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A Taxonomy of Singularities: Comparing the Literature on Systems of Accelerating Change



General Systems Theory



A Preliminary Singularity Taxonomy: Six Types



Singularity Research: A Multidisciplinary Activity


As we explore the technological singularity concept from a widely multidisciplinary, systems theory perspective, we should attempt to interrelate, if possible, known physical processes that appear to have similar accelerating dynamics. An early effort in this regard can be found below.


General Systems Theory

General systems theory is a conceptual paradigm which proposes a number of common processes and constraints that apply across a wide range of physical systems, from biology to economies, from molecules to minds. These commonalities arise in part because each system operates within the same universal parameters and boundary conditions, and each appears to be part of some common universally-guided developmental process.

Intrinsic to systems theory is the expectation that great conceptual and mathematical simplicities must emerge from cross-disciplinary comparisons. This curious phenomenon can be called the Cosmic Watermark Hypothesis, or Wigner's Ladder, (after the physicist Eugene Wigner) and has been observed by many careful thinkers since the dawn of natural philosophy.

There is a menagerie of singularity literatures that we may profitably consider. All singularities are models of a process of change that accelerates toward some perceptively 'infinite' or otherwise uncomputable, irreversible point, leading to a regime exhibiting new properties that are incomprehensible from within the dynamics of the original system, a state in which new global rules and dynamics emerge.



A Preliminary Singularity Taxonomy: Six Types

Briefly, the following six singularity types, in four apparent general classes (mathematical, physical, computational, and developmental), each have their own semi-independent literature. Comparing and contrasting them can be quite illuminating.

1. Mathematical singularities (the oldest singularity literature available) are systems of equations that lead, under certain conditions, to infinities, uncomputability, or irreversible emergences. Such singularities are hidden in the mathematics of Isaac Newton and Albert Einstein, and were uncovered by the impressive theoretical work of Karl Schwarzchild, Roger Penrose and Stephen Hawking, and have been more recently explored by Mikhail Zak's work with "terminal chaos" in differential equations, through singularities in topology theory, and within a number of other domains. Math has been called "the poetry from which physics springs," and most mathematics may reflect more the computational capacities of the human (or technological) mind than the objective nature of reality. Only a small subset of possible mathematical singularities reflect known physical processes. To our knowledge, so far there have been no published attempts to explain the coming technological singularity in the language of mathematical singularities, though such work is clearly needed.

2. Physical (a.k.a., "dynamical" or "discontinuous" or "disruptive") singularities include symmetry breaking, phase changes, punctuations, self-organized criticality, catastrophe points, and other rapidly emergent and discontinous nonlinear behavior in real physical systems, including the emergence of new physical law. Dynamically discontinous systems have been notably explored by Per Bak, Rene Thom, and various complexity and nonlinear systems theorists. Neural network theorists who model exponential convergence to new system equilibria, and the ecological psychology academic community also occasionally use physical singularity models. The study of singularities in a range of real and theoretical nonlinear dynamical systems (e.g., fluid dynamics, supernova) is both challenging and a frontier of physical theory. The discontinuity, punctuation, new equilibria emergence, or rupture-with-the-past aspect of the physical singularity is also an important component of the technological singularity proposal, as we will discuss.

3. Cosmological (a.k.a., "astrophysical" or "spacetime") singularities are perhaps most generally understood as a particular type of physical and mathematical singularity, though they also have computational and developmental singularity properties. They include black holes (primordial, quasar, stellar, supermassive, extreme), white holes (speculative), and Big Bangs, as well as now doubtful Big Crunches. They were incrementally deduced in theory by such notable scholars as John Michell, Subramanyan Chandrasekhar, Robert Oppenheimer, Roy Kerr, Edwin Salpeter, Yakov Zel'dovich, John Archibald Wheeler, Roger Penrose and Stephen Hawking. At the same time, both our universal Big Bang and black holes have been progessivelly experimentally confirmed by our intrepid astronomical community. We here at ASF would suggest that cosmological singularities may be our best current model for understanding the global attractors (endpoints, in this universe) for a variety of different processes of universal development, including, in the developmental singularity hypothesis, the destiny of cosmic intelligence itself. Such ideas, of course, still need extensive theoretical and empirical validation before they might move beyond the realm of informed speculation. Nevertheless, mounting circumstantial evidence suggests that such validation may soon be forthcoming.

