What is Mathematics: Gödel's Theorem and Around (Edition 2013)

Hyper-textbook for students in mathematical logic and foundations of mathematics. Edition 2013. ATTENTION! New Edition 2015 available at https://dspace.lu.lv/dspace/handle/7/5306.

PASSIM – an open source software system for managing information in biomedical studies

Abstract Background One of the crucial aspects of day-to-day laboratory information management is collection, storage and retrieval of information about research subjects and biomedical samples. An efficient link between sample data and experiment results is absolutely imperative for a successful outcome of a biomedical study. Currently available software solutions are largely limited to large-scale, expensive commercial Laboratory Information Management Systems (LIMS). Acquiring such LIMS indeed can bring laboratory information management to a higher level, but often implies sufficient investment of time, effort and funds, which are not always available. There is a clear need for lightweig…

Introduction to Mathematical Logic, Edition 2021

Textbook for students in mathematical logic. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional and predicate logic. Glivenko's theorem and constructive embedding. Axiom independence. Interpretations, models and completeness theorems. Normal forms. Tableaux and resolution methods. Herbrand's theorem. Sections 1, 2, 3 represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian).

Introduction to Mathematical Logic (Edition 2017)

Hyper-textbook for students in mathematical logic, Edition 2017

Comparing various concepts of function prediction. Part 1.

Prediction: f(m+1) is guessed from given f(0), ..., f(m). Program synthesis: a program computing f is guessed from given f(0), ..., f(m). The hypotheses are required to be correct for all sufficiently large m, or with some positive frequency. These approaches yield a hierarchy of function prediction and program synthesis concepts. The comparison problem of the concepts is solved.

Comparing various concepts of function prediction. Part 2.

Prediction: f(m+1) is guessed from given f(0), ..., f(m). Program synthesis: a program computing f is guessed from given f(0), ..., f(m). The hypotheses are required to be correct for all sufficiently large m, or with some positive frequency. These approaches yield a hierarchy of function prediction and program synthesis concepts. The comparison problem of the concepts is solved.

Computational complexity of prediction strategies

The value f(m+1) is predicted from given f(1), ..., f(m). For every enumeration T(n, x) there is a strategy that predicts the n-th function of T making no more than log2(n) errors (Barzdins-Freivalds). It is proved in the paper that such "optimal" strategies require 2^2^cm time to compute the m-th prediction (^ stands for expoentiation).

Digits of pi: limits to the seeming randomness

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$. In this article, first, another similar conjecture is explored - the seemingly almost random behaviour of digits in the base 3 representations of powers $2^n$. This conjecture seems to confirm well - it passes even the tests inspired by the Central Limit Theorem and the Law of the Iterated Logarithm. After this, a similar testing of the sequences of digits in the decimal representations of the numbers $\pi$, $e$ and $\sqrt{2}$ was performed. The result looks surprising: unlike the digits in the base 3…

Truth Demystified

For further development, see Karlis Podnieks in ResearchGate. How could we recognize truth, if we only have models, means of model-building, and the history of their evolution? Where is the truth in the cloud of models – with so many of them already gone with the wind? We can define truths as more or less persistent invariants of successful evolution of models and means of model-building. What is true, will not change in the future (for some time, at least). This approach to truth could be named demystified realism (or, demystified theory of truth) – the kind of realism based on a minimum of metaphysical assumptions. (“Robotic realism” also would be appropriate, but the term is occupied alr…

Methodological consequences of Gödel's incompleteness theorem

Published as: K. Podnieks. Methodological consequences of Gödel's incompleteness theorem. European Summer Meeting of the Association for Symbolic Logic, Berlin, 1989. The Journal of Symbolic Logic, Vol. 57, No. 1, March, 1992, pp.326-327

Towards a theory of inductive inference

On computation in the limit by non-deterministic Turing machines

Towards Model-Based Model of Cognition

If it's true that models are the ultimate results of cognition, then shouldn't we try reordering the field of the philosophy of cognition, starting with the notion of model? In this way, couldn't we obtain a unified and more productive picture - a model-based model of cognition?

Is Scientific Modeling an Indirect Methodology?

