Eric Steinhart

Nietzsche has a surprisingly significant and strikingly positive assessment of mathematics. I discuss Nietzsche's theory of the origin of mathematical practice in the division of the continuum of force, his theory of numbers, his conception of the finite and the infinite, and the relations between Nietzschean mathematics and formalism and intuitionism. I talk about the relations between math, illusion, life, and the will to truth. I distinguish life and world affirming mathematical practice from its ascetic perversion. For Nietzsche, math is an artistic and moral activity that has an essential role to play in the joyful wisdom.

I gather and constructively criticize Nietzsche’s writings on identity. Nietzsche treats identity as a logical fiction. He denies that there are any enduring things (no substances); he denies that there are any indiscernible things in any respect (no universals, no bare particulars). For Nietzsche, the world consists of durationless events bearing non-universal properties and standing to one another in non-universal relations. Events are bundles of tropes. Nietzsche even denies self-identity. His events are self-differing trope-bundles. I link Nietzsche’s denial of self-identity with modern treatments of paradox.

Teilhard is among the first to seriously explore the future of human evolution. He advocates both bio-technologies (e.g. genetic engineering) and intelligence technologies. He discusses the emergence of a global computation - communication system (and is said by some to have been the first to have envisioned the Internet). He advocates the development of a global society. He is almost surely the first to discuss the acceleration of technological progress to a Singularity in which human intelligence will become super-intelligence. He discusses the spread of human intelligence into the universe and its amplification into a cosmic-intelligence. His work has been taken up by Barrow and Tipler; Tipler; Moravec; and Kurzweil. Of course, Teilhard's Omega Point Theory is deeply Christian. For secular transhumanists, this may be difficult. But transhumanism cannot avoid a fateful engagement with Christianity. Christian institutions may support or oppose transhumanism. Since Christianity is an extremely powerful cultural force in the West, it is imperative for transhumanism to engage it carefully. A serious study of Teilhard can help that engagement and will thus be rewarding to both communities.

Neoplatonism is alive and well today. It expresses itself in New Thought and the mind-cure movements derived from it. However, to avoid many ancient errors, Neoplatonism needs to be modernized. The One is just the simple origin from which all complex things evolve. The Good, which is not the One, is the best of all possible propositions. A cosmological argument is given for the One and an ontological argument for the Good. The presence of the Good in every thing is Spirit. Spirit sits in the logical center of every body; it is surrounded by the regulatory forms of that body. Striving for the Good, Spirit seeks to correct the errors in its surrounding forms. To correct the errors in biological texts, modern Neoplatonists turn to the experimental method. This Neoplatonism is pantheistic not because of some theoretical definition of God but rather because of its practical focus on the shaping of Spirit.

You can survive after death in various kinds of artifacts. You can survive in diaries, photographs, sound recordings, and movies. But these artifacts record only superficial features of yourself. We are already close to the construction of programs that partially and approximately replicate entire human lives (by storing their memories and duplicating their personalities). A digital ghost is an artificially intelligent program that knows all about your life. It is an animated auto-biography. It replicates your patterns of belief and desire. You can survive after death in a digital ghost. We discuss a series of digital ghosts over the next 50 years. As time goes by and technology advances, they are progressively more perfect replicas of the lives of their original authors.

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals.

Ordinal polytheism is motivated by the cosmological and design arguments. It is also motivated by Leibnizian–Lewisian modal realism. Just as there are many universes, so there are many gods. Gods are necessary concrete grounds of universes. The god-universe relation is one-to-one. Ordinal polytheism argues for a hierarchy of ranks of ever more perfect gods, one rank for every ordinal number. Since there are no maximally perfect gods, ordinal polytheism avoids many of the familiar problems of monotheism. It links theology with counterpart theory, mathematics and computer science. And it entails that the system of universes has an attractive axiological structure

Nick Bostrom’s Simulation Argument (SA) has many intriguing theological implications. We work out some of them here. We show how the SA can be used to develop novel versions of the Cosmological and Design Arguments. We then develop some of the affinities between Bostrom's naturalistic theogony and more traditional theological topics. We look at the resurrection of the body and at theodicy. We conclude with some reflections on the relations between the SA and Neoplatonism (friendly) and between the SA and theism (less friendly).

