scholarly journals Realism and Knowledge First. Interview with Timothy Williamson

2022 ◽  
Vol 19 (3) ◽  
pp. 175-204
Author(s):  
I. E. Pris
Keyword(s):  
A Priori ◽  
Gettier Problem ◽  
Traditional Epistemology ◽  
Timothy Williamson ◽  
Skeptical Problem ◽  
Ill Posed ◽  
The Right ◽  
Metaphysical Realist ◽  
First Interview ◽  
The Gettier Problem

The renowned British philosopher Timothy Williamson talks about his philosophical views and main lines of research. Williamson is a metaphysical realist in a broad sense. Fir him there are true or false answers to questions about all aspects of reality. Classical logic is a universal true theory. Knowledge-first epistemology is an alternative to the traditional belief-first epistemology. The former takes the concept of knowledge as a basic concept, explaining other epistemic concepts, including belief, in its terms, whereas the latter does the opposite. Knowledge, not truth, is the fundamental epistemic good. The Gettier problem and the skeptical problem that arise within traditional epistemology are ill posed and therefore cannot be solved. Hybrid epistemological theories do not satisfy the principles of simplicity and beauty and are refuted by counter-examples. Epistemic contextualism is problematic, and relativism violates the semantics of the phenomena being explained. Knowledge does not entail knowledge about knowledge. Knowledge-how is a kind of knowledge-that. The distinction between a priori and a posteriori is superficial, and there are no analytical truths. The concept of qualia is unhelpful for solving the problems related to consciousness. The so-called “hard problem” of consciousness points to an area of conceptual confusions in which we do not know how to reason properly. Speculative metaphysics is quite a respectable enterprise. But progress in metaphysics is not automatic; it requires the right methodology.

scholarly journals Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter

2019 ◽  
Vol 54 (4) ◽  
pp. 408-416
Author(s):  
O. V. Matysik ◽  
V. F. Savchuk
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Regularization Parameter ◽  
A Priori ◽  
Operator Equations ◽  
Mathematical Economics ◽  
Prove Theorem ◽  
A Value ◽  
Ill Posed ◽  
The Right

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

Gettier Problem

2019 ◽  
Vol 56 (3) ◽  
pp. 58-75
Author(s):  
Timofey S. Demin ◽  
Keyword(s):  
Theory Of Knowledge ◽  
Gettier Cases ◽  
Gettier Problem ◽  
Strategy Analysis ◽  
The Right ◽  
The Mind ◽  
Open Question ◽  
Constituent Elements ◽  
Analysis Of Knowledge ◽  
The Gettier Problem

Theories, that answering the question “What is knowledge?” in analytic epistemology appears under the influence of Gettier cases – a way of refutation such theories of knowledge, that have truth and belief as constituent elements. In the paper were analyzed basic strategies of solving the Gettier problem. One way is to save the analysis of knowledge by changing the elements in order to avoid the Gettier problem. There are three possible ways of doing so: adding new elements to the justification, changing the justification on the other criteria or strengthen the justification in such a way, that it would resolve any possible Gettier cases. For each strategy analysis of the theories of knowledge is given. In the paper idea of the inescapability of Gettier cases for analysis of knowledge was supported by the argumentation of Linda Zagzebski. In that ground, the analysis of knowledge was refuted. From that perspective, two of the most influenced ways of answering the question “what is knowledge” was proposed. First, the irreducible theory of knowledge, where knowledge is a mere state of the mind. Second, rejection existence of the universal invariant of the knowledge in every case. There are multiple senses of what the knowledge is and none of them is prior to other. The author lives as the open question the right way to think about the knowledge. In the closing part of the paper, the author presents a perspective critique of the knowledge problem as the project of overrated significance, and argues for a need to create new arguments that supporting that problem.

Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Mikhail Y. Kokurin
Keyword(s):  
Hilbert Spaces ◽  
Total Error ◽  
A Priori ◽  
Stopping Rules ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
Irregular Operator ◽  
The Right

AbstractWe investigate a class of iterative regularization methods for solving nonlinear irregular operator equations in Hilbert spaces. The operator of an equation is supposed to have a normally solvable derivative at the desired solution. The operators and right parts of equations can be given with errors. A priori and a posteriori stopping rules for the iterations are analyzed. We prove that the accuracy of delivered approximations is proportional to the total error level in the operator and the right part of an equation. The obtained results improve known accuracy estimates for the class of iterative regularization methods, as applied to general irregular operator equations. The results also extend previous similar estimates related to regularization methods for linear ill-posed equations with normally solvable operators.