Imagine, quarks will be retained as a construct in all future physical theories. Do physicists need more than this kind of invariance to claim the "real existence" of quarks and believe in having a "direct representation" of them? If we consider modeling not as a heap of contingent structures, but (where possible) as evolving coordinated systems of models, then we can reasonably explain as "direct representations" even some very complicated model-based cognitive situations. Scientific modeling is not as indirect as it may seem. "Direct theorizing" comes later, as the result of a successful model evolution.

What is Mathematics: Gödel's Theorem and Around (Edition 2015)

Hyper-textbook for students in mathematical logic and foundations of mathematics. Edition 2015.

On the reducibility of function classes

Copy of a paper published 1972 in Russian.

Вокруг теоремы Геделя

Проведен методологический анализ природы математики. Показано, что сущность математического метода состоит в исследовании застывших моделей. Обоснована несостоятельность утверждений об ограниченности аксиоматического метода. Предлагается следующая методологическая оценка теоремы Геделя о неполноте: Всякая формальная теория с методологической точки зрения является моделью некоторой застывшей системы мышления. С учетом этого основной вывод из теоремы о неполноте можно переформулировать так: всякая достаточно всеобъемлющая, но застывшая система мышления неизбежно оказывается несовершенной – в ней содержатся либо противоречия, либо проблемы, для решения которых данной (застывшей!) системы недос…

Towards Integrated Computer Aided Systems and Software Engineering Tool for Information Systems Design

The paper starts with a brief overview of the current situation in the world of CASE tools for information systems. Then there follows the outline of the basic ideas and principles of integrated CASE tool GRADE. The most outstanding characteristics of GRADE are that the tool is based on a unified specification language GRAPES and that it supports all information system development phases including analysis, requirements specification, design and implementation.

Inductive inference of recursive functions: complexity bounds

This survey includes principal results on complexity of inductive inference for recursively enumerable classes of total recursive functions. Inductive inference is a process to find an algorithm from sample computations. In the case when the given class of functions is recursively enumerable it is easy to define a natural complexity measure for the inductive inference, namely, the worst-case mindchange number for the first n functions in the given class. Surely, the complexity depends not only on the class, but also on the numbering, i.e. which function is the first, which one is the second, etc. It turns out that, if the result of inference is Goedel number, then complexity of inference ma…

Platonism, intuition and the nature of mathematics

Platonism is an essential aspect of mathematical method. Mathematicians are learned ability " t o   l i v e " in the "world" of mathematical concepts. Here we have the main source of the creative power of mathematics, and of its surprising efficiency in natural sciences and technique. In this way, "living" (sometimes - for many years) in the "world"of their concepts and models, mathematicians are learned to draw a maximum of conclusions from a minimum of premises. Fixed system of basic principles is the distinguishing property of every mathematical theory. Mathematical model of some natural process or technical device is essentially a  f i x e d  m o d e l  which can be investigated indepen…

Comparing various types of limiting synthesis and prediction of functions

Introduction to Mathematical Logic (Edition 2014)

Sections 1, 2, 3 represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian).

Теорема Гёделя о неполноте

Published as: К. М. Подниекс. Теорема Гёделя о неполноте. "Математика XX века. Взгляд из Петербурга", МЦНМО, 2010, стр. 170-174.

Editor Definition Language and Its Implementation

Universal graphical editor definition language based on logical metamodel extended by presentation classes is proposed. Implementation principles of this language, based on Graphical Diagramming Engine are described.

Frege's Puzzle from a Model-Based Point of View

Every utterance comes from the world model of the speaker. More generally, every sentence comes from some kind of world model. It may be the world model of a (real or imagined) person, the world model represented in a novel, movie, scientific book, virtual reality, etc. In principle, even smaller informational units (stories, poems, newspaper articles, jokes, mathematical proofs, video clips, dreams, hallucinations, etc.) may introduce their own “partial world models” as small additions to “bigger” world models (regarded as background knowledge). Sometimes, sentences contain references to other world models. Trying to understand such sentences, we should identify, and keep separated, the wo…

Indispensability Argument and Set Theory

One may take several different positions with respect to the ontological status of scientific entities such as, for example, quarks (quarks can't be observed even in principle). Do quarks "really exist", or are they only a (currently successful) theoretical construct used by physicists in their models? Perhaps, the "least committed" position could be the formalist one: let us define the "real existence" of some scientific entity as its invariance in future scientific theories.

