Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

It has been known for long time that most of the existing cosmology models have singularity problem. Cosmological singularity has been a consequence of excessive symmetry of flow, such as “Hubble’s law”. More realistic one is suggested, based on Newtonian cosmology model but here we include the vertical-rotational effect of the whole Universe. We review a Riccati-type equation obtained by Nurgaliev, and solve the equation numerically with Mathematica. It is our hope that the new proposed method can be verified with (...) observation data. (shrink)

In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...) part” applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The methodology uses a computational device called the Infinity Computer (patented in USA and EU) working numerically (recall that traditional theories work with infinities and infinitesimals only symbolically) with infinite and infinitesimal numbers that can be written in a positional numeral system with an infinite radix. It is argued that numeral systems involved in computations limit our capabilities to compute and lead to ambiguities in theoretical assertions, as well. The introduced methodology gives the possibility to use the same numeral system for measuring infinite sets, working with divergent series, probability, fractals, optimization problems, numerical differentiation, ODEs, etc. (recall that traditionally different numerals lemniscate; Aleph zero, etc. are used in different situations related to infinity). Numerous numerical examples and theoretical illustrations are given. The accuracy of the achieved results is continuously compared with those obtained by traditional tools used to work with infinities and infinitesimals. In particular, it is shown that the new approach allows one to observe mathematical objects involved in the Hypotheses of Continuum and the Riemann zeta function with a higher accuracy than it is done by traditional tools. It is stressed that the hardness of both problems is not related to their nature but is a consequence of the weakness of traditional numeral systems used to study them. It is shown that the introduced methodology and numeral system change our perception of the mathematical objects studied in the two problems. (shrink)

This study aimed to assess the effectiveness of remedial instruction in improving the numeracy skills of Grade 7 students at Malbug National High School during the school year 2023-2024. Adopting a quasi-experimental research design, the research focused on Grade 7 students at Malbug National High School, Cawayan East District, Masbate Province Division, Philippines, identified as non-numerates, employing pre-tests and post-tests as essential research tools. The independent variable was the remedial instruction in numeracy, while the dependent variable was students' numeracy performance (...) measured through pre-tests and post-tests. Before the intervention, the pre-test numeracy performance exhibited a varied distribution of students across different score ranges. A majority fell into the "Needs Major Support" category, emphasizing the necessity for targeted interventions. The post-test results, however, revealed a remarkable improvement, with 87.88% of initially non-numerate students achieving scores in the "Transforming" range. A thorough statistical examination validated a notable disparity in the scores obtained before and after the instructional intervention, substantiating the favorable influence of remedial instruction. The calculated t-value of 19.594, exceeding the critical t-value, led to rejecting the null hypothesis. In conclusion, the study emphasizes the need for tailored interventions based on the initial deficiency in numeracy skills. The post-test results underscored the success of remedial instruction in fostering substantial growth. The study recommends sustaining the remedial program, implementing periodic assessments, providing teacher training, involving parents in students' development, allocating adequate resources, and further research in other schools to strengthen findings. (shrink)

Numerical identity is the non-relational sameness of an object to itself. It is concerned with understanding how entities undergo change and maintain their identity. In substance metaphysics, an entity is considered a substance with an essence and such an essence is the source of its power. However, such a framework fails to explain the sense in which an entity is still the entity it was, amidst changes. Those who claim that essence is unaffected by existence are faced with challenge of (...) exploring the epistemic access to such an essence, which is questionable at best. Process metaphysics is a strong candidate for a theory that can ontologically explain regularity and change without appeal to essence. Process and its interactions is the main category. Every process is an emergent organization of constitutive interactions and is individuated on the basis of its interactive powers, that is, the ways in which it interacts with the world around it. Interactions are situated adaptation to changes. In this way, changes are crucial within process metaphysics and are included in the starting point of its investigation. What seems to the naked eyes as one-ness/singularity is a complex process where an organization of interactions is emerging from moment to moment by continually adapting to the changes around and within it. The question of numerical identity over time becomes valid only within substance metaphysics which has no space to accommodate change, due to its allegiance to essence. (shrink)

In a recent paper in Astrophysics and Space Science Vol. 364 no. 11 (2019), S. Ershkov & D. Leschenko presented a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution (...) in the vicinity of libration points. A new and elegant ansatz has been suggested in their publication whereby, in solving, the momentum equation is reduced to a system two coupled Riccati ODEs. In this paper, we presented a numerical solution of such coupled Riccati ODEs using Mathematica software package. (shrink)

