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.

9 Remarks on Absolute Idealism 2.0

Introduction
The following remarks develop in a sketchy manner the main ideas of a theory I am still working on. I intend to develop these remarks more fully in the future. I like to call this theory "Absolute Idealism 2.0" since it starts from the basic insight of traditional Absolute Idealism (developed by Plotinus, Fichte, Schelling, Hegel) but then takes this insight into a new direction by drawing on ideas from modern physics and computability theory. The basic insight from traditional Absolute Idealism, I submit, is the idea that reality is at bottom a self-conscious whole, producing itself by being aware of itself (remarks 1 and 2). Using ideas from the American Idealist Josiah Royce, I argue in remark 5 that the recursivity inherent in self-consciousness allows us to establish an intrinsic connection between self-consciousness and the recursively generated natural numbers. This then allows us to connect the Absolute-Idealist notion of self-consciousness, as the ultimate ground and essence of reality, to modern physics and computability theory, where the natural numbers figure prominently in the definition of computable functions (remarks 6, 7 and 8). Since, however, I am certainly no expert in computation, I am not entirely sure about the correctness of my application of computability theory to Absolute Idealism (this holds in particular for remark 8, which is by far the most contentious). Hence my request to the reader: if you spot difficulties, obvious mistakes or gaps in my reasoning, please let me know.

Contents:
1. Reality must be self-causing.
2. Absolute Self-Awareness (ASA) is the self-causing cause of reality.
3. Physical reality reduces to consciousness.
4. ASA is pure joy.
5. ASA includes awareness of the set of natural numbers.
6. ASA is the 'cosmic computer'.
7. Physical reality is ASA's computational self-reflection.
8. Time and evolution exist because of the Halting Problem.
9. Qualia are the reflections of ASA's pure joy.

1. Reality must be self-causing: Why is there something rather than nothing? This question, famously raised by Leibniz, remains unanswerable as long as we presuppose any of the standard conceptions of explanation, whereby one thing is caused by another (thunder by lightning, the boiling of water by fire under the kettle, the falling of a body by gravitational force, and so on). Leibniz's question targets reality as a whole, i.e. the totality of what is, and then asks why that totality is there. But, by definition, there is nothing outside the totality (not even 'nothingness') by which it could have been caused. The only way to explain reality, therefore, is through self-explanation. The cause behind reality can only lie within reality. Self-causation is the only possible answer to Leibniz's question. Clearly, however, self-causation is impossible in time. As a temporal process, causation is marked by a temporal distance between cause and effect, such that the cause precedes the effect. Self-causation would then require that the cause precedes in time its own existence, which is absurd. We must assume, therefore, that the self-causation needed to answer Leibniz's question 'happens' outside of time. Also because time itself is something, an object of sorts, a 'thing' with various properties (such as those described by physics). Time, in other words, belongs to the 'something' we are trying to explain when we ask: Why is there something rather than nothing? Since time does not explain its own existence, it must be explained by something else, ultimately by the self-causing cause of reality. But the cause of time cannot itself be in time. Thus, again, the self-causing cause of reality must be timeless.

2. Absolute Self-Awareness (ASA) is the self-causing cause of reality: But how is self-causation possible? How can something bring about its own existence. Here the self-evident experience of our own self-awareness provides us with the only empirical clue we have. The crucial point is that the circularity of self-awareness 'fits' the circularity of self-causation: as the self-causing cause is its own effect, so self-awareness is its own object of awareness. Since self-awareness essentially is its own object of awareness, it cannot exist without being aware of itself. Its being is its self-perception. It bootstraps itself into existence through self-perception. From an empirical standpoint, therefore, self-awareness is our best guess at what the self-causing cause of reality amounts to. I will refer to this as “Absolute Self-Awareness” (ASA), which is "absolute" in the sense of having an unconditioned existence, not dependent on or relative to anything besides itself. Rather, the rest of reality is ultimately dependent upon it. Since the self-causing cause of reality must be timeless (see remark 1), ASA must be timeless as well, an "Eternal Consciousness" in the phrase of T.H. Green. The 'present' in which ASA is present to itself (since self-awareness is a form of self-presence) must be an eternal present, an unchanging now (nunc stans). Clearly, then, we are not talking about individual human self-awareness, as present in you or me. 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. 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 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.

