# Importance of standardization of definitions of mathematical terms

Don't we need to use internationally recognized standards in defining mathematical terms since differences in definitions play very much important role in finding a unique solution for a given problem and from where we can find those definitions ?
Here are few examples

1. Inclusion of 0 in set of natural numbers;
This was included upto 2015 in advanced level mathematics in my country but now they say not to include it. When I inquired the issue from a Cambridge examiner , he told since there is no agreement regarding it, questions not asked related to that issue in their exam papers.
2. Negative divisors;
In one of past papers in my country it was asked to find positive divisors and it implies that they believe there can be negative divisors too but when I asked about it from national institute of education an official told me normally they consider positive divisors only.
3. Negative prime numbers;
When we talk about primes normally we just say prime numbers and we don't specifically mention positive prime numbers but when I searched there are articles related to the existence of negative prime numbers too.
I suggest we better maintain one standard so that our students can go abroad and continue their studies without any issues related to this matter .
• (To me, this is not really a question about math education.) Jul 6, 2022 at 5:26
• I thought this issue is related to mathematics education because this has impact on performances of students in solving problems. If there is no such standard before answering we need to check the definitions used in particular exam body and this is discouraging students in participating international level mathematics competitions too. Jul 6, 2022 at 6:23
• Applying the definition of prime element in a ring to the ring of integers one finds that its primes are both positive and negative. The definition of prime element in a ring (if $p$ divides $ab$ then $p$ divides $a$ or $p$ divides $b$) is taught in any undergraduate mathematics degree and should be known to someone teaching prime numbers at any level. Jul 6, 2022 at 6:56
• Or: how about inculcating the skill/habit of reading and writing with context in mind, and teaching why and how it sometimes makes sense for definitions to vary, and that, even when it doesn't, that mathematics is a process of consultation whose universality comes more from its reasoning feature than from its standardised definitions ? Jul 6, 2022 at 10:40
• No offence intended, but I am under the impression that you are fighting a shadow here. I've seen students fail in mathematics at various different levels and for a number of reasons - but the fact that a few mathematical notions are not standardized was never one of those reasons. If we are able to teach our students that they need to understand rather than rote learn and that context is important, then they won't have problems with changing definitions. If, on the other hand, we are not able to teach this to our students, than we have much larger problems than a few varying definitions. Jul 6, 2022 at 17:41

IMHO one can tolerate such small differences in the mathematical language pretty easily (much easier than to tolerate the ambiguity of the natural language that lacks specifiers for almost any adjective, so people drive me nuts when saying "that is good/bad/beautiful" and I drive them nuts asking "good/bad for what purpose?" or "what is your definition of beauty?" in response) if one makes it a habit to explicitly state what certain words and symbols mean in the beginning of every article or assignment (for a class, you need to do it just once in a separate handout, or you can remind it five times in presentations after which everybody should get used to it).

The full standartization is, probably, impossible, because in reality it is rather convenient to assume slightly different conventions and notation in different situations. The argument about which one is the correct one is totally pointless as any argument about definition of words. One should just tell the students that when in doubt, they should ask what exactly is meant by this or that and assure them that such questions will always be answered. We all speak to each other like Humpty-Dumpty in Carrrol's "Through the looking glass" all the time and there is no way around it except asking "what exactly did you mean?" before making any strong comment or agreeing/disagreeing with the speaker.

So, I wouldn't bother too much about "international standartization" here. What I would rather like to see is sending a clear message to the students that the same word can have different meanings in different situations and that the speaker and the listener should agree on the underlying terminology before trying to convey any message.

As an anecdote that illustrates what happens if they don't, there is now a hot dispute in Russia about the following situation: On an exam a student was given a problem that literally translates as follows:

If you purchase 2 chocolate bars for 180 rubles, you receive one for free. How many chocolate bars can you receive if you have 360 rubles?

