The Russellian Influence on Hilbert and his School

The aim of the paper is to discuss the influence exercised by Russell's thought inGöttingen in the period leading to the formulation of Hilbert's program in theearly twenties. I show that after a period of intense foundational work, culminatingwith the departure from Göttingen of Zermelo and Grelling in 1910 we witnessa reemergence of interest in foundations of mathematics towards the end of 1914. Itis this second period of foundational work that is my specific interest. Through theuse of unpublished archival sources I will describe how Hilbert, Behmann, and Bernays,among others, were influenced by and reacted to the technical and philosophical thesespresented in Principia Mathematica. I also argue that there are some elements of continuity between Russell's approach and Hilbert's program as it was presented inthe early twenties.     

The Pragmatism of Hilbert's Programme

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics' assuggested by Penelope Maddy. The answer is ``no', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind' andrelating it to the ``one and only axiom' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper.     

Hilbert and the internal logic of mathematics

Hilbert's programme is shown to have been inspired in part by what we can call Kronecker's programme in the foundations of an arithmetic theory of algebraic quantities.While finitism stays within the bounds of intuitive finite arithmetic, metamathematics goes beyond in the hope of recovering classical logic. The leap into the transfinite proved to be hazardous, not only from the perspective of Gödel's results, but also from a Kroneckerian point of view.Hilbert's rare admission of a Kroneckerian influence does not constitute the basis of such a reconstruction; it is rather Kronecker's mathematical practice which is seen as a forerunner of Hilbert's endeavour in the foundations of mathematics.I am indebted to an anonymous referee for many helpful critical remarks.     

Hilbert's epsilon as an operator of indefinite committed choice

Hilbert and Bernays avoided overspecification of Hilbert's ε-operator. They axiomatized only what was relevant for their proof-theoretic investigations. Semantically, this left the ε-operator underspecified. After briefly reviewing the literature on semantics of Hilbert's epsilon operator, we propose a new semantics with the following features: We avoid overspecification (such as right-uniqueness), but admit indefinite choice, committed choice, and classical logics. Moreover, our semantics for the ε simplifies proof search and is natural in the sense that it mirrors some cases of referential interpretation of indefinite articles in natural language.     

Infinitary propositional normal modal logic

A logic with normal modal operators and countable infinite conjunctions and disjunctions is introduced. A Hilbert's style axiomatization is proved complete for this logic, as well as for countable sublogics and subtheories. It is also shown that the logic has the interpolation property.     

On interpreting Gödel's Second Theorem

Summary In this paper I have considered various attempts to attribute significance to G2.²⁵ Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.I would like to thank Dale Gottlieb, Stephen Barker, Tim McCarthy, Philip Kitcher, Michael Resnik and Richmond Thomason for extensive and helpful discussion of this work.     

On a hitherto unexploited extension of the finitary standpoint

P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary standpoint by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other (starting with trivial functions).     

Husserl's Two Notions Of Completeness

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Göttingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.     

The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program

After a brief flirtation with logicism around 1917, David Hilbertproposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays andWilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for everstronger and more comprehensive areas of mathematics, and finitisticproofs of consistency of these systems. Early advances in these areaswere made by Hilbert (and Bernays) in a series of lecture courses atthe University of Göttingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was theinvention of the -calculus, on which Hilbert's axiom systemswere based, and the development of the -substitution methodas a basis for consistency proofs. The paper traces the developmentof the ``simultaneous development of logic and mathematics' throughthe -notation and provides an analysis of Ackermann'sconsistency proofs for primitive recursive arithmetic and for thefirst comprehensive mathematical system, the latter using thesubstitution method. It is striking that these proofs use transfiniteinduction not dissimilar to that used in Gentzen's later consistencyproof as well as non-primitive recursive definitions, and that thesemethods were accepted as finitistic at the time.     

A multiprogram approach to real-time experimentation

A technique for implementing real-time experiments utilizing several separate and independent programs is discussed. The technique can be employed when the experimental procedure is divided into real-time and nonreal-time program units. These program units are executed in the most appropriate computing environment, and the resulting output is directed to a sequential file. The nonreal-time program issues commands to the real-time program by means of keywords that are inserted in the sequential file output at those points where real-time operations are required. These commands are interpreted by the real-time program which processes the sequential file. While the nonreal-time program is experiment-specific, the real-time program is independent of the particular experiment performed. The applications of this multiprogram approach to real-time experimentation are considered.     

Utilization of program evaluation information in public schools: Perspectives and guidelines for school psychologists

The ability of program evaluators in public schools to design evaluations the result in useful evaluation information is central to the advancement of the field of program evaluation. Unless evaluation information is utilized by program decision makers, the worth of an evaluation can be questioned. In this paper, the issue of utilization of program evaluation information in public schools is addressed. First, perspectives on utilization of evaluation information are provided that are derived from knowledge utilization theory and research. Second, a set of program evaluation guidelines are described that are intended to help enhance utilization of evaluation information by program decision makers. Third, a meta-evaluation of the application of the guidelines in public school settings is reported.     

