The Philosophical View of Mathematics: Truths and Fantasies
Mathematics, a subject that transcends practical applications and delves into the very essence of reality, invites a wide range of perspectives, from practical use to a profound exploration of the nature of existence itself. The mathematical community, including mathematicians, enthusiasts, and practitioners in STEM fields, often shares their views on the nature of mathematics and its role in understanding the world around us.
A Critical Examination of Mathematical Theorems and Proofs
Every theorem we accept in mathematics is based on a set of premises. These premises, when accepted, create a pressure that tests their validity. However, unlike other disciplines, mathematics often leaves proofs out of its discussions, only questioning them when a blatant contradiction or lacunae (a gap or omission) appears. This is a critical point, as these contradictions often come too late for the academic community to rectify the underlying issues. Our premises, therefore, must be questioned, as this is the only way to ensure the validity and robustness of our mathematical systems.
One of the most profound examples of this is G?del's Incompleteness Theorems, which demonstrated that within any sufficiently complex axiomatic system, there are statements that cannot be proven or disproven within the system itself. This raises deep philosophical questions about the nature of mathematical truth and the limits of mathematical logic.
Mathematics, much like the discovery of more dimensions in geometry, has been similarly revolutionized by G?del's findings. The discovery that the universe might be represented more accurately by non-Euclidean geometries, such as those with four dimensions, was a significant shift in our understanding of physical space. Similarly, G?del's work highlighted that our mathematical systems might need foundational overhauls to address truths that current systems cannot fully encompass.
The Question of Infinite Sets and their Relevance to Mathematics
Starting from theological debates about the infinite power of God and the infinite universe, the concept of infinite sets was posited as a practical mathematical aid. However, this idea is deeply controversial. The reliance on infinite sets, argued by some, is traced back to theological fantasies rather than empirical or theoretical evidence. Such concepts, it is suggested, should not influence mathematical methods, as they do not have a basis in observable reality.
Mathematics is often seen as a means to an end, initially a tool to avoid practical work or gain beneficial job prospects. This perspective has evolved over time, with many now viewing it as a fascinating arts subject that delves deeply into the patterns of nature and human thought. Mathematics interprets patterns in nature, and these patterns are then described using mathematical language. However, the accuracy of these descriptions is largely statistical and subject to test, but in the abstract realm of mathematics, one enters a world of absolute truths.
The Role of Theory and Practice in Mathematics
Physics, on the other hand, follows a different methodology. Scientific theories are developed by gathering evidence and formulating models to explain that evidence. This process is inherently probabilistic, as there is no absolute proof of the correctness of any theory, and experiments may not consistently yield the same results. Nonetheless, this flexibility allows for the possibility of disproving theories, a feature that adds to the robustness of scientific knowledge.
When it comes to mathematics, the interpreted patterns of nature or the human brain are analyzed and described. Despite the accuracy of these descriptions, they are ultimately approximate. Once patterns are established and described mathematically, one enters a purely idealized realm where theorems must be proven with iron-clad certainty. This is a realm where truth is absolute and unrestricted by empirical concerns. The outside world, while a source of inspiration, is of no consequence in this abstract mathematical framework.
Revisiting the Foundations of Mathematics: A Call for a New Revolution?
Despite the advancements, there are still foundational questions in mathematics that need addressing. The work on quantum mechanics in the mid-20th century suggested that the world is not as smooth as traditional mathematical frameworks assumed. The concept of the Continuum, for instance, which was introduced by Cantor, posited a consistent extension to the number line, aimed at creating a theory that would support smooth curves and continuous functions. However, this theory might not align with the empirical reality of the quantum world.
The problem of incorporating quantum sizes into general relativity remains a challenge, with one potential explanation being the inherent smoothness of the geometry used in general relativity. Perhaps the time has come for a new mini-revolution in mathematics, one that revisits the foundations of calculus and explores the implications of a more granular view of the universe. This would not only challenge our existing mathematical theories but potentially lead us to a more accurate and realistic understanding of the mathematical structures that describe our universe.
Mathematics, as a subject that seeks to uncover the truths of the universe, must constantly question its own foundations and methodologies. As we continue to explore the complex and uncharted territories of mathematical reality, the philosophical questions remain ever present, pushing us to refine our understanding and approach.