Vagueness in the Exact Sciences

Impacts in Mathematics, Physics, Chemistry, Biology, Medicine, Engineering and Computing

Apostolos Syropoulos, Basil K. Papadopoulos, Apostolos Syropoulos, Basil K. Papadopoulos

• Publisher: De Gruyter
• Year: 2021
• Print ISBN: 9783110704181
• eBook ISBN: 9783110704372
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

The book starts with the assumption that vagueness is a fundamental property of this world. From a philosophical account of vagueness via the presentation of alternative mathematics of vagueness, the subsequent chapters explore how vagueness manifests itself in the various exact sciences: physics, chemistry, biology, medicine, computer science, and engineering.

1 Vagueness from the philosophical point of view

Ken Akiba

Abstract

This paper offers philosophers’ viewpoint on the issues surrounding vagueness in a manner accessible to general readers and scientists. In particular, the paper aims to help answer the following two questions: First, wherein lies vagueness? And second, why do philosophers roundly dismiss fuzzy logic? On the first issue, two kinds of vagueness, worldly vagueness and representational vagueness, are distinguished.

This book is about vagueness in science. However, vagueness has traditionally been even a bigger subject in philosophy than in science. Over the years, but especially in the last 30+ years or so, philosophers have had much heated debate about the nature of vagueness, but the fruits of their discussion have not yet been widely shared in the scientific community. In this paper, I would like to convey, in a manner accessible to scientists and general readers, contemporary philosophers’ thoughts on vagueness, which should be helpful to those readers. I will stay away as much as possible from explicitly setting forth my own view; however, the way I present the subject matter will inevitably reflect my own viewpoint.

I will focus on two topics, or two questions. The first question is: Wherein lies vagueness? It seems that vagueness exists somewhere; otherwise, we would not even be talking about it. But where? In particular, does vagueness exist in the world itself, or does it exist only in our representation (perception, language, etc.) of the world? I cannot give a definitive answer to the question, but will try to make clear what the difference amounts to. The second question I will address is: Why do philosophers roundly dismiss fuzzy logic as a way of dealing with vagueness? While the use of fuzzy logic and fuzzy set theory is a widely celebrated treatment of vagueness in natural science and engineering, philosophers are generally very dismissive of their significance. But why? I will answer this question in the second section of this paper.

Throughout the paper, I will use George Klir and Bo Yuan’s Fuzzy Sets and Fuzzy Logic: Theory and Applications [14] as a point of reference. I chose this book because of its influence and popularity; however, whatever I say about the book is not specific to the book and can easily be generalized.

1.1 Wherein lies vagueness?

1.1.1 Introduction

Suppose an astronomer shows you a photo of an alien planet. But this photo is all foggy and blurry, and you cannot detect any solid object in it. What should you think about the photo? There are three possibilities. The first is that it is a crisp picture of the objectively blurry planet. The second is that it is a blurry picture of the objectively crisp planet. The third, the combination of the first two, is that it is a blurry picture of the objectively blurry planet. This distinction between an objectively blurry object and a blurry picture is a crucial distinction philosophers draw that is often neglected by nonphilosophers.

Let’s call vagueness of the first kind worldly vagueness (or vagueness in the world) and vagueness of the second kind representational vagueness (or vagueness in representation relations). Whether as a result of the worldly vagueness of the planet itself or as a result of the representational vagueness of the photo, there is no doubt that the picture in the photo is vague. For lack of a better name, I will call this vagueness apparent vagueness (or vague appearance). Then the above three views can be described, respectively, that the apparent vagueness of the picture is as a result of the worldly vagueness of the planet; as a result of the representational vagueness of the photo; and as a result of the combination of those two kinds of vagueness.

In the next two subsections, I will reverse what one may think to be the natural order of presentation, and first give a discussion about what a representation is. I will then give an admittedly sketchy account of the other side of the world-representation divide, objects in the world. You will see why this order of presentation is more convenient. In Subsections 1.1.4 and 1.1.5, I will illustrate, using Klir and Yuan’s aforementioned book as an example, how scientists are often confused about the general distinction between what’s in the world and what’s in a representation of the world, and the distinction between worldly and representational vagueness, in particular. The concept of disquotational truth will be introduced as a possible source of confusion.

1.1.2 Representations

What is a representation? Simply put, a representation is a copy. What is presented in the world (by God?) is represented, that is, presented again, in a representation. Copies, maps, and pictures are representations. Symbols, such as signs and signals, are also representations; for instance, a red light means “Stop!”, and smoke means “Fire!” But philosophers are particularly interested in two kinds of representation: linguistic and mental representations.

Linguistic expressions, sentences in particular, are representations. They try to copy reality. If I said “Lisa is a vegetarian,” and if Lisa indeed is a vegetarian, then the sentence succeeded in copying reality; if Lisa is not a vegetarian, the sentence failed to copy reality.

Some mental states, such as perception, beliefs, and thoughts, are representations, too. If I saw that the traffic light in front of me was green, and if it was indeed green, then my perceptual image succeeded in copying reality; if the traffic light was in fact red, the image failed to copy reality.¹ Beliefs and thoughts, unlike perception, are often not accompanied by images; still, if I believed or thought that New York was more populous than London, and if New York is in fact more populous than London, then my belief/thought succeeded in copying reality; if New York isn’t, it failed to copy reality. So beliefs and thoughts are also representations.

What are the essential features of a representation? It must have a propositional content, such as that Lisa is a vegetarian, that the traffic light in front of me is green, and that New York is more populous than London, and the content must be true or false, correct or incorrect, or accurate or inaccurate compared to reality. Sentences have linguistic content, and the said mental states have mental content.

Between linguistic and mental representations, it is usually held that mental representations are primitive and linguistic representations are derivative. Even before we acquired language, we could see, hear, think, and believe things. We invented language to think more clearly and convey our thoughts to others. However, language is much more tangible and easier to deal with than mental states. So philosophers tend to focus on language.

A linguistic expression has both form and content. The study of linguistic content is semantics (whereas the study of linguistic form is syntax or grammar). So the representation relations of linguistic expressions are their semantic relations.

A semantic relation connects linguistic expressions, themselves mere sounds or smidges of ink, with the world. Examples are meaning (x means y, where x is a linguistic expression and y is something in the world), expression (x expresses y), denotation (x denotes y), and reference (x refers to y). But the most important semantic relation is truth and falsity; other relations are important so long as they contribute to truth and falsity of sentences. For instance, if “Lisa” refers to my friend Lisa, “is a vegetarian” stands for the property of being a vegetarian, and my friend Lisa indeed has this property, then “Lisa is a vegetarian” will be true; otherwise, it will be false.

1.1.3 Objects in the world

But I’ve just stepped over the boundary and started talking about things in the world, such as the individual Lisa and the propert...

Table of contents

• Title Page
• Copyright
• Contents
• 1 Vagueness from the philosophical point of view
• 2 Vague mathematics
• 3 Vague theory of computation
• 4 Fuzzy hypotheses tests with crisp data using nonasymptotic fuzzy estimators
• 5 Fuzzy sets in quantum mechanics
• 6 Vagueness in chemistry
• 7 Vagueness in medicine: on indistinct phenomena, vague concepts, and uncertain knowledge
• 8 Vagueness in technology
• 9 The semiotic topos of vagueness
• Subject Index
• Person Index