Abstract
Mathematics is often perceived as a science solely concerned with quantitative aspects. However, even in mathematics qualitative aspects are fundamental, not only because they make it possible an informal analysis of a problem that may eventually be addressed quantitatively (and such an analysis is precious when teaching), but also because sometimes qualitative reasoning is precisely that best suited to investigate a situation. We show some examples drawn from notions of differential topology (transversality and general position).
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One of the ways to define a topology on a set X is to fix, for each point x in X, a family U(x) of subsets of X (which will be precisely the neighbourhoods of x in X) in such a way that the following axioms are satisfied: (1) X ∈ U(x), for all x ∈ X; (2) for all U ∈ U(x), we have that x ∈ U; (3) if U ∈ U(x) and U ⊆ V, then V ∈ U(x) too; (4) if U ∈ U(x) and V ∈ U(x), then U ∩ V ∈ U(x) too; (5) if U ∈ U(x), then there exists a subset V ⊆ U such that x ∈ V and V ∈ U(y), for all y ∈ V.
Of course, in saying “completely reconstruct the knot” we refer not so much to the curve itself, as to the equivalence class that interests us when we say that two knots are equal or different.
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Dedò, M. Q for Qualitative mathematics. Lett Mat Int 5, 167–171 (2017). https://doi.org/10.1007/s40329-017-0182-4
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DOI: https://doi.org/10.1007/s40329-017-0182-4