# Karlis PodnieksUniversity of Latvia | LU · Faculty of Computing, Institute of Mathematics and Computer Science

Karlis Podnieks

Dr. Math.

Trying to show: truth represents persistent invariants of successful evolution of models and means of model-building.

## About

77

Publications

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Introduction

Cognition is inherently limited because it means replacing of fragments of “external reality” (whatever that may mean) by another fragment of it. Consequently, it is impossible (for both human intelligence and AI) to create exact copies of sufficiently large fragments of reality. This limitation is not imposed by the constraints of humans minds or technological devices, but rather by the fundamental structure of the physical universe.

Additional affiliations

Education

September 1966 - February 1979

## Publications

Publications (77)

Introduction to mathematical logic. Part 2.Textbook for students in mathematical logic and foundations of mathematics.
CONTENTS.
Platonism, Intuition, Formalism.
Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory.
First order arithmetic. Hilbert's 10th problem.
Incompleteness th...

Of the substantial work done in the philosophy of modeling by Vaihinger (1876), Craik (1943), Rosenblueth and Wiener (1945), Apostel (1960), Minsky (1965), Klaus (1966) and Stachowiak (1973), only Vaihinger’s work has gained some recognition in the mainstream literature. However, the work of all these thinkers seems to contain original ideas worth...

NEW EDITION 2021: more and better motivations, chapter about tableaux method added, improved treatment of resolution method. [[[[[]]]]]Textbook for students in mathematical logic. Part 1. CONTENTS. Total formalization is possible! Formal theories. First order languages. Axioms of constructive and classical logic. Proving formulas in propositional a...

This paper contains a detailed exposition and analysis of The Philosophy of “As If“ proposed by Hans Vaihinger in his book published in 1911. However, the principal chapters of the book (Part I) reproduce Vaihinger’s Habilitationsschrift, which was written during the autumn and winter of 1876. Part I is extended by Part II based on texts written du...

The formalist philosophy of mathematics (in its purest, most extreme version) is widely regarded as a “discredited position”. This pure and extreme version of formalism is called by some authors “game formalism”, because it is alleged to represent mathematics as a meaningless game with strings of symbols. Nevertheless, I would like to draw attentio...

THIS IS PREPRINT! It has been substantially revised in the published version in Baltic J. Modern Computing, Vol. 9 (2021), No. 1, pp. 67–110 [[[[[ https://www.researchgate.net/publication/350207973_Philosophy_of_Modeling_in_the_1870s_A_Tribute_to_Hans_Vaihinger ]]]]]

The barometer explaining the storm? Moreover, what about the idea that the storm is CAUSED by a falling barometer? The motion of planets is caused by the Newtonian gravitation force acting instantly at a distance. Really? Shouldn't these two theses be set on a par - except that the second one was more productive in history?

Taken literally, Vaihinger's enthusiasm about contradictions ubiquitous in science and mathematics might seem discrediting him as a serious thinker. Wasn't he really?

The well-known "no miracles argument", proposed by Hilary Putnam in 1975, claims that realism "is the only philosophy that doesn't make the success of science a miracle". Interestingly enough, the possibility of such an argument was noticed by Hans Vaihinger almost one hundred years earlier - in 1876. Of course, Vaihinger rejected this argument fro...

Intended as part of my paper “Philosophy of Modeling in 1870s: a Tribute to Hans Vaihinger”. Vaihinger and Nietzsche, the two most radical followers of Kant's philosophy of cognition in the 19th century, both were promoting the idea that knowledge is invented, not discovered. “Truth” cannot be “proved” or verified in any absolute sense many people...

First deposited as https://philarchive.org/rec/PODEAU. [[[[]]]]
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 1 st c...

PREPRINT! You'd better to read the PUBLISHED version: https://www.researchgate.net/publication/327959480_Philosophy_of_Modeling_Neglected_Pages_of_History

I would propose reverting of the popular strategy: when we identify
some significant distinction in the human cognition, then the real thing is this very distinction, and not its correspondence to our intuitions. Identify a significant distinction, name it as XYZ, and prove that XYZ is more significant than the possibly corresponding mystical intui...

This is OLD Edition 2017. NEW Edition 2021 is available at https://www.researchgate.net/publication/349104699_Introduction_to_Mathematical_Logic_Edition_2021.

THIS IS WORKING PAPER! It has been substantially revised in the published version in Baltic J. Modern Computing, Vol. 9 (2021), No. 1, pp. 67–110 [[[[[ https://www.researchgate.net/publication/350207973_Philosophy_of_Modeling_in_the_1870s_A_Tribute_to_Hans_Vaihinger ]]]]]

What is true, will not change in the future (for some time, at least). Else, how could we benefit from knowing the truth? However, is truth any nobler than “something that will not change in the future”? I will show that we do not need it to be any nobler – if one adopts an appropriate model of cognition – the model-based model of cognition (MBMC)....

