Mathematical Entities Research Articles

Two-Track Depictions of Leibniz’s Fictions

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers as mathematical entities? Alice and Bob take on the labyrinth of the current Leibniz scholarship.

Continuous Abstract Data Types for Verified Computation

AbstractWe devise imperative programming languages for verified real number computation where real numbers are provided as abstract data types such that the users of the languages can express real number computation by considering real numbers as abstract mathematical entities. Unlike other common approaches toward real number computation, based on an algebraic model that lacks implementability or transcendental computation, or finite-precision approximation such as using double precision computation that lacks a formal foundation, our languages are devised based on computable analysis, a foundation of rigorous computation over continuous data. Consequently, the users of the language can easily program real number computation and reason about the behaviours of their programs, relying on their mathematical knowledge of real numbers without worrying about artificial roundoff errors. As the languages are imperative, we adopt precondition–postcondition-style program specification and Hoare-style program verification methodologies. Consequently, the users of the language can easily program a computation over real numbers, specify the expected behaviour of the program, including termination, and prove the correctness of the specification. Furthermore, we suggest extending the languages with other interesting continuous data, such as matrices, continuous real functions, et cetera.Abstract taken directly from the thesis.E-mail: sewonpark17@gmail.comURL: https://sewonpark.com/thesis

Social constructivism in mathematics? The promise and shortcomings of Julian Cole\u2019s institutional account

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.

Dread Data and Algorithms

The mystical aura surrounding algorithms we observed in the last years is currently more and more perceived as an annoyance that we must accept but not like. Nevertheless, algorithms are still praised as a panacea or are seen as the main culprit of «digital immaturity» (digitale Unmündigkeit). The current debate is less about the algorithms (as a mathematical entity) themselves than about their technical implementation in hardware and software, often used synonymously with «Artificial Intelligence». However, the enormous informational transformation potential of algorithms and data is already visible in the mathematical construct as we will see.

What Can Our Best Scientific Theories Tell Us About The Modal Status of Mathematical Objects?

Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, Blackwell, Oxford, 1989) each suggest that a consequence of the empirical methodology of indispensability arguments is that the resulting mathematical objects can only be said to exist (or not exist) contingently. Kristie Miller has argued that this line of thought doesn’t work (Miller in Erkenntnis, 77 (3), 335-359, 2012). Miller argues that indispensability arguments are in direct tension with contingentism about mathematical objects, and that they cannot tell us about the modal status of mathematical objects. I argue that Miller’s argument is crucially imprecise, and that the best way of making it clearer no longer shows that the indispensability strategy collapses or is unstable if it delivers contingentist conclusions about what there is.

Mathematical Indispensability and Arguments from Design

The recognition of striking regularities in the physical world plays a major role in the justification of hypotheses and the development of new theories both in the natural sciences and in philosophy. However, while scientists consider only strictly natural hypotheses as explanations for such regularities, philosophers also explore meta-natural hypotheses. One example is mathematical realism, which proposes the existence of abstract mathematical entities as an explanation for the applicability of mathematics in the sciences. Another example is theism, which offers the existence of a supernatural being as an explanation for the design-like appearance of the physical cosmos. Although all meta-natural hypotheses defy empirical testing, there is a strong intuition that some of them are more warranted than others. The goal of this paper is to sharpen this intuition into a clear criterion for the (in)admissibility of meta-natural explanations for empirical facts. Drawing on recent debates about the indispensability of mathematics and teleological arguments for the existence of God, I argue that a meta-natural explanation is admissible just in case the explanation refers to an entity that, though not itself causally efficacious, guarantees the instantiation of a causally efficacious entity that is an actual cause of the regularity.

The physics of conservation culturomics: the mass-energy-information equivalence principle to address misrepresented controversies

The application of the mass-energy-information equivalence principle developed after the experimentally demonstrated Landauer's principle on thermodynamics, entropy, and information is an unexplored but promising path in search of objectivity and compatibility between strict physical and mathematical entities and relative human behavior in biodiversity conservation issues. Conservation culturomics is proposed as the epistemic methodology and programme to trace the evolution in cultural human-nature relationships. Historically, controversies do persist between pro- vs. non- environmental opinions and policies. The proposed combination of physics and culturomics is feasible, although complex, multileveled, and depending on a series of academic, technical, and political prerequisites. In the era of staggering information technologies, Internet use proliferation and cultural relativism, reliable information on conservation knowledge vs. often unfounded story-tellings is a sine qua non for the development of badly needed modern global conservation strategies, targets, and goals.