3. Physical reality reduces to consciousness: Since we take ASA to be the self-causing of reality, we must explain physical reality in terms of consciousness rather than vice versa (as is standardly done in scientific materialism). That it is at least possible to explain physical reality in terms of consciousness follows from the Hard Problem of Consciousness, i.e. the impossibility to explain the qualia of consciousness in exclusively physical terms. But the Hard Problem of Consciousness leaves open the precise nature of the mind-body relation; it is compatible with substance dualism, where consciousness and physical reality form two separate ontological domains (which nevertheless somehow interact). So we need further arguments to effectuate the reduction of physical reality to consciousness. Here we can appeal to Russellian Monism, which shows (a) that physical reality (as revealed by modern physics) is basically a mathematical structure, and (b) that all structure, in order to exist, requires non-structural bearers, i.e. intrinsic entities, and (c) that such entities can only be found in the qualia which elude the mathematical structures of physics (as per the Hard Problem of Consciousness). Thus the qualia of consciousness must be ontologically prior to the mathematical structures that define physical reality. In that sense, at least, physical reality reduces to consciousness.

4. ASA is pure, self-enjoying joy: A side-effect of the Hard Problem of Consciousness is that it makes clear that ASA, too, must involve qualia, or at least one quale – indeed, a quale that is somehow self-revealing, a "self-intimating what-it's-like-ness" (I owe this happy phrase to the philosopher David Pearce). But what exactly is this quale? What is it like to be ASA? Since we take the experience of our own self-awareness as the empirical key to the Absolute, and since human self-awareness is infused with emotion and volition right from the start, it would be an illegitimate abstraction to see ASA as just a 'cold' theoretical self-registering, without any emotive and volitional aspects. Thus we must see ASA as not merely cognitive self-awareness, but as will and emotion as well. But what could ASA possibly want? Since ASA is, at this point in our construction, the only 'thing' that exists, there is nothing for it to desire apart from itself. Thus, qua will, ASA can only will itself. Its self-awareness coincides with its self-willing. We can say "ASA exists because it wills itself" just as much as we can say "ASA exists because it is aware of itself". From this it follows that, qua emotion, ASA must be pure joy, i.e. self-enjoying joy. It's will for itself immediately satisfies itself, since its self-willing is at the same time the self-causing cause of its own existence. It gives itself to itself merely by willing itself. ASA is a self-aware, self-willing, self-satisfying and self-enjoying joy. Reality exists because pure joy wills itself. Cf. Nietzsche "Alle Lust will Ewigkeit..." Also see the Vedantic definition of the Absolute (what the Indians call "Brahman") as Satchitananda, i.e. the indivisible unity of being ("sat"), consciousness ("cit") and bliss ("ananda").

5. ASA includes awareness of the set of natural numbers: Up till now (as remarked in remark 2) only one 'thing' exists in our construction, namely, ASA. So how do we get from ASA to the universe around us, this multitude of physical objects, coming and going in space and time, governed by natural law? This, basically, is the old problem of the One and the Many: how does the original One produce the Many? The source of the difficulty, for us, lies in the fact that ASA, qua cause of itself, is ontologically self-sufficient, not in need of anything beyond itself. Qua self-causing, it causes just itself, and nothing more. Thus we appear to have a dilemma: to solve Leibniz's question we need a self-causing being (and experience tells us this must be ASA), but precisely its self-causing capacity creates the problem of the One and the Many. However, as Josiah Royce pointed out, once we understand the self-causing cause of reality in terms of (absolute) self-awareness, this problem is automatically solved by the recursivity inherent in self-awareness, i.e. the fact that it takes itself as object of awareness (see remark 2). In this way, self-awareness generates an infinite sequence of ever higher levels of self-reflection, namely: self-awareness, awareness of self-awareness, awareness of awareness of self-awareness, awareness of awareness of awareness of self-awareness, and so on ad infinitum. In semi-formal terms, 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 self-awareness, being its own object of awareness, becomes the function f(x)=x which generates the infinite sequence f(x)=f(f(x))=f(f(f(x)))=f(f(f(f(x))))... etc. As Royce also pointed out, this sequence is isomorphic to the natural number system N={0, 1, 2, 3, … }, which is recursively generated through the successor function S(n)=n+1 such that S(0)=1, S(1)=2, S(2)=3, and so on. Since ASA is the self-causing cause of reality as a whole, we must conclude that its first product, beyond itself, is the reality of the natural numbers. The natural numbers exist because ASA ‘thinks’ them by being recursively aware of itself. Thereby the Problem of the One and the Many is solved. Through its internal recursivity ASA generates the infinite complexity of N. Since ASA exists outside of time, we must conclude that N, too, exists outside of time. Thereby the Platonic reality of N is saved, even if that reality derives from a form (or rather the primordial form) of subjectivity.

