stands in the way of the spiritual goal. The notion of Absolute Self-Awareness, then, signifies in the Eastern context not just the ultimate nature of reality, it also signifies the individual’s realization of the Absolute as his / her own innermost Self and as such the final Liberation from the suffering inherent in finite human existence. In Eastern spirituality, then, Absolute Self-Awareness is first of all not a theoretical concept (as it is in Western Absolute Idealism) but an experience or intuition, the experience of Enlightenment, the awakening to or realization of one’s true nature, the intuition of the Absolute as the core of one’s being. This is the experience that accomplishes the longed for Liberation from suffering. Here the Vedanta of the Upanishads provided the original template for all later Eastern spiritual traditions aimed at Enlightenment (even if these traditions criticized certain aspects of the Vedanta). For the Upanishadic sages, the key insight “Brahman is Atman” is not just a theoretical insight into the ultimate ground of reality, it is also the liberatory insight into the ultimate core of one’s own self, the realization “I am Brahman”, freeing one from the suffering inherent in finite human life. This comes out clearly in the Brihadaranyaka Upanishad, which we quoted earlier as clearly pronouncing the basic realization of the self-causing nature of Absolute Self-Awareness: “In the beginning this world was only Brahman, and it knew only itself (Atman), thinking: ‘I am Brahman.’ As a result, it became the Whole.” (1.4.9) This remarkable passage doesn’t stop here: it goes on pronouncing with equal clarity the spiritual significance of this realization: “If a man knows ‘I am Brahman’ in this way, he becomes this whole world. Not even the gods are able to prevent it, for he becomes their very Self (Atman)... He is the one who is beyond hunger and third, sorrow and delusion, old age and death.” (1.4.10, 3.4.2) Here the spiritual significance of Absolute Self-Awareness, the Liberation from the suffering inherent in finite existence, is clearly announced. The ultimate insight that explains everything? What all this makes clear is that the Enlightenment experience has both a theoretical and a practical value, indeed it is the ultimate accomplishment both philosophically and spiritually. Enlightenment is not just the insight that liberates from the confines of finite existence, it is also the insight that provides the ultimate epistemological foundation for the Absolute-Idealist worldview. This implies an extreme form of rationalism, such that in principle everything is explainable for us, finite human beings, because insofar as we are self-conscious beings we have a prereflective intuition of the nature of Absolute Self-Awareness as the self-causing cause of reality-as-a-whole. Looked at from the theoretical perspective, Enlightenment is the insight into the essential core of our own self-awareness as the absolutely free (i.e. self-causing) source of all reality, as the recursive fountainhead of all mathematics and thus of the physical universe as our own innermost mathematical self-image, and as the source of all morality qua self-recognition in others. Enlightenment, in short, is the ultimate insight that allows us to explain everything. But, so a critic might ask, does it even make sense to attempt an explanation of ‘everything’? Isn’t such an all-encompassing notion logically incoherent? I want to finish this post by taking a closer look at this objection and how Absolute Idealism can deal with it. Leibniz’s question, paradox, and self-awareness This objection has in particular been raised by analytic philosophers such as Alfred Ayer and Bertrand Russell: they argue that all-encompassing concepts like “everything” and “reality as a whole” lead to paradoxes of self-reference, akin to the paradoxes of the early ("naive") set theory developed by Cantor, Dedekind and Frege. In my view, however, this self-reference ceases to be paradoxical once we realize that Absolute Self-Awareness is the self-causing cause of reality and that self-reference belongs to the essence of self-awareness. Naive set theory is so-called because it allowed sets that are deemed "too big", such as the universal set: the set containing all sets, including itself. Thus the universal set is self-membered, and this leads – directly or indirectly – to paradoxes, such as Cantor's paradox, the Burali-Forti paradox and Russell's paradox. One could argue that Leibniz's question produces similar paradoxes because, in a way, it totalizes existence. By posing the question "Why is there something rather than nothing?", Leibniz invites us to look