What are authoritative publications regarding foundational mathematics?

Question

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related network standards as RFCs, for programming languages, frameworks, libraries the company/developers publish a documentation like Oracle's official Java documentation, Microsoft's MSDN for .NET and so forth).

For most mathematical concepts, one can find a definition in an introductory-level book on that subject. However, how can one know that this definition is universally accepted by most mathematicians around the world? Is there any central source for definitions regarding basic concepts of logic, set theory, abstract algebra etc.?

Or, put another way: How can one know that a description in a book for beginners was not intentionally simplified to facilitate understanding?

I would suppose one has to look up the original publication where a concept was first described (possibly in a way that would be considered "outdated" by now), or am I missing something here?

Answer 1

Perhaps the best way to find the "standard" definition of something is to look up a "standard" textbook on the subject (for instance, Walter Rudin's books on Analysis, or Munkres book on Topology). These textbooks have distilled many years of research into fairly accessible material, and looking up the original papers would be akin to re-inventing the wheel.

I am afraid there may not be a better solution to your problem - many fundamental concepts have many equivalent definitions that are best understood in the context within which they arose, so it might be hard to pin down one "dictionary".

Answer 2

There are no standards of that sort for mathematics, it's just not how work the field is done.

There are some "standard" axiomatic systems, like ZFC set theory. But even these vary in details from one author to another (for example, the set of axioms given by Jech is not the same as the set given by Kunen, although they yield equivalent systems).

The strange thing is that the "more foundational" you get, the less standardized things are. If you ask several PhD mathematicians about an abstract, advanced concept such as the definition of the fundamental group of a topological space, you are likely to get the same definition from all of them. If you ask them to define a group, you will get mostly the same definition. If you ask them to define a function, you will get several different definitions. If you ask them to define the number "3" you will get at least a few blank stares.

Every once in a while I see someone mention ISO standards such as ISO 80000-2. But these are essentially unknown in actual mathematical practice, to the point that the idea that they are in any way authoritative is amusing.

I would be willing to say the reason for this is that, at the professional level, each mathematician re-makes mathematics for herself as part of learning it. Some amount of standardization is necessary for communication, of course, but to really master an area of mathematics requires internalizing the definitions and theorems in a way that is hard to describe to someone who hasn't done it. If you ask a mathematician for the definition of something she has internalized, she is not likely to look it up, nor will she feel as if it is memorized - she will just "state it off the top of her head" based on her internal understanding of the definition. It is no surprise that, if many mathematicians do that, they will all achieve slightly different statements of the same mathematical concepts.