Still, my post on the analogy to solving simultaneous equations did prompt me to try to find my old writings from back when I was first trying to digest Objectivism, not just the material on that ABC stuff but everything I wrote just to see how badly I did go wrong (I knew I did, so I didn't go looking for it before). Almost all of it is gone, though, both on my computer and physically. All I have seem to have left from back then (the general late 90's period) are three of those diagrams (the date-stamp says it is actually circa 1998, not 1996 as I had thought), some other trivial writings, my notes from Understanding Objectivism, and notes from other courses such as Objective Communication. To be honest, I am not all that fussed about this precisely because I now recognise that my own thoughts were deeply flawed, and because today the only things I would find valid is precisely what does still exist: those UO and OC notes. I do recall dumping some stuff for that very reason, but perhaps I was more thorough about doing so than I actually recall being. I do know there was a hard-drive crash involved, as well. *shrug*
The reason why I dropped the analogy entirely was that it was an oversimplified notion of relationship. I realised that what I had in mind overlooked the richness of actual relationships in the real world, that the relationships between A and B&C can be very complex and may have very little at all to do with the relationships between B and C.
Now, all that being said, I reiterate my reversal of my previous outright dumping of the analogy. Why, I will show later. Still, I do see that I was indeed wrong to run with it as quickly I initially did. I was trying to eliminate the influence of consciousness altogether, which was in truth mostly motivated by a sense of defensiveness - but this I did not expressly realise until this week.
The problem with taking the analogy too far is lies in that original motivation, which is a fool's errand. There is no escaping the fingerprint of consciousness upon knowledge. The problem lies chiefly in not recognising the distinction between content and format, that consciousness is chiefly a supplier of the latter (both perceptually and conceptually), but also that at the conceptual level consciousness influences the quantity and depth of content provided as per each given consciousness's own purposes in gaining knowledge. That fact about influence upon concent itself redounds back on formatting at the conceptual level, because the topic of format is itself material of content, which then leads to the issue of method determining content and hence to an apparent chicken-and-egg paradox that only Objectivism can fully resolve. For more on that, see OTI, in the section where Dr Peikoff discusses the inductive basis for the principle that one needs a method of knowledge, why reference to the fallibility of volitional consciousness is nowhere near enough to support that principle, and who supplied what first pointers to the proper resolution of that paradox.
So, yes, I partially resurrect the analogy. Now, some concretisation for the reader of this idea of a conceptual equivalent of solving simultaneous equations is well in order - and, yes, I do know I ought to know much better than to bring this a bit late to the party.
It was in the process of working out concretes for something else that I realised I was looking at a concrete of my old analogy! (So, yes, I do have lots of concretes for myself, with the issue being not stating any at the time of writing about the principle, rather than not having them.) The truth is that one can objectively identify a significant relationship between B and C by abstracting from the relationships AB and AC. The validity of the analogy is tied up with recognising having been as rationalist in the rejection as I had been in the initial adoption. This is observable in those concrete examples of the method at work in the conceptual arena. I'll give you two, from the far opposite ends of the conceptual spectrum (it was the second of the two that prompted the resurrection.) A third that I will briefly mention here is how in a physics class I once took the solution to one complex equation was achieved by subtracting from it an equation from a related subject matter and working on the resulting simpler equation from there to identify what the relationship between elements of some real-world system are (I don't recall what - something QMish in optics and lasers maybe, but then again perhaps not).
The first is really one that Miss Rand herself gave: the relationship between a perceptual-level standard of physical measurement and the use of that standard to achieve conceptual comprehension of that which cannot be perceptually comprehended. The entire second half of chapter one of ITOE is about that issue, with her own concrete example being distance:
The purpose of measurement is to expand the range of man's consciousness, of his knowledge, beyond the perceptual level: beyond the direct power of his senses and the immediate concretes of any given moment. Man can perceive the length of one foot directly; he cannot perceive ten miles. By establishing the relationship of feet to miles, he can grasp and know any distance on earth; by establishing the relationship of miles to light-years, he can know the distance of galaxies.
That a given distance is ten miles is a physical fact. This not just that it is physically yay far from A to B, but that the numerical relationship of that distance is a certain number of multiples of the distance from A to C. Great - but has one thus excluded the influence of consciousness upon the identification of that fact? Absolutely not, because of the question of why on earth was the distance between A and C chosen to identify the multiples of it in the distance between A and B. The answer to that question is as Miss Rand identified, that of the point of using the distance AC as the reference is the particular needs of the consciousness using it. That distance is one directly comprehensible at the perceptual level by that consciousness, eg that point A is the heel of a typical foot and point C is the toe of the same, and so the influence of consciousness upon the conceptual expression of fact of AB's distance by means of presenting it a multiple of AC's distance (eg 52,800 feet) is now unmistakeable (and unavoidable):
It is here that Protagoras' old dictum may be given a new meaning, the opposite of the one he intended: "Man is the measure of all things." Man *is* the measure, epistemologically - not *metaphysically*. In regard to human knowledge, man has to be the measure, since he has to bring all things into the realm of the humanly knowable.