4. Computational (a.k.a., "cognitive", "simulational" or "informational") singularities can be considered as the informational dimension of mathematical, physical-dynamical, and cosmological singularities. They assume the proposition of the universe-as-a-computing-system. This assumption, also known as the "infopomorphic" paradigm, proposes that information processing is both a process and purpose intrinsic to all physical stuctures, and that the interrelationships between certain information processing structures can occasionally undergo irreversible, accelerating discontinuities. Many of our present information theory ideas were pioneered by such visionary thinkers as Alan Turing, John Von Neumann, and Claude Shannon, and have been extended by John Wheeler, Ed Fredkin, Stephen Wolfram, and other complexity and systems theorists, as well as by speculative philosophers and physicists like Frank Tipler.

Computational singularities occur when a mode of simulation or computation used by any discrete adaptive physical system undergoes an irreversible change in the way it processes information, experiencing a type of phase transition to a new regime. As one example, solitary insects simulate their external world in a particular algorithmic way. Social insects (such as bees, ants, or termites) add a whole new layer of simulation complexity, or "swarm computation," to the insect world view. The shift in reference frame between these two simulation systems apparently represents a computational singularity, one among a long chain of such singularities that have led to the developmental emergence of human consciousness within a very special class of Earth's physical-computational systems. Each operates in a relatively discrete computational domain, and organisms in one domain (say, an ant, or a chimpanzee) cannot meaningfully understand certain simulations occurring in another, once the latter's simulation system has become sufficiently quantitatively or qualitatively different. Also known as cognitive singularities, these play an important role in understanding how we will relate to the interior world of our increasingly self-aware technology in coming decades.

5. Developmental (a.k.a., "convergent," "hierarchical", "asymptotically accelerating," or "reproductive") singularities, are a refinement of the computational singularity model, which is itself an extension of the three previous types of singularities. Developmental singularities extend the "purposeful" nature of computational singularity models. They do this by assuming that not only is information processing a purpose of physical systems, but universal development is rigged so that continuous accelerating information processing will arise within a special subset of hierarchically emergent forms. An extension of the infopomorphic paradigm, developmental singularity models propose that that the universe "purposefully" self-generates a range of different semi-autonomous computing, universe-simulating "substrates" within its unfolding structure. Perhaps it does this in a developmentally-guided attempt to model itself more comprehensively over successive cycles, or perhaps for some other purpose yet unknown.

Examples of such hierarchically emergent structures would include our irreversible movement from galaxies to genes, from molecules to minds, or from cave painting to computers. Apparently, this special emergent subset is always subject to significant STEM compression with each new universal emergence, though this speculation remains to be rigorously critiqued. Developmental singularity elements or variations have been proposed by a number of systems thinkers, including Carl Sagan, Eric Chaisson, Andrei Linde, Valeri Frolov, Alan Guth, Lee Smolin, Martin Rees, Seth Lloyd, John Gribbin, Edward Harrison, Bela Balazs (see "The Role of Life in the Cosmological Replication Cycle," 2001), James Gardner (Biocosm, 2003), and John Smart (myself). The convergence, hierarchical emergence, asymptotically accelerating change, and reproductive components of the developmental singularity "meme complex" (idea set) are also all important aspects of the technological singularity idea.