Mark Balaguer. Metaphysics, Sophistry, and Illusion: Towards a Widespread Non-Factualism

As hardly needs to be said, the history of philosophy, Ancient, Medieval, and Modern, East and West, is littered with disputes about whether entities of certain kinds exist. In the present book Mark Balaguer aims to puncture at least many such debates. He argues that for many important questions of this kind — and so for others which presuppose the existence in question — there is simply no fact of the matter. He calls this view non-factualism. After a brief Chapter 1 introducing matters, the book has two parts. The first makes the case for non-factualism in detail for two specific questions: whether there are abstract mathematical entities, and whether there are partite objects. Chapter 2 argues that there are substantial issues here, and not merely verbal ones. It might be thought that non-factualism about mathematical entities undercuts both the objectivity and the applicability of mathematics. Chapter 3 argues that this is not the case. Such things can be accommodated by an appropriate version of fictionalism. Specifically, let |$M$| be some mathematical claim. Even if there is no fact of the matter about whether |$M$| is true, the conditional, is both true and suffices for these matters. A similar point is then made about realism concerning partite objects.

A Survey of CRF Algorithm Based Knowledge Extraction of Elementary Mathematics in Chinese

Chinese word segmentation is an important research direction in related research on elementary mathematics knowledge extraction. The speed of segmentation directly affects subsequent applications, and the accuracy of segmentation directly affects corresponding research in the next step. In the machine learning methods for extracting basic mathematical knowledge points, the Conditional Random Field (CRF) model implements new word discovery well, and is increasingly used in knowledge extraction of basic mathematics. This article first introduces the traditional CRF process of named entity recognition. Then, an improved algorithm CRF++for conditional field model is proposed. Since the recognition rate of named entities based on traditional machine learning methods is not high, a post-processing method for entity recognition that automatically generates a dictionary is proposed. After identifying mathematical entities, a pruning strategy combining Viterbi algorithm and rules is proposed to achieve a higher recognition rate of elementary mathematical entities. Finally, several methods of disambiguation after entity recognition are introduced.

STRATEGIES OF REDUCTION OF ABSTRACTION IN ABSTRACT ALGEBRA

This article is based on the study, which tries to unpack strategies of reduction of abstraction in learning abstract algebra from learners’ perspective. Ethnography was used to collect the required information. The study found the strategies of reduction of abstraction in abstract algebra are: making sense and meaning through previous experiences and existing knowledge an analogical creation of mental image, using first person language in course of doing mathematics by students as teachers do in the classroom for logical arguments, focusing on “symbol” or some mathematical entity to manage abstraction for their idiosyncratic understandings of abstract mathematical structure rather than the reflective thinking, using students own idiosyncratic figures to reduce the degrees of complexity of mathematical concepts. This study can lead teachers of abstract algebra to a new awareness of their teaching strategies and their practices.

Unpacking structural domain of mathematics to aid inquiry in physics: a pilot study

Analysing graphs, formulating covariational relationships, and hypothesizing systems’ behaviour have emerged as frequent objectives of contemporary research in physics education. As such, these studies aim to help students achieve these objectives. While a consensus has been reached on the cognitive benefits of emphasizing the structural domain of mathematical apparatus in physics instruction, the question remains open on how to accomplish this task in school practice. This study contributes to this discussion. It attempts to determine how high-school physics students perceive formulas as a means to understand covariational relationships of natural phenomena. It also explores the hypothesis that including more structural domains of mathematics makes physics content more attractive for students. A sample of 25 advanced high-school physics students was taught kinematics, dynamics, and energy sections considering functional representations of some laws and principles traditionally covered in those sections. At the concluding phase, these students were invited to contrast formulas and their covariational embodiments as a means of providing conceptual understanding. The analysis revealed that students considered formulas as algebraic tools to attain a specific value to a particular application or carry out algebraic algorithms to solve a physics problem. They found formulas converted to covariational relationships a better fit for the learning of general phenomenon behaviour and extracting new information, especially from a graphical representation of these relationships. These findings generated the conclusion that the structural domain of mathematics can aid physics understanding by alternatively presenting formulas as mathematical entities that are transferrable to functions. It was further concluded that such parallelism would benefit not only the depth of inquiry in physics but also a general students’ STEM disposition.

Metaphor and the Philosophical Implications of Embodied Mathematics.

Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.

Screw theory based method to formulate local Jacobian’ magnitude estimator contours for 6 DoF robots

Singularity prediction of Jacobian is inherently related to the design of any manipulator. Optimization of Jacobian is known to improve the joint rates and joint torques for the same set of end-effector twists and wrenches. Any mathematical entity which is an estimator of Jacobian can be used as a manifestation of the stated two schemes. It is required that online computation of Jacobian be implemented so that singular poses can be avoided and performance is maximized. We present an algorithm that relates to the determinant of the Jacobian for robotic manipulators. The algorithm developed is quicker to compute, and a strategy is developed to synthesize the local estimator contour. The contours change dynamically, henceforth optimal motor torques (in case of presence of interaction fields or system requiring force control) and optimal joint rates can be defined, depending on the control algorithm. The algorithm developed is independent of the structure of the matrix and is generalized to any six DoF manipulator structure used.