In 2014, Barzdins, Rencis and Sostaks introduced granular ontologies as a specific
organization of databases allowing for extremely fast processing of ad hoc queries, and proved
the following Granularity Theorem: Consider an ontology represented by graphical means of a
UML class diagram. Then, under certain restrictions on association multiplicity...

A comment on the paper:
Alexander Reutlinger, Dominik Hangleiter, and Stephan Hartmann. Understanding (With) Toy Models. The British Journal for the Philosophy of Science, 2016

An essay prepared and published online in 1999. Hegel died in 1831, Marx - in 1883. They would be pleased reading Gödel's paper in 1931, oder?
------- However, the idea that Hegel would enjoy Gödel's Theorem, was proposed much earlier - by J.N.Findlay in 1963:
“It is worth suggesting that the whole logical situation here presented furnishes a very...

A comment on the paper:
Francesca Pero and Mauricio Suárez. Varieties of Misrepresentation and
Homomorphism. European Journal for Philosophy of Science, 6(1), October 2015.
[https://www.researchgate.net/publication/282593826_Varieties_of_Misrepresentation_and_Homomorphism]

A comment on the paper:
Bryan Rennie. Myths, models, and metaphors: Religion as model and the philosophy of science. December 2009, Religion 39(4):340-347.

For a new and extended version see: Demystified theory of truth and understanding, Feb. 2017. //
What is true, will not change in the future (for some time, at least). Else, how could we benefit from knowing the truth? However, is truth any nobler than “something that will not change in the future”?

Draft (published online 25/11/2010). Superseded by: Demystified theory of truth and understanding, Feb. 2017. //
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...

Draft (published online 25/11/2010). Superseded by: Demystified theory of truth and understanding, Feb. 2017. //
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 introd...

Published in: Lecture Notes in Computer Science, Vol. 9118, Springer, 2015, pp. 58–69.
We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. ||n|| denotes the minimum number of 1's in the expressions representing n. The logarithmic complexity ||n||log is defined as ||n||/log3(n). The values...

Preprint deposited as http://arxiv.org/abs/1411.3911.
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 digit...

I propose a new argument in favour of the Dappled World Perspective, and show how this Perspective can be refined in the model-based model of cognition (MBMC), which I am trying to promote.

We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely...

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 articl...

For many years, I have been promoting the broadest possible notion of mathematical models. Manypeople think that mathematical models are built using well-known “mathematical things” such asnumbers and geometry. But since the 19th century, mathematicians have investigated various non-numerical and non-geometrical structures: groups, fields, sets, gr...

Preprint, deposited in PhilSci Archive, 5475 (2010). ///// Superseded by: Demystified theory of truth and understanding, Feb. 2017. ///// Main ideas published in: K. Podnieks, The Dappled World Perspective Refined. The Reasoner, Vol. 8, N 1, January 2014, pp. 3–4. /////
First, I propose a new argument in favor of the Dappled World perspective intr...

In Russian, for people trained somewhat in mathematics, but not in mathematical logic: a short (4 pages) explanation of Gödel's Incompleteness Theorem, its history and consequences for mathematics and computer science.

SIMBioMS is a web-based open source software system for managing data and information in biomedical studies. It provides a solution for the collection, storage, management and retrieval of information about research subjects and biomedical samples, as well as experimental data obtained using a range of high-throughput technologies, including gene e...

My general impression: despite many
brilliant insights, generated by philosophers for many
years, the ﬁeld (philosophy of cognition) remains unordered for too long a time. If it’s true that models are the ultimate results of cognition, then shouldn’t we try reordering the ﬁeld, starting with the notion of model? In this way, couldn’t we
obtain a u...

Do quarks “really exist”, or are they only an “indirect” entity introduced
by physicists? For the current purposes, this construct
works ﬁne, but will this situation continue in the future?
If not, quarks will be removed from the picture just as
ﬂogiston and aether were removed. But what if quarks
will be retained as a construct in all future physi...

How about the most fundamental mathematical entity—the famous unique “world of sets” to
which we ought to have ontological commitment and
that must be studied in set theory as the only structure
worth of consideration? And in which the famous Continuum Hypothesis must be either true, or false, independently of the ability of human mathematicians to...

Standalone Person Management Tool (PMT). PMT is intended for registering confidential information about the research subjects from whom samples have been taken. .zip file contains Windows executable, .xml database file and installation description in a README file.

