Abstract
When the results of partial mental adaptations fall into such opposition that thought is driven in different directions and disquiet mounts to the point that we consciously and deliberately seek for a thread to lead us through this confusion, then a problem has arisen. A stable and customary area of experience to which thoughts have adapted themselves rarely gives rise to problems; at least it would require great mental sensitivity to differences if here too problems were to arise. However, if the area of experience becomes wider through certain circumstances and thoughts come into contact with hitherto unknown facts to which they are inadequately adapted and if the thoughts modified by novel adaptation react on the results of earlier adaptations, then a host of new problems develops, as the history of civilization in general and of science in particular shows. Problems arise when thought and fact, or thought and thought no longer agree. We have not the power to adduce hitherto unknown facts that depend in unknown ways on circumstances within our sphere; they meet us against our will, without or against our expectations, and although they lie outside the scope of our work or investigation, they arise from chance, that is through circumstances that may not be without rule but beyond our ken and influence.
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Notes
James, Psychology, Vol. I, pp. 585 f.
Kepler, Ad Vitellionem paralipomena, 1604.
Kepler, Dioptrice, 1611.
Bretschneider, Die Geometrie und die Geometer vor Euklid, Leipzig 1870, p. 146.
Ibid., p. 147. For analytic and synthetic, see Euclid, Elements, XIII 1, in the edition by J. F. Lorenz, Halle 1798.
Newton, Arithmetica Universalis, 1732, p. 87.
Archimedes Werke (German translation by Nizze, Stralsund 1824). Cf. especially the piece on the quadrature of the parabola.
Hankel, Geschichte der Mathematik, Leipzig 1874, especially pp. 137–156.
Ofterdinger, Beiträge zur Geschichte der griechischen Mathematik (Programmabhandlung), Ulm 1860.
Mann, Abhandlungen aus dem Gebiet der Mathematik, (Festschrift for Würzburg University’s tercentenary) 1883;
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F. Klein, Ausgewählte Fragen der Elementargeometrie, Leipzig 1895.
F. Rudio, Geschichte des Problems der Quadratur des Zirkels, Leipzig 1892.
Abel, ‘Démonstration de l’impossibilité de la résolution algébrique des equations générales qui dépassent le quatrième degré’, Grelles Journal I, 1826.
One must of course be careful not to posit more principles than are necessary. Cf. Duhem, La Théorie physique, pp. 195 f.
Huygens, Traité de la lumière, 1690.
Newton, Optice, 1719.
Hooke, Micrographia, 1665.
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© 1976 D. Reidel Publishing Company, Dordrecht, Holland
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Mach, E., Hiebert, E.N. (1976). Problems. In: Knowledge and Error. Vienna Circle Collection, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1428-1_15
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