Mathematics is obviously important in the sciences. And so it is likely to be equally important in any effort that aims to understand God in a scientifically significant way or that aims to clarify the relations between science and theology. The degree to which God has any perfection is absolutely infinite. We use contemporary mathematics to precisely define that absolute infinity. For any perfection, we use transfinite recursion to define an endlessly ascending series of degrees of that perfection. That series rises to an absolutely infinite degree of that perfection. God has that absolutely infinite degree. We focus on the perfections of knowledge, power, and benevolence. Our model of divine infinity thus builds a bridge between mathematics and theology.

After rejecting substance dualism, some naturalists embrace patternism. It states that persons are bodies and that bodies are material machines running abstract person programs. Following Aristotle, these person programs are souls. Patternists adopt four-dimensionalist theories of persistence: Bodies are 3D stages of 4D lives. Patternism permits at least six types of life after death. It permits quantum immortality, teleportation, salvation through advanced technology, promotion out of a simulated reality, computational monadology, and the revision theory of resurrection.

A powerful argument against the resurrection of the body is based on the premise that all resurrection theories violate natural laws. We counter this argument by developing a fully naturalistic resurrection theory. We refer to it as the revision theory of resurrection (the RTR). Since Hick’s replica theory is already highly naturalistic, we use Hick’s theory as the basis for the RTR. According to Hick, resurrection is the recreation of an earthly body in another universe. The recreation is a resurrection counterpart. We show that the New Testament supports the idea of resurrection counterparts. The RTR asserts that you are a node in a branching tree of increasingly perfect resurrection counterparts. These ever better counterparts live in increasingly perfect resurrection universes. We give both theological arguments and an empirical argument for the RTR.

If the computational theory of mind is right, then minds are realized by machines. There is an ordered complexity hierarchy of machines. Some finite machines realize finitely complex minds; some Turing machines realize potentially infinitely complex minds. There are many logically possible machines whose powers exceed the Church–Turing limit (e.g. accelerating Turing machines). Some of these supermachines realize superminds. Superminds perform cognitive supertasks. Their thoughts are formed in infinitary languages. They perceive and manipulate the infinite detail of fractal objects. They have infinitely complex bodies. Transfinite games anchor their social relations.

Pantheism claims: (1) there exists an all-inclusive unity; and (2) that unity is divine. I review three current and scientifically viable ontologies to see how pantheism can be developed in each. They are: (1) materialism; (2) Platonism; and (3) class-theoretic Pythagoreanism. I show how each ontology has an all-inclusive unity. I check the degree to which that unity is: eternal, infinite, complex, necessary, plentiful, self-representative, holy. I show how each ontology solves the problem of evil (its theodicy) and provides for salvation (its soteriology). I conclude that Platonism and Pythagoreanism have the most divine all-inclusive unities. They support sophisticated contemporary pantheisms. (Published Online February 17 2004).

I discuss the view, increasingly common in physics, that the foundational level of our physical reality is a network of computing machines (so that our universe is ultimately like a cellular automaton). I discuss finitely extended and divided (discrete) space-time and discrete causality. I examine reasons for thinking that the foundational computational complexity of our universe is finite. I discuss the emergence of an ordered complexity hierarchy of levels of objects over the foundational level and I show how the special sciences study these emergent objects.

Many recent writers have developed a rich system of theological concepts inspired by computers. This is digital theology. Digital theology shares many elements of its eschatology with Christian post-millenarianism. It promises a utopian perfection via technological progress. Modifying Christian soteriology, digital theology makes reference to four types of immortality. I look critically at each type. The first involves transferring our minds from our natural bodies to superior computerized bodies. The second and third types involve bringing into being a previously living person, or person who has never existed, within an artificial digital environment. The fourth involves promotion of our lives into some higher level computational reality.