This, then, can be viewed as a very basic example of the analogy at work, as well as being a concrete of how all knowledge is relational. There exists two separate relationships, where, by a mathematical process, the nature of a third can be obtained. Feet to lightyears is of course somewhat elementary, but the principle is indefinitely extendable (eg that physics class, and the inventions that similar high-level instances have lead to such as in aerodynamics and electronics).
The other example, the one that got me to resurrect the analogy, originated in me figuring out how to reduce the express conceptual-level abstractions of existence and identity to the perceptual level. It is all very well to say that they are self-evident (which of course they), but that's an issue of validation, which presumes already knowing what they mean. The issue is best identified by the question of how one would teach what the words mean.
The following I am culling out of a larger piece of writing I'm still working on for my own sake - ie it is me going back to the start again and being thorough in grasping Objectivism through inductions of my own pursuit.
Begin with existence: how would you teach that? I would argue that trying to go straight to something along the lines of stuff being here and there, and there's this great big totality of all of what's here and there, is just going to lead you around in circles, because "here" and "there" already have definite meanings and trying to retask them is apt to confuse the living daylights out of a child who does not already have at least some knowing grasp on the issue of being qua being. Do remember that an explicit grasp is miles apart from the implicit grasp that everyone has - the former could not be achieved without the latter.
I came to the conclusion that explicit grasp the concept of to exist precedes explicit grasp of the concept existence. That is, one must first comprehend that X exists before one can grasp there is a grand totality of X and Y and Z. And, before one can do that, since we must start at the perceptual level, that means beginning by making the simplest observations about what X is, as part and parcel of learning all the individual first-level concepts of things and attributes of things etc. Here one has all those simple sentences that a normal three-year-old (and often younger) can grasp without difficulty: The apple is red. The sun is shining. The moon is rising. The dog is barking. And so on.
Therein lies the issue at hand: existence is intimately bound up with identity, and one cannot grasp the former except by abstrating from instances of the latter. The questions then arise: How does one mentally separate existence and identity? How would you point that out to a child?? Get this wrong and you have what I suspect is the reason why some Objectivists make the mistake of thinking that "Existence exists" can be deduced from "A is A".
The answer is to be found in the fact that, at the same time as the child is learning about things and their attributes, the child must also be learning about concepts relating to time. This, too is bound up with identity, since it revolves around issues of that which is, that which was, and that which will be, where in each case definite particulars are attached: the apple was in the bowl, is being eaten, and the core will be thrown in the bin ("or you're in big trouble, lad!") So, here we have more instances of multiple referents to the abstract concepts that a parent is trying to teach: more equations, as it were.
What is the answer itself? That in the process of teaching about temporal relations there will come a time that is directly about the issue of coming to be, being, and ceasing to be. The concept of to exist, of being qua being, can thus be pointed to at the perceptual level and thrown into express sharp relief by contrast to before something came to be and after it stopped being. The Gaussian Elimination, in effect, would be through showing a child this development of being across a variety of different things, and then drawing attention to the fact of things being. In concrete, this can be achieved by scratch-baking all sorts of things (stories from my boss about her baking things with her daughter fill me with delight), playing with construction toys like Lego (a favourite of my own youth), making snowmen ("CALVIN!!!!!") and sand castles - anything and everything where the child is hands-on involved in actually bringing into being something that did not before exist. On top of that you can add in reference to seeds germinating, flowers developing. Then back the way, refer also to the eating of foods, the breaking up of Lego spaceships to make other things, snowmen melting, sandcastles being washed away, and then also observe other events from afar like whole buildings being burned (I saw a large tenement fire when I was a kid in Glasgow), and so on.
The range of possible concretes - ie equations in this analogy - from which to obtain the contrast required to differentiate out and subsequently integrate the concepts of "to exist" and eventually "existence" is endless. The parallel with solving simultaneous equations is apt because all the issues are in reality bound up with each other simultaneously, a word expressly used by Dr Peikoff. The direct perceptions are the original equations to be worked from, first-level other lowerish-level abstractions (eg the concepts of attributes of things and actions, and also spatial and temporal concepts, quantities, and so on) are the smaller subsidiary equations, and from there one works towards the grand solutions. Finally achieving the express identification of what "to exist" means is akin to one's discovery of those first solution to one of the actual variables themselves in the Gussian Elimination process.
With the concept of "to exist" in place it then becomes possible to identify the concept of "existent," and from there to identify the concept of "existence" (in conjunction with following a similar path of particulars to higher abstractions in regards to the issue of location, to get to "everywhere" and hence to identify "existence" in terms of place). One can then take this solution and use it to isolate where-ever existence is bound up in more complex "equations" and find other solutions, ie to then be able to draw express attention to the concept of identity qua identity, as opposed to particular modes of identity, and similarly with consciousness. For instance, regarding the concept of identity, one is now able to teach to (and in a manner appropriate for) a child the idea that to be something is always to be something, such as how everything one sees has a colour of somesort, that all material stuff has some sort of weight to it, and so on to show that everything is always something in particular, even clouds and winds and piles of dirt and so on.
That should be more than enough to validate my assertion the analogy is not auomatically to be dismissed as rationalist.