Monday, August 20, 2018

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 BA = { 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:NN. (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:NN can be seen as the definition of a real number, and that each real number can be seen as the output of some f:NN 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:NN 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:NN defines some positive real number. Thus, as said, the set of all f:NN 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:NN, 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:NN 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:NN 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:NN 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:NN. 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:NN 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 Andre
ï 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:NN 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:NE 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…

-Baez, J.C & Stay, M. (2013), “Algorithmic Thermodynamics”,
-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.

Sunday, April 29, 2018

The Epistemological Nature of Early Modern Idealism

Modern Idealism, as developed by Berkeley and Kant and their successors, was mainly epistemological in nature. That is to say: the arguments they used to establish the central Idealist thesis – that reality exists only in or for the mind – were mainly epistemological arguments, based on analyses of knowledge and sense experience. They reasoned basically as follows: Since we can know reality only insofar as it is sensed and conceptualised by us, we literally can have no evidence of any reality beyond our sensations and concepts, and thus beyond our consciousness. All that we are justified in postulating, therefore, is the reality internal to our consciousness, the organized whole of sensations and concepts we ordinarily call “reality”. This, arguably, is the master argument for modern Idealism.

Although I am greatly attracted by Idealism, I am critical of modern, i.e. epistemological Idealism. In my view, we should accept Idealism on ontological grounds, i.e. because it offers the best explanation of reality, and not primarily on epistemological grounds. As I intend to argue in a following post, epistemology alone can never provide a sufficient justification for the Idealist thesis. In particular, the master argument for modern Idealism remains vulnerable to skeptical attacks. For from the mere fact that knowledge of objects is only possible of objects within consciousness, it does not follow that all objects are within consciousness; there might still be unknown or even unknowable objects

To this skeptical retort, modern Idealism has no satisfactory answer, precisely because it rests its case on our epistemological confinement to consciousness. If we are indeed trapped within the circle of consciousness, then – as the epistemological Idealist emphasizes – we cannot prove the existence of a reality outside of consciousness; but then neither can we disprove that existence. This is the weak spot of modern Idealism, the point at which it remains vulnerable to skeptical counter-attacks. To prepare the way for this critique of epistemological Idealism, this post explains why the Idealisms of Berkeley, Kant, and their successors took this epistemological form.

The Way of Ideas and Its skepticism
To understand why modern Idealism took this epistemological form, we have to place it in the context of its origination, namely, the Way of Ideas developed by Descartes, Locke and their followers, and the radical epistemological skepticism to which it led. As explained in a
previous post on this blog, the Way of Ideas led to skepticism because it had ‘imprisoned’ the knowing subject within the “circle of consciousness”, hiding external reality behind a “veil of perception”. As the Cartesian philosopher Arnauld put it: “We have no knowledge of what is outside us except by mediation of the ideas within us.” (Arnauld 1964 [1662]: 31) Thus arose the skeptical question: If all we know directly are the ideas within our consciousness, how can we know if these ideas correspond to a reality outside our consciousness, indeed, how can we know there is an external reality at all? We cannot, after all, step outside our consciousness in order to inspect its correspondence, or lack thereof, with external reality. This threat of skepticism was sharply felt by Descartes, Locke, and their successors, some of whom – most famously Hume – went on to argue that skepticism was indeed inescapable.

It was to counter this threat of skepticism that Berkeley and Kant developed their respective versions of Idealism. As both of them pointed out, the skepticism induced by the Way of Ideas turned on the assumption of a reality external to consciousness; strike that assumption, they argued, and the threat of skepticism vanishes. If reality is ‘just’ a product of the mind itself, then surely its knowability can pose no problem for us?

George Berkeley (1685 - 1753)
Berkeley’s Idealist Rescue of Common Sense
Berkeley had designed his Idealism particularly with the intent to save common sense from skepticism. Common sense says that the objects we perceive by our senses are indeed as we perceive them: they have the colours, smells, tastes, auditory and tactile qualities we perceive in them. The Way of Ideas, however, had placed all such “secondary qualities” within consciousness, locating the real object outside the latter, as the external cause of those sensations. When we eat an apple, for example, we see its redness, taste its sweetness, feel its smooth skin, etc. But according to the Way of Ideas, all these sensations are not qualities of the apple itself; the real apple is just some material structure in space and time of which we know nothing except what physical science tells us (and even the truth of physics became doubtful after Hume’s critique of causality). For Berkeley, this skeptical doubt concerning common sense, induced by the Way of Ideas, was absurd: 

Upon the common principles of philosophers, we are not assured of the existence of things from their being perceived. And we are taught to distinguish their real nature from that which falls under our sense. Hence arises Scepticism and Paradoxes. It is not enough that we see and feel, that we taste and smell a thing. Its true nature, its absolute external entity, is still concealed.” (Berkeley 1969 [1713]: 3)

It was therefore to redeem common sense that Berkeley argued for Idealism, which in his case amounted to the thesis that sensible objects do not exist unperceived: “Their esse is percipi,” as Berkeley famously put it (1995 [1710]: §3). Perceptible objects, he argued, are nothing but bundles of sensible qualities in consciousness. Thus, through his Idealism (or “Immaterialism” has he called it),
Berkeley could restore the common-sense belief that when we eat an apple, and see its redness, taste its sweetness, etc., we are eating, seeing and tasting the apple itself, not just its appearance as distinct from the real thing. The real apple, for Berkeley, is this bundle sensations; there is nothing beyond it. Berkeley made the same point by contemplating a cherry (fruit, apparently, lending itself very well for Idealist argumentation…):

