Leibniz and the spell of continuous

Author Information

• Hardy Grant

  • Born in Ottawa, Canada.

  • Degrees in mathematics from Queen’s University and McGill University.

  • Teaches at York University since 1965.

  • Interests include mathematics in cultural history, travel, birding, computer programming, and pre-1950s popular music.

Contextual Reference

• Philo Vance's Insight

  • Mentions the simplicity of 17th-century mathematics focusing on well-behaved functions.

  • Points out that continuous but nowhere differentiable functions were unknown at that time.

The Law of Continuity

• Significance for Leibniz

  • Leibniz: Pioneer of infinitesimal calculus and prominent philosopher.

  • Believed in mathematics as eternal truths representing an objective reality.

  • Saw mathematics as a model for inquiry in other fields and a source for understanding God’s creation.

• Key Ideas

  • Leibniz formulated the law of continuity: "Nature makes no leaps."

    • Movement occurs smoothly between the small to the great.

  • Critical statement from 1687 regarding continuity in function and results.

  • Hint of vagueness due to ambiguous terminology.

Interpretation of Continuity

• Functional Dependence

  • Leibniz's notion of continuity precedes rigorous definitions by Bolzano and Cauchy.

  • Acknowledged the limits where properties can be inherited by limits of sequences.

• Examples

  • Draws from regular polygons filling a circle and velocities approaching zero.

  • Dismisses discrepancies between sequences and their limits.

Philosophical Underpinnings

• Continuity and Aesthetics

  • Leibniz believed in an orderly and predictable universe reflecting a benevolent God.

  • Geometry viewed as the science of the continuous.

  • His admiration for contemporary mathematics emphasized continuous behavior of geometric objects.

Implications for Physical Reality

• Impact on Nature

  • Argues that physical experience reflects the continuity observable in mathematics.

  • Mathematics is foundational for understanding natural phenomena.

  • Dismisses chaos theory as merely apparent, reaffirming the law of continuity.

• Continuity as Overarching Principle

  • Regarded continuity as essential for any investigation or explanation within science.

  • Example provided contrasting Descartes' mechanics to elucidate the principle of continuity.

Biological Analogies

• Great Chain of Being

  • Leibniz connects biological diversity to mathematical continuity, asserting similarities among species.

  • Proposes that biological features transition smoothly across taxa, contradicting gaps in species.

Conclusion

• Legacy of Leibniz's Philosophy

  • His commitment to continuity shaped his worldview of a seamless and harmonious universe.

  • Anticipated challenges from later mathematical discoveries that contradicted his views on continuity.

References

1. Leibniz, New Essays Concerning Human Understanding.

2. Leibniz, Philosophical Papers and Letters.

3. Leibniz, Selections.

4. S. S. Van Dine, The Bishop Murder Case.