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This is long so...

TLDR: Proposing the math community steer away from the misleading term unique, with respect to functions and algebraic operations. Instead, use unambiguous. Why not? Analysis below.


I just started to get back into math after a year off, and after cracking open Pinter's book on intro abstract algebra I was reminded of a phrasing issue I've always had. The phrase "uniquely defined" or "uniquely mapped". For example, the operation $a*b$ defined by normal multiplication $ab$ on $\mathbb{R}$ is considered uniquely defined—the operation takes any ordered pair in $\mathbb{R}$ and produces one, and only one value in $\mathbb{R}$. As Pinter describes uniquely defined:

In other words, the value of $a*b$ must be given unambiguously.

After a few moments, it becomes clear to the experienced math reader that this is a defining property shared with functions—an operation can't have more than one output. We get it.

But as I was shaking off the cobwebs, I remembered this terminology being a small barrier for me when I first started learning about functions and operations (and other objects), one that, I'm ashamed to admit, slowed me down once again.

Reflecting over the past hour, I've begun to view my learning friction as the result of poor encoding—the the word unique doesn't align with intuition. When I first read the word unique I interpret it as well... the normal word unique. As Merriam-Webster defines it.

being the only one

Applying that to function/operation context, there's a few interpretations to be had, but one the first and most natural one that comes to my mind is that each ordered pair is assigned a value unique to that ordered pair. Which is stronger definition of uniqueness. Clearly, the example above, with normal multiplication, does not meet this stronger definition, since $(a,b)$ and $(b,a)$ map to the same value under normal multiplication on $\mathbb{R}$. In other words, $(4, 3)$ is not the "only one" since it shares a value with $(2,6)$, $(3, 4)$, etc.

I acknowledge that in this function/operation context there is an argument that mappings in their entirety are always unique, since we're using ordered values—one unique ordered pair, for example, maps to a value. But this is not how uniqueness is portrayed or described in textbooks, in textbooks emphasis is on function output or operation value being singular. And I further find this justification for the term unique to be weak, since it leans on the uniqueness of ordered values and not the combining process of the elements.

Of course, mathematical terms and definitions are not meant to align perfectly with normal language. So I'm not arguing that we restrict math terminology to only language aligned perfectly with intuition, however we do (and should) always make an effort to name things appropriately. I think that here, the term unique violates naming principles and is misleading. I propose that Pinter's own terminology better. Let's call this defining property of functions and operations unambiguousness. E.g., functions and operations are unambiguous. Unambiguity seems like the word we actually mean, which is probably why Pinter wrote it.

I wrote this because, anything that can be done to remove learning friction should be.

So... My Question

Can you see a good reason to not shift gears and teach functions/operations as unambiguous. Any obvious faults? And of course, I'd be telling students about classic terminology.

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    $\begingroup$ Perhaps it would be helpful to see the full quote in Pinter, as it's currently unclear what he's trying to express. There are other common terms for the things you're talking about. The fact that any input has one output makes a relation a function (as you said). The quality where any output has one input is usually called one-to-one (or an injection). Using "unique" or "unambiguous" for either of these would seem nonstandard. $\endgroup$ – Daniel R. Collins Jan 25 at 1:44
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    $\begingroup$ Sorry to be a bit harsh, but honestly, if you cannot grasp standard terminology that is no fault of the terminology and every fault of your own. You (or your students) will find even more words being used in ways contrary to their usual meaning as you continue in math, and that's something you have to get used to. $\endgroup$ – YiFan Jan 25 at 9:18
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    $\begingroup$ I'm not sure how it's clear that math teachers aren't using "unambiguous" already. I use a variety of descriptions: unambiguous, unique, predictable... and I relate it to the vertical line test, and we use the various terms when looking at functions expressed in various ways: formulas, lists of ordered pairs, graphs, etc. $\endgroup$ – G Tony Jacobs Jan 25 at 17:47
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    $\begingroup$ I think your question/objection is entirely reasonable. Why hijack one word and subvert its meaning, when there is already another word available whose colloquial sense is exactly right? And, no, the general trope that mathematics mangles natural languages is not a good enough excuse to keep doing it, especially when there's no need. $\endgroup$ – paul garrett Jan 25 at 21:35
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    $\begingroup$ @Namaste, well, I would tend to disagree completely with your account, but I guess if you have had that experience with usage, others will have as well. To me, unambiguous is unambiguous. :) "Unique" is so widely misused colloquially that I still do think that ordinary people would have no idea how to interpret its use in a stylized mathematical context. "Well-defined", similarly, has the weakness of having no colloquial counterpart, although it is less ambiguous than "unique", in fact, from my experiential viewpoint. (And "literal" is apparently problematical... Language drifts.) $\endgroup$ – paul garrett Jan 26 at 3:35
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I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way.

Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions about it.

Reflecting over the past hour, I've begun to view my learning friction as the result of poor encoding—the the word unique doesn't align with intuition.

I think you should be skeptical of this reflection. When we're learning, there's a common tendency to credit the very last thing that makes the idea fall into place, and miss that all the seemingly aimless struggling that came before it was actually what prepared us for that last insight.

Maybe if functions had been described to you using "unambiguous" rather than "unique", you would have understood them sooner - but I think it's more likely that you would have needed about the same amount of thinking about the idea to sort through all the possible misunderstandings, and some other aspect would appear to have been the last insight you needed.