“I see this cherry, I feel it, I taste it […]: it is therefore real. Take away the sensations of softness, moisture, redness, tartness, and you take away the cherry. Since it is not a being distinct from sensations; a cherry, I say, is nothing but a congeries of sensible impressions, or ideas perceived by various senses: which ideas are united into one thing (or have one name given them) by the mind; because they are observed together.” (Berkeley 1969[1713]: 117)

In his Three Dialogues between Hylas and Philonous, Berkeley makes his alter-ego Philonous (Greek for “Lover of mind”) respond as follows to the insensible matter beyond sensory experience defended by Hylas (Greek for “matter”):

“I am of a vulgar cast, simple enough to believe my senses, and leave things as I find them. To be plain, it is my opinion that the real things are those very things I see and feel, and perceive by my senses… A piece of sensible bread, for instance, would stay my stomach better than ten thousand times as much of that insensible, unintelligible, real bread you speak of… Away then with all that Skepticism, all those ridiculous philosophical doubts. What a jest is it for a philosopher to question the existence of sensible things, till he hath proved it to him from the veracity of God1; or to pretend our knowledge in this point falls short of intuition or demonstration! I might as well doubt of my own being, as of the being of those things I actually see and feel.” (Berkeley 1969 [1713]: 90-1)

As Berkeley admitted (see idem: 110), it is a bit strange to defend common sense by declaring that perceived objects exist only within the mind – a view that directly violates common sense, for which perceived objects ‘evidently’ exist outside the mind – but, according to Berkeley, it is the only way to save the reality of the sensible object within the context of the Way of Ideas. Idealism is the bitter medicine that common sense must take in order to cure it from the illness of skepticism.

Immanuel Kant (1724 - 1804)
Kant’s Idealist Rescue of Causality
For Kant, it was a different aspect of the skepticism induced by the Way of Ideas that brought him to accept Idealism. What worried him was not so much the affront to common sense as the affront to physical science presented by Hume’s skeptical attack on causality. As Kant noted in the Prolegomena, it was Hume’s attack on causality that first aroused him from his “dogmatic slumber” and stimulated the development of his “transcendental Idealism” (Kant 2001 [1772]: 5). Hume had shown, convincingly according to Kant, that our causal claims about reality are thoroughly unsupported by the sensations caused in us by external objects. We say, e.g., that fire causes smoke, but all the evidence we have is that sensations of smoke regularly follow sensations of fire. In the sensations themselves we find no reason why one should follow the other. Moreover, we cannot generalize from a finite number of past observations to universal claims: the fact that up till now sensations of smoke have followed sensations of fire does not guarantee that this will be so in the future as well (the problem of induction). As Hume put it:

Thus not only our reason fails us in the discovery of the ultimate connection of causes and effects, but even after experience has informed us of their constant conjunction, 'tis impossible for us to satisfy ourselves by our reason, why we should extend that experience beyond those particular instances, which have fallen under our observation.” (Hume 2003 [1739-40]: 66)

Kant was deeply disturbed by Hume’s attack on causality. His respect for the physical science developed by Copernicus, Galileo and Newton was so great that he simply could not stomach Hume’s dismissal of causal laws. The stunning success of the new science, especially Newton’s discovery of the laws of motion and gravitation, meant that Hume had to be wrong. And where he went wrong, according to Kant, was in his assumption that causality, if it exists at all, must be a feature of external reality, in other words, that causal connections must be connections between real objects, independent of our consciousness. But, as Kant argued, such external objects are “nothing to us”. Objects become something for us, i.e. they become accessible to us as experienceable and knowable objects, only if they conform to our forms of cognition, and causality is one such form. Raw sensations do not yet give us experiences of objects. The sensations have to be ordered by our forms of sensory intuition (space and time) and our forms of conceptual understanding (the categories, prime among which is causality); only then do we experience a single, ordered, integrated reality consisting of interconnected objects. This, according to Kant, explains our ability to make objective causal claims: because causality is not a feature of external reality but rather a cognitive form in our mind, a form to which objects must conform in order to become experienceable and knowable.

Kant’s Idealism, then, extends only to the forms of empirical reality, not to the sensory material structured by these forms. This is why Kant calls his philosophy “transcendental Idealism”, the term “transcendental” being his technical term for what pertains to the a priori forms of consciousness: “I call all cognition transcendental that is occupied not so much with objects but rather with our a priori concepts of objects in general.” (CPR: A12) In extension, Kant speaks of the “transcendental subject” as the subject who applies the a priori forms of cognition to the sensory material.

