We have seen that from the empty set we produce the first infinite set, the set of natural numbers. If you are a math student, beware of pitfalls like "teaching needs to be linked to the student's experience". Or pearls like: "the teaching of mathematics must start from the concrete experience of the student" and "the teaching must start from the student's daily experiences". If you believe these stories then you cannot learn what an empty set is and it will be a real scandal for your conscience. "to imagine" an infinite set. What "experience" or "experience" a person may have had with the "empty" or the "infinite"? Don't you think it takes a little imagination to learn these basic math concepts? Would your neurons collapse if you struggled a little to imagine the set of natural numbers?
Do you know anyone who has had an experience "concrete" how "infinite"? We think you will say no. We will be very surprised if one day someone appears stating that "had a concrete experience with emptiness and infinity". For there are schools that make plans that predict that learning a content can only occur if the student has previous experiences or experiences linked to this "new" content! Perhaps this is why we do not have the slightest hint of the existence of an elementary or middle school that allows the student to learn satisfactorily the concept of empty set and infinite set. There is an anecdote we once heard from the teacher who defined the empty set as "the set of hairs of a bald man". This anecdote has obvious variant versions that we don't need to repeat here. It will be very easy to understand why a person in adulthood may say that they do not like math. In fact she can't say that she doesn't like math because unfortunately she never had the opportunity to learn a bit of real math.
Mathematicians discovered in the twentieth century which axioms were sufficient to do mathematics more accurately. It is fascinating to follow the path that opens with the early axioms of Zermelo-Fraenkel set theory. Any of us can ask simple, natural questions in a logical order to pursue mathematical truths. With the emergence of the set of natural numbers we have endless ways of being curious and researching mathematical truths. But before we take an avenue "arithmetic", that is, by a way of curiosities about the natural numbers from the point of view of "operations", we still have a lot to have fun with curiosities about "sets".
We will use our "geometric intuition" to investigate the sets further. We will make some drawings to better imagine the properties of sets.
While certain teaching ideologies may be harmful, we must recognize that the ideology that preaches the use of intuition in mathematics is extremely useful. We will show in the next column examples of interesting questions that can be asked in mathematics with the help of "intuition", and how can we discover new mathematical truths through "good figures".
Our next step will be to define the operations of "intersection", in "unity" and from "symmetrical difference" of sets. We will begin a research method that will play an increasingly important role in our search for mathematical truths. It is the method of "intuitive anticipation" of truths and their later demonstrations.
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