At the end of the 19th century, Josiah Royce participated in what has come to be called the great debate (Royce, 1897; Armour, 2005).1 The great debate concerned issues in metaphysical theology, and, since metaphysics was primarily idealistic, it dealt considerably with the relations between the divine Self and lesser selves. After the great debate, Royce developed his idealism in his Gifford Lectures (1898-1900). These were published as The World and the Individual. At the end of the first volume, Royce added a Supplementary Essay. The Essay, continuing themes from the great debate, developed an intriguing mathematical model of the relations between the divine Self and lesser selves. The second volume went on to..

A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of gods. An alternative is that, for every ordinal, there is a god whose perfection is proportional to it. The n -th god actualizes the best universe(s) in the n -th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study.

Infinite machines (IMs) can do supertasks. A supertask is an infinite series of operations done in some finite time. Whether or not our universe contains any IMs, they are worthy of study as upper bounds on finite machines. We introduce IMs and describe some of their physical and psychological aspects. An accelerating Turing machine (an ATM) is a Turing machine that performs every next operation twice as fast. It can carry out infinitely many operations in finite time. Many ATMs can be connected together to form networks of infinitely powerful agents. A network of ATMs can also be thought of as the control system for an infinitely complex robot. We describe a robot with a dense network of ATMs for its retinas, its brain, and its motor controllers. Such a robot can perform psychological supertasks - it can perceive infinitely detailed objects in all their detail; it can formulate infinite plans; it can make infinitely precise movements. An endless hierarchy of IMs might realize a deep notion of intelligent computing everywhere.

Skeptical atheism is deeply concerned with the development of a true atheistic belief-system which competes with allegedly false theistic belief-systems. Spiritual atheists are concerned with building a successful atheistic culture to compete with an allegedly dysfunctional theistic culture. These atheisms are compared in terms of their epistemologies, metaphysics, axiologies, eschatologies, soteriologies, prosocial activities, and individual practices. Skeptical atheism is likely to remain a perpetually marginal community. Spiritual atheism may become a significant alternative to theism.

Structural idealism uses formal and computational techniques to describe an idealist ontology composed of God and a set of finite minds. A finite mind is a system of private intentional worlds. An intentional world is a connectionist hierarchy of intentional objects (propositions, concepts, sensible things, sensations). Intentional objects, similar to Leibnizian monads, are computing machines. To escape the egocentric predicament, Leibnizian relations of (in)compossibility exist between finite minds, linking them together into a constraint-satisfaction network, thereby coordinating their private intentional worlds.

Many religions and religious philosophies say that ultimate reality is a kind of primal energy. This energy is often described as a vital power animating living things, as a spiritual force directing the organization of matter, or as a divine creative power which generates all things. By refuting older conceptions of primal energy, modern science opens the door to new and more precise conceptions. Primal energy is referred to here as ‘spirit’. But spirit is a natural power. A naturalistic theory of spirit is developed using ideas from information theory and thermodynamics, such as the maximum entropy production principle. Spirit drives the evolution of complexity at all levels of existence.

I use modal logic and transfinite set-theory to define metaphysical foundations for a general theory of computation. A possible universe is a certain kind of situation; a situation is a set of facts. An algorithm is a certain kind of inductively defined property. A machine is a series of situations that instantiates an algorithm in a certain way. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g., Turing and super-Turing machines and more). There are physically and metaphysically possible machines. There is an iterative hierarchy of logically possible machines in the iterative hierarchy of sets. Some algorithms are such that machines that instantiate them are minds. So there is an iterative hierarchy of finitely and transfinitely complex minds.

