Eric Steinhart

Why Numbers Are Sets

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 Neumanns (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.

Logically Possible Machines

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.

Singularity Hypotheses: An Overview

Bill Joy in a widely read but controversial article claimed that the most powerful 21st century technologies are threatening to make humans an endangered species. Indeed, a growing number of scientists, philosophers and forecasters insist that the accelerating progress in disruptive technologies such as artificial intelligence, robotics, genetic engineering, and nanotechnology may lead to what they refer to as the technological singularity: an event or phase that will radically change human civilization, and perhaps even human nature itself, before the middle of the 21st century.

Emergent values for automatons: Ethical problems of life in the generalized Internet

The infrastructure is becoming a network of computerized machines regulated by societies of self-directing software agents. Complexity encourages the emergence of novel values in software agent societies. Interdependent human and software political orders cohabitate and coevolve in a symbiosis of freedoms.

Supermachines and Superminds

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.

Theological Implications of the Simulation Argument

Abstract 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 a_nities 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).

Infinitely Complex Machines

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.

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 (RTR). Since Hicks replica theory is already highly naturalistic, we use Hicks 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.

Pantheism and current ontology

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.

Analogy and Truth

I show how metaphors have non-trivial logical truth-conditions. Section 2 sets out non-trivial truth-conditions for assertions of analogy. Section 3 explains how utterances have both literal and metaphorical meanings. Section 4 develops intensional (possible worlds) truth-conditions for metaphors based on the existence of counterparts in analogous situations (parts of possible worlds).1 Appendix 7.1 lays out some of the formal extended predicate calculus (XPC) machinery for the analogical truth-conditions of section 4. It’s an appendix because it’s technical and depends on the discussion in Appendix 2.1.