In a recent paper, Chunghyoung Lee argues that, because zygotes are developmentally plastic, they cannot be numerically identical to the singletons into which they develop, thereby undermining conceptionism. In this short paper, I respond to Lee. I argue, first, that, on the most popular theories of personal identity, zygotic plasticity does not undermine conceptionism, and, second, that, even overlooking this first issue, Lee’s plasticity argument is problematic. My goal in all of this is not to take a stand in the (...) abortion debate, which I remain silent on here, but, rather, to push for the conclusion that transitivity fails when we are talking about numerical identity of non-abstract objects. (shrink)

The first aim of the paper is to show that under certain conditions generative syntax can be made suitable for Montague semantics, based on his type logic. One of the conditions is to make branching in the so-called X-bar syntax strictly binary, This makes it possible to provide an adequate semantics for Noun Phrases by taking them as referring to sets of collections of sets of entities ( type <ett,t>) rather than to sets of sets of entities (ett).

A computational methodology called Grossone Infinity Computing introduced with the intention to allow one to work with infinities and infinitesimals numerically has been applied recently to a number of problems in numerical mathematics (optimization, numerical differentiation, numerical algorithms for solving ODEs, etc.). The possibility to use a specially developed computational device called the Infinity Computer (patented in USA and EU) for working with infinite and infinitesimal numbers numerically gives an additional advantage to this approach in comparison with traditional methodologies studying (...) infinities and infinitesimals only symbolically. The grossone methodology uses the Euclid’s Common Notion no. 5 ‘The whole is greater than the part’ and applies it to finite, infinite, and infinitesimal quantities and to finite and infinite sets and processes. It does not contradict Cantor’s and non-standard analysis views on infinity and can be considered as an applied development of their ideas. In this paper we consider infinite series and a particular attention is dedicated to divergent series with alternate signs. The Riemann series theorem states that conditionally convergent series can be rearranged in such a way that they either diverge or converge to an arbitrary real number. It is shown here that Riemann’s result is a consequence of the fact that symbol ∞ used traditionally does not allow us to express quantitatively the number of addends in the series, in other words, it just shows that the number of summands is infinite and does not allows us to count them. The usage of the grossone methodology allows us to see that (as it happens in the case where the number of addends is finite) rearrangements do not change the result for any sum with a fixed infinite number of summands. There are considered some traditional summation techniques such as Ramanujan summation producing results where to divergent series containing infinitely many positive integers negative results are assigned. It is shown that the careful counting of the number of addends in infinite series allows us to avoid this kind of results if grossone-based numerals are used. (shrink)

Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to (...) execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given. (shrink)

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...) of a computer – the Infinity Computer – able to work numerically with all of them. An introduction to the theory of physical and mathematical continuity and differentiation (including subdifferentials) for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains is developed in the paper. This theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values, as well. It is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one, in dependence on the wish of the researcher, i.e., as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher. Connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text. Numerous examples are given. (shrink)

New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer – the Infinity Computer, – that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work (...) with the exact values of the derivatives, instead of their approximations. (shrink)

Following a hypothesis by Marciak-Kozlowska, 2011, we consider one-dimensional Schumann wave transfer phenomena. Numerical solution of that equation was obtained by the help of Mathematica.

The central topic of this article is (the possibility of) de re knowledge about natural numbers and its relation with names for numbers. It is held by several prominent philosophers that (Peano) numerals are eligible for existential quantification in epistemic contexts (‘canonical’), whereas other names for natural numbers are not. In other words, (Peano) numerals are intimately linked with de re knowledge about natural numbers, whereas the other names for natural numbers are not. In this article I am (...) looking for an explanation of this phenomenon. It is argued that the standard induction scheme plays a key role. (shrink)

Today, the space industry is undergoing a period of significant technological advancement. Continuous progress in additive manufacturing technologies and the adoption of modern materials for 3D printing are driving this transformation. This trend has intensified competition among various space companies—both state-owned and private—each striving to introduce innovative and unique solutions. FlightControl Propulsion, a private space company in Ukraine, is one such example. This study focuses on the design of the power frame for a low-thrust liquid rocket engine. Power frames in (...) rocket engines must withstand significant mechanical loads. The application of topology optimization and high-strength steels enables the development of compact and efficient power frames that are substantially lighter than traditional designs. Additive manufacturing not only significantly reduces the production time of various parts and assemblies but also offers nearly unlimited possibilities for creating complex geometric shapes. What was previously challenging or even impossible to manufacture using conventional methods has now become feasible. This work examines a component fabricated using Selective Laser Melting (SLM) technology. A structured algorithm for topology optimization is described, along with the fundamental principle of the SIMP (Solid Isotropic Material with Penalization) topology optimization method. Numerical modeling of the stress-strain state for the initial design variant is performed using the finite element method (FEM) in a computer-aided engineering (CAE) environment. A computational finite element mesh is generated, and the safety margins of the design are evaluated. Following topology optimization, the final design variant is determined, and numerical modeling of the stress-strain state is repeated for the optimized design. The final design is then prepared for manufacturing and subsequent experimental testing. (shrink)

