Dogmas and Mathematics

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?

• You might be looking for stuff more on the lines of Platonism, Realism, Formalism (or Finitism), ... – Git Gud Nov 4 '13 at 22:43
• What is your context? It sounds like you've been tasked to write an essay on a topic or something... it would help greatly if we knew both your background and that of the target audience, more details about what specifically you're supposed to research, related topics you've been studying, et cetera. – Hurkyl Nov 4 '13 at 23:00
• You may want to look at this article in the Stanford Encyclopedia of Philosophy: plato.stanford.edu/entries/philosophy-mathematics (Although note that most ideas in the philosophy of mathematics would not qualify as dogmas; mathematicians love to doubt things.) – Trevor Wilson Nov 4 '13 at 23:02
• Besides looking at non-mathematical (e.g. religious) beliefs, or at the formal axioms of mathematics, there is also the third option of looking at mathematicians' informal ideas about mathematics (their answers to questions like: what is a rigorous argument, what is a number, do mathematical objects really exist, is mathematics a process of invention or of discovery, what should the axioms of mathematics be, etc.) – Trevor Wilson Nov 4 '13 at 23:10
• Given that in math everything follows a rigorous way, I would define a dogma as one (or a set of) axioms one chooses to follow or develop, even though alternative axioms would also lead to consistent conclusions. 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

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.

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.)

• Yes, and thats why physicists are so far ahead of mathematicians. – Abdulh Khazzak Gustav ElFakiri Nov 5 '13 at 18:26
• @AbdulhKhazzakGustavElFakiri Good one! – zerosofthezeta Jan 29 '14 at 21:58