Ultimately, the necessity of the object to conform to our forms of cognition has to do with the fundamental role Kant accords to self-consciousness in experience and knowledge. This point is often described, rightly, as the cornerstone of Kant’s Idealism. According to Kant, a process or state in my consciousness counts as an experience or belief only if I can be aware of it as my experience or my belief, thus only if it belongs to the unity of my consciousness (a consciousness that forms a unity precisely because it is mine, i.e. because all episodes and states in it are related to me as their underlying subject). For a mental episode or state to be mine, then, I must as it were be able to prefix it with the qualifier “I think…”. By prefixing “I think…” to a mental content, such as an impression of redness, thus by thinking “I think (or rather I see) redness”, I indicate that the content belongs to the unity of my consciousness. As Kant puts it:

“The I think must be able to accompany all my representations; for otherwise something would be represented in me that could not be thought at all, which is as much to say that the representation would either be impossible or else at least would be nothing for me… The thought that these representations […] all together belong to me means, accordingly, the same as that I unite them in a self-consciousness […].” (CPR: B132, B134) 

According to Kant, the ultimate function of the forms of space and time and the categories of the understanding is to effectuate this unity of self-consciousness (a unity that Kant therefore calls “transcendental”, since it underlies the application of the transcendental forms of cognition). Thus only by placing all my mental episodes and states within a unified spatiotemporal network of causal relations can I recognize those episodes and states as mine, as belonging to my (self-)consciousness. The resulting integrated unity of empirical reality, then, is for Kant only a reflection or projection of the transcendental unity of self-consciousness unto the unorganized manifold of raw impressions. The unity of the object, and thereby the object as such (because there is no object without unity), is really a manifestation of the unity of the subject’s self-consciousness. In this sense, as later German Idealists would put it, the principle of subject-object identity is the central principle of Kant’s Idealism.

The Epistemological Nature of Idealism after Kant and Berkeley
Although the Idealisms of Berkeley and Kant differ greatly, they have roughly the same goal – to counter the epistemological skepticism engendered by the Way of Ideas – and use roughly the same strategy to achieve that goal, namely: argue that we can only know objects which are in or for consciousness, such that supposedly external reality falls away as irrelevant and unknowable, in which case the skeptical threat, too, falls away. The only reality left standing, then, is the reality inside consciousness. This, to repeat, is the master argument for modern Idealism – an argument either explicitly repeated or at least implicitly accepted by later German and British Idealists. They all stood on the shoulders of Berkeley and Kant, striving to improve or complete their ground-breaking but still imperfect Idealist systems (for the Germans, of course, Kant was more important, but the British Idealists drew on both Berkeley and Kant). As such, the German and British Idealists took over the epistemological agenda of Berkeley and Kant and remained within their epistemological mode of reasoning. For all of them, epistemology remained the prima philosophia, the foundational “first philosophy” that had to precede and ground all other theoretical endeavours. And even if later Absolute Idealists (such as Schelling, Hegel, Green and Bradley) went on to draw more ontological and metaphysical conclusions concerning the mind-dependence of reality, they did so ultimately because Idealist epistemology demanded it. As Frederick Beiser notes: “Although absolute idealism is indeed metaphysics, and in the very sense prohibited by Kant […], its metaphysics is necessary to solve the outstanding problem of Kant’s philosophy according to its own guiding principle.” (Beiser 2002: 369)

1. An obvious reference to Descartes’ appeal to God as the guarantor of the veracity of our perceptions, PS.

-Arnauld, A. (1964 [1662]), The Art of Thinking. Indianapolis: Bobbs-Merrill.

-Beiser, F. (2002), German Idealism: The Struggle against Subjectivism, 1781-1801. Cambridge, Massachusetts: Harvard University Press.
-Berkeley, G. (1995 [1710]), A Treatise Concerning the Principles of Human Knowledge. Indianapolis: Hackett Publishing.
-Berkeley, G. (1969 [1713]), Three Dialogues Between Hylas and Philonous. Chicago: Open Court.
-Kant, I (1998 [1781-87]), Critique of Pure Reason. Cambridge: Cambridge University Press.
-Kant, I. (2001 [1772]), Prolegomena To Any Future Metaphysics. Indianapolis: Hackett Publishing.

Thursday, April 5, 2018

The Problem of Skepticism in Early Modern Philosophy of Consciousness

This post is part of a larger project I am working on: a critique of the epistemologically motivated Idealisms of Berkeley, Kant, and the post-Kantians. I am greatly attracted to Idealism, but I think we should accept it primarily on ontological grounds, i.e. because Idealism gives the best explanation of why reality exists and why it is as it is. Hence my criticism of the Idealisms of Berkeley, Kant, et al., because for them Idealism was primarily epistemologically motivated, Idealism being their solution to the problem of epistemological skepticism as it arose within the early modern philosophy of consciousness advanced by Descartes, Locke, and their followers. As I will argue in a next post, modern Idealism, as an answer to this threat of skepticism, fails miserably (thus the only remaining reasons for accepting Idealism must be ontological). To prepare the way for this critique of epistemological Idealism, this post explains how the problem of skepticism arose in the early modern philosophy of consciousness, or the “Way of Ideas” as it was known to Descartes, Locke and their contemporaries. 

The Way of Ideas
There were two main, interconnected forces driving early modern philosophers towards the Way of Ideas and its epistemological centralization of consciousness. One of these forces was the desire for certain knowledge, which arose from the quarrels between the Church and the new natural science of Copernicus and Galileo, which rose all kinds of thorny issues concerning the authority of Faith and the powers of Reason. Here, famously, Descartes used the cogito ergo sum argument as a way to ground the certainty of knowledge on the self-evidence of consciousness’ knowledge of itself. Thus, the range of certain knowledge became limited to individual consciousness and its ‘contents’ (generically called “ideas” or “representations”; Kant spoke of “Vorstellungen”). According to the proponents of the Way of Ideas, then, the subject knows primarily what is inside the “circle” of his consciousness; only those contents are immediately present to it. All things outside consciousness are known mediately, by conjecture on the basis of what is inside consciousness (sensations, feelings, concepts, thoughts).