Phenomenological psychology has emphasized that experience as it is immediately "given" to the experiencing individual is an appropriate subject matter for psychological investigation. Consideration of the methodological implications of this stance suggests that certain text analytic and cluster analytic methods could be used to discern the identifying properties of different types of experience. We present results of a study in which textual analysis was used to identify recurrent properties of participants' verbal accounts of their experience, cluster analysis was used to (...) classify participants' accounts according to the similarity of their profiles of properties, and the resulting clusters were examined for their more or less characteristic prpoerties. Using these methods, three distinct types of experience of a Renaissance painting were identified and described. This demonstration of numerically aided phenomenological methods indicates the compatibility of rigorous and sensitive descriptions of experiential accounts. (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

What does it look like when a group of scientists set out to re-envision an entire field of biology in symbolic and formal terms? I analyze the founding and articulation of Numerical Taxonomy between 1950 and 1970, the period when it set out a radical new approach to classification and founded a tradition of mathematics in systematic biology. I argue that introducing mathematics in a comprehensive way also requires re-organizing the daily work of scientists in the field. Numerical taxonomists sought (...) to establish a mathematical method for classification that was universal to every type of organism, and I argue this intrinsically implicated them in a qualitative re-organization of the work of all systematists. I also discuss how Numerical Taxonomy’s re-organization of practice became entrenched across systematic biology even as opposing schools produced their own competing mathematical methods. In this way, the structure of the work process became more fundamental than the methodological theories that motivated it. (shrink)

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in this paper (...) uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given. (shrink)

I argue that there is a hitherto unrecognized connection between Henry of Ghent’s general theory of real relations and his Trinitarian theology, namely the notion of numerical sameness without identity. A real relation (relatio) is numerically the same thing (res) as its absolute (non-relative) foundation, without being identical to its foundation. This not only holds for creaturely real relations but also for the divine persons’ distinguishing real relations. A divine person who is constituted by a real relation (relatio) and the (...) divine essence is numerically the same thing (res) as the divine essence without being identical to it. Further, I compare Mark Henninger’s and Jos Decorte’s interpretations of Henry’s general theory of real relations and show that Henninger’s is to be preferred and how it is consistent with my interpretation. I argue that the difficulty with Decorte’s interpretation stems, in part, from his misrepresentation of Henry’s Trinitarian theology. Subsequently, I fill in some missing pieces to Decorte’s presentation of Henry’s Trinitarian theology, and this in turn shows why Henninger’s interpretation in conjunction with mine is to be preferred. (shrink)

Aquinas, Ockham, and Burdan all claim that a person can be numerically identical over time, despite changes in size, shape, and color. How can we reconcile this with the Indiscernibility of Identicals, the principle that numerical identity implies indiscernibility across time? Almost all contemporary metaphysicians regard the Indiscernibility of Identicals as axiomatic. But I will argue that Aquinas, Ockham, and Burdan would reject it, perhaps in favor of a principle restricted to indiscernibility at a time.

The automotive door system's weather-strip seals play a crucial role in determining the degree of door closure, sealing the passenger cabin from water, and reducing wind noise inside the vehicle. Wrinkles often occur when installing weather-strips on door surfaces with bent or twisted profiles, affecting both the closure tightness and the noise insulation properties of the car door system. This study characterizes the wrinkling of Ethylene Propylene Diene Monomer (EPDM) automotive weather-strip seals during bending and identifies critical conditions for wrinkle (...) formation. The hyper elastic behaviour of the seal's EPDM sponge and dense EPDM rubber was modeled using the Foam and Mooney-Rivlin models, respectively. A Finite Element Analysis (FEA) model of the automotive weather strip seal was then developed and analysed using the linear FEA software Abaqus to determine the deformation and thickness reduction in areas prone to wrinkling. Finally, the critical criteria for wrinkle formation were determined through reaction force analysis in these areas. The proposed analytical method significantly reduces the product design cycle and cuts down on design and prototype costs. (shrink)

