Two Flavours of Mathematical Explanation

2018 ◽  
Author(s):  
Mark Colyvan ◽  
John Cusbert ◽  
Kelvin McQueen
Keyword(s):  
Mathematical Explanation ◽  
Mathematical Theorem

A proof of a mathematical theorem tells us that the theorem is true (or should be accepted), but some proofs go further and tell us why the theorem is true (or should be accepted). That is, some, but not all, proofs are explanatory. Call this intra-mathematical explanation and it is to be contrasted with extra-mathematical explanation, where mathematics explains things external to mathematics. This chapter focuses on the intra-mathematical case. The authors consider a couple of examples of explanatory proofs from contemporary mathematics. They determine whether these proofs share some common feature that may account for their explanatoriness. The authors conclude with two plausible, but competing, accounts of mathematical explanation and suggest that there might be more than one kind of explanation at work in mathematics.

The Non-Causal Character of Renormalization Group Explanations

2018 ◽  
Author(s):  
Margaret Morrison
Keyword(s):  
Probability Theory ◽  
Renormalization Group ◽  
Dynamical Systems ◽  
Recent Literature ◽  
Mathematical Explanation ◽  
Causal Status ◽  
Physical Information ◽  
The Relationship ◽  
Structural Aspects

After reviewing some of the recent literature on non-causal and mathematical explanation, this chapter develops an argument as to why renormalization group (RG) methods should be seen as providing non-causal, yet physical, information about certain kinds of systems/phenomena. The argument centres on the structural character of RG explanations and the relationship between RG and probability theory. These features are crucial for the claim that the non-causal status of RG explanations involves something different from simply ignoring or “averaging over” microphysical details—the kind of explanations common to statistical mechanics. The chapter concludes with a discussion of the role of RG in treating dynamical systems and how that role exemplifies the structural aspects of RG explanations which in turn exemplifies the non-causal features.

The Effect of User Experience on Designing Interactive Tool: A Case Study of Learning Images Compression

2021 ◽  
Vol 9 (10) ◽  
pp. 1312-1320
Keyword(s):  
User Experience ◽  
Real Life ◽  
Mathematical Concept ◽  
Design Principles ◽  
Mathematical Explanation ◽  
Mathematical Concepts ◽  
Learning Platform ◽  
Learning Mathematics ◽  
The Common ◽  
Value Decomposition

Technology has significantly emerged in various fields, including healthcare, government, and education. In the education field, students of all ages and backgrounds turn to modern technologies for learning instead of traditional methods, especially under challenging courses such as mathematics. However, students face many problems in understanding mathematical concepts and understanding how to benefit from them in real-life. Therefore, it can be challenging to design scientific materials suitable for learning mathematics and clarifying their applications in life that meet the students’ preferences. To solve this issue, we designed and developed an interactive platform based on user experience to learn an advanced concept in the idea of linear algebra called Singular Value Decomposition (SVD) and its applicability in image compression. The proposed platform considered the common design principles to map between the provider in terms of clear mathematical explanation and the receiver in terms of matching good user experience. Twenty participants between the ages of 16 and 30 tested the proposed platform. The results showed that learning using it gives better results than learning traditionally in terms of the number of correct and incorrect actions, effectiveness, efficiency, and safety factors. Consequently, we can say that designing an interactive learning platform to explain an advanced mathematical concept and clarify its applications in real-life is preferable by considering and following the common design principles.

The Behavior of Number Concentration Tendencies for the Continuous Collection Growth Equation Using One- and Two-Moment Bulk Parameterization Schemes

2007 ◽  
Vol 46 (8) ◽  
pp. 1264-1274
Author(s):  
Jerry M. Straka ◽  
Katharine M. Kanak ◽  
Matthew S. Gilmore
Keyword(s):  
Growth Process ◽  
Number Concentration ◽  
Mathematical Explanation ◽  
Growth Equation ◽  
Parameterization Schemes ◽  
Distribution Shape ◽  
Bulk Parameterization

Abstract This paper presents a mathematical explanation for the nonconservation of total number concentration Nt of hydrometeors for the continuous collection growth process, for which Nt physically should be conserved for selected one- and two-moment bulk parameterization schemes. Where possible, physical explanations are proposed. The assumption of a constant no in scheme A is physically inconsistent with the continuous collection growth process, as is the assumption of a constant Dn for scheme B. Scheme E also is nonconservative, but it seems this result is not because of a physically inconsistent specification; rather the solution scheme’s equations simply do not satisfy Nt conservation and Nt does not come into the derivation. Even scheme F, which perfectly conserves Nt, does not preserve the distribution shape in comparison with a bin model.

scholarly journals Star Wars - an episodes battle

2020 ◽  
Vol 8 (6) ◽  
pp. 367-380
Author(s):  
Patrícia Nunes da Silva ◽  
Monica Almeida Gama ◽  
André Luiz Cordeiro dos Santos
Keyword(s):  
Mathematical Explanation ◽  
Star Wars ◽  
Market Experiment

Mlodinow (2008) proposed a crazy market experiment: to release the same film under two titles: Star Wars: Episode A and Star Wars: Episode B. Their marketing campaigns and distribution schedule are identical except by their titles on trailers and ads. He looks at the first 20,000 moviegoers and record the film they choose to see. He claims it is most probable the lead never changes, and it is 88 times more likely that one of the two films will be int the lead through all 20,000 customers than it is that the lead continuously seesaw. We present a detailed mathematical explanation for Mlodinow claims.

Small Steps and Great Leaps in Thought

Reasoning ◽  
2019 ◽  
pp. 152-177 ◽  
Author(s):  
Joshua Schechter
Keyword(s):  
Modus Ponens ◽  
Single Step ◽  
Step Counts ◽  
Mathematical Theorem ◽  
Relevant Rule

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? This chapter canvasses several possible explanations. It argues that the most promising approach is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought.

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