The other force that drove early modern philosophers to embrace the Way of Ideas was the atomism – or “corpuscular philosophy” – of the new natural science. Reviving (and transforming) the atomism of Democritus, the proponents of the new science advanced the hypothesis that all natural phenomena are explainable in terms of tiny particles of matter, “corpuscles”, interacting mechanically in space. This, however, led to the question of how to explain sensory qualities such as colour, smell, sound, and taste, which are notoriously subjective. What colour something appears to have or how it sounds, tastes or smells can differ from person to person, depending on one’s physical constitution and the surrounding environment (thus, a thing’s colour changes with the light falling on it; things can taste and smell differently when you are sick, etc.). However, like the atoms of Democritus, the corpuscles of the new science were supposed to exist objectively, independently of our consciousness of them. They were, moreover, supposed to be so small as to be imperceptible and thus as being in themselves without colour, taste, smell, etc. Hence, like Democritus, the corpuscularians – including Descartes and Locke – concluded that such sensory qualities were merely the effects in our minds of the collisions of corpuscles on our sense organs. Such sensory qualities, then, are only subjective and do not reveal the objective qualities of the corpuscles, which consist merely of solidity, spatial form and position, and motion. This distinction between subjective and objective qualities became known as the distinction between secondary and primary qualities. Whereas the primary qualities, such as spatial position and motion, are objective, measurable, and mathematizable, and thus are crucial to natural science, the secondary qualities convey no trustworthy information about the reality outside our consciousness.

The general picture that thus arose was of a knowing subject locked inside his “circle of consciousness”, with external objects impinging on it from the outside, causing perceptions within the circle. “We have no knowledge of what is outside us except by mediation of the ideas within us,” as the Cartesian philosopher Arnauld (1964 [1662]: 31) summarized it. Such was the overall conceptual framework within which the Way of Ideas operated. And although this focus on consciousness was partly motivated, notably in Descartes, to provide a secure foundation for knowledge, the irony of the situation was that the Way of Ideas ended up fostering a radical epistemological skepticism. For if certainty pertains only to what is inside consciousness, how then can we know what is outside consciousness, the external reality? If all we know with certainty are the contents of consciousness, how can we know that these contents correspond to external objects? After all, as the problem was frequently put, we cannot step outside our consciousness in order to inspect its correspondence, or lack thereof, with external reality. 

The Veil of Perception and the Cartesian Circle
The problem is sometimes put in terms of a veil-of-perception theory which has been attributed to Descartes, Locke, and other philosophers of the Way of Ideas. On this theory, our sensory experiences of external objects do not give us cognitive access to these objects but rather form a ‘veil’ or ‘screen’ hiding them from our view. So the medium we use to know external objects, our sensations and ideas, blocks our very access to them. Thus Barry Stroud describes Descartes’ sceptical conclusion in his First Meditation as “implying that we are permanently sealed off from a world we can never reach”: “We are restricted to the passing show on the veil of perception, with no possibility of extending our knowledge to the world beyond. We are confined to appearances we can never know to match or deviate from the imperceptible reality that is forever denied to us.” (Stroud 1984: 33-4) Similar veil-of-perception theories have been attributed to Locke, Berkeley and Hume (cf. Bennett 1971).

The radical nature of the epistemological problem created by this veil-of-perception theory is well illustrated by the desperate solution offered to it by Descartes. In his Meditations on First Philosophy he famously argued that the only way to ‘pierce through’ the veil of perception, in order to reach the objects in themselves, is by evoking God, whose goodness would guarantee the veracity of our perceptions, such that “all things which I perceive very clearly and distinctly are true” (Descartes 1996 [1641]: 24). But to this solution, of course, the skeptic can easily respond by asking how Descartes can know for sure that God exists. If our ideas form a screen between us and external reality, then surely they would also screen us from the true nature of God, if He exists at all. Descartes had an answer to this, but few would find it convincing. It could even be argued that it is downright circular. Descartes argued that we find within our minds an idea of an infinite being, thus an idea which we as finite beings cannot possibly have produced; thus, it can only have been put in our minds by our Creator, “like the mark of the workman imprinted on his work” (Descartes 1996 [1641]: 35). Descartes’ assumption, however, that a finite being cannot form any idea of infinity, is rather questionable. Therefore Descartes also had recourse to a version of the ontological proof of God’s existence. But, as was already pointed out by critics in Descartes’ time, this makes his argument for the veracity of clear and distinct ideas rather circular. For Descartes cannot know that this proof of God does not contain any error unless he assumes that his clear and distinct perception of the steps of his reasoning guarantees that the proof is correct. So Descartes has to presuppose the veracity of clear and distinct ideas in order to prove the existence of God, which he then invokes as the guarantee of this very veracity – a conundrum known as the “Cartesian circle”.

