Thursday, February 17, 2011

Simultaneous equations and induction

(Update for direct-hitters to this post: concretisation is here.)

A long time ago (circa 1996) I thought up an analogy in reasoning to the mathematical methodology of simultaneous equations. I eventually dropped it, however, because I thought it was too rationalistic (plus because I was being ridiculed for it - though somewhat deservedly at the time, too, I must add). I now realise that I was definitely on to something important back then and that I was wrong to drop it entirely.

What I had in mind back then was quite simple: if one knew the relationship between A and B, and one knew the relationship between A and C, where for instance A was oneself (ie one's consciousness), then one could identify the relationship between B and C as it stood independently of A (ie, contra Kant, conceptually able to identify things in themselves). Working on that principle, one can then tackle more and more complex relationships, succesively busting down complex equations into lesser ones and from there to the identification of individual variables. I had neat diagrams to go with it, too, such as:

The reason why I think that there is something valid in this for all thought in general (I've never rejected the fact that it was a definitely valid method of approaching the quantitative aspects of reality) is because of its connection to Mill's methods (particularly difference and agreement) and their connection in turn to the general cognitive processes of differentiation and integation. It is quite valid to view systematic methodologies for solving them, such as Gaussian Elimination, as being advanced implementations of Mill's methods to the mathematical realm. That is, what is GE but a complex method for the progressive discovery of residues? Taking it back to reality itself, what is the method of concomitant variations but the means of identifying the equations themselves? There's even analytical software available for that purpose - that is what all those statistical packages amount to, with the R-squared variables and the like indicating how much confidence one can place in the equations so generated from a mass of raw input data.

The other reason why I am reversing my past dropping is that Miss Rand noted that to invalidate concept formation one first had to invalidate algebra. The solving of simultaneous equations is an element of the subject of algebra! It is one of the methods that is directly applicable to reality itself. The fact that there really is a strong mathematical element to consciousness, along with that express identification, clearly suggests to me that the method of solving simultaneous equations ought be investigated by the epistemologist. But I am neither a professional mathematician nor a professional philosopher, so I will leave this for those professional epistemologists to develop.

That being said, I know what I know, I do know that I am on to something valid and that it is not to be dismissed as inherently rationalistic. "Everything in reality is simultaneous" notes Dr Peikoff in many places, and that hierarchy in epistemology existed because we had to approach reality in that fashion: knowledge is built on previous knowledge. I strongly suspect that this is also a tie-in with the spiral theory of knowledge, in particular applications, in that the more one knows the more one can go back and refine more detail out of equations that one had already partly solved in the past. Thus I will use it when and how I think appropriate, being careful of course not to get over-enthusiastic and rationalist about it.

Dr Peikoff also notes that science was nothing more than the conceptual unravelling of perceptual data. Miss Rand noted that cognition was a mathematical process. I would suggest, then, in combining the two along with recognition of Mill's methods and their derivatives (particularly in mathematics), is that science in large part consists of identifying the figurative or literal equations in reality or fragments thereof and also of contrasting various equations against each other to isolate ever smaller valid sub-equations and ultimately to identify the root irreducible variables.



  1. You should check out category theory. In fact, based on your diagram, I have a hard time believing that you haven't already. Here's a really good book on it that is even accessible to a layman. .
    Chapter 6 makes good comparisons of thinking and reality; objective and subjective.

    Category theory is the closest math comes to being philosophy.

  2. I never heard of category theory. The only formal philosophy class I took was an introductory course in 1997 or so, as an elective. It covered just the big figures of the western tradition, and only loosely (the textbook was Ed Miller's "Questions that matter" 4th Ed). The only philosophical-mathematical issue I have heard of was that someone once mentioned predicate calculus to me, but I never followed up on that.

    Anyway, I thought up the SE analogy while I was still in school, during my engineering mathematics classes (we did SE by the ton). My first application of the principle to philsophy was to the mechanisms of sensory perception. I still think that is a very valid application, but my motive at the time was to try to prove the validity of the senses thereby. Now I know that such an approach is wrong (the validity of the senses as such is validated simply by recognising it is integral to the fact that one is able to identify that existence exists), even though it is later useful in examining those mechanisms after the validation.

    As to maths and philosophy, there are works by Pat Corvini available from the Ayn Rand Bookstore, but I've never looked at them. Thank you for the Amazon link, though I doubt I will be getting around to that in the foreseeable future.


  3. Incidentally, another connection between my schooling and philosophy, again relating to perception (and also thought of at about the same point in time) was the notion of the sense organs as transducers. They are devices for converting energy in one form into energy in another form, where that latter form is one directly accessible by consciousness itself. I do recall now that I got somewhat pummelled for that by another because it lead to an implicit humunculus notion. I later found that General Semantics beat me to the punch, too (I also vaguely recall subseqently looking at what GS had to say overall and then backing away from it with a shudder - don't press me for details, though!)


  4. Sorry. From where did you get the diagram at above?

    1. Sorry, I haven't been to blogger in a long time, didn't see your comment.

      I created the diagram myself, again circa 1996. It is one of a small set I made for an article I was writing at the time.