# Is there any example of definition-construction difference in engineering mathematics tools?

@Martin Brandenburg, in his answer to one of my questions, stated that the construction of a mathematical notion is not the same as its definition.

The construction is not identical to the definition of something. The definition can be really abstract, whereas the construction must be concrete, often as concrete as possible in order to work with it.

He then stated an example of that difference:

For example, you can define the completion of a metric space via its universal property, but its construction is done via equivalence classes of Cauchy sequences. Also here, there is no equivalence relation in the definition of the completion of a metric space.

In another related Q&A, here, @Asaf Karagila also explains the difference, but I still cannot wrap my head around it (Maybe because I am not a mathematician.). So, is there any example of definition-construction difference regarding tools in applied mathematics (for example, those of real analysis)?

• The natural numbers can qualify -- eg you can define a set of axioms that natural numbers satisfy (for example en.m.wikipedia.org/wiki/Peano_axioms) and then construct natural numbers out of sets (en.m.wikipedia.org/wiki/Von_Neumann_ordinal) Commented Jan 17, 2023 at 2:24
• "Definition" here should probably read "specification". So in the physical world, the specification might be a door more than 6 feet and 6 inches high: if you constructed a 6 feet 7 inch door you would have met that specification. Commented Jan 17, 2023 at 2:46
• @RobArthan However, in these examples the object defined is unique in a very strong sense, while in your example, many different doors would still meet the specification. Commented Jan 17, 2023 at 5:18
• @TorstenSchoeneberg: I think that is a moot point. You (and the OP) might like to look at Benecarraf's paper What numbers could not be for the difference between constructions and specifications. I suspect the confusion here is between defining entities by construction (e.g., constructing the reals as equivalence classes of Cauchy sequences) and defining properties (e.g., the property of being a complete linearly ordered field). The latter specifies the real numbers, the former defines a system that meets that specification. Commented Jan 23, 2023 at 22:44
• @TorstenSchoeneberg: the relevance of my comment was meant to be that Benecarraf shows how two very different constructions satisfy the same specification. Commented Jan 27, 2023 at 20:33

Since you said "real analysis", here is one way to "define" differentiation (on polynomials $$\in \mathbb R[x]$$, say) rather abstractly, versus just "constructing" it:
Definition: $$D$$ is the operator $$D: \mathbb R[x] \rightarrow \mathbb R[x]$$ which satisfies i) $$D(a\cdot p+q) = a\cdot D(p) +D(q)$$ for all $$a \in \mathbb R$$ and polynomials $$p$$ and $$q$$, ii) $$D(p\cdot q) = p\cdot D(q) + D(p) \cdot q$$ for all polynomials $$p,q$$, and iii) $$D(x) = 1$$. (One can check this defines $$D$$ uniquely, i.e. if for two such operators $$D_1, D_2$$ satisfying these conditions one can show $$D_1 = D_2$$.)
Construction: $$D(\sum_{k=0}^n a_k \,x^k) := \sum_{k=1}^n k a_k \,x^{k-1}$$.