The Problem of Primary and Secondary Qualities in Democritus, Locke, and Berkeley
The Way of Ideas, then, fostered epistemological skepticism by imprisoning the knowing subject within the circle of his consciousness, hiding external reality behind a veil of perception. It is often said that this type of skepticism was exclusively modern and cannot be found in premodern times. This is by and large true, but not entirely. It is true that for Pyrrhonism, the dominant form of epistemological skepticism in antiquity, the gap between what is in consciousness and what outside it didn’t matter much (Pyrrhonism was mainly concerned with showing that we can have no definitive criterion of truth, since every proof of such a criterion must either be circular or presuppose another criterion of truth, for which then the same problem arises). Nevertheless, the problem of the gap between consciousness and external reality was not completely unknown in classical philosophy, as shown by the remarkable case of Democritus, the "laughing philosopher". Not only did Democritus, with his atomism, anticipate the modern scientific worldview, he also anticipated the modern distinction between primary and secondary sensory qualities, as well as the epistemological skepticism induced by this distinction. In one of the few surviving fragments of his work, Democritus stages a striking dialogue between the Intellect and the Senses:

“Intellect: By convention there is sweetness, by convention bitterness, by convention colour, in reality only atoms and the void.
 Senses: Foolish intellect! Do you seek to overthrow us, while it is from us that you take your evidence?” 

In other words: if the secondary qualities do not convey objective information about the atoms, how can we ever know about them? How, in particular, can we know their primary qualities, since we cannot experience a thing’s spatial position and motion apart from its colour, sound, etc. If we disregard all secondary qualities, external objects become utterly unobservable to us. This means, as Democritus realized, that the atomic theory undermines the very credibility of the empirical evidence on which it rests. Democritus’ point was later repeated by early modern philosophers, notably Berkeley in his critique of Locke. 

Locke conceded that secondary qualities give us no insight into the true nature of external objects, but like Descartes he remained steadfast that we can nevertheless know these objects by observing their primary qualities, e.g. spatial position and motion. Thus Locke claimed that “the ideas of primary qualities of bodies are resemblances of them, and their patterns do really exist in the bodies themselves, but the ideas produced in us by these secondary qualities have no resemblance to them at all” (Locke 1996 [1689]: 51). Berkeley objected – much as Democritus had argued some 2000 years earlier – that we can observe a thing’s primary qualities only through its secondary qualities, and thus that our beliefs about the primary qualities of external objects are as problematic as the secondary qualities we attribute to them. Thus, Berkeley writes: “In short, extension, figure, and motion, abstracted from all other qualities, are inconceivable. Where therefore the other sensible qualities are, there must these be also, to wit, in the mind and nowhere else.” (Berkeley 2003 [1710]: 35)

Locke and the Problem of the ‘Thing in Itself’
In this way, however, Berkeley only aggravated a skepticism that was already present in the inaugurators of the Way of Ideas. We have already seen how Descartes felt the sceptical challenge and how he attempted to meet it by invoking God as the guarantor of the veracity of his “clear and distinct ideas”. Locke, too, felt this challenge. Although Locke thought (pace Berkeley) that we can know an external thing’s primary qualities, he also thought that we could not know what that thing is in itself, independent from its relation to us and other objects. Primary qualities, after all, are thoroughly relational, pertaining to a thing’s position in space and motion relative to other things. But what is an external thing in itself, apart from those relations? This, as Locke conceded, we cannot know, since we are ‘locked’ (pun unintended) inside our consciousness and cannot inspect objects as they exist outside of consciousness. Thus, what a thing is in itself, what the Aristotelians called its “substance”, was for Locke merely a “supposed” something “I know not what” (Locke 1996 [1689]: 123). For Locke, therefore, even the new natural science, despite its huge empirical success in the work of Galileo and Newton, yielded only opinion, not knowledge. Such sceptical modesty concerning the success of the new physics was in fact widely shared in early modernity, even by those who were directly involved in the development of the new science, such as Mersenne and Gassendi in France and John Wilkins in England. For all of them, our ‘imprisonment’ in consciousness precluded any knowledge about the true nature of external reality. 

Hume’s Critique of Causality
The authority of epistemological skepticism was further cemented by David Hume, who specifically undermined the causal claims of natural science, i.e. the claim that the scientist’s “laws of nature” refer to real causal connections within external reality. Hume followed Locke in holding that all belief begins with “impressions”, i.e. sensations, passions, emotions, which are the primitive imprints of external objects on our passive sensibility. We then form “ideas” which are the recollections of these impressions, their “faint images” or “copies” in memory. Hume argued that what guides us in these recollections of impressions, and thus in the formation of ideas, is the associative law of similarity: impressions which are sufficiently similar to each other get mutually associated, and thus form an idea. For example, our sense impressions of particular fires start over time to evoke recollections of each other due to mutual association, and this gives us the general idea of fire. Finally, beliefs emerge because these ideas, too, get linked to each other on the basis of association. To give an obvious example: in the past we have often experienced one sort of impression, e.g. of smoke, as immediately following upon another kind of impression, e.g. of fire, and this causes the general idea of smoke to become associated with the general idea of fire. This, according to Hume, is the full extent of what we mean when we say “fire causes smoke”. There is nothing more to our concept of causality, according to Hume, than this regular, inductively based association of one idea with another:

