Changes

==Hilbert==

==Hilbert==

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<p>Finally, let us recall our real subject and, so far as the infinite is concerned, draw the balance of all our reflections.  The final result then is:  nowhere is the infinite realized;  it is neither present in nature nor admissible as a foundation in our rational thinking &mdash; a remarkable harmony between being and thought.  We gain a conviction that runs counter to the earlier endeavors of Frege and Dedekind, the conviction that, if scientific knowledge is to be possible, certain intuitive conceptions <nowiki>[</nowiki>Vorstellungen<nowiki>]</nowiki> and insights are indispensable;  logic alone does not suffice.  The right to operate with the infinite can be secured only by means of the finite.</p>

<p>Finally, let us recall our real subject and, so far as the infinite is concerned, draw the balance of all our reflections.  The final result then is:  nowhere is the infinite realized;  it is neither present in nature nor admissible as a foundation in our rational thinking &mdash; a remarkable harmony between being and thought.  We gain a conviction that runs counter to the earlier endeavors of Frege and Dedekind, the conviction that, if scientific knowledge is to be possible, certain intuitive conceptions [Vorstellungen] and insights are indispensable; logic alone does not suffice. The right to operate with the infinite can be secured only by means of the finite.</p>

<p>The role that remains to the infinite is, rather, merely that of an idea &mdash; if, in accordance with Kant's words, we understand by an idea a concept of reason that transcends all experience and through which the concrete is completed so as to form a totality &mdash; an idea, moreover, in which we may have unhesitating confidence within the framework furnished by the theory that I have sketched and advocated here. (p.&nbsp;392).</p>

<p>Hilbert (1925), &ldquo;On the Infinite&rdquo;, pp.&nbsp;369&ndash;392 in Jean van Heijenoort (1967/1977).</p>

 

<p>Hilbert (1925), "On the Infinite", pp. 369&ndash;392 in Jean van Heijenoort (1967/1977).</p>

 

==Hilbert and Ackermann==

==Hilbert and Ackermann==