Abstract
Of all the leading philosophers Immanuel Kant was undoubtedly the one who assigned the greatest importance to the part played by intuition in what we call knowledge. He observed that two opposite factors are basic to our knowledge: a passive factor of simple receptivity and an active factor of spontaneity. In his Critique of Pure Reason, at the beginning of the section entitled ‘Transcendental Theory of Elements; Part Two: Transcendental Logic,’ we find the following:
“Our knowledge comes from two basic sources in the mind, of which the first is the faculty of receiving sensations (receptivity to impressions), the second the ability to recognize an object by these perceptions (spontaneity in forming concepts). Through the first an object is given to us, through the second this object is thought in relation to these perceptions, as a simple determination of the mind. Thus, intuition and concepts constitute the elements of all our knowledge…”.
That is to say, we conduct ourselves passively when through intuition we receive impressions, and actively when we deal with themin our thought. Further, according to Kant, we must distinguish between two ingredients of intuition.
First published in Krise und Neuaufbau in den exakten Wissenschaften, Fünf Wiener Vorträge, Leipzig and Vienna, 1933.
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Notes
H. Mark, Die Erschütterung der klassischen Physik durch das Experiment and H. Thirring, Die Wandlung des Begriffssystemes der Physik.
For further details see W. Heisenberg, Die physikalischen Prinzipien der Quantentheorie (Leipzig, 1930) [E. T. The Physical Principles of the Quantum Theory (Chicago, 1930)] and Thirring’s lecture quoted in Note 1.
K. Menger, Die neue Logik [E. T. ‘The New Logic’ as Chapter I of K. Menger, Selected Papers in Logic and Foundations, Didactics, Economics in this Collection].
The main work here is Whitehead-Russell, Principia Mathematica (Cambridge, 11910–3, 21925). Presentation for the general reader in B. Russell, Introduction to Mathematical Philosophy (London, 1919) [G. T. Munich, 1923].
The considerations about ‘velocity’ and ‘slope’ that follow will be found in greater detail in Hahn-Tietze O, Einführung in die Elemente der höheren Mathematik (Leipzig, 1925) pp. 153ff., 190ff.
See H. Hahn, Jahresber. d. Deutschen Math.-Vereinigung 26 (1918) p. 281,
and L. Bieberbach, Differential- u. Integralrechrung I (Leipzig, 1917) p. 104, for a precise mathematical treatment of what follows.
The reader may turn to H. Hahn, Monatshefte f. Math. u. Phys., 16 (1905) p. 161, which deals with motions (or curves) that assume infinite velocities (or slopes of infinite value).
By ‘motion’ is here meant a change of place that proceeds continuously, i.e. one in which between two instants sufficiently close to one another the point in motion passes through arbitrarily few positions. Such a point will make no ‘jumps’.
Detailed discussion in H. Hahn, Theorie der reellen Funktionen (Berlin, 1921) p. 146,
K. Menger, Kurventheorie (Leipzig, 1932) p. 10.
This question was answered in 1913–14 by H. Hahn and S. Mazurkiewicz. Later accounts in F. Hausdorff, Mengenlehre 2 (Berlin, 1927) p. 205;
H. Hahn, Reelle Funktionen (Leipzig, 1932) p. 164;
K. Menger, Kurventheorie (Leipzig, 1932) p. 31.
L. E. J. Brouwer, Math. Annalen 68 (1910) p. 427. In the present discussion we make use of an intuitive interpretation suggested by the Japanese mathematician Wada. A point is called a ‘boundary point’ if in each of its neighbourhoods lie points of various countries. Three countries meet in a boundary point if in each of its neighourhoods points are to be found of each of the three countries.
A proof by H. Hahn in Monatsh. f. Math. u. Phys. 19 (1908) p. 289.
Lilly Hahn, Monatsh. f. Math. u. Phys. 25 (1914) p. 303;
N. J. Lennes, American Journal of Mathematics 33 (1911) p. 37.
Detailed presentation in K. Menger, Dimensionstheorie (Leipzig, 1928).
Detailed presentation in K. Menger, Kurventheorie (Leipzig, 1932).
It should properly be called the postulate of Eudoxus. Eudoxus’ dates are 408–355 B.C.; Archimedes’, 287–212 B.C.
The first to investigate thoroughly the properties of non-Archimedean spaces was the Italian mathematician G. Veronese, Grundzüge der Geometrie von mehreren Dimensionen und mehreren Arten gradliniger Einheiten (Leipzig, 1894).
On non-Archimedean number systems see H. Hahn, S.-B. Wien (Math.-Nat. Wiss. K1.) 116 (1907) p. 601.
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Hahn, H. (1980). The Crisis in Intuition. In: McGuinness, B. (eds) Empiricism, Logic and Mathematics. Vienna Circle Collection, vol 13. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-8982-5_7
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