This paper discusses the ecological case for epistemic innocence: does biased cognition have evolutionary benefits, and if so, does that exculpate human reasoners from irrationality? Proponents of 'ecological rationality' have challenged the bleak view of human reasoning emerging from research on biases and fallacies. If we approach the human mind as an adaptive toolbox, tailored to the structu...

We start with irrationality proofs. Historically, the first ones concerned irrational algebraic numbers, like the square roots of non square positive integers. Next, the theory of continued fraction expansion provided a very useful tool. Among the first proofs of irrationality for numbers which are now known to be transcendental are the ones by H. Lambert and L. Euler, in the XVIIIth century, f...

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...

produces ‘good’ rational approximations to log a. There are several ways of perfoming integration in (1) in order to show that the integrals lies in Q log a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and ...

produces ‘good’ rational approximations to log a. There are several ways of performing integration in (1) in order to show that the integral lies in Q log a+ Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and l...

The story exposed in this paper starts in 1978, when R. Apéry [Ap] gave a surprising sequence of exercises demonstrating the irrationality of ζ(2) and ζ(3). (For a nice explanation of Apéry’s discovery we refer to the review [Po].) Although the irrationality of the even zeta values ζ(2), ζ(4), . . . for that moment was a classical result (due to L. Euler and F. Lindemann), Apéry’s proof allows ...