Knowledge, Benign Falsehoods, and the Gettier Problem

2017 ◽  
Author(s):  
Claudio de Almeida
Keyword(s):  
Theory Of Knowledge ◽  
False Beliefs ◽  
Gettier Problem ◽  
Inferential Knowledge ◽  
The Gettier Problem

Contrary to millennial thought, inferential knowledge does seem to arise in certain cases of reasoning to which false premises are evidentially essential. The phenomenon refutes all of the well-known epistemologies that account for inferential knowledge. I offer an explanation of the phenomenon based on a fairly conservative revision to the defeasibility theory of knowledge, and explain why Peter Klein’s proposed solution fails. The explanation put forward here aims at giving us these two highly desirable results: (a) something we have never had and may not have noticed we needed, a defeasibility theory that is compatible with epistemological fallibilism, and, (b) within this revised, fallibilistic version of the defeasibility theory, an explanation of the benign/malignant distinction for false beliefs that completes the defeasibilist resolution of the Gettier Problem.

Sed ubi Socrates currit? On the Gettier Problem before Gettier

2017 ◽  
Author(s):  
Risto Hilpinen
Keyword(s):  
Twentieth Century ◽  
True Belief ◽  
Belief System ◽  
Gettier Problem ◽  
Current Belief ◽  
The Future ◽  
Late Nineteenth ◽  
The Gettier Problem ◽  
The Way ◽  
Forward Looking

Medieval philosophers presented Gettier-type objections to the commonly accepted view of knowledge as firmly held true belief, and formulated additional conditions that meet the objections or analyzed knowledge in a way that is immune to the Gettier-type objections. The proposed conditions can be divided into two kinds: backward-looking conditions and forward-looking conditions. The former concern an inquirer’s current belief system and the way the inquirer acquired her beliefs, the latter refer to what the inquirer may come to learn in the future and how she can respond to objections. Some conditions of knowledge proposed in late nineteenth- and twentieth-century epistemology can be regarded as variants of the conditions put forward by medieval authors.

Explaining Knowledge

2017 ◽  
Keyword(s):  
State Of The Art ◽  
The State ◽  
Methodological Aspect ◽  
Gettier Problem ◽  
Contemporary Epistemology ◽  
The Gettier Problem

This is an edited collection of twenty-three new papers on the Gettier Problem and the issues connected with it. The set of authors includes many of the major figures in contemporary epistemology who have developed some of the well-known responses to the problem, and it also contains some younger epistemologists who bring new perspectives to the issues raised in the literature. Together, they cover the state of the art on virtually every epistemological and methodological aspect of the Gettier Problem. The volume also includes some skeptical voices according to which the Gettier Problem is not deeply problematic or some of the problems it raises are not genuine philosophical problems.

scholarly journals Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Rujia Li ◽  
Liangcai Cao
Keyword(s):  
Phase Retrieval ◽  
Dynamic Range ◽  
A Priori ◽  
Measured Intensity ◽  
Physical Constraints ◽  
Coherent Diffraction Imaging ◽  
Effective Dynamic ◽  
Ill Posed ◽  
Retrieval Problem ◽  
Diffraction Imaging

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.

On Fourier Series in Eigenfunctions of Elliptic Boundary Value Problems

2003 ◽  
Vol 10 (3) ◽  
pp. 401-410
Author(s):  
M. S. Agranovich ◽  
B. A. Amosov
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Elliptic Boundary ◽  
A Priori ◽  
Adjoint Problem ◽  
Elliptic Boundary Value ◽  
Right Hand ◽  
The Difference ◽  
The Right

Abstract We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿2(Ω) corresponding to this problem has an orthonormal basis {𝑢𝑙} of eigenfunctions, which are infinitely smooth in . However, the system {𝑢𝑙} is not a basis in Sobolev spaces 𝐻𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻𝑡 (Ω) one can explicitly construct a function 𝑢0 ∈ 𝐻𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢0 in the functions 𝑢𝑙 converges to this difference in 𝐻𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢0. The function 𝑢0 is obtained by applying some linear operator to these right-hand sides.

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