Joel David Hamkins delivers a startling correction to the history of calculus: the "ghosts of departed quantities" that philosophers once mocked were not mathematical nonsense, but the very engine of profound discovery. In this excerpt from his lectures on the philosophy of mathematics, Hamkins argues that the intuitive use of infinitesimals drove centuries of progress long before their foundations were rigorously fixed, challenging the notion that sound logic must precede scientific advancement.
The Productive Power of Intuition
Hamkins begins by dismantling the caricature of early calculus as a period of "bumbling nonsense." He reminds us that despite the lack of formal rigor, the infinitesimal conception was "extremely fruitful and led to many robust mathematical insights, including all the foundational results of calculus." This is a crucial historical pivot. We often assume that mathematics is a linear march toward precision, but Hamkins illustrates that the messy, intuitive work of slicing volumes into "infinitesimally thin disks" or treating curve segments as hypotenuses of tiny triangles yielded correct results long before the epsilon-delta methods arrived to clean up the mess.
"On the contrary, it was a time of enormous mathematical progress and deep insights of enduring strength."
This observation forces a philosophical pause. If the foundations were shaky, how did the structure hold? Hamkins suggests that the resolution of these foundational issues via limits was valuable, but not strictly necessary for the initial explosion of mathematical knowledge. The argument here is compelling because it validates the heuristic power of intuition in science, suggesting that getting the answer right can sometimes precede understanding why the answer is right.
The 1961 Turn: Robinson's Gift
The narrative shifts dramatically to 1961, when Abraham Robinson introduced nonstandard analysis, effectively resurrecting the infinitesimal on a rigorous footing. Hamkins describes this as a "joke that mathematical reality has played on both the history and philosophy of mathematics." The hyperreal number system, denoted as ℝ*, extends the real numbers to include actual infinite and infinitesimal quantities. Unlike the vague "ghosts" of the past, these are mathematically defined entities where a number like δ is "positive, being strictly larger than 0 but smaller than every positive real number."
The brilliance of this system lies in the "transfer principle," which guarantees that any statement true for real numbers is also true for their hyperreal counterparts. This allows mathematicians to compute derivatives without the convoluted limit processes of standard analysis. Instead of wrestling with epsilon and delta, one simply calculates with an infinitesimal and then takes the "standard part" of the result.
"The use of the standard-part operation in effect formalizes how one can correctly treat infinitesimals—it explains exactly how the ghosts depart!"
This is a masterful reframing. Hamkins shows that the "ghosts" didn't need to be exorcised; they just needed a proper address. The elegance of nonstandard analysis lies in its ability to restore the intuitive language of Leibniz and Newton while satisfying the rigorous demands of modern logic. Critics might argue that this approach adds a layer of complexity by requiring a new number system, but Hamkins counters that for many problems, it actually simplifies the conceptual burden.
The Problem of "The" Hyperreal Numbers
However, Hamkins does not shy away from the philosophical thorns in this beautiful garden. While the hyperreal numbers are a powerful tool, they lack a unique identity. Unlike the real numbers or natural numbers, which have categorical characterizations (meaning there is essentially only one such structure), the hyperreal numbers do not. There are many different structures that satisfy the axioms of nonstandard analysis, and they are not all isomorphic.
"It seems incorrect, and ultimately not meaningful, to refer to 'the' hyperreal numbers, even from a structuralist perspective."
This is a profound critique of mathematical language. We speak of "the" real numbers as if they are a singular, fixed entity, but Hamkins points out that the hyperreals are a family of structures. The fact that mathematicians routinely refer to "the" hyperreal numbers suggests a pragmatic blindness; the differences between these non-isomorphic models simply don't matter for the calculus problems they are trying to solve. Hamkins suggests this lack of categoricity might explain why some mathematicians hesitate to embrace nonstandard analysis—it resists the clean, structuralist view that many prefer.
"We just do not seem entitled, in any robust sense, to make a singular reference to the hyperreal numbers."
Yet, Hamkins notes a potential resolution in the surreal numbers, a class-sized field that does offer a categorical characterization. This hints that the "joke" of mathematical reality might have a punchline we haven't fully appreciated yet. The tension between the utility of the hyperreals and their lack of uniqueness remains a fascinating open question in the philosophy of mathematics.
Bottom Line
Hamkins' strongest argument is his rehabilitation of the infinitesimal, proving that mathematical intuition can outpace foundational rigor without collapsing. His biggest vulnerability lies in the unresolved philosophical discomfort regarding the non-uniqueness of the hyperreal system, a gap that challenges our desire for a singular mathematical truth. Readers should watch for how this tension between pragmatic utility and structural purity continues to shape the future of mathematical logic.