3 ms·

Does anyone know of any mathematical objects that have been created intentionally to defy intuition?

• Twisol 8y ago
  Oh sure. Topology has a wonderful cavalcade of such objects. Wikipedia mentions a book that collects many (https://en.wikipedia.org/wiki/Counterexamples_in_Topology https://en.wikipedia.org/wiki/Counterexamples_in_Topology). A particularly nice example is the Alexander horned sphere. https://en.wikipedia.org/wiki/Alexander_horned_sphere https://en.wikipedia.org/wiki/Alexander_horned_sphere
  • JJMcJ 8y ago
    There is also a book "Counterexamples in Analysis", and other counterexamples books.
• soVeryTired 8y ago
  I don't know whether many mathematical objects are defined to defy intuition per se, but counterintuitive results often occur through an effort to prove a statement along the lines of "Not all objects X have property Y" (see the sibling comment's counterexamples in topology). One nice example of "pathological" behaviour came out of Cantor's study of the continuum [0]: he gave an example of a function that is continuous and has zero derivative almost everywhere (i.e. everywhere except at a set of isolated points), but that manages to increase from 0 to 1. https://en.wikipedia.org/wiki/Cantor_function https://en.wikipedia.org/wiki/Cantor_function
• xorand 8y ago
  There is intuition and educated intuition. Analysis and topology counterexamples are usually met with high disbelief. And then there is intuitionistic logic https://en.wikipedia.org/wiki/Intuitionistic_logic https://en.wikipedia.org/wiki/Intuitionistic_logic
• dubya 8y ago
  Calculus has a bunch of nice counterexamples to early intuitions. The Weierstrass function and the Devil's staircase come to mind.