6. Technological (a.k.a., "human competitive" or "effective machine consciousness") singularities, are commonly considered a composite of developmental, physical, and computational singularity types, though they also have mathematical and even cosmological singularity properties (e.g., Chaisson's Phi trends). The coming emergence of human-surpassing artificial intelligence, the "A.I. singularity," represents the usual meaning of the term "singularity" when used generically by futurists. In our research so far, the technological singularity proposal can be deconstructed into at least seven discrete concepts. Note that five of these are easily understood within the developmental singularity idea set, which is itself an amalgam of other singularity types. While there are other qualities we could consider, these seven may provide a useful introduction to the nature of the coming technological singularity:

1) "singular" human-competitive A.I. emergence (a.k.a. the "A.I. singularity," can be seen as hierarchical emergence, a developmental singularity)
2) discontinuity (or "punctuation," also a property of physical singularities)
3) unknowability (a property shared with computational singularities)
4) convergent (predictably emergent, a property of developmental singularities)
5) hierarchical (adding to a chain of previous emergences, a property of developmental singularities)
6) 'instantaneity' (or more accurately: a sustained period of increasingly rapid acceleration in special emergent systems (think of ovulation, at the cellular level), also a property of developmental singularities)
7) reproductive (able to recreate itself with important new variations, also a property of developmental singularities)

Singular, discontinous, unknowable, convergent, hierarchical, instantaneous, and reproductive all denote interesting properties associated with a number of other known singularities/phase changes in general. So while the technological singularity concept is a complex amalgam, in the process of unpacking it we find only a small kernel of unique new information has been added to the conceptual landscape.

The word "singular", in the first component, refers to the new, unique, and one-time-only nature of the emergence of any developmental singularity. This aspect is perhaps the simplest and most frequently overlooked dimension of the technological singularity concept. Yet this contains the only truly unique idea that the hypothesis proposes, namely that general technological intelligence must soon capture and permanently exceed even the highest-level features of human biological intelligence and autonomy.

In futurist and transhumanist literature, this specific idea has also been called the "A.I. singularity." This simple outcompetition does not itself comprise a singularity, but it is the technological singularity's most original contribution to the general singularity literature discussed above. The A.I. singularity idea proposes that there is one imminent singular time on Earth when technological intelligence will surpass human biological intelligence as the dominant form of local computation.

It will be difficult to know exactly when this competitive and permanent event will have occurred, so various measures have been proposed. A machine solution to a generalized Turing Test of "effective consciousness" (regardless of the actual inner subjective state of the system) is most commonly suggested as a useful indication of its arrival, though there are problems with this measure, as might be expected.

Note also that the A.I. technological singularity idea does not require that the imminent runaway of local intelligence must have some universal significance, but leaves that issue as an outstanding question. Prominent explorers and advocates of the technological singularity idea have been John Von Neumann, I.J. Good, Hans Moravec, Vernor Vinge, Danny Hillis, Eliezer Yudkowsky, Damien Broderick, Ben Goertzel, John Smart, a small number of other futurists, and most eloquently to date, Ray Kurzweil, in his book summary, "The Law of Accelerating Returns".



Singularity Research: A Multidisciplinary Activity

Mathematical singularities have been studied for over a century within theoretical mathematics. Physical and cosmological singularities have also been studied for over seventy years within the applied disciplines of general relativity, mathematical physics, and quantum physics, whose equations treat particles as singular "point masses," and require singularity-like "renormalizations." In the last three decades, quantitative efforts to study various mathematical, physical-dynamical, and computational singularity processes have emerged in the fields of complexity, non-linear science and theoretical computer science. Recently, discoveries of underlying simplicity in network theory (see Duncan Watts, Small Worlds, 1999 or Albert Barabasi, Linked, 2002) have also helped us understand that small linkage changes in networks can create major, accelerating changes in the way the network processes information, e.g., allowing a computational singularity, or new "small world" to emerge.

For cognitive singularities within biological organisms, the disciplines of evolutionary psychology, ecological psychology, and cognitive science are also promising new areas of exploration. These fields are also usefully informed, unlike most philosophy of mind, with an increasingly developmental perspective. Finally, the A.I. singularity can be explored within the many pastures of artificial intelligence, including but not limited to such disciplines as cognitive science, computer science, information theory, and computational neurobiology.