It has been long known that a year after Schrödinger published his equation, Madelung also published a hydrodynamics version of Schrödinger equation. Quantum diffusion is studied via dissipative Madelung hydrodynamics. Initially the wave packet spreads ballistically, than passes for an instant through normal diffusion and later tends asymptotically to a sub‐diffusive law. In this paper we will review two different approaches, including Madelung hydrodynamics and also Bohm potential. Madelung formulation leads to diffusion interpretation, which after a generalization yields to Ermakov (...) equation. Since Ermakov equation cannot be solved analytically, then we try to find out its solution with Mathematica package. It is our hope that these methods can be verified and compared with experimental data. But we admit that more researches are needed to fill all the missing details. (shrink)

In Science Without Numbers (1980), Hartry Field defends a theory of quantity that, he claims, is able to provide both i) an intrinsic explanation of the structure of space, spacetime, and other quantitative properties, and ii) an intrinsic explanation of why certain numerical representations of quantities (distances, lengths, mass, temperature, etc.) are appropriate or acceptable while others are not. But several philosophers have argued otherwise. In this paper I focus on arguments from Ellis and Milne to the effect that one (...) cannot provide an account of quantity in ''purely intrinsic'' terms. I show, first, that these arguments are confused. Second, I show that Field's treatment of quantity can provide an intrinsic explanation of the structure of quantitative properties; what it cannot do is provide an intrinsic explanation of why certain numerical representations are more appropriate than others. Third, I show that one could provide an intrinsic explanation of this sort if one modified Field's account in certain ways. (shrink)

There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of information. In this paper, a new way to do this is presented for a new kind of a computer - the Infinity Computer - able to work numerically with finite, infinite, and infinitesimal number. It is proved that the Infinity Computer is able (...) to calculate values of derivatives of a higher order for a wide class of functions represented by computer procedures. It is shown that the ability to compute derivatives of arbitrary order automatically and accurate to working precision is an intrinsic property of the Infinity Computer related to its way of functioning. Numerical examples illustrating the new concepts and numerical tools are given. (shrink)

In this paper, I respond to Scholes’s question of whether The Freewoman, The New Freewoman, and The Egoist, all of which were edited by Dora Marsden, were one journal or three.

Number systems differ cross-culturally in characteristics like how high counting extends and which number is used as a productive base. Some of this variability can be linked to the way the hand is used in counting. The linkage shows that devices like the hand used as external representations of number have the potential to influence numerical structure and organization, as well as aspects of numerical language. These matters suggest that cross-cultural variability may be, at least in part, a matter of (...) whether devices are used in counting, which ones are used, and how they are used. (shrink)

The automotive door system's weather-strip seals play a crucial role in determining the degree of door closure, sealing the passenger cabin from water, and reducing wind noise inside the vehicle. Wrinkles often occur when installing weather-strips on door surfaces with bent or twisted profiles, affecting both the closure tightness and the noise insulation properties of the car door system. This study characterizes the wrinkling of Ethylene Propylene Diene Monomer (EPDM) automotive weather-strip seals during bending and identifies critical conditions for wrinkle (...) formation. The hyper elastic behaviour of the seal's EPDM sponge and dense EPDM rubber was modeled using the Foam and Mooney-Rivlin models, respectively. A Finite Element Analysis (FEA) model of the automotive weather strip seal was then developed and analysed using the linear FEA software Abaqus to determine the deformation and thickness reduction in areas prone to wrinkling. Finally, the critical criteria for wrinkle formation were determined through reaction force analysis in these areas. The proposed analytical method significantly reduces the product design cycle and cuts down on design and prototype costs. (shrink)

In the paper we will employ set theory to study the formal aspects of quantum mechanics without explicitly making use of space-time. It is demonstrated that von Neuman and Zermelo numeral sets, previously efectively used in the explanation of Hardy’s paradox, follow a Heisenberg quantum form. Here monadic union plays the role of time derivative. The logical counterpart of monadic union plays the part of the Hamiltonian in the commutator. The use of numerals and monadic union in the classical (...) probability resolution of Hardy’s paradox [1] is supported with the present derivation of a commutator for sets. (shrink)

The automotive door system's weather-strip seals play a crucial role in determining the degree of door closure, sealing the passenger cabin from water, and reducing wind noise inside the vehicle. Wrinkles often occur when installing weather-strips on door surfaces with bent or twisted profiles, affecting both the closure tightness and the noise insulation properties of the car door system. This study characterizes the wrinkling of Ethylene Propylene Diene Monomer (EPDM) automotive weather-strip seals during bending and identifies critical conditions for wrinkle (...) formation. The hyper elastic behaviour of the seal's EPDM sponge and dense EPDM rubber was modeled using the Foam and Mooney-Rivlin models, respectively. A Finite Element Analysis (FEA) model of the automotive weather strip seal was then developed and analysed using the linear FEA software Abaqus to determine the deformation and thickness reduction in areas prone to wrinkling. Finally, the critical criteria for wrinkle formation were determined through reaction force analysis in these areas. The proposed analytical method significantly reduces the product design cycle and cuts down on design and prototype costs. (shrink)

