A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems X and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified.

A new approach based on inventory control using interval differential equation with application to manufacturing system

<p style='text-indent:20px;'>Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued average profit of the proposed model has been obtained in parametric form and it is maximized by centre-radius optimization technique. Then to validate the proposed model, two numerical examples have been solved using MATHEMATICA software. In addition, we have shown the concavity of the objective function graphically using the code of 3D plot in MATHEMATICA. Finally, the post optimality analyses are carried out with respect to different system parameters.</p>

On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic

We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.

Stability of Number-Processing Based on Information Granularity Associative with Parietal Cortex

Stability, the ability to automatically extract and produce the efficient and accurate results of a defined problem without making epistemic assumptions, is discussed here as a possible memory system for understanding complex cognitive functions of the arithmetical learning. Stability is of top priority because it may typify organization of granule (knowledge-based information unit) structure. Memory efficiencies are that they depend on both linguistic factors and exposure to arithmetic training during granule formation or consolidation, supporting the idea of analog coding of numerical representations. Neuroimaging studies suggest that the parietal lobe as a potential substrate for a domain-specific representation of numeric quantities and associative memory mechanisms in stability, and results from these studies indicate that there may be the organization of number-related processes of stability in the parietal lobe. Stability seems to depend on the automatic information-processing system's response to experiential knowledge combining granularity (degree of detail or precision), maturational constraints, spatial factors (mental number line) and linguistic factors, making it an ideal candidate for understanding how these interactions play out in the cognitive arithmetic system.

VME Bus-Based Multiprocessor Parallel Data Acquisition System

Synchronization data acquistition system with multiple processors is the trend of the modern manufacturing equipment development. Complex high-precision equipment even has hundreds of sensors and they are acquiring data at the same time, the sensors are distributed in dozens of data acquisition cards. The key technology is the acquisition system enable to synchro-gather data from data acquisition cards which base on data bus between them and then sent the gather data to arithmetic system. In order to synchronic data integration, it is required a high-speed and low-latency data channels and a suitable data protocol between data acquisition cards and main control card. This paper introduce a new type of synchronous data acquisition system, through the self-defined data bus and a specific memory allocation mechanism, the data acquisition system can sent the integrate data to arithmetic system through main control card after it get the data from acquisition cards, the transfer delay of data exchange is nanoseconds.

Jako Jan jechał samotrzeć przez las półczwarta dnia - czyli o dawnych sposobach wyrażania relacji ilościowych

A comparison of old and present indexes for quantitative assessment shows very significant lexical and formal differences. A quantification manner, inherited from the Proto-Slavic language, was unstable and multidimensional, which was apparent in Old Polish texts, whose old complex structures already had a synthesized form, but nevertheless, their inflections still revealed a primary structure. Lack of formal stabilization, which was still noticeable in the Middle Polish period, made numerals and numerical structures directly reflect mathematical operations they had been founded upon. Complex numerical structures often expressed arithmetical operations explicite: 1) addition, e.g. thirty and five, two and twenty, 2) subtraction, e.g. a hundred without one, 3) multiplication, e.g. three-fold one hundred thousand. An essential feature of such structures was stimulation of utterance recipient’s mental activity. One-word numerals in a form of composites, which were constructed on a basis of a mechanism of subtraction, i.e. collective numerals like samotrzeć ‘one in a group of three’, and partitive like półczwarta ‘four minus half, enforced similar behavior. Disappearance of the above mentioned structures and lexemes occurred in New Polish period in result of numerous linguistic tendencies, among which the most vital trend in the history of a language was communication improvement and reduction of mental activity to a necessary minimum, as well as in effect of a tendency to simplify a system. Only the items that directly determine a sequence of numbers in an arithmetic system have remained in the Polish language from a formally and formatively varied class of lexemes naming numbers, i.e. cardinal numbers have specialized as numerical quantifiers supported only residually by collective numbers with greatly restricted context.