at reality as a whole, the totality of what exists, in order to find the cause or reason explaining this totality. But in conceiving this totality, aren't we relapsing into the naiveté of early set theory? Aren't we allowing a set that is "too big"? This was indeed the main objection raised by logical positivism against Leibniz’s question: it is meaningless because it leads to paradoxes of self-inclusion. Thus Alfred Ayer: "Supposing you asked a question like 'Where do all things come from?' Now that's a perfectly meaningful question as regards any given event. Asking where it came from is asking for a description of some event prior to it. But if you generalize that question, it becomes meaningless. You're then asking what event is prior to all events. Clearly no event can be prior to all events. Because it's a member of the class of all events it must be included in it, and therefore can't be prior to it." (Ayer quoted in Holt 2013: 24) Bertrand Russell too noted the paradoxical self-referentiality of the philosophical concept of reality-as-a-whole: “The comprehensive class we are considering, which is to embrace everything, must embrace itself as one of its members. In other words, if there is such a thing as “everything,” then “everything” is something, and is a member of the class of “everything”.” (Russell 1919: 136) Thus Russell was suspicious of all-embracing philosophical concepts such as reality-as-a-whole: “The difficulty arises whenever we try to deal with the class of all entities absolutely [...]; but for the difficulty of such a view, one would be tempted to say that the conception of the totality of things, or of the whole universe of entities and existents, is in some way illegitimate and inherently contrary to logic.” (Russell 1903: 362) With the concept of reality-as-a-whole out of the window, however, Leibniz's question can no longer be posed. If the concept of reality-as-whole is logically incoherent, then the question why that whole exists must be illogical as well. In response to this criticism we only have to point out that the set-theoretical paradoxes are defused by the phenomenon of self-awareness. For what appears as paradoxical in the foundations of mathematics – namely, self-reference – actually is a living reality in the phenomenon of self-awareness. Why then should we reject self-reference as paradoxical, and banish it from the foundations of mathematics, when self-reference is a clearly a bona fide aspect of reality, an aspect of which the existence is attested – with Cartesian self-evidence – by the undeniable phenomenon of self-awareness? Thus it becomes clear how the Absolute-Idealist view of reality as essentially a form of self-awareness – namely, Absolute Self-Awareness – saves Leibniz’s question from Russell's criticism. If we take reality as such to be self-awareness, then the self-inclusion of the totality of what exists ceases to be paradoxical, because such self-inclusion is to be expected of self-awareness. This self-inclusion is the inherent recursivity of self-awareness, which necessarily involves awareness of self-awareness, and awareness of awareness of self-awareness, and so on without end. In other words, self-awareness must include itself as one of the objects of which it is aware. Thus we can compare self-awareness to a ‘magical matryoshka’, a Russian nesting doll that somehow contains itself: if one opens up the doll, one finds the same doll inside… In short, then, the Absolute Idealist conception of self-awareness does not just enable us to answer Leibniz's question, it also enables us to pose that question in a meaningful way. It shows that the self-inclusion of the totality of what exists – a totality presupposed by Leibniz's question – is not a senseless violation of logic, because it belongs to the living essence of reality qua Absolute Self-Awareness. References -Chalmers, D. J. (1996), The Conscious Mind: In Search of a Fundamental Theory, Oxford University Press.   -Holt, J. (2013), Why does the world exist?, Profile Books. -Kant, I. (1781/’87 [2009]), Critique of Pure Reason, Cambridge University Press. -Plotinus, Enneads, translation by A.H. Armstrong, Loeb edition. -Royce, J. (1899 [1959]), The World and The Individual, First Series: The Four Historical Conceptions of Being, Dover Publications. -Russell, B. (1903 [1964]), The Principles of Mathematics, George Allen & Unwin. -Russell, B. (1919 [1970]), Introduction to Mathematical Philosophy, George Allen and Unwin. -Schelling, F.W.J. (1800 [2001]), System of Transcendental Idealism, translated by Peter Heath, University Press of Virginia. -Upanishads, translation by Patrick Olivelle, Oxford University Press, 2008.

