"I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers"

Georg Cantor, a pioneering mathematician known mostly for developing set theory and introducing the concept of transfinite numbers, made profound contributions to the understanding of the mathematical infinite. His quote reflects the revolutionary nature of his work and the intellectual opposition he faced.

In this statement, Cantor acknowledges that his concepts were in contrast to the mainstream mathematical viewpoints of his time. Traditionally, the infinite was a mystical idea, typically perceived as something that might not be carefully evaluated within the framework of mathematics. Numerous mathematicians and thinkers concerned the infinite as an ill-defined or unreachable concept, more the domain of approach than mathematics. Cantor's work, nevertheless, looked for to formalize the unlimited and bring it into the realm of mathematics through a methodical development of set theory.

By stating that he placed himself "in a certain opposition" to commonly held views, Cantor is subtly indicating the questionable nature of his concepts. His introduction of various sizes or levels of infinity, such as countable and vast infinities, challenged the prevailing notion of the infinite as a monolithic entity. Moreover, his idea of transfinite numbers and the hierarchy of infinities offered a revolutionary structure that extended beyond the arithmetic of limited numbers into a brand-new and mostly uncharted area.

Cantor's work drew apprehension and criticism, especially from those who abided by more classical views of mathematics. Some peers even viewed his theories as esoteric or alarmingly speculative. Regardless of this opposition, Cantor continued, inspired by a deep conviction in the validity and sophistication of his mathematical developments.

Hence, this quote encapsulates Cantor's awareness of the obstacles he faced due to his unique ideas about the mathematical infinite. It highlights his function as a leader willing to advance a strong vision that eventually laid the groundwork for modern mathematical analysis and had an enduring impact on mathematics, philosophy, and reasoning.