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An Introduction To Non-aristotelian Systems And General Semantics.

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MATHEMATICS AND THE WORLD                 255
analysis merges with general semantics; and life itself becomes a physico-chemical colloidal occurrence. The language of 'degree' has very important relational, quantitative, and order implications, while that of 'kind' has, in the main, qualitative implications, often, if not always, concealing relations, instead of expressing them.
The current definition of 'number', as formulated by Frege and Russell, reads: 'The number of a class is the class of all those classes that are similar to it'.1 This definition is not entirely satisfactory: first, because the multiordinality of the term 'class' is not stated; second, it is A, as it involves the ambiguous (as to the order of abstractions) term 'class'. What do we mean by the term 'class' ? Do we mean an extensive array of absolute individuals, un-speakable by its very character, such as some seen aggregate, or do we mean the spoken definition or description of such un-speakable objective entities? The term implies, then, a fundamental confusion of orders of abstractions, to start with - the very issue which we must avoid most carefully, as positively demanded by the non-identity principle. Besides, if we explore the world with a 'class of classes'., and obtain results also of 'class of classes', such procedure throws no light on mathematics, their applications and their importance as a tool of research. Perhaps, it even increases the mysteries surrounding mathematics and conceals the relations between mathematics and human knowledge in general.
We should expect of a satisfactory definition of 'number' that it would make the semantic character of numbers clear. Somehow, through long experience, we have learned that numbers and measurement have some mysterious, sometimes an uncanny, importance. This is exemplified by mathematical predictions, which are verified later empirically. Let me recall only the discovery of the planet Neptune through mathematical investigations, based on its action upon Uranus, long before the astronomer actually verified this prediction with his telescope. Many, a great many, such examples could be given, scientific literature being full of them. Why should mathematics and measurement be so extremely important? Why should mathematical operations of a given Smith, which often seem innocent (and sometimes silly enough) give such an unusual security and such undeniably practical results ?
Is it true that the majority of us are born mathematical imbeciles ? Why is there this general fear of, and dislike for, mathematics? Is mathematics really so difficult and repelling, or is it the way mathematics is treated and taught by mathematicians that is at fault? If some light can be thrown on these perplexing semantic problems, perhaps we shall face a scientific revolution which might deeply affect our educational