Sample management database. .zip contains sql version of the database, documentation and the files necessary for the installation of the system.

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 ar...

Tim Berners-Lee and co-authors in their seminal paper "The Semantic Web", published in 2001, outlined their vision about the future Semantic Web. But today we are still far from the implementation of this vision. Despite fundamental achievements, like definition of OWL (Web Ontology Language) and rapid progress of RDF/OWL content creation, storage...

How to determine, is a proposed model transformation correct, or not? In general, the answer may depend on the model semantics. Of course, a model transformation is "correct", if we can extend it to a "correct" instance data transformation. Where should model semantics be defined? Assume, model syntax and semantics are defined in the same meta-mode...

The foundation of a generic modelling tool is its flexible diagramming facility. The paper proposes a universal graphical editor definition language based on logical metamodel extended by presentation classes. Some more advanced diagram definition facilities such as patterns and diagram cores are also introduced. Implementation principles of this l...

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.

The paper presents some discussion on current issues of modeling languages and tools. Two areas are considered: business modeling and Unified Modeling Language based modeling (object modeling). The application areas well covered by existing languages and tools and those where further development is required are pointed out. Some of the authors' exp...

Full text available at http://link.springer.com/chapter/10.1007/978-1-4471-1486-4_1 .
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...

Joint English translation of two papers: (1) K. M. Podnieks. Probabilistic synthesis of enumerated classes of functions. Dokl.Akad.Nauk SSSR, 1975, Vol. 223, N5, pp. 1071–1074 (in Russian). (2) K. M. Podnieks. Probabilistic program synthesis. In: Theory of Algorithms and Programs, Vol. 3, Latvia State University, 1977, pp. 57–88 (in Russian). [[[[[...

Textbook for students in mathematical logic and foundations of mathematics. Axiomatic set theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem. Around Ramse...

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...

Paper published in Russian as: K. M. Podnieks. Platonism, intuition and the nature of mathematics. In: Semiotika i informatika, Moscow, VINITI, 1990, Vol. 31, pp. 150–180. ]]]]][[[[[
The same text in Russian was published as Chapter 1 of the book: K. Podnieks. What is Mathematics? Gödel’s Theorem and Around. Zinatne Publishers, Riga, 1992. Online...

Platonist attitude of mathematicians to objects of their investigations is determined by the very nature of the mathematical method. The evolution of greek mathematics led to mathematical objects in the modern meaning of the word: the ideas of numbers, points, straight lines etc. s t a b i l i z e d and thus - were distracted from their real source...

Some function f is known to belong to a countable set of total functions g_n from N to N. Prediction: f(m+1) is guessed from given f(0), ..., f(m). Any values g_n(x) are available on request. The hypotheses are required to be correct for all sufficiently large m. Deterministic, probabilistic and "voting" prediction strategies are considered. Theore...

Paper published as: K. M. Podnieks. Computational complexity of prediction strategies. In: Theory of Algorithms and Programs, Vol. 3, Latvia State University, 1977, pp. 89–102 (in Russian).
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...

Published as: K. M. Podnieks. Probabilistic program synthesis. In: Theory of Algorithms and Programs, Vol. 3, Latvia State University, 1977, pp. 57–88 (in Russian). [[[[[]]]]]
The following model of inductive inference is considered. Arbitrary
numbering tau = {tau-0, tau-1, tau-2, ...} of total functions N->N is fixed. A "black box"
outputs the val...

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 an English translation, see Section 6.2 of my book "What is Mathematics? Godel's Theorem and Around."

Published as: K. M. Podnieks. Probabilistic synthesis of enumerated classes of functions. Dokl. Akad. Nauk SSSR, 1975, Vol. 223, N5, pp. 1071–1074 (in Russian), English translation in: Soviet Math. Dokl., 1975, Vol.16, N4, pp. 1042–1045. Proofs were published as: K. M. Podnieks. Probabilistic synthesis of programs. In: Theory of Algorithms and Prog...

Original title:
К.М.Подниекс. Сравнение различных типов предельного синтеза и прогнозирования функций (статья вторая). Ученые записки Латвийского государственного университета, 1974, том 233, стр. 35-44.
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 hyp...

Published as: K.M.Podnieks. Probabilistic prediction of computable functions. Scientific Proceedings of Latvia State University, 1975, Vol. 233, pp. 57–76 (in Russian). [[[[[]]]]]
The value f(m+1) is guessed from given f(0), f(1), ... f(m). Let tau(n, x) be a recursive function, set tau_n =lambda-x tau(n, x). There is a probabilistic strategy R_ta...

