All, One, Infinite: The Interpenetration and Opposition Between the Conceptual and Subjective Empirical Aspects of Mathematics

The philosophical exploration of the concepts of unity and infinity has always had a significant influence on mathematicians. Hegel posits that the achievement and realization of the infinite state of completeness within the finite realm is contingent upon the use of pure intellect and conceptual frameworks, rather than relying on perceptual intuition. Similar to Hilbert's exploration of the concept of infinity in his dissertation On the Infinite, it may be seen that the actualization of infinity is unattainable in reality. Instead, the finite can only comprehend and achieve infinity via abstract and purely conceptual reasoning, rather than through any kind of perceptual experience of the physical world. Hence, the attribute of ultimate perfection characterizes the concept of "true infinity," but any attempt to attain such perfection within limited boundaries is certain to be unsuccessful. This study delves into the exploration of the historical beginnings of the concept of infinity in the field of mathematics. The concept of absolute perfection is unattainable in the real world due to its reliance on the presence of infinite entities.