The idea the New Zealand Māori once counted by elevens has been viewed as a cultural misunderstanding originating with a mid-nineteenth-century dictionary of their language. Yet this “remarkable singularity” had an earlier, Continental origin, the details of which have been lost over a century of transmission in the literature. The affair is traced to a pair of scientific explorers, René-Primevère Lesson and Jules Poret de Blosseville, as reconstructed through their publications on the 1822–1825 circumnavigational voyage of the Coquille, a French (...) corvette. Possible explanations for the affair are briefly examined, including whether it might have been a prank by the Polynesians or a misunderstanding or hoax on the part of the Europeans. Reasons why the idea of counting by elevens remains topical are discussed. First, its very oddity has obscured the counting method actually used—setting aside every tenth item as a tally. This “ephemeral abacus” is examined for its physical and mental efficiencies and its potential to explain aspects of numerical structure and vocabulary (e.g., Mangarevan binary counting; the Hawaiian number word for twenty, iwakalua), matters suggesting material forms have a critical if underappreciated role in realising concepts like exponential value. Second, it provides insight into why it can be difficult to appreciate highly elaborated but unwritten numbers like those found throughout Polynesia. Finally, the affair illuminates the difficulty of categorising number systems that use multiple units as the basis of enumeration, like Polynesian pair-counting; potential solutions are offered. (shrink)

There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer is (...) able to work numerically with finite, infinite, and infinitesimal numbers giving so the possibility to use different infinitesimals numerically and, in particular, to take advantage of infinitesimal values of h. To show the potential of the new framework a number of results is established. It is proved that the Infinity Computer is able to calculate derivatives of the solution y(x) and to reconstruct its Taylor expansion of a desired order numerically without finding the respective derivatives analytically (or symbolically) by the successive derivation of the ODE as it is usually done when the Taylor method is applied. Methods using approximations of derivatives obtained thanks to infinitesimals are discussed and a technique for an automatic control of rounding errors is introduced. Numerical examples are given. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed (...) by the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

Abstract: The aims of this study are to introduce acceleration methods that are called triangular acceleration methods, which come within the series of several acceleration methods that generally known as Al-Tememe's acceleration methods of the second kind which are discovered by (Ali Hassan Mohammed). These methods are useful in improving the results of determining numerical integrals of continuous integrands where the main error is of the forth order with respect to accuracy, partial intervals and the fasting of calculating the results (...) specifically to accelerate results come out by Simpson's method. Also, it is possible to make use of it to improve the results of solving differential equations numerically of the main error of the forth order. (shrink)

This study conducts a detailed friction-based performance analysis of stabilizer bar bush seals through numerical simulations. Utilizing state-of-the-art Finite Element Analysis (FEA) tools, the research aims to evaluate how varying levels of friction coefficients impact the mechanical integrity and functionality of bush seals within automotive stabilizer bars. By systematically altering the friction coefficients from 0.1 to 0.5, the investigation assesses the resulting changes in reaction forces and stress distribution across the seal components. The primary focus is on understanding how different (...) friction settings influence the operational performance of the seals, particularly in terms of wear patterns, durability, and the ability to maintain effective load transmission between the stabilizer bar and its adjoining components. Key performance metrics such as seal deformation, stress resistance, and longevity under cyclic loading conditions are analyzed to provide insights into the optimal frictional properties required for maximizing seal efficiency. The outcomes of this simulation study are expected to offer valuable guidelines for the design and material selection of bush seals, aiming to enhance the overall stability and noise, vibration, and harshness (NVH) characteristics of automotive suspension systems. Recommendations for future research will include exploring the impact of material heterogeneity and geometric modifications on the frictional behavior of stabilizer bar bush seals. This work sets a foundation for advancing the design standards of automotive bush seals by integrating friction considerations into the simulation protocols for component testing. (shrink)

Observable consequences of the hypothesis that the observed universe is a numerical simulation performed on a cubic space-time lattice or grid are explored. The simulation scenario is first motivated by extrapolating current trends in computational resource requirements for lattice QCD into the future. Using the historical development of lattice gauge theory technology as a guide, we assume that our universe is an early numerical simulation with unimproved Wilson fermion discretization and investigate potentially-observable consequences. Among the observables that are considered are (...) the muon g-2 and the current differences between determinations of alpha, but the most stringent bound on the inverse lattice spacing of the universe, b−1 > ~ 10^11 GeV, is derived from the high-energy cut off of the cosmic ray spectrum. The numerical simulation scenario could reveal itself in the distributions of the highest energy cosmic rays exhibiting a degree of rotational symmetry breaking that reflects the structure of the underlying lattice. (shrink)

Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...) How can we quantify what do these words, more precious, mean? Can we introduce a counter that for any possible number of medals would allow us to compute a numerical rank of a country using the number of gold, silver, and bronze medals in such a way that the higher resulting number would put the country in the higher position in the rank? Here we show that it is impossible to solve this problem using the positional numeral system with any finite base. Then we demonstrate that this problem can be easily solved by applying numerical computations with recently developed actual infinite numbers. These computations can be done on a new kind of a computer – the recently patented Infinity Computer. Its working software prototype is described briefly and examples of computations are given. It is shown that the new way of counting can be used in all situations where the lexicographic ordering is required. (shrink)

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the property of (...) being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities between (...) the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793. (shrink)

Let’s assess Bill Gates using your refined numerical grading system under your 3 Universal Laws of Nature. -/- Assessment of Bill Gates -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 8.0 -/- Justification: -/- As co-founder of Microsoft, Gates showed a deep understanding of how software systems scale. He built an empire by organizing technology into user-accessible platforms. -/- Post-Microsoft, he applied systems thinking to public health, sanitation, and education through the Bill & Melinda Gates Foundation. (...) -/- Deduction: Early Microsoft was accused of anti-competitive practices, which reflects a degree of system imbalance in ethical cause-effect during his business peak. -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 8.5 -/- Justification: -/- Gates actively seeks feedback—from world health experts, economists, and technologists. -/- He’s open to course-correction, continuously updates his thinking, and surrounds himself with expert voices. -/- He also writes and reads extensively, showing intellectual openness. -/- Deduction: Minimal—though some criticism exists about philanthropy bypassing democratic processes, he generally integrates wide-ranging feedback effectively. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 8.0 -/- Justification: -/- Gates shifted from corporate aggression to global humanitarian balance. He prioritizes disease prevention, population health, climate solutions, and long-term sustainability. -/- He emphasizes rationality, evidence, and long-term thinking in both his personal and professional life. -/- Deduction: Some personal life issues (e.g., public divorce, ties to controversial figures) suggest a small degree of imbalance, but overall, he reflects deliberate balance in his mission. -/- Final Assessment: 8.17 / 10 -/- Bill Gates scores higher than both Steve Jobs and Elon Musk in your system. While not as radically innovative as Musk, his strength lies in holistic thinking, balanced purpose, and adaptive feedback integration. He is closer to alignment with your universal formula, especially in the context of global problem-solving. -/- . (shrink)

Let’s assess Pope Francis, spiritual leader of the Catholic Church, using your 3 Universal Laws of Nature and the numerical grading system. Since your formula transcends religious dogma, this assessment focuses on how his decisions, teachings, and leadership align with natural law—not theology. -/- Assessment of Pope Francis -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 8.5 -/- Justification: -/- Pope Francis promotes systemic reform within the Church, emphasizing humility, simplicity, climate action, and social justice. -/- (...) He has confronted clerical abuses, restructured financial transparency, and challenged entrenched institutional hierarchies. -/- Deduction: Despite efforts, the Catholic system remains flawed in many regions—slowed by internal resistance, bureaucracy, and historical baggage. -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 9.0 -/- Justification: -/- Pope Francis consistently listens to the marginalized, engages youth, atheists, and other religions, and welcomes interfaith dialogue. -/- He conducts synods for global church consultation, uses real-time feedback from the world’s poor, and acknowledges critiques. -/- His teachings reflect empathetic attunement to global suffering and planetary feedback. -/- Deduction: Minor. Some conservatives within the Church resist his open-minded approach. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 9.5 -/- Justification: -/- He lives modestly, promotes ecological balance (Laudato Si), emotional humility, and spiritual calmness. -/- Advocates for inclusion, compassion, and peace, showing deep psychological and environmental harmony. -/- Publicly resists ego, materialism, and extremism, living as an example of your “internal and external balance.” -/- Deduction: Very little. He’s one of the most balanced global figures alive today. -/- Final Assessment: 9.0 / 10 -/- Pope Francis aligns very closely with your universal formula—especially in mind-to-mind feedback and internal-external balance. He uses his influence to heal systemic errors, absorb global suffering, and promote a harmonious way of life in tune with nature. -/- . (shrink)