Original title:
К.М.Подниекс. Сравнение различных типов предельного синтеза и прогнозирования функций. Ученые записки Латвийского государственного университета, 1974, том 210, стр. 68-81.
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 requ...

Duplicate of:
https://www.researchgate.net/publication/316715107_On_computation_in_the_limit_by_non-deterministic_Turing_machines

Duplicate of:
https://www.researchgate.net/publication/316715194_On_speed-up_of_function_synthesis_and_prediction

Original title:
Я.М.Барздинь, К.М.Подниекс. К теории индуктивного вывода. Proceedings of 2nd Symposium and Summer School on Mathematical Foundations of Computer Science, Strbske Pleso, High Tatras, Czechoslovakia, September 3–8, 1973, pp. 9–15.

Original title: Р.К.Шранка, К.М.Подниекс. Надежность релейно-контактных схем при достаточно надежных контактах. Автоматика и вычислительная техника, 1973, 4, 44-48. [[[[[]]]]] It is shown that, for sufficiently reliable contacts, the reliability of S - P circuits that implement an n-position symmetrical quasi-monotonic function with indexes k and l...

Original title: Р.К.Шранка, К.М.Подниекс. Надежность релейно-контактных схем при достаточно надежных контактах. Автоматика и вычислительная техника, 1973, 4, 44-48. [[[[[]]]]] It is shown that, for sufficiently reliable contacts, the reliability of S - P circuits that implement an n-position symmetrical quasi-monotonic function with indexes k and l...

Deterministic stochastic matrices (DSM) consist of 0 and 1 only. Decomposition of a stochastic matrix (SM) means representation of it as a linear combination of DSMs with positive coefficients. In the classical (non-constructive) setting, any m x n SM can be represented as a linear combination of maximum m(n–1)+1 DSMs. In the constructive setting,...

N – the set of all natural numbers, F – the set of all total functions N→N, A , B⊆F. We say that A is m-reducible to B (A≤ m B), iff there is a recursive operator M such that f ∈A ↔ M (f)∈B for all f ∈F. Similarly, 1-reducibility, tt-, btt-, 1tt-and Turing reducibility can be introduced. Table of contents. 1. Introduction. 2. Definitions of reducib...

Deterministic stochastic matrices (DSM) consist of 0 and 1 only. Decomposition of a stochastic matrix (SM) means representation of it as a linear combination of DSMs with positive coefficients. If no additional conditions are set, any m x n SM can be represented as a linear combination of maximum mn–m+1 DSMs. However, let us require that decomposit...

Deterministic stochastic matrices (DSM) consist of 0 and 1 only. Decomposition of a stochastic matrix (SM) means representation of it as a linear combination of DSMs with positive coefficients. In the article, a natural stability measure of a decomposition is introduced and a general upper bound for it is proved.
Original title: К.М.Подниек. О ме...

In the article, some information is obtained about the set of non-isolated cut-points of a stochastic (probabilistic) automaton having commutative transition matrices. In particular, the set is shown to be nowhere dense in [0, 1].
Original title: К.М.Подниек. О точках сечения некоторых конечных вероятностных автоматов. Автоматика и вычислительная...

## Questions

Questions (3)

Couldn't it be the following one: using one-s, additions and multiplications only, is (1+1)...(1+1) the shortest possible expression for a power of 2? See F26 in Guy's book and https://oeis.org/A005245.

His paper shows impressive "fluency" in the philosophy of modeling:

MATHEMATICAL MODELS IN BIOLOGICAL THEORY

J. A. RAFFERTY

American Scientist

Vol. 38, No. 4 (OCTOBER 1950), pp. 549-567, 579

The question is asked on p.809 of Feller's 1945 paper THE FUNDAMENTAL LIMIT THEOREMS IN PROBABILITY, see http://www.ams.org/journals/bull/1945-51-11/S0002-9904-1945-08448-1/S0002-9904-1945-08448-1.pdf

"Our theorems give precise theoretical information concerning the probable amplitude of the oscillations of Sn(x) as a function of n. It would be of considerable theoretical and practical interest to have more information as to the frequency or wave-length of these oscillations. What can be said concerning the frequency with which Sn(x) changes sign? Many similar questions can be raised, but again very little is known in this direction. (However, in the special case (4) some interesting results were recently obtained by Erdös; they are not yet published.) These questions are related to the iterated logarithm, that is to say, they are of the measure-theoretic or "strong" type. However, there are many open questions of the "weak" type, which are really problems concerning distribution functions and can be formulated in terms of Fourier analysis."