text: Degrees of certainty Physicist Lawrence M. Krauss suggests that the need for identifying degrees of certainty is under-appreciated in various domains, including policy-making and the understanding of science. This is because different goals require different degrees of certaintyand politicians are not always aware of (or do not make it clear) how much certainty we are working with.
  Rudolf Carnap viewed certainty as a matter of degree ("degrees of certainty") which could be objectively measured, with degree one being certainty. Bayesian analysis derives degrees of certainty which are interpreted as a measure of subjective psychological belief.
  Alternatively, one might use the legal degrees of certainty. These standards of evidence ascend as follows: no credible evidence, some credible evidence, a preponderance of evidence, clear and convincing evidence, beyond reasonable doubt, and beyond any shadow of a doubt (i.e. undoubtablerecognized as an impossible standard to meetwhich serves only to terminate the list).
  If knowledge requires absolute certainty, then knowledge is most likely impossible, as evidenced by the apparent fallibility of our beliefs.
  Foundational crisis of mathematics
  The foundational crisis of mathematics was the early 20th century's term for the search for proper foundations of mathematics.
  After several schools of the philosophy of mathematics ran into difficulties one after the other in the 20th century, the assumption that mathematics had any foundation that could be stated within mathematics itself began to be heavily challenged.
  One attempt after another to provide unassailable foundations for mathematics was found to suffer from various paradoxes (such as Russell's paradox) and to be inconsistent.