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References to Heraclitus and his simile of the river into which one cannot step twice occur frequently in the poems of Jorge Luis Borges. Borges understood the river to represent both time and the constantly changing nature of human existence. In this essay I argue that while Borges embraced the reality of change, he also struggled against the threat it posed to our sense of ourselves as enduring entities. For Borges, the similarity between our personal existence and the flowing of a river cannot be the whole story. Both the poetic art and its practitioner have a claim on immortality.