“We have no other notion of cause and effect, but that of certain objects, which have been always conjoin’d together, and which in all past instances have been found inseparable. We cannot penetrate into the reason of the conjunction. We only observe the thing itself, and always find that from the constant conjunction the objects acquire an union in the imagination. When the impression of one becomes present to us, we immediately form an idea of its usual attendant […].” (Hume 2003 [1739-40]: 67)

For Hume, then, the necessity we associate with the laws of causality, i.e. the idea that if one thing happens then another thing must happen, is nothing but the strength of this association, the power exerted by habit over the workings of our minds. We project this feeling of necessity onto the world, seeing the connection between one object and another as a necessary link between cause and effect. But, according to Hume, this is just an illusion, albeit a very powerful one. If we analyse our ideas more closely, Hume argued, we find no intrinsic connection between them that could substantiate a causal claim, such as that fire causes smoke. Imprisoned as we are within the circle of consciousness, we cannot know the real causal connections between external objects, if there are any at all. All we can know, Hume concludes, are the impressions and ideas of those objects within consciousness, and the merely associative connections between those impressions and ideas. Thus the causal laws of natural science evaporate into subjective feelings of necessity as we have been habituated to associate one idea with another.

The Mind-Body Problem and the Crisis of the Causal Theory of Perception
In sum, the Way of Ideas fostered epistemological skepticism by imprisoning the knowing subject with the circle of consciousness, hiding external reality behind a veil of perception. But it fostered such skepticism also in another (though closely related) way, namely, by inviting the mind-body problem. For how can mind interact with the external and supposedly material world if they are so very different, as the Way of Ideas suggests? The external world, after all, insofar as we can know it, is knowable only through its primary qualities, such as solidity, spatial position, and motion. For all we know, therefore, external reality is nothing but solid bodies interacting mechanically in space. Hence, of course, Descartes’ definition of the external world in terms of “res extensa”. But consciousness is very different from this world of extension, since ideas appear to have no solidity, no weight, no well-defined spatial position (if ideas can be said to be in space at all, they must be somewhere in my head, but where exactly?), and they do not interact by bumping into each other as material bodies do. Moreover, the conscious subject appears to have the capacity for free will, but free will seems impossible in a material world governed by causal determinism (pace Hume). Thus, consciousness appears to be in an entirely different realm of being, the immaterial realm of “res cogitans” as Descartes put it. Locke, too, drew the conclusion that mind must be immaterial, and thus categorically different from the material world which we can know through its primary qualities.

But, to repeat, if mind and matter belong to ontologically distinct realms, how can they possibly interact? Descartes wavered on this question, sometimes allowing mind-body interaction in the pineal gland, at other times doubting the possibility of such interaction; to Princes Elisabeth of Bohemia, with whom Descartes corresponded extensively, he admitted that this problem vexed him greatly and that he had no good solution to it. Locke was more resolute in that he openly declared the problem insoluble, there being no possibility for mind and matter to interact, except through divine intervention. As Locke argued, all you can get from spatial form and motion are other spatial forms and motions, and since the contents of consciousness are neither spatial forms nor motions, they cannot be caused by matter; nor can they exert causal influence on matter. However, since mind and matter obviously do interact, Locke felt compelled – much like Descartes in his solution to the problem of skepticism – to invoke God, who must have “superadded” mysterious properties to material objects, over and above their essential primary qualities, rendering them capable to cause sensations and ideas. Thus in his Essay Concerning Human Understanding, Locke writes:

“[B]ody as far as we can conceive being only able to strike and affect body; and Motion according to the utmost reach of our Ideas, being able to produce nothing but Motion, so that when we allow it to produce pleasure or pain, or the Idea of a Colour, or Sound, we are fain to quit our Reason, go beyond our Ideas and attribute it wholly to the good pleasure of our Maker.” (Locke 1996 [1989]: 237)

Obviously, no skeptic will be persuaded by this appeal to God in order to explain mind-body interaction. That Locke feels compelled to invoke divine intervention in this context only goes to show the deepness of the problem. And apart from being an ontological problem concerning the place of mind in the material world, it is also an epistemological problem, and one that aggravates the skepticism already induced by the Way of Ideas. For insofar as the causal interaction between matter and mind becomes mysterious, it becomes equally mysterious how perceptions can convey information about external objects. For here the only possible theory seems to be some version of the causal theory of perception, such that perceptions carry information about external objects because they have been caused by these objects, i.e. by the impingements of material objects on our external sense organs. Locke accepted a causal theory of perception, and he used it to explain how we can know external objects. Although the secondary qualities caused in our minds by external objects do in no way resemble those objects, as Locke admits, the situation is different with the primary qualities, i.e. with our perceptions of solidity, spatial position, figure, motion, etc. Here, according to Locke, our perceptions do resemble the objects by which they have been caused. By causing perceptions in us, then, external objects convey to us information about their primary qualities. And, for Locke, this is the only way we can know external objects, since according to him all knowledge starts with sensory impressions, the mind being a tabula rasa prior to experience. Hence the dire consequences of the mind-body problem. If the causal interaction between mind and matter becomes mysterious, to such an extent even that we need to invoke divine interaction to explain it, then clearly the causal theory of perception is of little help in explaining the veracity of our perceptions. Due to the mind-body problem, then, the epistemic position of the subject under the Way of Ideas deteriorates even further: not only is the subject shielded from external objects by a veil of perception, imprisoned in the circle of consciousness; the only way for external objects to pierce through that veil – by causing perceptions in us that resemble their primary qualities – falls away by being a complete mystery. And even if we accept mind-body interaction as an unexplainable yet undeniable given, we still have the problem raised by Berkeley (following Democritus) that we really have no perception of primary qualities apart from secondary qualities… 

