Mathematics is the doctrine of the laws in the world of numbers, of functions, and of geometric diagrams. ... Also the Faustian desire to comprehend what holds the world together in the innermost, that is deeply embedded inside us, extends itself to those worlds of pure thought. The more that knowledge of mathematics advances, it becomes a so much more powerful tool in the hands of man for the gigantic tasks that the natural sciences and technology will present to it. Thus it can serve the human world much more thoroughly and widely. In its striving for this powerful ability and rewarding mastery lies the driving incentive to research in mathematics. To this comes the wonderment before the beauty and elevation of the mathematical world, before its crystalline clarity and inviolable rigour.
By each of the steps from an intuitively conceived idea, to outline, through many revisions, to improvement, to extension and to reorganisation into a final form, there is something of the way of nature, of the intuitive mental grasp, of the whole of the inherent dynamics, just before the beauty is lost in the breadth, while the beauty in the small is perhaps attained. If this process is carried to extremes there will remain in the end a lifeless, dead image of the emaciated structure: Definition, Theorem, Proof, repeated over and over again. And it is in this flagrant manner that the already named life law of mathematics is broken. ... [The 'Elements' of Euclid are a prime example of a] lifeless final state. The attraction of this sort of thing, complete in its external form but internally a lifeless composition after formalisation and axiomatisation, has been resurrected for the first time since Euclid in this century. As we explained already, this endeavour blooms most at this time in North America. But it has also taken a grip here, in part because of our great Hilbert's life work, who in his 'Grundlagen' directly follows Euclid, in part though the American taste for European mathematics, but mostly because of itself. In this I see a tragedy living within every mathematician. The urge, in itself a healthy thing, to create what is seen and recognised as perfectly structured, ends up, if one works it through to its final consequence, with the total death of all that had life in the original proper work that abounded with effervescent life.