R. L. Goodstein

The Decision Problem

Alan Turing’s 1936 paper “On Computable Numbers with an Application to the Entscheidungsproblem” [3] proved most influential not only for mathematical logic, but also for the development of the programmable computer, and together with work of Alonzo Church (1903–1995) [1, 2] inaugurated a new field of study, known today as computability. Recall that Turing’s original motivation for writing the paper was to answer the decision problem of David Hilbert (1862–1943), which asked whether there is a standard procedure that can be applied to decide whether an arbitrary statement (within some system of logic) is provable. A previous project examined the construction of Turing’s “universal computing machine,” which accepts the instructions of any other machine M in standard form, and then outputs the same sequence as M . The concept of a universal machine has evolved into what now is known as a compiler or interpreter in computer science, and is indispensable for the processing of any programming language. The question then arises, does the universal computing machine provide a solution to the decision problem? The universal machine is the standard procedure for answering all questions that can in turn be phrased in terms of a computer program. First, study the following excerpts from Turing’s paper [3, p. 232–233]

AXIOMATIC THEORY OF SETS

This chapter discusses the axiomatic theory of sets. It adds universal quantification denoted by “∀ x ” (read as “for all x”) and existential quantification, denoted by “∃#” (read as “there is an x”) to the logical operations. Thus, given some property Px of a real number x it writes (∀x) Px to denote the false sentence “for all x , x is an even prime” and it writes (∃x) P ( x ) to denote the true sentence “there is an x such that x is an even prime.” Clearly, (∀ x ) Px→ Py is true for every property P , and any y and so is the sentence P y →(∃ x )P x . Primitive sentences are of the form x ∈ y, and every sentence is formed from primitive sentences by quantification, negation, disjunction, conjunction, and equivalence.

Remarks on the Foundations of Mathematics

Wittgensteins work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

The Mathematical Mind

ion and Imagination: Montessori tells us that Abstraction and Imagination “... play a mutual part in the construction of the mind’s content. ... Of its nature, the mind not only has the power to imagine (i.e., to think of things not immediately present), but it can also assemble and rearrange its mental content, extract – let us say – an “alphabet of qualities” from all those numberless things that we meet in the outside world. 4 Exactness: Montessori highlights the significance of Exactness for the Abstractions which form the basis of the Imagination, stating that “... abstract ideas are always limited in number, while the real things we encounter are innumerable. These limited abstractions increase in value with their precision. In the world of the mind, they come to have the value of a special organ, an instrument of thought which serves to give us our bearings in space, just as a watch gives us our bearings in time.” 5 Montessori seems to summarize the action and interaction of these Tendencies in the following: If we study the works of all who have left their marks on the world in the form of inventions useful to mankind, we see that the starting point was always something orderly and exact in their minds, and that this was what enabled them to create something new. Even in the imaginative worlds of poetry and music, there is a basic order so exact as to be called “metrical” or measured. 6 is that as the mind matures from age six onwards, it can intentionally decide to explore and analyze specific experiences in the environment to the exclusion of other available experiences, whereas the first plane mind seems to lack such discriminatory specificity. 3 The Absorbent Mind p. 185 4 The Absorbent Mind p. 184 5 The Absorbent Mind p. 184 6 The Absorbent Mind p. 185. To be ‘mathematical’, then, is not necessarily the same as ‘being good at math’; and it does not necessarily the same as having well-developed skills in arithmetic: instead, to be mathematical is a natural and universal characteristic of the human mind itself – a birthright of every child and the characteristic mind of every human being, of every time and culture. The Mathematical Mind Montessori Northwest Primary Course 37, 2013-14

Elementary Differential Equations

1. Introduction to Differential Equations. 2. First-Order Equations. 3. Second and Higher-Order Linear Differential Equations. 4. Some Physical Applications of Linear Differential Equations. 5. Power Series Solutions of Differential Equations. 6. Laplace Transforms. 7. Introduction to Systems of Linear Differential Equations and Applications. 8. Numerical Methods. 9. Matrix Methods for Systems of Differential Equations. 10. Nonlinear Equations and Stability. 11. Fourier Series and Boundary Value Problems. 12. Partial Differential Equations. Appendices.

The Elements of Mathematical Logic

An excellent introduction to mathematical logic, this book provides readers with a sound knowledge of the most important approaches to the subject, stressing the use of logical methods in attacking nontrivial problems. It covers the logic of classes, of propositions, of propositional functions, and the general syntax of language, with a brief introduction that also illustrates applications to so-called undecidability and incompleteness theorems. Other topics include the simple proof of the completeness of the theory of combinations, Churchs theorem on the recursive unsolvability of the decision problem for the restricted function calculus, and the demonstrable properties of a formal system as a criterion for its acceptability. 1950 ed