Construction of the continuum

By Lario Sinigaglia

The nature of the 'continuum' is a problem of great profundity that philosophers and mathematicians have been studying for more than two millennia. The problem is abstract, but it relates to concrete situations and is probably connected with the profound nature of the human mind. The author comes to the conclusion, through concrete and abstract examples, that the continuum is not found in what exists, but is made up of the synthesis between what exists and its hidden potential.

• Language: English
• Publisher: Youcanprint
• Release date: Mar 7, 2018
• ISBN: 9788827817667

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PREFACE

The subject of my work is the nature of the continuum and its relationship with discontinuity.

I have used some common experience as examples, and also the rational, irrational, and real numbers about which, despite some brief explanation in the text, some basic understanding is required. The topic is covered in compulsory education, but is rarely explored. Yet, scholars have dedicated themselves to it for about 2,500 years. My work will have achieved its purpose if the reader comes away with some depth of understanding of the subject matter and its connection with everyday facts. Basic notions of rational, irrational, and real numbers are widely and freely available on the web.

Lario Sinigaglia

THE CONTINUUM – CHAPTER I

The concept of continuum is paradoxical because it is completely missing from our understanding; that is to say: the continuum does not exist.

However, it does exist. But it is hard to describe because what every description captures is an individual.

Let us dwell a bit on the individual and leave behind the continuum while anticipating only one conclusion: ‘continuum’ is that which cannot be singled out.

By contrast, 'individual' is that which is detectable, that which is circumscribed by a description of any type (verbal, graphic, formal mathematical), which makes it possible to assign ‘predicates’ to a determined individual, i.e. of properties or of acts and to nothing else.

We must allow a very broad notion of what is meant by 'individual'. These include:

a) People, animals, objects;

b) Concepts, propositions (sentences), various texts;

c) Representations of every type;

d) Etc.

Natural languages distinguish homogeneous entities (for example, air, water, sand, geometric spaces) from those which are complex (for example, people, artefacts, geometric figures) and usually the former are 'uncountable' while the latter are 'countable'.

The first are divisible at will while maintaining their own nature while the latter are usually indivisible. Some of the former are indefinitely divisible.

Natural languages therefore attribute a kind of continuity, and therefore of ‘non-individuality’, to the first type, while those of the second type are discontinuous and with individuality.

In