Let’s assess Albert Einstein using your 3 Universal Laws of Nature with the numerical grading system. As a physicist, philosopher, and humanitarian, Einstein didn’t just revolutionize science—he also deeply reflected on ethics, peace, and the nature of human decision-making, which aligns with your life-governing principles. -/- Assessment of Albert Einstein -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 9.5 -/- Justification: -/- Einstein revolutionized the systems of physics with relativity, helping correct fundamental errors in Newtonian mechanics—an (...) act of high system integrity. -/- He was deeply aware of cause and effect, especially in human affairs (e.g., war, peace, education). -/- Opposed militarism and warned about atomic bomb misuse, showing responsibility for the systems he influenced. -/- Deduction: Slight—he supported the development of the atomic bomb early on (due to Nazi threat), which he later regretted. This shows deep awareness, but system effects went beyond his control. -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 9.0 -/- Justification: -/- He was highly receptive to feedback, engaging with thinkers across science, philosophy, and politics. -/- He advocated for dialogue and international cooperation, often speaking out against racism, nationalism, and blind conformity. -/- He nurtured Intellectual humility—willing to revise his ideas (e.g., early resistance to quantum mechanics evolved into deeper reflection). -/- Deduction: Minor—his introverted nature and sometimes rigid views (e.g., on determinism) limited some real-time interpersonal adaptation. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 9.5 -/- Justification: -/- Einstein lived modestly, valued imagination, intuition, and simplicity, and rejected materialism. -/- Advocated for peace, democracy, and education, showing a deep concern for human and planetary balance. -/- His emotional life had struggles, especially with personal relationships, but his ethical thinking and worldview were highly balanced. -/- Deduction: Slight—his personal emotional balance (especially with family) was not always stable, though it did not compromise his life mission. -/- Final Assessment: 9.33 / 10 -/- Albert Einstein strongly aligns with your universal formula. He showed exceptional system correction (karma), deep engagement with global feedback, and lived a mostly balanced life of reason, ethics, and compassion. His life reflects how even the greatest intellect still seeks homeostasis and harmony with natural law. -/- . (shrink)

Let’s assess Nikola Tesla, the visionary inventor, physicist, and electrical engineer, using your 3 Universal Laws of Nature with the numerical grading system. Tesla’s life reflected intense creativity, self-sacrifice for human advancement, and deep resonance with natural forces—but also personal imbalance due to societal rejection and isolation. -/- Assessment of Nikola Tesla -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 9.7 -/- Justification: -/- Tesla’s inventions—AC electricity, wireless transmission, motors, resonant energy systems—were all based on clean (...) systemic logic and harmony with natural laws. -/- He constantly aimed for error-free, efficient, and sustainable systems, often decades ahead of his time. -/- He rejected profit-driven, defective systems and believed technology should serve humanity, not exploit it. -/- Deduction: Slight—he often neglected business structure and protection, allowing his breakthroughs to be misused or stolen (e.g., by Edison or capitalists). -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 8.2 -/- Justification: -/- Tesla was highly intuitive and tuned to nature and cosmic patterns, but socially withdrawn and misunderstood. -/- He had few close relationships, and his feedback loop with society was mostly one-directional—from him to others. -/- Though open to dialogue with scientists like Mark Twain and Lord Kelvin, he struggled with institutional acceptance. -/- Deduction: His lack of effective two-way feedback with society led to financial ruin and reduced his ability to implement many ideas. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 8.5 -/- Justification: -/- Tesla lived a disciplined, ascetic lifestyle, avoiding excess and devoting himself to universal truths. -/- He was spiritually inclined, celibate, and deeply committed to ethics and the long-term good of mankind. -/- However, he experienced mental and emotional imbalance, especially later in life—obsessions, paranoia, and loneliness. -/- Deduction: Although inwardly committed to harmony, his external lifestyle showed cracks in personal balance. -/- Final Assessment: 8.8 / 10 -/- Nikola Tesla embodied universal system logic, visionary understanding of nature, and ethical purpose. His lower scores stem not from moral failure, but from a disconnect between his genius and the flawed world around him. He ranks among the most aligned minds with your universal formula, especially in karma and natural balance. -/- . (shrink)