Some Thoughts on the Mathematical Unfolding of Absolute Self-Awareness

In various posts on this blog I have sketched the rough outlines of a contemporary version of Absolute Idealism, which I like to call – for lack of a better term – “Absolute Idealism 2.0”. The philosophical tradition of Absolute Idealism, stretching from the Upanishads in the East and Plotinus in the West to the German and British Idealists, can be summarized by the claim that everything exists because it is thought and/or experienced by an Absolute Mind, which in turn exists because it thinks/experiences itself. Thus, the Absolute Mind makes itself exist by being aware of itself, and it should as such be defined as Absolute Self-Awareness (ASA). This self-causing capacity of ASA (developed especially by Plotinus and Fichte) is in my view one of the strong features of Absolute Idealism, as it provides a possible (and, perhaps, plausible) answer Leibniz’s famous question why something exists rather than nothing.

This answer, however, is only worth anything if the concept of ASA can also explain why reality is the way it is. For we do not just want to explain the existence of reality; we also want to explain its nature. Why did reality take the form of this universe we see around us, developing in space and time, governed by physical laws? This is where Absolute Idealism 2.0 comes in. Taking its cue from modern physics, which shows the thoroughly mathematical nature of physical reality, Absolute Idealism 2.0 stresses the intimate connection between mathematics and the structure of (absolute) self-awareness. In earlier posts I already developed some ideas about this connection (see here, here and here). This post takes these ideas to a (somewhat) higher level.

I will end with some speculations about a mathematical solution to the problem of evil (the theodicy problem): given the randomness of by far the most real numbers, is it possible that the Absolute simply ‘lost itself’ in what Leibniz called the “labyrinth of the continuum”? Does this explain why the universe is not perfect, despite being the mathematical image of ASA?

ASA’s awareness of the natural numbers and real numbers
The basic idea is that ASA, due to its inner recursivity, generates an infinite sequence of reflection levels (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, …}. Presupposing a structuralist account of mathematics (such that mathematical objects are numerically identical iff they are isomorphic), we can conclude that the natural numbers exist because ASA, through its inner recursivity, thinks them. N, then, is ASA’s first creation beyond its immediate self-awareness.

This idea, that ASA through its inner recursivity generates a sequence isomorphic to N, was first put forward systematically by the American Idealist Josiah Royce, influenced by Dedekind’s notorious Gedankenwelt proof of the existence of infinity (see the “Supplementary Essay” in Royce 1959 [1899]). Anticipations of this idea, however, can already be found in the Neoplatonic philosopher Plotinus (as I explain more fully here). Virtually the same idea was later developed by the Husserlian phenomenologist and mathematician Oskar Becker, who shows in some detail how the inner unfolding of self-awareness exhibits the same principles as the ones used by Cantor in his construction of the transfinite hierarchy (see Becker 1973 [1927]).

It is sometimes objected that this infinity of levels of self-awareness is humanly impossible: we can be aware that we are self-aware, and perhaps we can also be aware of this awareness of our self-awareness, but this is where the buck stops for most of us. Russell, for example, comments as follows on Dedekind’s idea that self-awareness implies infinitely many reflection levels: “Now it is plain that this is not the case in the sense that all these ideas have actual empirical existence in people’s minds. Beyond the third or fourth stage they become mythical.” (Russell 1970 [1919]: 139)

In response to this objection, it should be remembered that we are not speaking of human self-awareness, but of absolute self-awareness (ASA) qua self-causing cause of all reality. The assumption that this ASA exists is admittedly not a matter of course, and I can see why a philosopher like Russell would reject that assumption out of hand (after all, Russell and Moore started analytic philosophy as a revolt against the Absolute Idealism of their teachers). Nevertheless, the idea that self-awareness has a self-causing capacity can be defended, and I see no other equally plausible answer to Leibniz’ question “Why does reality exist?” on the table. Once we accept the assumption that ASA is the self-causing cause of reality, then the above objection to the infinity of levels falls away. For, surely, such infinite complexity would be no problem for the Absolute, i.e. that which explains everything else? We should also keep in mind here that, since self-causation is obviously impossible in time, the ASA can only exist timelessly. So the infinite hierarchy of reflection levels cannot be conceived as a merely potential infinity, unfolding in time; it must be conceived as a timelessly existing actual infinity, accomplished ‘at once’ by the ASA, in the nunc stans of its timeless reality.

