What are the dogmas that restrict or promote the development of mathematics?
I know that a dogma is a set of beliefs that is accepted by the members of a group without being questioned or doubted. However, dogmas are usually religious, and it is here that I am stuck. Should I be looking for religious dogmas that impacted mathematics, or should I be looking at the dogmas (axioms) of mathematics?
The main dogma of mathematics is that what is acceptable must be demonstrably true; that is, it must be proved. Any assertion, no matter how plausible, that has not been proved has the lower status of being a conjecture. Exactly what constitutes a proof evolves over time. This evolution is slow: for example, most proofs from 100 years ago would be pretty much acceptable today. The present system of axioms for mathematics, which is accepted by most mathematicians, is ZFC, which combines the systems of Zermelo and Fraenkel with the axiom of choice. So every accepted statement in the main body of mathematics must, at least ultimately and in principle, be deducible from the ZFC axioms. (Arguably there are some exceptions at the frontiers of research in mathematics.)
Since the OP mentioned in a comment that his course is currently in late 16th century, I would like to mention an issue related to Simon Stevin (1548 – 1620). Stevin promoted the idea that every number is representable by means of an unending decimal representation, whether it is rational or irrational (other terms used at the time were surds, etc.). In this way he was arguably the founder of the real number system (or perhaps the Stevin number system). There is a common belief (I am not sure if this qualifies as a dogma) that this number system was only constructed around 1870. A recent article argues that Stevin deserves more credit as a pioneer of the common number system.
Generally speaking, the term dogma is emotionally loaded and implies disapproval to a certain extent, especially when speaking about a scientific endeavor that is supposed to be free of preconceptions. If it is to be understood as a set of assumptions, either explicit or implicit, then one can meaningfully speak of such a phenomenon in the context of a scientific enterprise.
One such assumption is the idea that the evolution of mathematics has a certain predetermined outcome. This is related to Platonist views commonly held in the community. In the context of the evolution of analysis, for example, the assumption is that the said outcome is the framework developed by the "great triumvirate" (as historian Boyer put it) of Cantor, Dedekind, and Weierstrass, involving an Archimedean field of scalars, with the background of ZFC (see for example John Bentin's answer). A recent article argued against such a predetermined course, called the A-track ("A" for Archimedes), and outlined a parallel track, the B-track ("B" for Bernoulli), involving an infinitesimal-enriched continuum arguably relied upon by Leibniz, Euler, Cauchy, and others. See Is mathematical history written by the victors?
The parallel track (though not the B-terminology) was evoked in Felix Klein's famous book "Elementary mathematics from an advanced viewpoint", as well as by historian H. Bos in his seminal work on Leibniz that can be found here.
In this sense, one could say that Euclidean Geometry is based on the dogma that two parallel lines have no point of intersection.
In a similar reasoning, one could say that Religion is a set of theorems based on the axiom that God exists.
– Leo Azevedo Jan 29 '14 at 22:04