Deep below in Galileo's soul there lived, as it does in every human being, the intuitive knowledge, acquired in early childhood, that part of nature's order is recordable in the conceptual language of mathematics. In order that this intuition should rise sufficiently far into his conscious mind to guide him, as it did, in his observations, the veil of oblivion which otherwise separates our waking consciousness from the experiences of earliest childhood must have been momentarily lightened. Unaware of all this, Galileo was duly surprised when in the onlooker-part of his being the truth of his intuition was confirmed in a way accessible to it, namely through outer experiment. Yet with the veil immediately darkening again the onlooker soon became subject to the illusion that for his recognition of mathematics as a means of describing nature he was in need of nothing but what was accessible to him on the near side of the veil.