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to look up the original publication where a concept was first described (possibly in a way that would be considered "outdated" by now), or am I missing something here?

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    $\begingroup$ Are you looking for standardized mathematical foundations? I don't think you will ever find them. $\endgroup$
    – Asaf Karagila ♦
    Aug 10, 2013 at 17:53
  • $\begingroup$ @AsafKaragila Bourbaki might not agree with you. :-) $\endgroup$
    – Jay
    Aug 10, 2013 at 18:36
  • $\begingroup$ @Jay: Are there any of them alive today? Besides almost every mathematician alive today would agree that Bourbaki was an excellent endeavor, but it is ultimately outdated. $\endgroup$
    – Asaf Karagila ♦
    Aug 10, 2013 at 18:50
  • $\begingroup$ @AsafKaragila I believe Alexander Grothendieck was in Bourbaki and is still alive, though not doing mathematics. Given any approach, that approach will, in all likelihood be eventually be outdated. The smiley was meant to indicate a comment that was not made in an entirely serious vein. $\endgroup$
    – Jay
    Aug 10, 2013 at 19:01
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    $\begingroup$ In CS, there is no standard operating system, but any widely applicable OS must be able to accomplish certain "standard" tasks. In MS Windows and Mac OS X, for example, you must be able to do the same basic functions. It is like this for the various axiomatic foundations of mathematics. There is no single "standard" mathematical foundation. But, as with operating systems, I suppose one day in the distant future, there may be a global standard for mathematical foundations with many if not all of the current ones being found to be dead-ends. $\endgroup$
    – Dan Christensen
    Aug 12, 2013 at 15:20

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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.

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  • $\begingroup$ +1 for a generic female mathematician. It helps! I'll just add that those standards exist so people can build things that work together without ad-hoc human communication/intervention, a solution unnecessary in math. Also, mathematicians are like implementations--how often are standards implemented exactly according to the specs? This variety/flexibility is good and efficient. If I want to know common or uncommon definitions, I sample the internet. $\endgroup$
    – Rachel
    Aug 11, 2013 at 18:54
  • $\begingroup$ -1 for a generic female mathematician. Most mathematicians are men. If the information here comes as accurate, it consequently comes as preferable to reflect the reality for the majority. $\endgroup$
    – Doug Spoonwood
    Aug 11, 2013 at 22:40
  • $\begingroup$ @DougSpoonwood It sounds like you are saying that generic pronouns for members of a group should always be chosen according to the majority sex for that group. That is not a very good rule, for several reasons. $\endgroup$
    – Trevor Wilson
    Aug 12, 2013 at 2:16
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    $\begingroup$ I have seen the comments above. I would like to gently encourage others to simply not respond to them any further, or to use meta (only if absolutely necessary) for extended discussion. $\endgroup$
    – Carl Mummert
    Aug 12, 2013 at 2:18
  • $\begingroup$ @TrevorWilson No, I didn't say always. As a general rule, maybe. In this case though, yes. Actually, it does come as easy to make the case for such a rule in general, because it involves a choice to reflect the reality as it stands. In other words, reflecting that reality comes as honest. You're bluffing about the "several reasons". $\endgroup$
    – Doug Spoonwood
    Aug 12, 2013 at 2:45
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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".

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    $\begingroup$ Also, because mathematics is written by humans and intended to be read by other humans, minor discrepancies don't matter as much, unlike with computers. The human brain is extremely flexible and forgiving :) $\endgroup$
    – Ted
    Aug 10, 2013 at 18:04
  • $\begingroup$ @Ted but human brain fills blanks with its own conceptions and those conceptions can differ from men to men. This results in many definitions not being equivalent. $\endgroup$
    – Trismegistos
    Aug 12, 2013 at 7:32

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