-Arnauld, A. (1964 [1662]), The Art of Thinking. Indianapolis: Bobbs-Merrill.
-Bennett, J. (1971), Locke, Berkeley, Hume: Central Themes. Oxford: Oxford University Press.
-Berkeley, G. (2003 [1710]), A Treatise Concerning the Principles of Human Knowledge. Mineola: Dover Publications.
-Descartes, R. (1996 [1641]), Meditations on First Philosophy. Cambridge: Cambridge University Press.
-Hume, D. (2003 [1739-40]), A Treatise of Human Nature. Mineola: Dover Publications.
-Locke, J. (1996 [1689]), An Essay Concerning Human Understanding. Indianapolis: Hackett Publishing Company.
-Stroud, B. (1984), The Significance of Philosophical Scepticism. Oxford: Oxford University Press.

Friday, March 30, 2018

Absolute Idealism 2.0 and Plotinus

In various posts on this blog I have sketched the rough outlines of a contemporary version of Absolute Idealism – ‘Absolute Idealism 2.0’ – which is both ontological and mathematical in nature. It is ontological, not epistemological, in nature in that its main motivation is to explain reality rather than just our knowledge of reality. Its fundamental concept is the ontological self-grounding of self-consciousness, i.e. the idea that self-consciousness – due to its circular, self-referential nature – grounds its own existence and is in that sense causa sui. This makes possible, in my view, an Absolute-Idealist answer to the most fundamental question of ontology, namely, Leibniz’ question: “Why is there something rather than nothing?” Here Absolute Idealism can answer: there is something, rather than nothing, because self-consciousness is causa sui. In my view, this ontological prioritization of self-consciousness as the explanation of reality as a whole – including physical reality – is confirmed by recent developments in the philosophy of mind (notably the Hard Problem of Consciousness) and of physics (Russellian Monism, the role of observation in quantum mechanics, the anthropic principle, and Wheeler’s idea of the self-observing universe).

Metaphysics continuous with science
Obviously, this self-consciousness I appeal to in order to explain reality as a whole is not the individual, finite self-consciousness embodied in physical organisms. Rather, it is a universal, infinite, absolute self-consciousness that is ontologically prior to time and space. I consider this assumption of an absolute self-consciousness as a metaphysical hypothesis that is justified to the extent that it helps us to explain reality. It is, therefore, a form of metaphysics, but one that aims to be continuous with science. In my view, Absolute Idealism is justified only insofar as it accords with the scientific world view. This also explains the mathematical orientation of my approach to Absolute Idealism. Physics, after all, shows that mathematics is the deep structure of physical reality. Thus, the Absolute-Idealist explanation of reality as a whole in terms of absolute self-consciousness can only work if it also explains this ontologically fundamental role of mathematics.

Royce’s mathematical view of the Absolute
In my view, we find the required link between mathematics and absolute self-consciousness by focussing on the recursivity of the latter, i.e. on the fact that self-consciousness, in being its own object of awareness, is also aware of its self-awareness, and aware of that awareness of its self-awareness, and aware of the awareness of that awareness of its self-awareness, and so on ad infinitum. As the American Idealist Josiah Royce has pointed out, this infinite recursion of self-consciousness is isomorphic to the recursion that defines the natural number system
(i.e. the recursive successor function S(n)=n+1, which starting with n=0 generates 1, 2, 3 …). In this way, we can see the absolute self-consciousness, through its inner recursivity, as aware of all natural numbers. From here, as I have argued in different posts, it is only a small step to seeing the absolute self-consciousness as a ‘cosmic computer’, given the fact that computation is standardly understood in terms of mappings from to .

The Absolute as ‘cosmic computer’
Since physics shows the basic computability of all physical processes, we can view the physical universe as a privileged subset of all the computations going on in the absolute self-consciousness. But why is this subset privileged? Why does the absolute self-consciousness ‘think’ the computations that constitute this universe rather than any other universe? Two facts suggest an answer: (1) the anthropic principle in physics, which points out that the universe seems ‘just right’ for the evolution of life, and (2) the tautological fact that the aim of absolute self-consciousness is to attain complete knowledge of itself. Thus, it stands to reason that insofar as the absolute self-consciousness computes at all, it pays special attention to those computations that “simulate” intelligent, self-aware organisms. For by focusing its attention on those computations – e.g. the computational structure of the human brain – it sees its own essence reflected in the medium of mathematics. This gives us the following hypothesis: the universe is that proper subset of computations in which the absolute self-consciousness sees its own essence best reflected. It is, to repeat, only a hypothesis, which becomes acceptable only insofar as it enables us to explain reality, in conformity with the scientific world view.

