There is an almost ubiquitous misconception about what mathematics really is, and it’s a misconception that genuinely beckons a correction. I would take a guess that if the average person was asked “What is mathematics?”, they would respond with something along the lines of “well, it’s a bunch of rules that help you find certain numbers”. While this may have been a correct answer long ago, it is far from correct today.
Fortunately, what it actually is can be summarized very succinctly. Mathematics is simply the process of making assumptions and proving what follows. Hence, we all do math on a daily basis—either when talking to one another, or when thinking to oneself: “given what I know, I think that…”. This is math.
This is also why math courses through calculus are terrible—as they are absurdly misleading. Current curriculum is libelous to the discipline of mathematics and its participants, and action needs to be taken to address this.
At this point, high school language arts and composition courses may teach more math than actual math courses. Fundamental to mathematics is logic and its application in the context of sets. Logic was essentially nonexistent when I was in high school. Yet math and language arts classes implicitly assume that students have a solid understanding of it when they are asked to make arguments. Granted, as we are logical entities cognitively, we are trivially masters of logic. But in terms of conveying it communicably, improved training is necessary. I feel the overlooking of this necessity is a grave miscalculation that has hindered scientific thinking (an ability from which every citizen of the world can drastically benefit) for far longer than it should have. This needs to change.
I thought I’d attempt to come up with a consistent convention of classifying mathematical statements. To do so we will simply think of the collection of statements as a partially ordered set where is interpreted as . And for each statement/node/point there will be two -valued functions that represent the number of incoming and outgoing edges (i.e. immediate antecedents and consequences). A point will be called a corollary if it has one incoming edge. A point will be called a theorem if its set of successors is a “large” subset of the poset (even if it is a corollary). A point will be called a lemma if at least one of its immediate successors is a theorem. A point will be called an axiom if it has zero incoming edges. And a point will be called a proposition if it is none of the above.