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    $\begingroup$ +1 for "When we're learning, there's a common tendency to credit the very last thing that makes the idea fall into place". I have definitely had professors give strange little analogies or anecdotes which are tangentially related to the topic at hand, with the expectation that they will be completely illuminating. I think that comes from this phenomenon, and it is good advice to monitor myself for this kind of behavior. $\endgroup$ – Steven Gubkin Jan 24 at 21:16
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    $\begingroup$ Henry, you have a point, but it does not apply here. This is not a "credit the very last thing that makes the idea fall into place". As implied I recall having confusion upon my first math encounter with unique, many years ago. But after some exposure my brain adjusted correctly. Yet now, after taking a long hiatus, I re-read the term and initial confusion I had long ago resurfaced. This time the confusion was very short-lived because I already understood the concept; the trouble was connecting the word—unique—to my notion. A few minutes later I see Pinter's clarification in the text. $\endgroup$ – Zduff Jan 24 at 21:42
  • $\begingroup$ Why not just test the hypothesis of the original poster by seeing if this particular hangup is resolved by trying it out? $\endgroup$ – The_Sympathizer Jan 26 at 14:02
  • $\begingroup$ In terms of relating sth to sth, I find the word unambiguous much clearer than unique or uniquely defined. When I was a kid learning math (which I actually really liked, opposed to the greater part of other kids in the school), one of the hardest word to grasp was .. "defined". Math/phys teachers loved it. Definitions were all over the place. But when it came to functions, that uniquely defined collided on semantical layer with the notion of a "definiton" (like, Ohm's law, etc). When I noticed it's just about unambiguously mapping input to output, it was "hell, it's just that?" $\endgroup$ – quetzalcoatl Jan 26 at 23:32
  • $\begingroup$ (cont') Of course, then I noticed that Ohm's law is a function :) But that was afterwards. Most probably the core of the problem was that everyone used the word "definition", "defines" etc assuming it is clear and it never really was explained what that word means. It's a pretty rare word, scarcely used in a normal life outside of a scientific discussion. I mean, in my native language at least ("definicja", "definiuje"). On the other hand, "unambiguous" ("jednoznaczny") as complicated word as it is, it actually is a common word present in everyday use, so I see kids grasping it sooner. $\endgroup$ – quetzalcoatl Jan 26 at 23:39
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I personally use the following terminology:

A relation $R \subset A \times B$ is said to be single-valued if $(a,b_1) \in R$ and $(a,b_2) \in R$ implies $b_1 = b_2$.

A relation $R \subset A \times B$ is said to be total if for all $a \in A$ there exists $b$ in $B$ with $(a,b) \in R$.

A relation which is both single valued and total is a function.

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    $\begingroup$ I like this terminology. But I do think it's a bit steep for first-time learners. $\endgroup$ – Zduff Jan 24 at 18:12
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    $\begingroup$ @Zduff I think these are subtle concepts, and anything less than the full logically correct definitions will lead to "hazy" understanding of the function concept. I think that is appropriate! These definitions should not (imho) be part of someones first introduction to the function concept. A math major does need to be capable of handling these things eventually. I teach this stuff in discrete math, a sophomore level class, at my university. $\endgroup$ – Steven Gubkin Jan 24 at 19:20
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    $\begingroup$ I like this answer a lot, but I think it is not terribly accessible to say, high school students, nor Freshman in college studying College Algebra (not to be mistaken with abstract algebra). It is a entirely accurate, but students at the level I mention may have had no exposure to elementary set theory, nor elementary order-theory. It is a very concise, but abstract definition that will alienate most students first learning about functions and operations on numbers. Indeed, this answer fits best on math.se. But not so much for teaching students first encountering functions/operations. $\endgroup$ – amWhy Jan 25 at 20:08
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In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.

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    $\begingroup$ It's hard for me to believe that word choice has no effect on connecting intuition to concept. $\endgroup$ – Zduff Jan 25 at 17:28
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    $\begingroup$ It's hard for me to believe that one can simply extract one detail out of all the ways that German math education differs from, say, American, and without any kind of study, confidently assert that it's the reason. I'm not saying word choice has no effect - that would seem naive - but I'm not going to just say "this agrees with my thesis, so it must be right and significant!" - that would also be naive. $\endgroup$ – G Tony Jacobs Jan 25 at 17:45
  • $\begingroup$ No. We can create an intentionally misleading or confusing name for a concept. This shows directly that there's a relationship between name and definition. We might need to study the significance of the effect depending on name and definition, but it's obvious that there's a relationship. $\endgroup$ – Zduff Jan 26 at 19:04
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    $\begingroup$ Right, and mathematicians generally agree, in my experience that the term "normal" is overused and other words like "imaginary" can be problematic for students. And you forget that I stated in the above post that I agree it's unwise to restrict naming to perfect alignment with intuition. However, we can produce clear examples of bad naming, e.g., "Definition 1". So naming matters, I believe good names serve as mnemonics to what they refer. This topic is a close relative certain some notation types—consider the contour integral symbol which connotes the action. $\endgroup$ – Zduff Jan 26 at 23:43
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    $\begingroup$ Naming is crucial for learning and understanding. The sole existence of a Neurolinguistic-Programming/etc proves that. One of the bad things you can do to a young kid learning your native language is to misname, mispronounce, or misexplain something. If I say that dogs don't like cats because they are "D" and are after "C", they will believe. If I say "play with" or "hate" instead of "like", they will get the same thing differently. We learn basing on words we know. Instead of warping an intuitive words into new unrelated meaning, it would be better to use a new, artificial new word instead. $\endgroup$ – quetzalcoatl Jan 26 at 23:51

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