Understanding the Universe: A Discussion on Comprehensive Knowledge

Nobody ever truly completely understands anything. Indeed, if we waited until such a state was attained, our species would have died out long ago. Instead, people often need only a minimum understanding of certain aspects to function effectively in their daily lives. For example, a child can ride a bicycle without delving into the laws of physics. Conversely, many scientists may not be adept at riding bicycles or driving cars but can explain the principles of chemical combustion in great detail. This fluctuation in expertise highlights the complexity of human knowledge acquisition and its application in practical contexts.

Mathematics: A Craft, Not a Divine Calling

Mathematics is not a fixed, immutable science, but rather a human craft. It evolves through trial and error, much like other forms of human activity. The notion that mathematics is a divine calling has been misinterpreted over the decades. Books such as Imre Lakatos' Proofs and Refutations (1976) and Philip J. Davis and Reuben Hersh's The Mathematical Experience (1981) shed light on this perspective. These texts dismantle the idea that mathematics is a permanent, unchanging truth, demonstrating its dynamic nature.

The historical development of mathematics provides a clear contrast. Initially, arithmetic and geometry had direct connections to everyday life—counting fingers, building clay pots, and making shadows. However, as society advanced, mathematics became increasingly abstract, leading to its perception as unrelated to practical life. This abstraction, coupled with the division of labor between academic disciplines, has contributed to the widespread belief that mathematics is a realm of abstract, and therefore irrelevant, knowledge.

Language and Perception

The words we use often reflect these perceptions, with terms like "concrete" being metaphors rather than literal references to tangible things. For example, "tangible proof" and "rock-bottom" are linguistic constructs that do not reflect the actual nature of proof or the physical world. Even the phrase "down to earth" is a metaphor, which is ironic given its context. Similarly, "rectilinear motion" is as much a metaphor as "salami on rye," yet no one complains about the use of metaphors in language.

The division of labor in academia, driven by economic considerations of speed, power, and profit, has further exacerbated this issue. The compartmentalization of education into discrete, mutually exclusive domains has led students to feel that certain subjects are irrelevant unless they are directly applicable to their major. This approach has eroded the interdisciplinary connections that underpinned the early development of mathematics and other sciences.

Philosophical Implications

The quest for comprehensive knowledge has also been challenged by philosophers. Ludwig Wittgenstein, in his Remarks on the Foundations of Mathematics, questioned the necessity of foundational knowledge in mathematics. He argued that just as a house is built step by step, mathematics must be developed gradually, with tools and linguistic apparatus evolving with each step. Just as a house always needs repairs, mathematics is an ongoing process, never to be completed.

William James, reflecting on the nature of knowledge, posited that intellectual endeavors should not end in dogma but in openness. His famous last words, as he neared death, are a testament to this idea: 'There is no conclusion. What has concluded that we might conclude in regard to it? There are no fortunes to be told, and there is no advice to be given. Farewell!'

Albert Einstein, similarly, acknowledged the wonder of comprehending the universe but never stopped striving to fathom its secrets. Despite his friendship with Kurt G?del, who challenged his views on the nature of time, Einstein maintained that the universe was comprehensible, despite its incomprehensibility.

For further exploration, the works of Douglas Hofstadter's G?del, Escher, Bach: An Eternal Golden Braid (1999) and Palle Yourgrau's A World Without Time: The Forgotten Legacy of G?del and Einstein (2005) offer valuable insights. These texts delve into the interplay between mathematics, art, and the nature of reality, providing a deeper understanding of the quest for knowledge.

Ultimately, the quest for comprehensive knowledge is a Sisyphean task. Between the divine omniscience of gods and the baseless ignorance of the garbage bin lies a spectrum of possibilities. Cultivating an understanding that recognizes the value of a balanced approach to knowledge, rather than an all-or-nothing stance, can guide us in our pursuit of truth and understanding.