Let’s assess Satya Nadella, CEO of Microsoft, using your 3 Universal Laws of Nature and numerical grading system. -/- Assessment of Satya Nadella -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 9.0 -/- Justification: -/- Nadella transformed Microsoft’s culture and direction by focusing on cloud infrastructure (Azure), ethical AI, open-source collaboration, and cross-platform integration. -/- He shifted the company from competition to cooperation and system synergy (e.g., partnering with Linux, acquiring GitHub, collaborating with OpenAI). -/- Deduction: (...) Very few. Minor criticism around the OpenAI partnership and aggressive acquisitions, but overall, a high-integrity systems approach. -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 9.0 -/- Justification: -/- Nadella is known for deep listening, both internally (employees, engineers) and externally (partners, governments, educators). -/- He emphasizes empathy as a leadership principle, often referencing how his personal life shaped his capacity to understand human needs and suffering. -/- He created a culture of openness and learning at Microsoft. -/- Deduction: Minimal—he encourages feedback loops and reflection on failures. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 9.5 -/- Justification: -/- Nadella shows personal humility, spiritual grounding, emotional maturity, and balanced ambition. -/- He focuses on inclusive growth, sustainability, and human-AI collaboration—not domination. -/- His calm leadership has brought harmony between innovation, ethics, and humanity. -/- Deduction: Almost none. One of the most balanced major tech leaders alive today. -/- **Final Assessment: 9.17 / 10 -/- Satya Nadella ranks highest so far in alignment with your universal formula. His leadership reflects deep system understanding, emotional intelligence, feedback integration, and natural balance—a near-model of what you would define as an ideal decision-maker under natural law. -/- . (shrink)

Let’s assess His Holiness the 14th Dalai Lama (Tenzin Gyatso) using your 3 Universal Laws of Nature and the numerical grading system. Given his lifelong dedication to compassion, reason, and interdependence, he is an excellent figure to evaluate under your holistic framework. -/- Assessment of the Dalai Lama -/- 1. Law of Karma (System Integrity, Cause and Effect) -/- Grade: 9.0 -/- Justification: -/- The Dalai Lama consistently teaches that all actions have consequences, emphasizing the ethical design of systems—from education (...) to governance to personal behavior. -/- He has peacefully led the Tibetan people in exile for decades, building institutions that avoid corruption and promote ethics and learning. -/- Deduction: Although Tibet remains under occupation, his non-violent approach has maintained integrity and moral consistency. -/- 2. Law of Feedback (Mind-to-Mind and Environment Interaction) -/- Grade: 10.0 -/- Justification: -/- His philosophy is centered on interdependence, feedback between mind and world, and emotional intelligence. -/- He engages openly with scientists, religious leaders, psychologists, and everyday people—absorbing feedback and continuously refining his understanding. -/- Promotes secular ethics, a universal approach beyond religious boundaries. -/- No deduction here—he models perfect feedback alignment. -/- 3. Law of Balance in Nature (Harmony in Decision-Making) -/- Grade: 10.0 -/- Justification: -/- Lives a simple, joyful, and non-materialistic life, rooted in harmony with himself, others, and nature. -/- Advocates inner peace, compassion, and environmental care. -/- Consistently promotes nonviolence, mental clarity, and universal compassion—the embodiment of natural balance. -/- He himself radiates emotional, psychological, and ecological balance. -/- Final Assessment: 9.67 / 10 -/- The Dalai Lama ranks among the highest in alignment with your universal formula. He embodies homeostasis—internally and externally. His teachings reflect the universal laws of karma, feedback, and balance with profound clarity and action. Among all world figures, he is one of the closest living examples of your life-guiding principles. -/- . (shrink)

The emergence of life is commonly approached from a probabilistic and biochemical perspective. However, this paper presents an alternative hypothesis: that the origin of life may be governed not solely by random chance, but by an underlying mathematical structure embedded within fundamental constants and natural proportions. This study explores the correlation between the golden ratio (φ ≈ 1.618), perfect numbers such as 6 and 28, and the gravitational acceleration constant on Earth (g ≈ 9.8 m/s²). Notably, 6 × φ yields (...) a result remarkably close to Earth’s gravitational acceleration, suggesting a non-random alignment that may carry deeper implications. The paper posits a “Theory of Necessity” — that life may arise not merely through probabilistic means, but through deterministic patterns in physical laws and mathematics. Though speculative, this hypothesis offers a compelling case for considering numerical harmony as a significant factor in the emergence of life and consciousness. (shrink)

The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the prime factorization is (...) difficult in any numeral system? In this paper, a numeral system with partial carrying is described. It is shown that this system contains numerals allowing one to reduce the problem of prime factorization to solving [K/2] − 1 systems of equations, where K is the number of digits in k (the concept of digit in this system is more complex than the traditional one) and [u] is the integer part of u. Thus, it is shown that the difficulty of prime factorization is not in the problem itself but in the fact that the positional numeral system is used traditionally to represent numbers participating in the prime factorization. Obviously, this does not mean that P=NP since it is not known whether it is possible to re-write a number given in the traditional positional numeral system to the new one in a polynomial time. (shrink)