CHAPTER XIV ON THE NOTION OF INFINITY
The questions on which there is disagreement are not trivialities; they are the very roots of the whole vast tree of modern mathematics. (22)
E. T. BELL
The task of cleaning up mathematics and salvaging whatever can be saved from the wreckage of the past twenty years will probably be enough to occupy one generation. (22) E. T. bell
The intention of the Hilbert proof theory is to atone by an act performed once for all for the continual titanic offences which mathematics and all mathematicians have committed and will still commit against mind, against the principle of evidence; and this act consists of gaining the insight that mathematics, if it is not true, is at least consistent. Mathematics, as we saw, abounds in propositions that are not really significant judgments.
<549) HERMANN WEYL
An objectivated property is usually called a set in mathematics. (549)
If the objects are indefinite in number, that is to say if one is constantly exposed to seeing new and unforseen objects arise, it may happen that the appearance of a new object may require the classification to be modified, and thus it is we are exposed to antinomies. There is no actual (given complete) infinity. («7) h. poincare
The structural notion of 'infinite', 'infinity', is of great semantic importance and lately has again become a subject of heated mathematical debates. My examination of this subject is from the point of view of a -system, general semantic, and a theory of sanity which completely eliminates identification. In Supplement III, I give a more detailed analysis of the problem already anticipated by Brouwer, Weyl, Chwistek, and others. These problems are not yet solved, because mathematicians, in their orientations and arguments, still use el, A 'logic', 'psychology', and epistemology, which involve and depend on the 'is' of identity, making agreement impossible.
Mathematical infinity was first put on record by the Roman poet, Titus Lucretius, who, as far back as the first century B.C., wrote very beautifully about it in his De Rerum Natura.1 As the author was a poet, and his work poetry, a few privileged literati had great pleasure in reading it; but this discovery, not being rigorously formulated, remained inoperative, and so practically worthless for mankind at large, for 2000 years. Only about fifty years ago, mathematical infinity was rediscovered by mathematicians, who formulated it rigorously, without poetry. Since then, mathematics has progressed with all other sciences in an unprece-