Solving Unresolved Problems with Classical Methods in Contemporary Mathematics: Possibilities and Limitations

Scott Watters questions the feasibility of solving unsolved problems in contemporary mathematics using classical methods in his work. While it is common to use modern mathematical techniques, there are instances where classical approaches can still be relevant. This article explores the rationale behind using classical methods and challenges the necessity of complex proofs.

The Dominance of Contemporary Mathematical Techniques

One reason why classical methods are infrequently used for solving unsolved problems in contemporary mathematics is the pervasive use of contemporary mathematical language and techniques. Most open areas of research are framed within the context of advanced mathematical concepts. For example, addressing questions related to topology requires a solid understanding of topological techniques, much like contemporary problems often necessitate specific modern approaches. This structural dependence on contemporary mathematics often precludes the application of classical methods, even when they might be applicable.

Additionally, most mathematicians are initially trained in classical techniques before advancing to more modern methodologies. Consequently, any problem that can be solved using classical techniques is typically resolved relatively quickly and does not reach the status of an open problem. This educational and practical bias towards modern methods further restricts the likelihood of classical methods being used for contemporary unsolved problems.

Proving Results with Good Ideas: The Role of Novel Approaches

Many mathematical results are established through ingenious ideas, often involving innovative perspectives. Reducing a problem to more well-known and simpler methods, such as linear algebra, is a common technique. The humor among mathematicians about reducing problems to linear algebra underscores how often this tactic is employed. However, the extent to which a problem must be simplified before it can no longer be considered a proof using classical results is sometimes ambiguous.

The statement that complex problems require lengthy proofs often overlooks the fundamental simplicity underlying nature. Despite the complex universe, the rules governing it are not infinitely complex. This observation is echoed through various examples, such as DNA and the periodic table, which demonstrate remarkable simplicity in their structure, given the immense complexity of the organisms and elements they compose.

Conjectures on the Complexity of Natural Laws

Scott Watters conjectures that in an infinite universe of infinite complexity, the guiding laws are finite. This idea is supported by the observation that the complexity of fundamental building blocks (such as DNA) vastly outstrips the complexity of the entities they help form. Similarly, the periodic table, which categorizes fundamental elements, is far less complex compared to the natural world it represents.

These examples suggest a finite set of underlying laws guiding the infinite complexity of the universe. Since mathematics is inherently coupled and self-consistent, it implies that the complexity of natural laws is much less than the complexity of the universe itself. This finite set of laws can explain everything from the tiniest particles to the largest galaxies. Hence, if these fundamental laws are as few and concise as this hypothesis suggests, why would it take hundreds or thousands of pages to prove a fundamental concept?

Scott argues that if a few equations can explain everything, then proving such concepts should not be unnecessarily complex. The length of the proof should be reflective of the complexity of the underlying laws, not an inherent trait of the problem itself. This perspective questions the current practice of writing extensive proofs and highlights the potential for more efficient proof methods.

Drawer empathy, comments, and feedback about this view can help gauge its rarity or acceptance in the mathematical community. This exploration challenges conventional wisdom and invites further discussion on the utility and necessity of complex proofs in contemporary mathematics.

References and Further Reading

For further discussion and exploration, readers are encouraged to explore the works of mathematicians and philosophers who delve into the nature of mathematical proofs and the underlying laws of the universe. Key figures include Bertrand Russell, Alfred North Whitehead, and contemporary math educators specializing in the simplification of complex mathematical concepts.