The mess behind the models: Too many of the COVID-19 models led policymakers astray. Here's how tomorrow's models will get it right

If you wanted to “flatten the curve” in 2019, you might have been changing students' grades or stamping down a rug ripple. Today, that phrase refers only to the vital task of reducing the peak number of people concurrently infected with the COVID-19 virus. Beginning in early 2020, graphs depicting the expected number of infections spread through social networks, much like the virus itself. We've all become consumers of epidemiological models, the mathematical entities that spit out these ominous trend lines. Such models have existed for decades but have never received such widespread attention. They're informing public policy, financial planning, health care allocation, doomsday speculation, and Twitter hot takes. In the first quarter of 2020, government leaders were publicly parsing these computational speculations, making huge decisions about whether to shut down schools, businesses, and travel. Would an unchecked outbreak kill millions, or fizzle out? Which interventions would help the most? How sure could we be of any forecast? Models disagreed, and some people pointed to whichever curve best supported their predilections. It didn't help that the researchers building the models were still figuring out what the heck they were doing.

Denotational recurrence extraction for amortized analysis

A typical way of analyzing the time complexity of functional programs is to extract a recurrence expressing the running time of the program in terms of the size of its input, and then to solve the recurrence to obtain a big-O bound. For recurrence extraction to be compositional, it is also necessary to extract recurrences for the size of outputs of helper functions. Previous work has developed techniques for using logical relations to state a formal correctness theorem for a general recurrence extraction translation: a program is bounded by a recurrence when the operational cost is bounded by the extracted cost, and the output value is bounded, according to a value bounding relation defined by induction on types, by the extracted size. This previous work supports higher-order functions by viewing recurrences as programs in a lambda-calculus, or as mathematical entities in a denotational semantics thereof. In this paper, we extend these techniques to support amortized analysis, where costs are rearranged from one portion of a program to another to achieve more precise bounds. We give an intermediate language in which programs can be annotated according to the banker's method of amortized analysis; this language has an affine type system to ensure credits are not spent more than once. We give a recurrence extraction translation of this language into a recurrence language, a simply-typed lambda-calculus with a cost type, and state and prove a bounding logical relation expressing the correctness of this translation. The recurrence language has a denotational semantics in preorders, and we use this semantics to solve recurrences, e.g analyzing binary counters and splay trees.

General definitions for the union and intersection of ordered fuzzy multisets

Since its original formulation, the theory of fuzzy sets has spawned a number of extensions where the role of membership values in the real unit interval $[0, 1]$ is handed over to more complex mathematical entities. Amongst the many existing extensions, two similar ones, the fuzzy multisets and the hesitant fuzzy sets, rely on collections of several distinct values to represent fuzzy membership, the key difference being that the fuzzy multisets allow for repeated membership values whereas the hesitant fuzzy sets do not. But in neither case are these collections of values ordered, as they are simply represented through multisets or sets. In this paper, we study ordered fuzzy multisets, where the membership value can be an ordered $n$-tuple of values, thus accounting for both order and repetition. We present some basic definitions and results and explore the relation between these ordered fuzzy multisets and the fuzzy multisets and hesitant fuzzy sets.

An Extra-Mathematical Program Explanation of Color Experience

ABSTRACT In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call ‘extra-mathematical’ explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain between the PEs’ mathematical explanantia and physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization.

Path Integrals and Holism

This paper argues that the path integral formulation of quantum mechanics suggests a form of holism for which the whole (total ensemble of paths) has properties that are not strongly reducible to the properties of the parts (the single trajectories). Feynman’s sum over histories calculates the probability amplitude of a particle moving within a boundary by summing over all the possible trajectories that the particle can undertake. These trajectories and their individual probability amplitudes are thus necessary in calculating the total amplitude. However, not all possible trajectories are differentiable, thus suggesting that they are not physical possibilities, but only mathematical entities. It follows that if the possible differentiable trajectories are taken to be part of the physical system, they are not sufficient to calculate the total probability amplitude. The conclusion is that the total ensemble is weakly non-supervenient upon the physically possible trajectories.

Planar semilattices and nearlattices with eighty-three subnearlattices

Finite (upper) nearlattices are essentially the same mathematical entities as finite semilattices, finite commutative idempotent semigroups, finite join-enriched meet semilattices, and chopped lattices. We prove that if an nelement nearlattice has at least 83 · 2 n−8 subnearlattices, then it has a planar Hasse diagram. For n > 8, this result is sharp.

La naturaleza de las entidades matemáticas. Gassendi y Mersenne: objetores de Descartes

La naturaleza de las entidades matemáticas ha sido un problema filosófico recurrente en diversas épocas; aquí mostraré que fue una pieza clave en la definición de las posturas ontológicas durante la Modernidad temprana. La piedra de toque para la fundamentación de los conocimientos científicos fue el carácter que se atribuyó a las entidades matemáticas —y, en general, a las entidades abstractas, incluidas las lógicas— en la filosofía natural. Expongo dos posiciones de la Modernidad: la que defendió René Descartes, quien las concibió como entidades perennes, inherentes a la propia constitución y funcionamiento de la mente y la de autores como Pierre Gassendi y Marin Mersenne, quienes defendieron el origen empírico e instrumental de esas entidades.