Designer babies? I can't figure out where the transhumanists are coming from. It's a creepy world, that's for sure.
I guess we are facing reality squarely. I think the H+ folks are similar to the gorts who want to colonize Mars. I don't even want to contemplate the contents of their minds or why they think there is any meaning whatsoever in such idiotic schemes.
As for joining me in this mathematical sojourn (I had to look up the meaning of this word, sojourn), rest assured that it is very far removed from such ambitious projects as transhumanism or human colonization of outer space.
Of course, there are those, including myself at times, who might think I really am crazy or stupid for trying to force myself to find pleasure in studying things I find particularly difficult.
I am glad you are open to discussing the subtle difficulties in studying mathematics, and that, like me, you can imagine dropping out of the workforce to just spend your life trying to understand the underlying structure of mathematics. They should be careful who they teach certain things to, for we can really go off the deep end studying and thereby become quite useless to the captains of industry.
I am going to give an example of just how the novelty of the formal and rigorous presentation appears so strange and, well, difficult.
You understand that I had been getting back into Linear Algebra and Calculus, working with Gaussian elimination, studying Infinite Series, exploring the techniques of integration. Well, these methods are relatively straightforward even though they can appear intimidating. There is the use of abbreviations which expand into more fundamental operations. That's neither here nor there. It's just that when I studied calculus in 1994 at a local community college, my interest in mathematics was restored, rejuvenated.
Anyway, there is something about constructing proofs for some of the basic operations with the real number system that are requiring me to "start over" and move in baby steps. It is so formal that I can see why, as a teenager, I might have been overcome with grief over how something so simple, like addition, had suddenly become so very complicated. The formality and rigor was lost on me. It annoyed me that something I had become so familiar with had suddenly been transformed into what I found to be unnecessarily complicated.
For instance, in proving the conditional statement, "If 7 = 5 + 2, then 7 + (-2) = 5."
Clearly, we can see that the hypothesis, "7 = 5 + 2," and the conclusion, "7 + (-2) = 5," are both true, so the conditional statement as a whole is true. And yet, have you ever been exposed to a formal proof of such a simple statement?
I am only half joking when I say that the level of detail reminds me of god damn assembler language low-level instructions.
I have to find a way to become less annoyed by this requirement for rigorous proof, for this has formed some kind of mental block against the rigor of pure maths. I may have to "train my brain" to be able to make this great leap into pure mathematics. It's a paradox that I have to "go back in time" in order to move forward.
It transforms the familiar process of addition into a
weird series of statements that take slow motion photographs of the minute details of the thought processes involved; hence, the formality.
An example - a proof of the above conditional statement:
If 7 = 5 + 2, then 7 + (-2) = 5(1): 7 = 5 + 2 [Hypothesis]
(2): 7 + (-2) = (5 + 2) + (-2) [Substitution principle]
(3): (5 + 2) + (-2) = 5 + [2 + (-2)] [Associative axiom of addition]
(4): 7 + (-2) = 5 + [2 + (-2)] [Transitive property of equality]
(5): 2 + (-2) = 0 [Axiom of additive inverses]
(6): Therefore, 5 + [2 + (-2)] = 5 + 0 [Substitution principle]
(7): 7 + (-2) = 5 + 0 [Transitive property of equality]
(8'): 5 + 0 = 5 [Axiom of zero (the additive identity)]
(9): Therefore, 7 + (-2) = 5 [Transitive property of equality]
This is called a direct proof. To make the proof shorter and easier to follow, steps involving substitution and the properties of equality are not stated. Steps 2 to 9 could be replaced by the following chain of equations;
7 + (-2) = (5 + 2) + (-2) [Substitution principle]
= 5 + [2 + (-2)] [Associative axiom of addition]
= 5 + 0 [Axiom of additive inverses]
= 5 [Axiom of zero]
Therefore, 7 + (-2) = 5 [Transitive property of equality]
This proof is based on the field axioms.
Is it any wonder they no longer teach mathematics this way in the high schools? In fact, this kind of teaching was only implemented for a couple decades. I was at the tail end of it ... I never wanted mathematics to be like that, but this is evidently the language of actual mathematicians as opposed to "math educators". There was none of that in the community college when I took calculus and physics, and only something similar in a course called Mathematical Reasoning at the state university.