Georg Cantor (1845 - 1918)

I note here in passing that this idea of an infinite hierarchy of reflection levels inside the ASA (a hierarchy which even extends into the transfinite, as Oskar Becker argues) fits Cantor’s original vision of transfinite set theory wonderfully well. Cantor was a deeply religious man, interested in theology and metaphysics no less than in mathematics. For him, the existence of the transfinite hierarchy was guaranteed by God, in whose mind all the infinite sets exist as separate ideas. These sets, as Cantor wrote, “exist in the highest degree of reality as eternal ideas in the Intellectus Divinus” (quoted in Dauben 1979: 228). Obviously, other mathematicians generally disapprove of Cantor’s theological views (sometimes interpreting them as signs of Cantor’s mental illness, which had a touch of religious insanity). For most mathematicians, transfinite set theory can do just fine without a grounding in theological metaphysics. But from the perspective of Absolute Idealism 2.0, Cantor’s theological views are not so strange. One could even say that by arguing for the self-causation of ASA and its inherent recursivity (which generates the infinite hierarchy of reflection levels), we give a philosophical foundation to Cantor’s belief in the existence of the transfinite hierarchy in the Divine Mind.

Be that as it may, the next step is the realization that ASA, through its awareness of the natural numbers, is also aware of all possible mappings from the natural numbers to the natural numbers, i.e. ASA is aware of all total functions f:N→N. (Formally, the set of all functions from A to B is defined as B^A = { f : f є P(AXB) and f is single-valued}.) To see why, we need to keep in mind what ASA essentially is, namely, absolute self-awareness. From this it follows that on each reflection level n from N ASA is aware of its identity with itself on every reflection level m from N (with the possibility that n=m). Such an awareness of self-identity between different reflection levels n and m, then, amounts to a mapping from n to m, that is, a function f such that f(n)=m. And since, as indicated, this holds for all n and m from N, it follows that ASA ‘performs’ or ‘executes’ all total f:N→N. (When I speak of functions in the following, I always mean total functions as opposed to partial functions; for the distinction see here.)
 
Now, the set of all f:N→N is basically the set of all (positive) real numbers R⁺, i.e. the positive continuum (cf. Burrill 1967). This follows from the facts that each f:N→N can be seen as the definition of a real number, and that each real number can be seen as the output of some f:N→N as it progressively evaluates its domain N. This turns on the fact that each real number can be defined as a natural number (i.e. the integer part) followed by a unique and infinite decimal expansion, for example, π=3.141592654…. The point is that among all the f:N→N there is at least one f that outputs π as it progressively evaluates N. That is: there is at least one f such that f(0)=3, f(1)=1, f(2)=4, f(3)=1, and so on. Thus, one possible definition of π is in terms of this f, namely: π=f(0).f(1)f(2)f(2)f(3)f(4)f(5)…

In this way, each positive real number can be defined in terms of some f:N→N. And conversely, each f:N→N defines some positive real number. Thus, as said, the set of all f:N→N is basically identical with the set R⁺. This, of course, requires the convention that for each such f we see f(0) as the integer part of the real number defined by f, but this is unproblematic. There is, however, one minor complication with this definition of R⁺ in terms of all f:N→N, namely: it implies that different functions sometimes define the same real number. For example, we saw that π is defined by the function f such that f(0)=3, f(1)=1, f(2)=4, f(3)=1, f(4)=5,… But there is also another function (let’s call it g) from the set of all f:N→N that outputs π as follows: g(0)=3, g(1)=1415, g(2)=9, and so on. Thus π can also be written as g(0).g(1)g(2)g(3)… In fact, it is easy to see that infinitely many functions from the set of all f:N→N define the same real number.

To avoid such multiple definitions of the same real number, the definition of R⁺ in terms of functions on N is usually limited to all f:N→{0, …, 9}. In this way, each positive real number is defined by only one such f. This is admittedly much more economical, but not strictly necessary. What matters is that the set of all f:N→N basically is (i.e. defines) the set R⁺. I will stick to this latter definition of R⁺ because it fits the above account of ASA as generating N through its inner recursivity. It makes little sense to say that ASA, through this recursivity, generates only reflection levels 0 to 9 and then stops, or that ASA indeed generates all reflection levels n from N but is only aware of its interlevel self-identity on the first 10 levels (and thus of all f:N→{0, …, 9}). No, ASA generates all reflection levels n from N and is aware of its interlevel self-identity on all these levels, thereby performing all f:N→N. As we have seen, this means that ASA is also aware of all positive real numbers, i.e. the set R⁺. The fact that multiple f’s from the set of all f:N→N then define the same real number is irrelevant; it is a redundancy built into the nature of ASA.

