Many definitions in mathematics are "fully crystalized". Sometimes the form of these definitions might be somewhat baffling to the uninitiated.

For example, the definition of a relation from $X$ to $Y$ as a subset of $X \times Y$ is extremely economical, and makes formulating and proving theorems about relations very efficient.

This same economy is a disadvantage when teaching the concept. I want my students to understand that "x is a brother of y" is an example of a relation on the set of people. In this context, saying that the relation "is a brother of" actually is a set of ordered pairs of people seems philosophically disturbing. I would rather speak of this set of ordered pairs as a kind of ledger sheet which records all of the brother relationships between people. For instance (Steve, Kayla) will be on that list because Steve (me) is the brother of Kayla (my sister), but (Kayla, Steve) will not be on this ledger sheet. If I accurately record all of the information about all pairs of people in this list, then I can replace the relation "x is a brother of y" with the equivalent relation "(x,y) is on the 'brother of' ledger sheet". Since any relation can be so translated, we skirt the issue of "actually" defining what a relation is by instead using this construction as a proxy.

I think that this kind of definition is quite common. We often use an easily available "proxy" for a concept to provide an economical definition. We then need to mentally translate each time we use the concept. This skill at conceptual translation is important to develop, but seems difficult to assess or help students to attain.


  1. Has this conceptual skill of translating between concepts and their proxies been identified and studied in the literature?
  2. What are the advantages and disadvantages of straying from the "official" definitions to allow easier assimilation? For instance, defining a relation instead as a two variable predicate might be "closer" to the students native conception of the concept.
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    $\begingroup$ What a great question! $\endgroup$ Apr 2, 2021 at 14:05
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    $\begingroup$ Would you say that “to illustrate that a crystallized definition encapsulates all the properties of the intuitive definition and also has advantages for formal operations/proofs” is what is meant by “motivating the [crystallized] definition” of a mathematical object? $\endgroup$
    – Steve
    Apr 2, 2021 at 18:12
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    $\begingroup$ I don't personally feel that a relation is a set of ordered pairs, but rather that relations can be modeled by sets of ordered pairs (for the purposes of proofs). I find it particularly troubling when my college algebra textbooks "define" functions in this way, since I don't think it helps my students develop their intuitions at their level. $\endgroup$
    – TomKern
    Apr 3, 2021 at 1:22
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    $\begingroup$ Related question: what is the squaring function ? How do we define squaring ? Almost any domain you choose is to small minded to capture the totality of the concept of squaring. (this example is from my brother) $\endgroup$ Apr 3, 2021 at 17:23
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    $\begingroup$ For what it is worth, inside my head where the students aren't looking, I think of a relation not at a subset of $X\times Y$ but as a function $X\times Y \to \{\text{True}, \text{False}\}$. i.e. there is some process which determines if the statement "$A$ is the brother of $B$" is true of false. The set is then the support of this function. $\endgroup$
    – Adam
    Apr 5, 2021 at 2:18

2 Answers 2


Your question makes me think of the definition of linear (in)dependence used in Lay's Linear Algebra. It does not feel natural, and yet it's the simplest definition to work with.

definition of linear (in)dependence

I start with a more natural definition, that if one vector can be made as a combination of others in the set (ie it depends on those others), then the set is linearly dependent. And then I show that the two definitions are equivalent. It's still confusing for students.

I believe it always helps to connect with what students already know. So I believe that this is helpful. But I do not have solid evidence to back up that belief.

  • $\begingroup$ This was exactly what I was doing recently in making this comment. $\endgroup$
    – ryang
    Apr 3, 2021 at 3:36
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    $\begingroup$ Totally agree that the linear algebraic definition of "linear dependence" is more naturally stated in terms of writing one vector in the set as a linear combination of the remaining vectors. However, even so, is $\{0 \}$ a linearly dependent set of vectors ? $\endgroup$ Apr 3, 2021 at 17:18
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    $\begingroup$ Also, and much more tangentially, it is interesting to note Lay's definition works for modules whereas the more "natural" definition fails due to the existence of zero-divisors. Failure to realize this wasted a few of my days in a research project. For example, over $\mathbb{Z}/6\mathbb{Z}$ the vectors $(2,0), (0,3)$ are linearly dependent since $3(2,0)+2(0,3) = (0,0)$. Indeed, even $\{ (2,0) \}$ is linearly dependent set since $3(2,0)=(0,0)$. $\endgroup$ Apr 3, 2021 at 17:21
  • $\begingroup$ Yes, as I stated above, his definitions is better to work with. And the zero vector is the main reason I can think of, too. Thanks for the extension. It's intriguing. $\endgroup$
    – Sue VanHattum
    Apr 4, 2021 at 3:45

Another example of a crystalized definition is Dedekind's approach to defining finite. First we define a set $S$ be infinite if it is equivalent to a proper subset, i.e., if there is an injection $f:S\to S$ that is not a surjection. Only then do we define what is means for a set to be finite. A set is finite if it is not infinite. "Hilbert's Hotel" is a meme for this approach.

Obviously, people think only about finite sets in the early stages of their cognitive development. Later they learn about big numbers (the "number of grains of sand found in every region whether inhabited or uninhabited"--Archimedes, The Sand Reckoner), and infinite sets.

Only after this development occurs, either individually or in mathematical culture, do we see the brilliance of Dedekind's approach to defining finite, which is to first characterize infinite sets.

Hans Freudenthal, a Dutch mathematician who took an interest in mathematics education in the 1970s and founded the journal Educational Studies in Mathematics, wrote about this definition of Dedekind in his book Didactical Phenomenology of Mathematical Structures. He says that this definition "presupposes a well-stuffed arsenal of mathematical strategies."

  • $\begingroup$ Dedekind's definition of finite set does not actually guarantee that the sets are actually finite. In the absence of the axiom of choice there exists infinite sets that are Dedekind-finite. Most definitions of a finite set characterise finite sets directly as a set that is in bijection with the set of natural numbers that are less than $n$ for some natural number $n$. $\endgroup$
    – user19584
    Mar 1 at 21:35

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