Six Math Essentials

Set theory and logic stay below numbers, but they do not belong to quantitative reasoning.<p>By definition, quantities, either discrete or continuous, as already classified by Aristotle, are the properties that can be expressed by numeric values, as a result of counting for discrete quantities, or as a result of measuring for continuous quantities.<p>Properties that are not quantities, but which can be compared, to determine which is greater or less, or if they are equal, are qualities (also by definition; there are also properties that can neither be expressed by a number nor compared, so they are neither quantities nor qualities, e.g. the properties of belonging to a certain subset or of having a certain name).<p>The simplest numbers, i.e. cardinal numbers a.k.a. natural numbers, are equivalence classes of sets, so indeed the theory of numbers is built upon that of sets (a.k.a. multitudes in older language), but this does not contradict in any way the title from the book, because quantitative reasoning refers to only a part of mathematics, not to all of it.<p>In medieval education, the subjects of learning were partitioned into 3 trivial sciences, which were those that do not need quantitative reasoning, which were grammar, logic and rhetoric, plus 4 mathematical sciences, which were those that require quantitative reasoning, which were arithmetic, geometry, music (i.e. harmony) and astronomy.<p>Therefore the original meaning of the word &quot;mathematics&quot; referred to the sciences that need quantitative reasoning, while logic and set theory were outside &quot;mathematics&quot; (at that time the primitive form of set theory was included in &quot;logic&quot;, i.e. the concepts of set operations and of logic quantifiers).<p>Much later, towards the end of 19th century, due to their importance for describing mathematics in a rigorous way, mathematical logic and set theory became included in &quot;mathematics&quot;.<p>Humans have learned how to handle big numbers only very recently, not much before the invention of writing and the Bronze Age (the &quot;Bronze&quot; age happened only several thousands years later after the first use of agriculture and after the first use of metals like lead, copper and gold).<p>On the other hand, small numbers are perceived directly by the senses and recognized, not only by humans, but also by many animals.

&gt; I.e., when teaching kids advanced math at an early age it&#x27;s often better to skip numbers altogether and start with set theory or logic.<p>What do you mean by &quot;often better&quot;. Has it even ever been tried out?