Does infinity exist? - Mathematician explains | Joel David Hamkins and Lex Fridman

Quick Overview

The discussion concludes that mathematical objects, like numbers, do not possess a physical existence in space and time, contrasting with physical objects like tables or bicycles, despite the intuitive desire to anchor abstract concepts in reality, which is a source of profound mystery in the philosophy of mathematics.

Key Points: Mathematical ontology, the study of what mathematical entities exist, is a central, unresolved philosophical question. The guest, Joel David Hamkins, leans toward a structuralist view, suggesting that numbers exist only as parts of mathematical structures, not as independent objects. Hamkins referenced Gottlob Frege's attempt to reduce mathematics to logic, which failed due to Russell's Paradox, highlighting foundational difficulties. The Cantor-Hume Principle defines number equivalence through one-to-one correspondence, which applies to both physical collections and abstract mathematical sets. The crucial distinction is that abstract objects like numbers lack spatial/temporal location, unlike physical objects (e.g., the number 4 is not on the desk like an apple). The discussion explored the difficulty in providing a satisfactory account of the existence of abstract objects without resorting to a Platonic realm or an equally mysterious physical reality.

Context: This segment of the Lex Fridman Podcast features an interview with mathematician Joel David Hamkins, focusing on the philosophy of mathematics, specifically the ontological status of mathematical objects like numbers, sets, and structures. The conversation delves into whether these abstract entities exist independently or only as components within larger mathematical frameworks, contrasting this with the tangible existence of physical objects.

Detailed Analysis

The conversation centers on mathematical ontology, questioning whether mathematical entities like numbers, sets, and structures exist, and if so, in what manner. Hamkins leans toward a structuralist perspective, suggesting that mathematical objects do not exist independently in the physical world or in any separate Platonic realm, but rather exist only as components within mathematical structures. He notes that this view contrasts with the intuitive desire to anchor abstract concepts in some form of reality. Hamkins cites Gottlob Frege's foundational project to reduce arithmetic to logic, which ultimately encountered failure due to Russell's Paradox, indicating deep foundational issues. The discussion clarifies the Cantor-Hume Principle, which states that two collections have the same number if their elements can be matched one-to-one without any remainder. While this principle applies to both physical and abstract collections, the crucial difference is that abstract objects lack spatial or temporal location, making their existence fundamentally different from physical objects. Hamkins emphasizes that while we experience physical reality through objects like bicycles or apples, we lack a satisfying account for the existence of abstract mathematical objects that is not deeply mysterious, suggesting that the nature of their existence remains unclear.