Patterns in the continuum and algorithmic information theory
The next step is somewhat more speculative, but not unreasonable. We have established that ASA is aware of all positive real numbers. So now what? What does ASA ‘do’ with the real numbers? What does the continuum ‘mean’ to ASA? Because the essence of ASA is to be aware of itself, it must use its awareness of R⁺ to further increase its self-awareness. This, it seems to me, can only mean that ASA looks for patterns (i.e. ordered number sequences) in the continuum in which it recognizes itself, i.e. patterns that somehow mirror its own nature.

What does this mean? It basically means that there are algorithms that mirror the nature of ASA, for example the algorithms inherent in the functioning of the human brain. We know from algorithmic information theory (developed around 1970 by Andreï Kolmogov and Gregory Chaitin, among others) that a number sequence is patterned (i.e. ordered, regular, as opposed to random) iff there is an algorithm, shorter in length than this sequence, which outputs this sequence. This is a definition of what order is. The shorter the algorithm, the more ordered the sequence it outputs. If for some sequence S no algorithm shorter than S can be given, then S is random. In that case, the only way to describe S is simply to reproduce S in full. S is not algorithmically compressible in that case, i.e. it contains no regularity that allows the formulation of a rule (i.e. algorithm), shorter than S itself, for the generation of S.

The number π provides a good example of a sequence that is highly ordered in the sense of algorithmic information theory. This may come as a surprise, since π is often considered to be a typically random number, whose decimal expansion evinces no clear order. It is true that π is a normal number, i.e. an irrational number whose decimal expansion features all possible number strings with equal frequency irrespective of the chosen base, which is a kind of statistical randomness. Nevertheless, the normality of a number does not per se imply its algorithmic randomness, as is shown by the computability of π. For, as is well-known, there are a number of relatively short algorithms that calculate π’s decimal expansion up to its n-th digit for some arbitrary n. From the perspective of algorithmic information theory, then, π is in fact highly ordered, since some arbitrarily long (but obviously still finite) stretch of it its decimal expansion can be generated by an algorithm much, much shorter than this string. On second thought, this is really not so surprising. For as we all learn in high school, π is just a circle’s circumference divided by its diameter. If one were to live forever and continued this division endlessly, one would eventually calculate every digit of π. Hence the computability of π and hence its orderedness in the sense of algorithmic compressibility.

Algorithmic compressibility offers an objective and universal measure of order. This can be seen from two facts: (1) that the thermodynamic concept of entropy can also be understood in terms of algorithmic compressibility (see Baez & Stay 2013), and (2) that the algorithmic compressibility of any sequence is more or less invariant between different formal languages. To make the intuitive concept of algorithm precise, after all, we need to unpack it in terms of some formal language, such as the language of Turing machines, lambda calculus, or programming languages such as Pascal, C or LISP. Algorithms, therefore, are notation dependent, relative to some formal language. One of the strengths of the notion of algorithmic compressibility is that such differences between formal languages are more or less irrelevant to it: the algorithmic compressibility of some sequence in a formal language is the same (up to an additive constant) as its algorithmic compressibility in any other formal language. This means that algorithmic compressibility is indeed a universal and objective measure of order.

For algorithmic information theory, then, each ordered sequence of numbers represents the shortest algorithm that outputs it. This enables us to make sense of the above claim that ASA recognizes itself in some patterns in the continuum, for we can now unpack this as the claim that the algorithms represented by these patterns mirror ASA’s essence. It stands to reason that these are the algorithms that simulate intelligent agency, e.g. the algorithms that describe the functioning of human brains (and the functioning of intelligent organisms in general). We know from physics that physical reality is thoroughly computable (i.e. algorithmic). Moreover, the anthropic principle in cosmology tells us that the universe is surprisingly well-suited for the evolution of life, and thus of those physically realized algorithms that mirror ASA’s essence. Perhaps, then, we can explain the universe as that hugely complex pattern in the continuum (which, remember, exists in our view only as the structure of ASA’s self-awareness) in which ASA sees its essence best reflected? The universe, then, would simply be an extremely complicated pattern in the recursive unfolding of ASA’s self-awareness, namely, that pattern whose (shortest) algorithm simulates intelligent agency to the highest degree.

