否定證明(Proof by negation,又或者 Proof by contraposition)係一種利用換質換位(Contraposition)邏輯嚟去證明一啲定理嘅證明方法[3]。「換質換位」指嘅簡單啲講係話「」同「」呢兩句嘢喺邏輯上係有關嘅:如果「 暗示(Imply) 」係啱嘅,噉如果 唔係真,噉 都唔會係真。例如係以下呢個論證:
Clapham, C. & Nicholson, JN. The Concise Oxford Dictionary of Mathematics, Fourth edition. "A statement whose truth is either to be taken as self-evident or to be assumed. Certain areas of mathematics involve choosing a set of axioms and discovering what results can be derived from them, providing proofs for the theorems that are obtained."
Buss, Samuel R. (1998), "An introduction to proof theory", in Buss, Samuel R., Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, Elsevier, pp. 1–78, ISBN 9780080533186. See in particular p. 3: "The study of Proof Theory is traditionally motivated by the problem of formalizing mathematical proofs; the original formulation of first-order logic by Frege [1879] was the first successful step in this direction."
"What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way, there were no theorems only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were coins of the realm ant the conventions of that day dictated that journals only published theorems", David Mumford, Caroline Series and David Wright, Indra's Pearls, 2002.
"Mandelbrot, working at the IBM Research Laboratory, did some computer simulations for these sets on the reasonable assumption that, if you wanted to prove something, it might be helpful to know the answer ahead of time."A Note on the History of Fractals Archived 2009-02-15 at the Wayback Machine.