Leibniz himself denied that infinitesimals had any fixed ontological status, treating them as useful fictions governed by the law of continuity rather than as genuine magnitudes of a specific kind.

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Reasons For

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Reason for

1.Leibniz's extensive use of infinitesimals in calculus while denying their reality suggests he viewed them pragmatically as computational tools.
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2.The law of continuity allows infinitesimals to produce correct results without requiring commitment to their actual existence as magnitudes.
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3.Treating infinitesimals as fictions avoids metaphysical problems about infinite divisibility that plagued contemporary mathematics.
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Reasons Against

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Reason against

1.Leibniz's monadic metaphysics suggests he may have endorsed infinitesimals as real features of his actual infinite metaphysical structure.
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2.His private correspondence reveals commitments to infinitesimal reality that contradict the 'useful fiction' interpretation of published work.
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3.If infinitesimals are mere fictions, the law of continuity cannot explain why they yield empirically accurate physical predictions reliably.
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Key Terms

Genuine magnitudes(as what Leibniz said infinitesimals were NOT): Real, actual sizes or quantities that truly exist as mathematical objects in their own right.
Law of continuity(as the rule Leibniz said infinitesimals follow): A principle suggesting that nature doesn't make sudden jumps—instead, changes happen smoothly and gradually, without gaps.
Leibniz: Leibniz is a German philosopher and mathematician from the 1600s-1700s who developed calculus (a powerful math tool for measuring change and areas) independently around the same time as Isaac Newton. He's famous for creating much of the notation we still use in mathematics today and for arguing that everything in the universe follows logical principles. His ideas profoundly influenced modern science, mathematics, and philosophy, making him one of history's most important thinkers.
Ontological status(in metaphysics (the study of what exists)): What kind of thing something is considered to be or how real it exists—for example, whether something is a physical object, a concept, a property, or something else entirely.
Useful fictions(what Field claims mathematical statements are): Things we treat as true and useful for practical purposes, even though we don't think they actually exist in reality—like how we might use a fictional character's perspective to understand human nature.
infinitesimals(Peirce's philosophy of mathematics and foundations of calculus): Quantities that constitute the 'glue' causing points on a continuous line to lose their individual identity, thereby grounding the concept of a true continuum

Connections

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Truth & Knowledge1 linked Philosophy of Language1 linked

His private correspondence reveals commitments to infinitesimal reality that con...If infinitesimals are mere fictions, the law of continuity cannot explain why th...

Details

Type: claim
Perspectives: 2 (1 for, 1 against)
Edits: 1 edit