Meaning: Abraham Robinson, a prominent mathematician, raised an intriguing concept in the quote: the notion of mathematical truths that may be inherently unknowable. This idea challenges the widely held belief that all mathematical truths can be discovered or proven. To grasp the significance of this quote, it is essential to understand the context in which it was made and the implications it carries for the field of mathematics.

Mathematicians have long debated the nature of mathematical truths and the philosophical foundations of their discipline. The Platonist view, which Robinson alludes to, is based on the philosophy of Plato, who posited that mathematical objects and truths exist independently of human thought and experience, in a realm of ideal forms. According to this perspective, mathematicians are not creators of mathematical truths but rather discoverers of pre-existing truths.

Robinson's reference to a "small minority of mathematicians" who entertain the idea of unknowable mathematical facts suggests that this concept is not widely embraced within the mathematical community. This is notable because mathematics is often seen as a realm of certainty and provability, where logical reasoning and rigorous proofs lead to incontrovertible truths. The notion of unknowable mathematical facts challenges this foundational assumption, raising profound questions about the limits of human understanding and the nature of mathematical reality.

The idea of unknowable mathematical truths has implications for various branches of mathematics, such as set theory and logic. In these areas, mathematicians grapple with questions of the nature of infinity, the continuum hypothesis, and the limits of formal systems. Gödel's incompleteness theorems, formulated by the mathematician Kurt Gödel in the 20th century, demonstrated that within certain formal systems, there exist statements that are true but unprovable within that system. This result shook the mathematical community and challenged the notion of complete and consistent formal systems.

Robinson's assertion that even Platonist mathematicians are divided on the concept of unknowable mathematical facts reflects the deep philosophical and epistemological implications of this idea. If there are indeed mathematical truths that lie beyond the reach of human knowledge and understanding, it calls into question the very foundations of mathematics as a field of inquiry. It raises fundamental questions about the relationship between mathematical objects and human cognition, the nature of truth, and the limits of what can be known or proven.

The debate surrounding unknowable mathematical facts also intersects with broader philosophical and scientific discussions about the nature of reality and the limits of human knowledge. It touches on issues of epistemology, the philosophy of mathematics, and the nature of truth itself. The implications of this concept extend beyond the boundaries of mathematics and have reverberations in fields such as philosophy, logic, and computer science.

In conclusion, Abraham Robinson's quote about the possibility of mathematical facts that are true but unknowable challenges deeply held assumptions about the nature of mathematics and the limits of human knowledge. The idea raises profound philosophical questions and has far-reaching implications for the foundations of mathematics as a discipline. By highlighting the existence of a minority of mathematicians, even among Platonists, who entertain this notion, Robinson invites us to reconsider our understanding of mathematical truth and the boundaries of human cognition in relation to the abstract realm of mathematics.