Unfortunately, neither the full definition of the technological singularity nor the main elements of the developmental singularity hypothesis are yet studied in formal academic programs. That intellectual oversight is a state of affairs that our new nonprofit, the Acceleration Studies Foundation, hopes will change in coming years. Nevertheless, there are both specialist and generalist degree programs that would provide valuable preparation for such work. In preparation for generalist study of accelerating change and phase transitions (singularities), the disciplines of future studies, science and technology studies (STS), systems theory, complexity studies, information studies, evolutionary and developmental biology, evolutionary and biologically-inspired computation, and the new field of astrobiology are all promising avenues for wide-ranging exploration of universal accelerating change.

Mathematical singularity inquiries, though the oldest singularity literature in existence, still remain in an early and fractionated stage of exploration, though there have been promising long-duration conferences (e.g., Singularity Theory, July-Dec, 2000) which demonstrate the continued development of this field. Likewise, physical singularity processes, though extensively explored by complexity researchers and physicists in recent decades, remain a rich and poorly synthesized area of investigation. One early attempt at a mathematical and physical singularity unification of sorts, "Catastrophe Theory," emerged in the 1960's. Rene Thom made perhaps the first systematic mathematical categorization of various classes of physical systems where small changes in the control variables might lead to radical, discontinuous change in system properties. "Catastrophe" was therefore used by Thom as a very limited subset of its most general definition (e.g., "catastrophism" as violent and sudden feature change). As catastrophists know, such change can be as much a mechanism of emergent, constructive phenomena, as of destructive ones. Furthermore, many events that appear destructive on one level, such as natural catastrophes, or even natural selection, are quite constructive on another, as in the preservation of life, and the acceleration of computational abilities amongst a special subset of the survivors.

So while catastrophe theory was a good initial effort at developing a kind of physically-based, mathematically-grounded phase transition paradigm, it soon fell out of favor for not being sufficiently predictive. This is probably because, as presently formulated, it does not help to show where and when catastrophes catalyze the development of new complexity. In other words, it has become clear in recent years that many common "catastrophes" lead statistically to the accelerating development of all the most interesting and complex phenomena in our universe.

Consider the unmistakable acceleration of various types of chemical, biological and social complexity that has been observed immediately after, for example, a supernova (creating chemical complexity), a meteor impact (catalyzing biological complexity), or ice age (catalyzing social complexity). Such catastrophes may be either externally imposed (e.g., the K-T meteorite), or internally generated (e.g., the Permian extinction), and all are co-evolutionary in interesting ways (in a plague, for example, both the host and the pathogen are modified substantially).

Consider even the minor catastrophes of civilized life. We invariably do our best work under deadline, and even our office workflow and environment move rapidly to the point of "self-organized criticality," where a small system change can cause a "catastrophic" result (see Mark Buchanan's Ubiquity, 2002). Systems theories incorporating self-organization, complexity, self-organized criticality, cosmology, evolutionary developmental biology, and some of the more recent integrative non-linear science, such as ecological psychology, are now leading areas for deepening our understanding of the nature and role of singularities in universal change.

Don't worry if much of this is still unclear to you. It is the nature of the subject, and we are only now beginning to uncover the full meaning, variety, and interrelationships of singularity models. There may be several more (or less) varieties than the four fundamental types (mathematical, physical, computational, and developmental) of singularity that we have proposed above, and all are likely to be unified by a future generalized theory of information.

It is interesting and potentially very important that both the computational and developmental singularity models give us a way to place our present rapidly-developing technological intelligence into a universal context. In particular, developmental singularity models can propose, in a testable and statistically predictable manner, what must occur in our local environment soon after our machines wake up, due to universal — and still-very-poorly-understood — developmental physical constraints on all information processing systems.

Though accessible for over a decade, developmental singularity ideas are still largely new to those who write about the technological singularity. Much work remains to be done to highlight their potential utility among those who presently study the development of science and technology.

I am hopeful that my forthcoming book on this topic, Journey and Destiny , will be at least partially useful in that regard. Until my book is available, you may wish to investigate the developmental singularity hypothesis in the speculative section of this site.