I teach senior mathematics. But I am no ordinary "teacher". I teach understanding. The concepts I introduce to learners are well defined. Using well-defined concepts, those learners can construct real objects. What this means, is that those objects exist, have always existed and shall always exist, whether or not they thought of them.
In fact, there is no such thing as new knowledge and inventions. Future inventions have always existed. Those we are not aware of are just the ones we have yet to think about.
Knowledge is discovered, never invented.
I cannot stress enough the importance of well-defined concepts. It is out of ideas and concepts, that real and useful objects are reified (constructed). Whatever one imagines is real, if and only if, whatever one imagines is well defined.
Humans learn by a process I call "inferential suspension in knowledge acquisition". Whenever a human meets a new concept for the first time, he has no clue what it means. So he mulls over the knowledge (if any) he already has, and attempts to determine if there are any links with existing knowledge. If there are no links, he stores whatever he can - the sound of the word, a picture, a smell, an experience, to go to the most primitive. If machines are ever to think like humans, they will be designed with "inferential suspension in knowledge acquisition".
So what exactly does this mean? Well, we learn by producing inferences. However, when we can't produce inferences, the next best thing is to store (or suspend) the concept until such time that inference can be made.
If you had to ask a robot today, "What is a magnitude?", it will most likely respond with "I do not understand". No matter how many times you ask it, it will respond the same way. It cannot learn from concepts that have not been reified. Software must already be in place that tells it what is the reification of the concept. For example, you cannot tell a robot:
1. 0 is a natural number.
You first have to define "natural number" for it. But you can tell a human, and even though he has no clue, he will suspend inference until such time as he can make sense of the concept.
2. The successor of 0 is a natural number.
Gee, the robot will simply do nothing with that statement. The robot will need a precise definition of what is successor. Even an intelligent human will not understand what is a successor, except that it indicates something that comes after.
To cut a long story short, we always start with a concept and then construct objects (tangible or intangible) from that concept.
The thought paradigms of the Ancient Greeks with respect to the concept of number:
Assume that they knew NOTHING about number, because they really didn't!
1. They noticed that lines were not the same length or just not the same. They noticed that certain objects were heavier than others or took up more space. But we will stay with lines for the rest of this comment. Being curious as intelligent humans are, they began to ask, How long? How short?
2. They began to compare lines. But how could they notice anything without the concept of 'difference'. The first thing one looks for, is not addition, but difference. The Greeks did not ask at first, what is the sum of these two lines, they asked, what is the difference between these two lines. The sum came only once difference had been established. Difference is what results from a comparison of objects.
3. But what is length? What is mass? What is volume? What do all these have in common? Well, they are all related by the idea of 'size' or 'magnitude'. So, length was first established as a magnitude. The Greeks knew nothing else except that length was a magnitude.
4. Where to begin? They started by formalising comparison of magnitudes. If x and y are magnitudes, then x : y means x compared with y. They called x:y a ratio of magnitudes. But all they could tell at this stage was whether x is shorter or longer than y, provided they could tell x is not equal to y. That's all they could conclude. This type of comparison is a form of measurement - QUALITATIVE measurement. It is measurement without numbers. As any astute person will realise, qualitative measurement isn't very useful. For example, if you know that x and y are different magnitudes, then you still can't say how much shorter or longer one is from the other. There is no way to tell.
5. The Greeks began to superimpose lines on top of other lines and noticed that some lines fit exactly into other lines, that is, they have the same magnitude. It was by doing this, that they arrived at a very special ratio, that is, x:x. This ratio became known as the unit or 1 much later. This incredible breakthrough made it possible to tell the difference in terms of units, provided both lines are measurable in whole units.
6. The natural numbers are composed of units, multiples of units, that is, one or more units arranged adjacently to make up the length (magnitude) of a given line. But then a new problem arose: what if the lines were not composed of whole units? Well, that lead directly to the concept of rational number, that is, the comparison of natural numbers. A rational number is a ratio of natural numbers. For example, 2 units : 3 units is a ratio, but it's quite different to a fraction, that is, 2/3 which expresses a part of a unit, rather than the comparison of 2 and 3 units respectively.
7. So now, the Greeks were able to perform measure with numbers. All the magnitudes they had measured thus far were measurable. It did not take long before they stumbled onto magnitudes that refused to be measured. These magnitudes were called incommensurable magnitudes, example: sqrt(2), pi, e, etc. Incommensurable magnitudes could only be measured as approximations, using rational numbers. There are no such objects as irrational numbers. They don't exist! Neither in theory nor otherwise. Rational numbers are VERY REAL objects. There is NO VALID construction of objects that measure incommensurable magnitudes exactly (read as: there are no irrational numbers, only magnitudes that refuse to be measured).
The Ancient Greeks were brilliant beyond belief. The following ratio describes their intelligence compared to modern academics very precisely:
Modern academic IQ/ Ancient Greek IQ = Baboon IQ / Modern Human IQ
If I compiled an exam on the Elements, Apollonius's Conics and the Works of Archimedes, I don't know of anyone who would pass it!! :-) So what can we conclude from all this? Which is the best way to learn about numbers?
Well, we know that numbers cannot be constructed using set theory unless one already assumes they exist and a lot of their properties are already in place. Even then, there is no valid construction that does not include knowledge from the Elements, or prior knowledge about natural numbers.
Rather than build on the light and beauty of Greek mathematics, morons chose the poisonous and rotten ways of men like Cantor, Hilbert, Peano, ZF, Russell and all the other baboons.
They abandoned true knowledge for the fantasies of Cantor's paradise. It did not occur to them to look at the type of men who created this rot. Cantor was insane. Hilbert was an idiot. Peano was a juvenile when it came to constructing anything. Russell was a tobacco monkey. Most of these men were disgustingly filthy. How could they have had good minds?
And yet, their rot has been espoused. This is similar to artists abandoning classical art for the rot called "modern art".
A good mathematician is like a fine artist: the objects arising from concepts in a mathematician's mind are only as appealing as they are well defined.
To educate the future, we must be like Renes Des Cartes, who returned to the sound foundations of the Ancient Greeks.
Without DesCartes, much of the math today would not exist because the Greeks did not use coordinate systems. Yet what DesCartes accomplished made it possible for all to learn about conics, functions and much more.
Whilst finite set theory is useful, infinite sets are purely imaginary. They don't exist because infinity does not exist. No one can reify infinity. It is an ill-formed concept existing only in dysfunctional minds. Learners never understand such concepts. Their educators never understand such concepts. Yet they think it reasonable to build knowledge on such shaky foundations.
It is time to return to the path of light and beauty.
You can espouse well-defined concepts or you can follow the rot of Cantor, Peano, Dan Christensen (a contributor on sci.math), etc.