Towards a General Definition of Modeling

What is a model? Surprisingly, in philosophical texts, this question is asked (sometimes), but almost never – answered. Instead of a general answer, usually, some classification of models is considered. The broadest possible definition of modeling could sound as follows: a model is anything that is (or could be) used, for some purpose, in place of something else. If the purpose is “answering questions”, then one has a cognitive model. Could such a broad definition be useful? Isn't it empty? Can one derive useful consequences from it? I'm trying to show that there is a lot of them.

The double-incompleteness theorem

Let T be a strong enough theory, and M - its metatheory, both are consistent. Then there is a closed arithmetical formula H that is undecidable in T, but one cannot prove in M neither that H is T-unprovable, nor that H is T-unrefutable. For English translation and proof, see K. Podnieks What is mathematics: Godel's theorem and around.

Prediction of the next value of a function

The following model of inductive inference is considered. Arbitrary set tau = {tau_1, tau_2, ..., tau_n} of n total functions N->N is fixed. A "black box" outputs the values f(0), f(1), ..., f(m), ... of some function f from the set tau. Processing these values by some algorithm (a strategy) we try to predict f(m+1) from f(0), f(1), ..., f(m). Upper and lower bounds for average error numbers are obtained for prediction by using deterministic and probabilistic strategies.

Explanation and Understanding in a Model-Based Model of Cognition

This article is an experiment. Consider a minimalist model of cognition (models, means of model-building and history of their evolution). In this model, explanation could be defined as a means allowing to advance: production of models and means of model-building (thus, yielding 1st class understanding), exploration and use of them (2nd class), and/or teaching (3rd class). At minimum, 3rd class understanding is necessary for an explanation to be respected.

The Nature of Mathematics – an interview with Professor Karlis Podnieks

Many people think that mathematical models are built using well-known “mathematical things” such as numbers and geometry. But since the 19th century, mathematicians have investigated various non-numerical and non-geometrical structures: groups, fields, sets, graphs, algorithms, categories etc. What could be the most general distinguishing feature that would separate mathematical models from non-mathematical ones? I would describe this feature by using such terms as autonomous, isolated, stable, self-contained, and – as a summary – formal. Autonomous and isolated – because mathematical models can be investigated “on their own” in isolation from the modeled objects. And one can do this for ma…

On speeding up synthesis and prediction of functions

Inductive inference of recursive functions: Complexity bounds

This survey includes principal results on complexity of inductive inference for recursively enumerable classes of total recursive functions. Inductive inference is a process to find an algorithm from sample computations. In the case when the given class of functions is recursively enumerable it is easy to define a natural complexity measure for the inductive inference, namely, the worst-case mindchange number for the first n functions in the given class. Surely, the complexity depends not only on the class, but also on the numbering, i.e. which function is the first, which one is the second, etc. It turns out that, if the result of inference is Goedel number, then complexity of inference ma…

Towards a General Definition of Modeling

What is a model? Surprisingly, in philosophical texts, this question is asked (sometimes), but almost never – answered. Instead of a general answer, usually, some classification of models is considered. The broadest possible definition of modeling could sound as follows: a model is anything that is (or could be) used, for some purpose, in place of something else. If the purpose is “answering questions”, then one has a cognitive model. Could such a broad definition be useful? Isn't it empty? Can one derive useful consequences from it? I'm trying to show that there is a lot of them.

The Formalist Picture of Cognition. Towards a Total Demystification.

This paper represents a philosophical experiment inspired by the formalist philosophy of mathematics. In the formalist picture of cognition, the principal act of knowledge generation is represented as tentative postulation – as introduction of a new knowledge construct followed by exploration of the consequences that can be derived from it. Depending on the result, the new construct may be accepted as normative, rejected, modified etc. Languages and means of reasoning are generated and selected in a similar process. In the formalist picture, all kinds of “truth” are detected intra-theoretically. Some knowledge construct may be considered as “true”, if it is accepted in a particular normativ…