How Has Mathematics Gotten So Abstract? Mathematics, once regarded as a natural science closely tied to the physical world, has evolved into an abstract discipline deeply rooted in formal logic and set theory. This shift stems from challenges such as the paradoxes involving infinity and the quest to ground mathematical concepts independently from experience. --- Historical Context and Intuition Traditionally, math was about observing real-world phenomena, like counting and measuring, without questioning why operations like 2 + 2 = 4 held true. Zeno’s paradox of Achilles and the tortoise illustrates the difficulty of applying naive intuition to infinite processes. Calculus resolved such paradoxes by showing infinite sums can converge to finite values, but introduced issues with infinitely small quantities (infinitesimals), raising foundational questions. --- Peano Arithmetic: Formalizing Numbers and Addition Giuseppe Peano introduced an axiomatic system defining numbers starting from a single "zero" element and a "successor" function \(S\). Numbers are built as successive applications of \(S\) to zero: \(1 = S(0)\), \(2 = S(S(0))\), etc. Addition is recursively defined: \(a + 0 = a\) \(a + S(b) = S(a + b)\) For example, \(2 + 2\) is computed by recursively applying these rules without relying on inherent meaning of numbers or operators. A C code example demonstrates implementing Peano arithmetic using linked lists encoding successor relations. --- Set-Theoretic Construction of Numbers Beyond Peano's approach, numbers can be modeled as sets: \(0 = \{\}\) (empty set) \(1 = \{0\}\) \(2 = \{0, 1\}\) \(3 = \{0, 1, 2\}\), and so on. Each ordinal number is a set containing all preceding ordinals, constructed using the successor operation: \[ S(n) = n \cup \{n\} \] This explicit set-based approach reveals deeper mathematical structure but is abstract and less practical. --- Infinite Ordinals and Non-Intuitive Arithmetic The set of all natural numbers, \(\mathbb{N} = \{0, 1, 2, 3, \ldots\}\), forms the infinite ordinal \(\omega\). Important properties: \(\omega \neq \omega + 1\): adding one to an infinite ordinal creates a strictly larger set. Addition with infinite ordinals is not commutative: \(1 + \omega = \omega\) \(\omega + 1 > \omega\) Larger ordinals like \(\omega \cdot 2\) illustrate infinite sequences of successor discontinuities. --- Cardinalities: Measuring Size of Sets Besides ordinals, cardinality measures "size" based on one-to-one correspondences between sets. For finite sets, cardinality corresponds to counting. For infinite sets: The set of natural numbers and the set of even numbers have the same cardinality (\(\aleph0\)) due to a bijection, despite one being a strict subset. Real numbers have a larger cardinality than natural numbers (the cardinality of the continuum). Cantor's diagonal argument proves there are more real numbers than natural numbers by constructing a real number not in any supposed list. --- Philosophical and Practical Implications Infinite sets and their arithmetic have bizarre, counterintuitive properties. Some mathematicians and philosophers reject the existence of actual infinities or set theory itself. Nonetheless, infinite mathematics remains indispensable for rigorous foundations and advanced theories. Calculus often taught without full rigor in limits and infinitesimals illustrates the tension between intuition and formalism. --- Additional Notes from Author’s Commentary Infinite cardinal numbers such as \(\aleph0\) are abstract measures, not simple set sizes. Infinite cardinals and ordinals are distinct concepts; cardinals classify classes of sets. Many infinite ordinals share the same cardinality \(\aleph_0\); e.g., \(\omega\), \(\omega +1\), \(\omega \cdot 2\). Larger infinities exist with unknown relationships to