The Viability of Organizational Wellness Programs: An Examination of Promotion and Results

Wellness programs in contemporary organizations are increasing in number, but attendance is low and results often are difficult to track. We examined participant and program characteristics in 2 organizations that offered extensive wellness activities. One organization considered its program successful and had been in operation for 10 years, while the other organization's program ran for 2 years and ended because of a lack of funding. In the successful program, wellness program attendance showed a relationship to time employees took off for sick days, and improved fitness produced similar results. In the less successful operation, interviews were held with mangers and employees regarding promotion and program support. Suggestions are provided for future wellness programs.     

Training school personnel to implement a universal school-based prevention of depression program under real-world conditions

The present study evaluated the impact of a universal prevention of depression program [the Resourceful Adolescent Program (RAP)] when implemented under real-world conditions in a school setting. Prior research has found the RAP program to be beneficial for high-school students when the program was implemented by university staff selected, trained, and supervised by a research team. The present study evaluated the RAP program when implemented by existing school personnel. Separately, we measured the impact of a training program for facilitators, the quality of subsequent program implementation, and the student's response to the RAP Program. Results showed that, in response to the training program, facilitators believed they had acquired the knowledge and confidence to implement the program and that the quality of program implementation was acceptable. The study did not demonstrate a beneficial impact of the RAP program for the students. The results raise important questions regarding the extent of training and ongoing supervision facilitators require if the beneficial outcomes for students are to be maintained when interventions are implemented under real-world conditions in school settings.     

Comprehensive School Counseling Programs: A Review for Policymakers and Practitioners

This article describes components of effective school counseling programs that have emerged from 30 years of empirical research and professional standards. Results are summarized in seven sections: core principles of school counseling programs, program resources, program interventions, program evaluation, program renewal, written policies, and program climate.     

Perinatal Positive Parenting: A program of primary prevention through support of first-time parents

This article provides an overview of the Perinatal Positive Parenting program (PPP). This primary prevention program, developed at the Institute for Family and Child Study at Michigan State University, was designed to meet the parenting support and informational needs of first-time parents. The basic tenets of the PPP program are discussed and overviews of the in-hospital, at home, and parents' support group functions are provided. The evaluations of the primary demonstration of PPP are presented and the PPP videotapes and other materials developed for use with program participants are outlined.     

Skill development and cognitive approaches to educating young children

A variety of early childhood education programs have been established in the last decade that differ markedly in their philosophical and theoretical foundations. This article illustrates how distinct foundations are reflected in all program variables in a unified program. Two unified early childhood education programs are contrasted: a skill development program based upon behaviorist theory, and a cognitive growth program based upon cognitive developmental theory. Implications of unified programs for classroom behavior and the school psychologist are discussed.     

The Origin and Evolution of the Behavior Analysis Program at the University of Nevada,Reno

The origins of the Behavior Analysis program at the University of Nevada, Reno by way of a self-capitalized model through its transition to a more typical graduate program is described. Details of the original proposal to establish the program and the funding model are described. Some of the unusual features of the program executed in this way are discussed, along with problems engendered by the model. Also included is the diversification of faculty interests over time. The status of the program, now, after 25 years of operation, is presented.     

Decision analysis: An approach for multidisciplinary teams in planning special service programs

A decision analytic approach for use by multidisciplinary teams (MDTs) in planning special service programs is described, and formative evaluation information on its application by MDTs in planning individualized education programs (IEPs) is presented. Decision analysis is applied following initial determination of program goals and objectives. In using the approach, an MDT engages in a six-step process: (a) delineation of program alternatives, (b) determination of program outcomes, (c) assessment of probabilities, (d) assessment of utilities, (e) determination of overall program values, and (f) selection of the program to be developed. Utilization of the approach allows a complex program planning situation to be disaggregated into its essential elements, with discussion of program alternatives occurring relative to specific decision criteria. Strengths and limitations of the approach also are discussed, and future directions for research are briefly noted.     

Multicultural Peer Counseling: A Developmental Perspective and Rationale

This article focuses on the creation and growth of a peer counseling program assisting ethnic minority college students. A brief rationale of the program is offered. Problems are described and recommendations are made for subsequent program deuelopment.     

学思维网络活动对小学生创造性的影响:认知风格的调节作用

以89名小学生为被试,采用实验组对照组前后测实验设计,考察学思维网络活动对培养小学生创造性思维和创造性倾向的影响,以及认知风格的调节作用。结果发现：（1）学思维网络活动能有效促进小学生创造性思维以及创造性倾向的想象力和好奇心的发展;（2）学思维课堂活动和学思维网络活动对于培养小学生的创造性思维和创造性倾向具有一致的效果;（3）认知风格在学思维网络活动和学思维课堂活动对小学生创造性思维的影响中起调节作用：对于场依存学生,学思维网络活动能更大程度地提高其流畅性和独创性的表现。