I have found myself doing exactly that at times, plugging in numbers. For instance, if I am asked to prove by counterexample that an assertion is false, my first impulse is to plug numbers in.
For all x, u, y, and v in {real numbers}: if x < u and y < u, then x*y < u*v.
The trick not to try to find a case where x*y > u*v, but simply a case where x*y is not less than u*v; in other words, showing a case where x*y = u*v is sufficient in order to contradict the conclusion, x*y < u*v.
So we can plug in negative and positive ones.
let x = -1 and u = 1
let y = -1 and v = 1
then x < u, y < v; and we have x*y = (-1)*(-1) = 1 and u*v = (1)*(1) = 1;
hence xy = uv, and we have proven by counterexample that the assertion is false.
These are the easiest kind of proofs where we just have to show a counterexample, and it invariably involves plugging in numbers.
I try not to type up too much about specific exercises since I don't want to bore you with tedium.
What the boy in the market place is doing is actually the definition of multiplication, where multiplication is a series of additions.
I would be stumped on how to go about teaching arithmetic to children. There were supposedly educational experiments in Africa, where "it takes a village to raise a child". What they have been doing there for a long time is having children teach children. Their philosophy is that a child a little beyond the level being taught has a clearer idea of the nature of the conceptual mental blocks that can cause difficulties, and they are far more patient in helping a child to gain insight.
My nephew was home-schooled, and he would experience a mental block when looking at the formal definition of absolute value. You see, when he saw "-x", he read "negative x", but the definition means the opposite sign of x. So, here, "-x" is a positive number.
You know, abs(x) = x if x > 0 and -x if x < 0.
say x = -5
then abs(-5) = 5; and I could see how frustrated my nephew would become because, in his mind, x is 5 and -x meant -5. In the definition, -x means -(-5) = 5.
He would become quite angry.
When I was stuck in a "welfare compound" in 2003, after graduating from the university, unable to find any work, and after receiving some kind of "note" from a psychiatrist notifying the officials that I was no longer required to "seek employment", that I had been diagnosed as bipolar with rapid cycling between mania and chronic depression, even as I was hitting the bottle (vodka) regularly, on days when I was sober, I would tutor an entire family of Puerto Rican brothers, from grade 3 to senior in high school. They were all genuine biological brothers. I am not using the word the way "Jesus" would refer to everyone as "Brother".
I was able to help the senior in high school the most, with the algebra and trigonometry, but algebra stretches throughout the grades, so the 7th and 8th grader also got much from my tutoring. We were all in the same welfare motel ... for over a year. One of the brothers called me el maestro. The two youngest ones didn't get much out of it. The youngest one was too full of pep and vinegar ... paradoxically, he was like "the boss" ... a real character.
It is fascinating for me to consider youth in India who might not make use of multiplication by 10 (and just adding the 0 to the number) since I, and all the world, including the Arabian Algebraists, know that the concept of 0 and the number system based on 10 comes directly from India.
I guess people forget that mathematical symbols and the sophistication of manipulations belongs to the realm of culture and is not something people are born with. I suppose any kind of education, even one filled with years of being shuffled through corridors like sheep, is not something to take for granted.
And yet, when we are really honest, we must admit that "counting" and the sense of "sets" have to be inherent in the human brain. We have to suspect that chimpanzees know the difference between 2 bundles of bananas and 4 bundles of bananas.
How many leaves does the squirrel need to build a sufficient nest? Do they have a language? Can one female squirrel signal to her daughter that she needs "at least 3 more loads of leaves"?
The weed man is very good with arithmetic. He knows multiplication by ounces, grams, and pounds. He does not care at all about how one proves that 1 < 0 is a false assertion. He knows 100 dollars is not less than 0 dollars.
When you refer to something as Dionysian Mathematics, I take this to mean "free play" as opposed to rigor, learning by discovery as opposed to proving theorems and corollaries.
Maybe the Apollonian way is the way of rigor, formalism, proofs, and Dionysian is trial and error, free-play, exploration and discovery.
I think my experiment with going back 33 years to engage with a text that made me feel totally inadequate has to do with seeing if, after much more experience with applied mathematics and computations, using algorithms, vectors, trigonometric functions, etc, I might be less intimidated, or at the very least, more careful and attentive to the rigor and formalism.
The thing is, I am old enough to be patient and not too defensive about protecting a fragile ego. I can admit to myself when formalism and rigor are psychologically painful. I can laugh it off and realize that deep down inside I am a brute who is impatient with such rigor.
Perhaps I long to develop a bit of sophistication ... and I can tell the youth that it took me a lifetime to develop just a small degree of mathematical maturity. Meanwhile there will be those math wizards who are very comfortable with rigor and formalism, and they might be like that at an early age. I still think more has to do with culture and environment. If they have been groomed for this ...
I was reading online a math text where the author suggested one implore "God" to help one understand. I didn't want to look any further. How could I? How can one sit there writing a mathematics text and suggest to the reader to pray to God to "help us understand the math"?
