Exploring Infinite Divisibility: Mathematics, Metaphysics, and the Nature of Matter - with Einstein

The question of whether matter is infinitely divisible or fundamentally granular has intrigued thinkers for centuries. Philosophers like Leibniz and Herbart, alongside mathematicians like Poisson, have contributed significantly to this debate, proposing that matter either consists of indivisible “monads” or can be broken down infinitely into smaller parts. Einstein’s reflections offer a critical lens through which to evaluate these ideas.

Einstein quotes Herbart approvingly, stating, “It is wrong that matter ultimately consists of matter again; its true components are simple (simple beings, substances, monads).” This argument emphasises that the infinite divisibility of matter leads to an unsolvable paradox: if matter were endlessly divisible, we would never reach a fundamental “last part.” Einstein critiques this idea as a limitation of human thought rather than a reflection of physical reality, suggesting that such a division is a mathematical abstraction rather than a metaphysical truth.

Leibniz took this further by proposing that matter is composed of simple, indivisible substances he called monads—metaphysical points of force and activity rather than physical particles. Poisson and others, however, treated the concept of the “infinitely small” as a useful mathematical fiction, allowing for convenience in calculations but leaving the metaphysical questions unresolved.

Einstein appears to straddle these views, acknowledging the practical utility of infinite divisibility in mathematics but rejecting its physical application to matter. He argues that it is erroneous to project the imperfections of human conceptual frameworks—such as the notion of infinite divisibility—onto the nature of objects themselves.

This tension between metaphysical simplicity and mathematical abstraction continues to resonate in contemporary science. For instance, quantum mechanics reveals that matter has a granular structure at subatomic scales, yet mathematical models often rely on continuous variables. Einstein’s critique remains relevant, urging us to distinguish between the tools we use to understand the universe and the universe’s inherent properties.

Ultimately, the debate between infinite divisibility and fundamental granularity underscores the interplay between metaphysics and mathematics, inviting us to question whether the infinite is a concept of the mind or a feature of reality. 

Einstein’s boyhood comments on this can be found as comments written in the margins of Einstein's copy of Lübsen 1869, which is signed "J Einstein."

In detail his comments read: 

Leibnitz applied this part of a finite magnitude, which continued into infinity, also to matter, in order to arrive at the true components of the same, and Herbart rightly says: "Even before the first specific section was passed through the present lump, the infinite possibility lies in the day that one could carry out this same section in many infinite ways differently. With this, the whole infinite tealing is really accomplished at once; and one has reached the last parts, namely in thought, what was the only thing that mattered. These last parts cannot be matter" (because otherwise one would always have to repeat these countless parts countless times again, which is unrhymed). From this one should now immediately conclude, as Leibnitz already concluded: It is wrong that matter ultimately consists of matter again; its true components are simple (simple beings, substances, monads). And so it is according to the truth." (Herbart's Metaphysics.) 

It is wrong to deduce from the imperfection of our thinking on that of the objects.

Whether one, like Leibnitz, Poisson, Herbart u. A., the infinitely small

Seriously for a truly indivisible element or like others, and thereby, as one thinks, to eliminate every metaphysical difficulty, only wants to take for a useful fiction, in order to initiate the calculation comfortably and quickly, is always indifferent for the calculation, since one like the other leads to the goal.