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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.
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CHAPTER ONETHE CONTINUUM
CHAPTER TWODESCRIPTION AND DEMONSTRATION
CHAPTER THREERECONSTRUCTION OF THE WHOLE
CHAPTER FIVETHE NUMERIC CONTINUUM
CHAPTER SIXDOES THE FIELD OF REAL NUMBERS EXIST?
CHAPTER SEVENRECENT PROPOSALS
CONSTRUCTION OF THE CONTINUUM
Title | Construction of the Continuum
Author | Lario Sinigaglia
ISBN | 9788827817667
© All rights reserved by the author
No part of this publication may be reproduced without prior consent of the author.
Via Roma, 73-73039 Tricase (LE) - Italy
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.
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;
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 the following, we uncouple ourselves from natural languages, and we will not make any distinction between homogeneous and non-homogeneous entities, in the sense that we will deem them both 'individuals'.
Recognising the discontinuum is a natural faculty of the senses and of the mind, which gives an order and a sense to discontinuity.
However, the mind, and not the senses, has a further requirement, perhaps typically human: that discontinuity is only a part of a larger unit, for the most part not perceptible, at least by the senses.
Such a unit would be 'the whole' and not just 'a part' and it would have the condition of 'continuity', while the perceptible part has the condition of 'discontinuity'.
The completion ‘to the whole’ is the traditional role of religions, which is carried out with an extraordinary wealth of arguments, which the unfortunate race of women and men has often misused.
We will stop much before that, and will endeavour to show that there is also an unnatural faculty of the mind, and therefore typically human, which consists of attempting partial reconstructions ‘of the whole from which the part originates’.
We will describe it using the natural language of examples of ‘retrieval from the whole’.
But there will only be a few simple mathematical examples that ‘demonstrate’ and not only describe this activity and its problems.
Because this is an important aim of this short work: to show that the formal language of logic and mathematics demonstrates the fact, it doesn’t just describe it.
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