Mathematical proof seems attractive to some, yet impenetrable to others. In this paper a theory is suggested involving “cognitive units” which can be the conscious focus of attention at a given time and connections in the individual’s cognitive structure that allow deductive proof to be formulated. Whilst elementary mathematics often involves sequential algorithms where each step cues the next,...

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...

In a recent paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! and on the same lines a simple criterion for some fast converging series representing irrational numbers was given. In the present paper, we give some generalizations of our previous results. 1 Irrationality criterion Our considerations in [3] lead us to the following criterion for irrationality, where x...

We show how one can use Hermite-Padé approximation and little q-Jacobi polynomials to construct rational approximants for ζq(2). These numbers are qanalogues of the well known ζ(2). Here q = 1 p , with p an integer greater than one. These approximants are good enough to show the irrationality of ζq(2) and they allow us to calculate an upper bound for its measure of irrationality: μ (ζq(2)) ≤ 10...