And yet, now, at age 50, I am really curious to see if such formality and rigor might fill in many gaps and serve as a bridge to engaging with mathematics on a deeper, structural level.
The preceding proof suggests the following theorem.
For all real numbers b and c, (b + c) + (-c) = bb and c are real numbers [Hypothesis]
b + c is a real number [Closure axiom of addition]
-c is a real number [Axiom of additive inverses]
(b + c) + (-c) = b + [c + (-c)] [Associative axiom of addition]
= b + 0 [Axiom of additive inverses]
= b [Axiom of zero]
Therefore, (b + c) + (-c) = b [Transitive property of equality]
Then we can use this theorem to prove other theorems. This theorem we just proved is called the Cancellation Property of Addition, so, once proved, we can just list [Cancellation property of addition] whenever we use the fact that "if a + c = b + c, then a = b".
This all begs the question, "Why bother?" - and I confess to not being able to give you, or anyone else for that matter, a good reason why I even bother. George Carlin quit school in 8th grade. He wanted to "burn down the math building". Maybe I am a miserablist, and since I find this stuff rather depressing, it keeps me grounded. In other words, I am only happy when I am depressed. I have to wonder if I even "like" mathematics. Sometimes I suspect that I do not like it at all, not at this level of rigor and formality. Hence, the inevitable identity crisis which is sure to ensue.
Ironically, I WANT to like it! Does this make sense? A great deal of the material I skip because it is not necessary for me to just work through exercises for the hell of it. I only work through the exercises that are challenging to me, such as the ones involving proofs or the extras at the end.
Well, Holden, whether I like it or not, this It's something I want to force myself to face ... a spiritual exercise which demands great humility and patience. The parts that are not challenging make me doubt this decision, thinking I might have wasted money on the books, but when I come upon any exercises requiring these direct proofs, I am reassured that this was a good move. It's not the kind of thing you can just track down on the internet.
I still experience a conscious resistance to such formality and rigor, but now I have become stubborn in the opposite direction. Now I want to force whatever part of me that is repulsed by the level of detail to actually slow down and "get into it". It takes a certain amount of faith or "hope" that in the process of finally taking such matters seriously, inner transformations will occur in my head which might crystallize into the attainment of that mythological awareness
they call "mathematical maturity".
The only way I am able to take my present obsessions seriously is to be so detached from mainstream society, and to kind of "pretend" I am undergoing a mysterious initiation into a secret order of "monks" who are called to see behind the curtain, to firmly grasp that the mathematical operations we take for granted are based upon foundational axioms dealing with that set of real numbers represented by the number line. These same axioms can be used to determine if sets other than numbers are groups or fields. That's why it is presented as such, as the underlying unifying structure is transferable to other areas of mathematics.
I think I have repressed my early exposure to this kind of rigor since it proves to be so useless in day to day practical existence - in "getting things done".
Now I am hoping I can not only accept the rather uselessness of this formal rigor, but even try to embrace the uselessness, or at least try to get a little psyched about the weirdness of it.
There is definitely a sense of "defamiliarization" in that we are not accustomed to stating such minute details about operations we just take for granted.
Oh well, Holden.
I think that what we may have in common when it comes to mathematics is that we are honest about our contradictory and conflicting feelings toward it as a whole.
It hurts my brain to revisit this material, which is, I suspect, all the more reason I am so obsessed with focusing on this for a few years before devoting my mind to "advanced calculus and computational physics". For all I know, my study of this foundational material will be it for me before I die in some "vehicular misadventure".
So, I can't become too concerned about the time spent going through the more challenging exercises in this special series of old neoteric high school textbooks.
Like I said, the only way I have been able to remain committed to this endeavor is to proceed as though I were some kind of holy and deranged madman.
Don't think that I take our communications for granted. If it were not for you, I really would have no one to complain to about
my struggles. Not only does no one care, but hardly anyone, including current students of so-called "engineering" mathematics, would encourage me to pursue these nagging details the way you do. I think that you have an intuitive appreciation for how significant it is for me to come to terms with these frustrations with mathematics at this point in my life. I feel that it makes no sense to proceed until I can unify mathematics in my own mind, in my own little world.