The student answered $$2$$ because her idea was that "receive" and "purchase" are two mutually exclusive words. The official answer was $$6$$ because the examiners considered these words synonyms. It cost the student a gold medal and the dispute was taken all the way up to the Ministry of Education and is still ongoing.

So, just keep in mind and make it clear to others that everyone will speak like Humpty-Dumpty no matter what even in scientific communication and that the prudent thing to do as a listener would be to take the approach he recommended to Alice, though, as we all remember, she wasn't very fond of him or his way of speaking in the end :-)

• as you suggested individually we can do our best but isn't it much better to have a ruling about this such as you have to use definitions provided by a particular source only because if we have that kind of system specially for department of mathematics in national institutes there will be no disputes. I agree as you mentioned there may be unavoidable issues related to wordings of a statement because if it is not clear enough you can describe it in your own way. Jul 6, 2022 at 13:40
• Why would you convey nuts to people in vehicles? What do you mean by "certain" words? I find that people who are most certain are often wrong, and I'd imagine it's the same with words! Yeah, I can see why doing this to people would annoy them. Jul 6, 2022 at 17:42
• @JanakaRodrigo "but isn't it much better to have a ruling about this such as you have to use definitions provided by a particular source". My answer to that is "no". Some freedom of language has to be allowed to accommodate for various circumstances. For instance, it is convenient to think that the primes are positive in the ring of integers but if you consider an arbitrary ring that doesn't allow an order, then it stops making any sense and in view of this more general definition you'll be forced to accept that the primes in $\mathbb Z$ can be negative to avoid a definition conflict. Jul 6, 2022 at 20:28
• @Tierry Why would you convey nuts to people in vehicles? Yep, this was exactly my point. What do you mean by "certain" words? In that sentence "certain" stands as a synonym for the quantifier "some" with the connotation that a vague consensus about the set of words satisfying the property is expected. Yeah, I can see why doing this to people would annoy them I can see it too. The problem is that the alternative may be much worse, IMO, leading to plenty of unnecessary confrontation and empty but rather heated arguments. Of course, you shouldn't "do it to people" on every occasion :-) Jul 6, 2022 at 20:43

It is impractical to provide a worldwide standard because many conventions are already in use, and this is driven by the utility of one convention over another in context.

For example, when discussing natural numbers, whether 0 or 1 is included can affect some formulae. If you want "n!" to count the number of subsets of a set of size n for all natural numbers, then 0 should be included since sets of size 0 certainly make sense. However to define the harmonic sequence as 1/n for all natural numbers, we would wish to exclude 0. This is a simple example, but should get across that in different disciplines, the choice of which precise definition to use is dependent on which is more useful.

Instead, it is more valuable to teach students to parse new definitions when needed, than to teach one unified, unerring definition. Indeed, the practice of mathematics is widely regarded by mathematicians not as a set of facts, but, in brief, the skill of drawing logical conclusions from assumptions, and to this purpose, manipulating expressions when necessary. The skill of parsing definitions, then, is key in the practice of mathematics, for it is necessary for drawing any conclusion from the assumption that "an object is an X," for any mathematical definition of the word X. In practice, definitions are given or made accessible when needed, especially when they may differ in convention from another subfield.

But not only this; let's consider your "student draws an inadequate conclusion in a standardized exam due to unclear definitions" example. If the examining body had simply said the definition of what they were asking the students to compute, or (if they wanted all students to know the definition ahead of time) gave a booklet of the definitions the examining body would always use, then a student with the skill to parse definitions would be able to adapt to whichever definition the examining body chose to use, even if they had learned a slightly different convention in school. And a student going abroad who learns to parse definitions will be able to adapt, and this is a much more realistic expectation than getting all universities across the globe to agree with each other.

• yes I agree with you, if there is no ruling to use such a set of definitions for all , examination bodies should provide what they accept in their exams but unfortunately it is not happening in my country for local exams . Jul 7, 2022 at 0:23