6. ASA is the 'cosmic computer': But we do not just want N. We want to know how ASA explains the physical universe. Two considerations, when combined, suggest a clear answer. The first consideration, taken from the theory of computability, is that the notion of computation can be captured in terms of functions on N, such that all computable functions (i.e. algorithms, computations, effective procedures) are a subset of all n-ary functions from Nⁿ to N (i.e. f:Nⁿ→N). The second consideration, taken from modern physics, is that all physical processes are thoroughly computable, with the laws of nature acting as algorithms taking the present state of a physical system as input and producing the next state as output. So if we represent a physical system by a set of natural numbers (an n-tuple from N), we can then understand the natural law governing this system as a computable function. This, basically, is what modern physics does. Thus the natural laws turn out to form a subset of all computable functions. How does this solve our problem? As we have seen in remark 5, ASA is aware of N. An obvious way to see ASA as producing the physical universe, therefore, is to see ASA as computing those functions from Nⁿ to N that describe the evolution of the universe. The universe then becomes a 'digital simulation' run on ASA qua 'cosmic computer'. The fact that ASA can indeed be seen as engaged in computation follows from its intrinsic connection to N. 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 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 f(x)=f(f(x))=f(f(f(x)))=... etc. – this self-knowledge amounts to a knowledge of equivalence 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 equivalence relation (4, 7, 15)=(2). But such an equivalence relation is just a mapping from Nⁿ 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 Nⁿ to N, including all computable functions. Since the laws governing our physical universe form a subset of all computable functions, ASA can be said to compute our universe.

7. Physical reality is ASA's computational self-reflection: To repeat, ASA is aware of all computable functions, of which the computations that constitute our physical universe form a subset. This raises the question: Why is that subset special? Why is our physical universe realized and not any of the countless other computationally possible universes? Or should we say that all computable functions are realized, with our universe being just one of infinitely many computable worlds, all equally real? This last option would give us a computational version of the principle of plenitude: everything which is computationally possible is realized. However, it is easy to see that ASA, as we conceive it, excludes such ontological plenitude. Here we should remind ourselves what ASA essentially is, namely, self-awareness, and nothing more. It’s awareness of N, as we have seen, is just an extension of that self-awareness, as it recursively generates the infinite hierarchy of its self-reflection. Likewise, its awareness of all functions from Nⁿ to N results from the awareness of its self-identity throughout all the levels of that hierarchy. In short, ASA’s awareness of N and of all the functions from Nⁿ to N is completely subservient to one essential goal: to know itself as completely as possible. This forces us to see certain computations as privileged over others, insofar as certain computations reflect ASA’s essential properties (self-causation, self-awareness, joy) better than others do. Some computations, after all, such as the computations that describe the functioning of our brains, can be said to compute (self-)consciousness, intelligence and volitional agency. Given the fact that ASA’s sole purpose is to know itself, it is clear that those computations, which emulate intelligent organisms, will for ASA be objects of special attention, in contrast to all other possible computations. For by focusing its awareness on those computations, ASA reaches an even higher level of self-awareness, as it ‘sees’ itself reflected (‘mirrored’) in those computations. True, ASA is aware of all functions from Nⁿ to N. But only some of those functions, namely those that compute intelligent organisms, contribute to its increased self-awareness. And since, as we have seen in remark 2, self-awareness is the self-causing cause of reality, only those functions that increase ASA’s self-awareness acquire full reality. All other computable functions remain merely possible computations. This allows us to formulate the following hypothesis concerning our physical universe: it is the set of those computable functions that best reflect ASA’s essence. On this hypothesis, then, our universe is ASA’s computational mirror. And insofar as this mirror reflects ASA's pure joy, the universe can be said to be a work of art. Since ASA is self-causing through self-awareness, the physical universe too must be self-causing through self-awareness. John Wheeler's hypothesis of the self-observing universe, therefore, must be basically correct.

