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.