Things seem more natural after sleep. I refer to math constructions like “Let’s build,” “Suppose we have,” and other proposals. Happily proceeding from definition to definition, and then to proofs, is the wall to climb. Thinking about math before doing a pullup: yes, let’s have an ordinal system and here are its properties.
Thinking in layers, we go from elements organized into groups discretized into classes, and each class representing some number of elements symbolized by digits, and these are called cardinal numbers, but the numerals themselves are actually natural numbers. And the endless list you built before is yet another layer; on top is the ordinal system.
This has the property of “before” and “after,” “precedes” and “follows.” On top of this (again) we have assemblages of men, a few of whom could have coincident heights – prohibited in the two levels beneath – an imposition of ranking that we had to explicitly define as possible before.
HB Fine’s book “A College Algebra” is available free on Google Play! I bought an old edition because computers are distracting. But you can see how algebra is constructed, step by step, from theorems and proofs. This matter is relegated in later books as exercises, which without this context represented walls and time sinks.
Armed with the practice of this book’s exposition, I expect later mathematic encounters to be either more pleasant or more familiar. I did not see “conjugate” in the index, but hopefully Fine’s algebra covers certain of those division eliminations that let us simplify to elegant terms.
Without exercises in the first part, it is best to practice and memorize everything. I have to turn it into doctrine. Perhaps it will replace certain other conceptions and prejudices, and allow me respite to focus on abstractions.
At this rate, I will regurgitate everything I’ve ever learned, and advance the timeline farther than I have years to claim. I guess it could be worse. Here is the construction of a number system.
First we have elements, and we select them gradually into a growing group: one in one, representing any such class of one elements, and designated with the cardinal number as its size “1.” Then we add a second element and the group grows, and this we represent with the digit-symbol “2.” And we go forever.
Each cardinal number is finite. By induction, first 1: there is no part of one element that would be its equivalent. The next (2) is finite because adding one element to a finite group – the group containing one element – does not disturb the property of the new group being finite . So the group II is finite because I is, and III is finite because II is, etc.
All cardinals are in this list. We can take a group and mark each element within it, and the last mark will be finite. That would be the cardinal number. If our counting never ended, the group would be equivalent to a part of itself, and therefore would be infinite.
No numbers are equal to each other. Any two finite groups are either equivalent or one is less than the second. Our construction assures us the latter, that each group is a part of its subsequent.
Having to prove if a < b and b < c, then a < c is me staring down a acid-free page and willing the ghost of genius to unlock something profound in the symbols themselves. That’s the first mistake.
These symbols, not even defined until after sets were defined, after cardinal numbers were described, and after equivalence was conceived in English, only represent those meanings before: of the sub-set, of belonging in groups.
Fifty years earlier, an author wrote:
If the first of three groups be a part of the second, and the second a part of the third, then the first is also a part of the third.
Not having a solution manual feels like the demand for garbage collectors baked in to C: it’s the wrong kind of want for the context. It does make it difficult not having a “compiler” for math problems. Here is your input (the problem), but no real sense of the output. Except for proofs, but I could not begin with those.
Actually, there are two methods. One involves a time machine, to collect books from the 1900s. One such is “College Algebra” by Henry Fine. The second is to consult a discrete math textbook that suggests proof strategies. But even that ends with “proofs are an art” and “there is no one way to reach a proof.” The advice is to do more proofs; just as in programming, we have to achieve a certain level of reasoning fluency.
Suppose proofs are actually the specific steps to get from A to B. Could you have not encountered a part of this problem while noodling with the basic concepts before the exercises? The great sin is treating it as a fantasy novel of pure exposition, I guess. But it is difficult to do much when all one knows is printing things to the screen.
Some further perversions on the idea of learning: I dug through a bibliography and ordered a few more old books. These approach the mechanics of high school busywork, and maybe there will be some magic concepts therein that prove (ha) this wisdom. I don’t think forever is enough time to clear the stack while the heap keeps growing.
I can’t trust a notebook to order my thoughts. The answers could be lost; the index is not searchable; and I wanted to learn enough LaTeX to make things neat for future me. Recursive projects are the worst.
I don’t like that math has not a compiler. There’s no sense of whether my “syntax” is wrong or right. There are no formal checks. This continues to the present. I go along because this is the convention and harder to argue than feelings on technological preference.
Algebra, not like in school, starts with sets. These elements associate to elements, set to set, a creative correspondence: chairs to students, objects to positive integers. Counting them, the highest ordinal number becomes the cardinal number, and this is what we see, naked, in every arithmetic operation thence.
The abstraction elevates to the sky: each number represents an abstraction of its class of sets; that is, 3 represents three of a set of anything; we free ourselves from leaks. We define equivalence and relations, addition, subtraction, multiplication and division.
Even the early problems ask for proofs. I’m tempted to jump back into discrete to plow through the proof tips section. But that would start a prerequisite cascade: I would begin with the first sentence of the first chapter, starting over, and exhaust months.
I thought probability theory would be easy, but I was wrong. It feels like black magic, even when you have the solution. My prerequisites were weak sauce, so I knew I had to fix it. I decided to work through “Fundamental Concepts of Algebra.”
Where am I going to find the time to grind through these books? I also need to apply the stuff, so that my programming does not atrophy. I’ve pondered moonlighting as a sys admin at home. Wouldn’t that be a great second job; coming up with learning materials and infrastructure for mom and pop, typical users, and gleaning killer UX experience?
The answer is yes! Of course. I could create a whole ecosystem here. To them, I could be Apple and Google. I could be their rockstar. I could show them my talent. My skills would grow with their computer literacy.
Back to the math: so, you can take any set of objects – logs, men, or rocks – and establish a one-to-one correspondence with the set of positive integers, finite to fit the former’s length, and that cardinal value – that *number* – associates concrete notions with the abstract. Now we can operate on them with all the fine finesse of mathematics.