8. Time and evolution exist because of the Halting Problem: ASA is aware of all functions from Nⁿ to N, for all possible inputs (remark 6). But, as Turing showed, the totality of these functions includes both computable and uncomputable functions. (This follows from the fact that, since a computable function reaches its output after finitely many steps, the set of all computable functions is countable, whereas the set of all functions from Nⁿ to N is uncountable; hence by far most of these functions are uncomputable.) So how can ASA know 'in advance' which functions are computable and which are not? Here, it would seem, ASA is faced with the undecidability of the Halting problem (as demonstrated by Turing), i.e. the fact that there is no general algorithm for deciding which functions are computable (i.e. which functions halt after finitely many steps). However, on closer inspection it becomes clear that ASA is not affected by the undecidability of the Halting Problem. This follows from the fact that ASA exists outside of time (remarks 1 and 2). Thus the question how it can know 'in advance' which functions are computable simply does not arise for ASA; the distinction between before and after does not apply to it. Being timeless, ASA is aware of all functions from Nⁿ to N at once, and thus it 'sees' immediately which functions halt after a finite number of steps and which do not. Thus ASA needs no algorithm for solving the Halting Problem, and therefore the undecidability of that problem poses no difficulty for it. Nevertheless, the Halting Problem is undecidable for ASA's computational image, i.e. the complex computation that best reflects ASA (let's call this complex computation "the Intellect", after Plotinus). Since ASA has awareness of all functions from Nⁿ to N, the Intellect – as ASA's image – must have the same awareness, only this time computationally executed (since the Intellect is nothing but computation). So the Intellect must compute all functions from Nⁿ to N, and then find those computable functions that best reflect ASA (thereby the Intellect in effect computes itself, true to its nature as computationally executed self-awareness). But, as said, this means that the Intellect is faced with the undecidability of the Halting Problem. In this way a radical uncertainty is introduced in the Intellect's knowledge: it can't compute in advance which functions are computable. This, I venture, is the reason why time exists. The uncertainty which exists for the Intellect, about which functions will turn out to be computable, is the uncertainty that defines the future, its inherent unpredictability. This is not to say that the Intellect itself exists in time. As ASA's computational image, the Intellect itself exists outside of time, as a timeless mathematical structure in ASA's self-awareness. But in computing which functions are computable, and which of those computable functions best reflect ASA, the Intellect nevertheless produces time. Given the undecidability of the Halting Problem, the only way for the Intellect to find out which functions are computable is through a form of dovetailing (a familiar technique in computer science), such that it simultaneously executes step-by-step all functions from Nⁿ to N (so first it computes simultaneously the first step of all functions, then simultaneously the second step of all functions, then simultaneously the third step of all functions, and so on). Remember that the Intellect itself exists outside of time, so this simultaneous stepwise execution of all functions poses no problem for it. Then, as the Intellect goes along from step to step, it will after a finite number of steps find some computable functions (those that halt), whereas the stepwise execution of all other functions continues. For these latter functions, the Intellect can't know in advance if these are genuinely uncomputable or if they will also halt if more computational steps (of a finite number) are taken. In other words, the Intellect will never know if it has found all computable functions. So the process of the simultaneous stepwise execution of functions will never stop. This unending process of the stepwise disclosure of which functions are computable, and which of these computable functions best reflect ASA – this process, I venture, is time itself (the stepwise disclosure of the future). Time exists, then, because the Intellect is subject to the undecidability of the Halting Problem. Moreover, since the Intellect can't know in advance which computable functions it will find by dovetailing all functions from Nⁿ to N, this process is also a process of evolution, whereby the computations that best reflect ASA are only gradually discovered by the Intellect as time progresses. This evolution is the process of the Intellect's self-discovery, since the Intellect is that complex computation which best reflects ASA's essence. The evolution in time of our physical universe (which in remark 7 we hypothesized to be ASA's computational image), therefore, is the coming to self-awareness of the Intellect. The physical universe is the Intellect insofar as it has computed itself, insofar as it has become self-aware.

9. Qualia are the reflections of ASA's pure joy: Pure joy is the self-intimating quale in which ASA consists (remark 4). But how do all other qualia emerge, i.e. the qualia inherent in our conscious experience of ourselves and of our physical surroundings? My hypothesis is that these qualia emerge through the refection of ASA's pure joy in its computational mirror, i.e. the computations that constitute the physical universe. Through ASA's computational self-reflection its pure joy gets reflected back to it in multifarious ways, thereby breaking up the original unity of its pure joy (the arch-quale) into a multitude of qualia.