Closeness to Neoplatonism
Looking for historical precedents of this approach to Absolute Idealism, we arrive first and foremost at Neoplatonism, especially as developed by Plotinus. Plotinus was unique among the Neoplatonists in that he accorded a fundamental role to self-consciousness in the self-causation of the Absolute, i.e. “the One” in his terminology. According to Plotinus, the One is the consciousness it has of itself and as such it exists because it is conscious of itself. Thus, Plotinus writes that the One "so to speak looks to himself, and this so-called being of his is his looking to himself, he as it were makes himself […]." (Ennead VI.8.16, 19-23) In my view, this insight into the ontologically self-grounding nature of the absolute self-consciousness is precisely what we need to answer Leibniz’ question as to why there is something rather than nothing. In this respect, then, Plotinus is a major inspiration for my approach to Absolute Idealism.

The mathematical aspect of Neoplatonism
But not only that; the insight into the link between mathematics and absolute self-consciousness can also be found already in Plotinus. This is, perhaps, not so surprising, given the well-known influence of Pythagoreanism on (Neo-)Platonic thought. The Pythagorean idea that numerical relations and geometrical forms are constitutive of reality was already dear to Plato himself, and only gained importance with the further development of Platonism. Thus the “emanation” of reality from the One was for all Neoplatonists also a mathematical process, a multi-leveled unfolding of increasing multiplicity out of a primordial unity. Plotinus was not unique in this. Neither was he unique in his technical development of mathematical ideas (in this respect, in fact, Plotinus was rather weak). He was unique, however, in the connection he forged between the self-consciousness of the One and the mathematical unfolding of emanation. Here he virtually anticipated Royce’s insight into the infinite recursivity of absolute self-consciousness as the generative source of the natural number system.

Plotinus and Royce
This becomes clear when Plotinus writes about the second hypostasis, Intellect, which is the first self-image generated by the self-consciousness of the One: “[W]hen it sees itself it does so not as without intelligence but as thinking. So that in its primary thinking it would have also the thinking that it thinks […].” (Ennead II.9.1, 49-59) Plotinus then goes on, in the same passage, to argue that we should not stop here, we should rather add “another, third, distinction in addition to the second one which said that it thinks that it thinks,” namely, “one which says that it thinks that it thinks that it thinks”. And then Plotinus asks rhetorically: “And why should one not go on introducing distinctions in this way to infinity?” Thus Plotinus clearly indicates that the recursion involved in Intellect’s self-thinking is endless and as such generates infinite multiplicity. In this way, one can say, the self-thinking of Intellect amounts to an endless self-multiplication.

In this way, Plotinus clearly anticipated Royce’s insight into the link between the natural number system and the infinite recursivity of absolute self-consciousness. In fact, I think that Plotinus took this insight a great deal further than Royce did. For Royce, this insight remained something of an afterthought – quite literally, as his ideas about the mathematical nature of absolute self-consciousness were only expressed in the “Supplementary Essay” to his The World and the Individual. Royce never fully embraced a Neopythagorean, mathematical view of the universe. Plotinus, of course, did embrace such a view, given his Neopythagorean commitments. For this reason, too, my approach to Absolute Idealism owes more to Plotinus than to Royce (the other reason being Plotinus’ insight into the self-causing nature of absolute self-consciousness, which is more or less lacking in Royce).

The self-reflection of the Absolute in Neoplatonism
There is also a third reason why I like Plotinus. Earlier I said that we can, perhaps, explain the physical universe as the computational self-image of absolute self-consciousness, i.e. as its self-reflection in the medium of mathematics. The fact of the matter is that this emphasis on creation as a self-imaging or self-reflection of the Absolute is also thoroughly Neoplatonic in nature. Emanation is for Plotinus essentially a process of imaging and re-presentation, where a higher reality creates a lower reality as its own image (thus material Nature is the image of Soul, which in turn is the image of Intellect, which finally is the image of the One). In this way, of course, Plotinus takes over, and develops further, the Platonic theory of participation, where empirical particulars are seen as the images or shadows of ideal archetypes.

Plotinus systematizes the Platonic theory by seeing the One as the ultimate archetype that creates, in successive stages, its own images (Intellect, Soul, Nature). Although Plotinus remains frustratingly implicit about this, it seems clear to me that this theme of imaging is intimately related to the self-consciousness of the One. That is to say: because the One is essentially self-consciousness, it creates images of itself, images in which it reflects itself and through which it enhances its own self-awareness. This seems to me the most logical interpretation of Plotinus’ theory of emanation, where each lower hypostasis is the image of the preceding hypostasis: this entire sequence of images is nothing but the unfolding of the primordial self-consciousness which is the self-caused essence of the One.

Neoplatonism as Absolute Idealism
One possible misunderstanding should be avoided: Plotinus' claim that each hypostasis produces an image of itself should not be understood as meaning that this image exists independent or outside of its source. For Plotinus makes it quite clear that each later hypostasis exists only inside the preceding hypostasis. Thus, Nature exists inside Soul, which in turn exists inside Intellect, which finally exists inside the One. In this way Plotinus can say that “all things belong to It [i.e. the One, PS] and are in It” (Ennead, V.4.2). In this way, Plotinus transformed Platonism in a thoroughgoing monism where only the One really exists and all other levels of reality are somehow produced inside the One as the Hen Kai Pan (“All-In-One”). Thus it becomes clear that Plotinus’ Neoplatonism is essentially a form of Absolute Idealism, since the One is for Plotinus nothing but the consciousness it has of itself. The entire sequence of self-images produced by the One should be seen as a sequence internal to the One, an internal unfolding of the One's self-contemplation.