Did God lose Himself in the “labyrinth of the continuum”?
A second reason why I like this theory is that it enables us to explain why the universe is not perfect, despite being the mathematical image of ASA (or ‘God’ if you prefer). For, as Turing showed (as part of his proof of the undecidability of the halting problem), by far most of the real numbers are uncomputable and therefore transcendental. This means that their decimal expansions cannot be generated by any algorithm. Thus, from the perspective of algorithmic information theory, their decimal expansions are totally random. In being aware of the continuum, therefore, ASA is aware of something that is for the most part unordered, a kind of primordial chaos. ASA’s attempt to find patterns in the continuum (in order to mirror itself in those patterns) must therefore be extraordinarily difficult, indeed virtually impossible, since the ordered part of the continuum is infinitesimally small compared to the unordered part. In fact, if one could randomly pick out a real number (say, by pricking somewhere in the real number line with an infinitely sharp needle), the probability of getting an uncomputable number is approximately 1 (cf. Chaitin 2005: 113)! Perhaps this explains why the universe, despite being an image of ASA, is not perfect? It must, after all, be close to impossible for ASA to find order in the continuum.

Since, as we have seen, R⁺ and the set of all f:N→N are basically the same set, the fact that most real numbers are uncomputable also means that most of the f:N→N are uncomputable. To see why most of the real numbers are uncomputable, remember that the notion of algorithm is always relative to some formal language. This language must have a finite set of basic symbols (i.e. a vocabulary) and a finite set of syntactical rules for the combination of these symbols into larger expressions. This means that the language can generate only a countably infinite number of expressions, since we can list them in order of length (i.e. we can have a bijection f:N→E where E is the set of all expressions generatable in the language). Since the set of algorithms is a proper subset of the set of all expressions generatable in this language, the set of all possible algorithms too must be countably infinite. So if we assume, for contradiction, that all positive real numbers are computable, then R⁺ must be countably infinite as well. But we know this is not the case, given Cantor’s proof of the uncountability of the real numbers: already in the unit interval [0,1] there are uncountably many numbers (in fact, as Cantor’s sun theorem shows, there are as many reals in [0,1] as in the entire continuum!). Thus, the set of real numbers is said to be “maximally larger” than the countable set of all possible algorithms. So there simply aren’t enough algorithms to compute all the real numbers; by far most of the real numbers are uncomputable and have therefore totally random decimal expansions.

Could this, perhaps, explain why the universe is imperfect, despite being (on our account) the mathematical self-image of God, i.e. self-causing Absolute Self-Awareness? Having generated the continuum through the recursivity of its self-awareness and its interlevel self-identity (which, as we have seen, gives all f:N→N and thus all real numbers), ASA looks for those patterns in the continuum in which it can mirror its own essence (which is self-awareness), only to find that patterns form an infinitesimally small portion of the continuum, since almost all real numbers are uncomputable. So ASA’s trying to find its own image in the continuum is a bit like trying to find a needle in a haystack… only much more difficult! As said, the probability of randomly selecting a computable number out of the continuum approaches zero. One could say that ASA, trying to see its own mathematical mirror image, instead lost itself in the “labyrinth of the continuum” (as Leibniz called the complex of unsolved problems and paradoxes surrounding the real numbers). And still, we are here, there is this ordered universe in which we find ourselves. True, it is not perfect, that is, it is the not the true image of the Absolute, but still it is there and it is computable. So, despite its near impossibility, the Absolute must nevertheless have succeeded in finding order in the arch-chaos of the continuum which the Absolute had itself created. It’s a bit like that old question: what happens when an unstoppable force meets an immovable obstacle? Well, what happens is the creation of this refractory miracle which we call the universe…

 References
-Baez, J.C & Stay, M. (2013), “Algorithmic Thermodynamics”, http://math.ucr.edu/home/baez/thermo.pdf
-Becker, O. (1973 [1927]), Mathematische Existenz: Untersuchungen zur Logik und Ontologie mathematischer Phänomene. Tübingen: Max Niemeyer Verlag.
-Chaitin, G. (2005), Meta Maths: The Quest for Omega. London: Atlantic Books.
-Burrill, C. (1967), Foundations of Real Numbers. New York: McGraw-Hill.
-Dauben, J.W. (1979), Georg Cantor: His Mathematics and Philosophy of the Infinite. Cambridge, Mass.: Harvard University Press.
-Royce, J. (1959 [1899]), The World and The Individual, First Series: The Four Historical Conceptions of Being. New York: Dover Publications. 
-Russell, B. (1970 [1919]), Introduction to Mathematical Philosophy. London: George Allen and Unwin.