Metaphors have computable semantics. A program called NETMET both generates metaphors and produces partial literal interpretations of metaphors. NETMET is based on Kittay's semantic field theory of metaphor and Black's interaction theory of metaphor. Input to NETMET consists of a list of literal propositions. NETMET creates metaphors by finding topic and source semantic fields, producing an analogical map from source to topic, then generating utterances in which terms in the source are identified with or predicated of terms in the topic. Given a metaphor, NETMET utilizes if-then rules to generate the implication complex of that metaphor. The literal leaves of the implication complex comprise a partial literal interpretation.

It has often been said that metaphors are based on analogies, but the nature of this relation has never been made precise. This article rigorously and formally specifies two semantic relations that do obtain between some metaphors and analogies. We argue that analogies often provide conditions of meaningfulness and truth for metaphors. An analogy is treated as an isomorphism from a source to topic domain. Metaphors are thought of as surface structures. Formal analogical conditions of meaningfulness and truth are fully and rigorously worked out for several grammatical classes of metaphors. By providing analogical meaningfulness and truth conditions for metaphors, this article shows that truth-conditional semantics can be extended to metaphors.

The game of life is an excellent framework for metaphysical modeling. It can be used to study ontological categories like space, time, causality, persistence, substance, emergence, and supervenience. It is often said that there are many levels of existence in the game of life. Objects like the glider are said to exist on higher levels. Our goal here is to work out a precise formalization of the thesis that there are various levels of existence in the game of life. To formalize this thesis, we develop a set-theoretic construction of the glider. The method of this construction generalizes to other patterns in the game of life. And it can be extended to more realistic physical systems. The result is a highly general method for the set-theoretical construction of substances

An Omega Point Theory says that reality is making progress from some initial state to some final state. It moves from some Alpha Point (the initial state) to some Omega Point (the final state). The progress is an increase in some quality. For example, reality is making progress from the chaotic to the orderly; or it is making progress from the simple to the complex; or from the mindless to the mental; or from evil to good. Here we focus on the Omega Point theory of Peirce. An Omega Point Theory is an evolutionary cosmology – it says that the universe is evolving from the Alpha Point to the Omega Point. Peirce refers to his evolutionary cosmology as a Cosmogonic Philosophy would suppose that in the beginning – infinitely remote – there was a chaos of unpersonalized feeling, which being without connection or regularity would properly be without existence. This feeling, sporting here and there in pure arbitrariness, would have started the germ of a generalizing tendency. Its other sportings would be evanescent, but this would have a growing virtue. Thus the tendency to habit would be started; and from this, with the other principles of evolution, all the regularities of the universe would be evolved. At any time, however, an element of pure chance survives and will remain until the world becomes an absolutely perfect, rational, and symmetrical system, in which mind is at last crystallized in the infinitely distant future. (Peirce, Collected Papers , 6.33)

Many writers advocate the development of new and more naturalistic religions.1 Perhaps these new religions will emerge from religious naturalism. Peters believes that religious naturalism “could lead to a new significant form of organized religion with a structured community, ritual practices, and ways of moral living.”2 However, at the present time, religious naturalism is not a nature-centered religion. The features mentioned by Peters are mainly missing.3 At the present time, the most significant effort to derive a nature-centered religion from religious naturalism is found in Crosby. Over the course of several books, Crosby lays out his metaphysical theory of nature.4 He uses that theory to develop a...

Our digital technologies have inspired new ways of thinking about old religious topics. Digitalists include computer scientists, transhumanists, singularitarians, and futurists. Digitalists have worked out novel and entirely naturalistic ways of thinking about bodies, minds, souls, universes, gods, and life after death. Your Digital Afterlives starts with three digitalist theories of life after death. It examines personality capture, body uploading, and promotion to higher levels of simulation. It then examines the idea that reality itself is ultimately a system of self-surpassing computations. On that view, you will have infinitely many digital lives across infinitely many digital worlds. Your Digital Afterlives looks at superhuman bodies and infinite bodies. Thinking of nature in purely computational terms has the potential to radically and positively change our understanding of life after death.