At that point I become more than a little disgusted.
Not understanding something is nobody's fault.
Does one care to understand? Does one care to learn a rule that will make their daily calculations easier?
What is the motivation to learn anything, for that matter?
I am certain that the Division of Vocational Rehabilitation did not invest in my further education so that I could take up mathematics and computer programming as hobbies that would motivate me to resist the strong compulsion to drink myself into oblivion. They intended for me to become trained and ... EMPLOYED. The State wants us EMPLOYED.
And yet there are many who study things like foreign languages simply to improve their minds ... culture?
Maybe my interest in mathematics is a longing for culture. I want to receive a small kernel of the culture of humanity, the culture of mathematics ... as a human being on this planet, as an aging human being on this planet. I do not want to be a barbarian. Although, I recall, as a teenager, I often envied aliterate peoples who made their living fishing or hunting/gathering. There is great irony in imagining Albert Einstein foraging for edible roots in the ground and constructing a shelter out of leaves and twigs. These are the kinds of things that had me in such a confused state as a teenager. I began to suspect that non-alphabetic "uncivilized" peoples knew a great deal more about actual living than the most well-trained scientists of the Industrial World.
And so I study ... the language of mathematics. Free-play or rigor, by discovery or by proof ...
I can't remember the last time I went fishing. At least I grew some tomatoes this past summer.
Maybe my ancestors have only recently crawled out of the caves of Europe. So be it. Now I live in a world in between ... not knowing enough about hunting, gathering, or even fishing ... and not efficient enough to earn a living as some kind of scientist. My only option was to stand in line for one of the redundant jobs required "semi-skilled" labor. This world is Hell, but I hide away and live in a world of exercises and solutions ... textbooks ... searching the internet for information and guidance ... When the last fish is dead, all money will be worthless. I depend on exploited labor to knock the sacred cow on the head so I can fry its liver in a wad of butter. Are we no longer barbarians simply because of mechanized agriculture? Maybe we are more barbaric than ever. As John Trudell used to say, "Civilization is not civilized."
The Greeks and even the Romans may have viewed the Teutonic Tribes of Northern Europe as barbarians, but look how they took to philosophy a couple thousand years later!
Alcohol ... I do not possess the gene to break down the alcohol. I am susceptible to being destroyed by it ... rapidly.
Anyway ... Your example just got me thinking about how culture is passed through the generations, and that it is not something passed on through the DNA. A child might be born in the land where so much sophisticated thinking developed, but the demands of his daily existence may force him to get by with rudimentary education.
If one's parents require him or her to go into the market to sell product, then there will be trouble for him or her if they are caught hiding in the hills with math books or any kind of literature at all, I suppose.
I knew a cop who would beat the crap out of his kid when the kid got overly enthusiastic about Calculus. He didn't want his kid "thinking he was smart". The cop did not respect such education and most likely found it effeminate, or at least not very "macho" or "practical".
Some other parent might shelter a child from the world specifically to allow them to study in a fortress of solitude.
Maybe it is a matter of luck and fate how much "culture" we are exposed to.
It is said that Industrial Civilization is culture-destroying, and yet we witness how much culture is acquired through conquerors via militaristic endeavors.
As far as money goes, I pay close attention to my arithmetic and never try anything "fancy".
The inner barbarian doesn't fuuck around when it comes to dollars and cents.
That's when decimal place errors can really be devastating.
So I certainly applaud careful arithmetic. That is preferable to Gaussian formulas.
Let me ask you. Do you prefer pencil or paper when working with math? Not just arithmetic, but any kind of math. Do you write down the linear equations for a linear programming problem in pencil or pen?
I always use pencil for anything other than arithmetic, and I frequently make use of erasers.
I am error-prone.
Now, the type of math I am studying at the moment requires a great deal of thought before the pencil hits the paper. It's not the kind where I can compute or calculate then check for errors.
Actually, as I have said before, this kind of math that requires proofs is not my cup of tea at all; but I am forcing myself to RELEARN HOW TO THINK, and how to develop patience for SLOW THINKING as opposed to quick calculations which crunch numbers through a machine (algorithm).
Again, I appreciate these conversations.
Maybe your Dionysian Mathematics is a call for a less formal, less rigorous, almost blasphemous free-style mathematics. I don't blame you for rejecting the edifice of pure formal mathematics. I do enjoy learning as much of their language as possible. My goal is not to become a mathematician, but merely to be able to read enough mathematical notation so as to be able to study a few of the most fundamental subjects in applied mathematics ... In other words, I humbly submit that I intend to use mathematics, not to "do" mathematics. Although, I would like to be able to read and write a few basic proofs.
I have a similar attitude toward computer programming. I only ask of myself to be able to write command line programs that perform some kind of numerical computation.
In the meantime, may we not believe the Great Lie that this is "civilization" ...