“I’m never going to use that!” Variations of this war cry are frequently lodged in protest against that Torquemada-ish, 7th level of hell known as “high school algebra.” I am inclined to sympathize with this lament, but not for the reasons one might suppose. The problem, you see, lies not in the pragmatics – the usefulness (or lack thereof) of the subject – but rather in the pedagogical techniques and intentions with which it is presented. Algebra, as it is almost universally taught in secondary school, is little more than a peculiarly mangled arithmetic. Algebra as it ought to be taught is relational logic, it is algebraic reasoning.
So what does the above mean?
Just from the words, “logic of relations” or “relational logic” rather obviously have to do with relations and the logical connections between them. But what I wish to press here is a rather larger claim: it is the idea that relations as such deserve to be taken seriously, and on their own account, and not just as a parasitic short-hand for talking about self-identical, and as such, unrelated, “things”. Thus, in saying that, “Jack is taller than Fred,” “Jack” and “Fred” are taken as existing in a way that is sufficient unto themselves (they are what I called above, “self-identical”.) The relation of “is taller than” is generally treated as a short-hand way of talking about the only two “real” things in the example sentence above. The relation “is taller than” has no standing of its own, and is entirely parasitic upon Jack and Fred (and their particular “standing” of one being taller than the other.)
I’d like to suggest the rather “radically other” position that relations have standing of their own; that “is taller than” has characteristics of its own which persist regardless of whether Jack and Fred are there to exemplify them. In a very real sense, Jack’s and Fred’s own reality is parasitic upon the relations that qualify Jack and Fred in just and only the ways they are qualified. Justifying this claim would require a tour through the minutiae of metaphysics that would easily cover 1,000 densely written pages of manuscript. I could recommend reading Alfred North Whitehead‘s magnum opus, Process and Reality, but that is already over 300 pages of what is widely considered to be one of the five most difficult texts in the entire Western Philosophical canon. Much of the secondary literature that is intended to guide one through this difficult work are either impenetrably difficult themselves &/or patently wrong.
I will, however, offer the following thought experiment. Suppose you have Jack and Fred before you; surely they are just “there”, robust and real, the self-identical persons and things that they happen to be? But isn’t the relational fact that Jack “is taller than” Fred equally present? Indeed, how could you subtract that fact from the combination “Jack and Fred,” and still have “Jack and Fred”? Oh sure, you could imagine a world where Fred “is taller than” Jack, or they are the same height. But that requires that you have the relation “is taller than” in your toolbox of concepts (along with “Jack” and “Fred.”) Even “the same height” requires that you have “is taller than” (regardless of which side of the relation Jack and Fred stand) because the concept of “same” is meaningless except within a context of relative – which is to say, relational – differences. But let me repeat, “is taller than” is every bit as present as Jack and Fred within your experience. So how is “is taller than” just magically less real, even though it is just as “there”?
(The above is a VERY quick gloss of William James’ notion of “radical empiricism.” Because this is (a) such an enormously important idea, and (b) James is such a fabulous writer, I encourage the reader to learn more about the idea from James himself.)
There are two points from the above I’d like to particularly highlight. The first is that relations and relational connections deserve to be treated seriously on their own account, and not dismissed as merely secondary adjuncts to the “really important,” self-identical “things.” The second is the relational concept emphasized above, and which I repeat here: the concept of “same” is meaningless except within a context of relative – which is to say, relational – differences.
Algebra – and hence, algebraic reasoning – is that mode of formal thought which endorses the first claim, and focuses upon the second. Which is to say, algebraic reasoning takes relations seriously, and chiefest amongst those relations is that complex/meta-relation of same/different. The “tools of the trade” here are the various sorts of “morphisms” with which one relational structure is compared to another, and they do this by highlighting the kinds of relational connections that are emphasized or that are dismissed in developing kinds of relative sameness &/or difference. When this study is conducted without any specific reference to another thing or structure, it is considered genuinely “abstract,” thus giving the field its name of “abstract algebra.” So, for example, one can study and characterize the various ways a wine glass is the same or different: Is it the same glass when I rotate it, or move it from one end of the table to the other? How about when I hit it with a hammer (all the glass is still there)? These are examples of reversible and irreversible changes, which correspond to “groups” and “monoids” respectively, which are, in turn, important topics in abstract algebra.
There are a number of good texts out there for anyone interested in pursuing abstract algebra in more detail (I am myself quite impressed by Judson’s book.) However, for persons not acclimated to the abstract approach, there are easier introductions. And for this, one can scarcely do better than George Boole. Boole’s classic An Investigation of the Laws of Thought, from 1853, is a marvelously gentle introduction to the basic forms of algebraic thinking in that it is the first systematic development of what came to be known as “algebraic logic.” Algebraic logic focuses on the ways in which relational structures and collections of propositions can be evaluated as “the same” – the term often evoked here is “equivalence classes.” Of the other major branches of formal logic, model theory is very closely related to algebraic logic, while proof theory is at quite a distant remove. (Perhaps this is why proof theory is such a poor tool for teaching philosophy students logic.) An author contemporary with Boole, Josiah Royce, produced a handy little primer for introducing students to this sort of algebraic logical reasoning. (Conversely, a contemporary introduction, that emphasizes the mathematical aspects of algebraic logic, may be found HERE. This latter booklet, by Andréka and Sain, makes the relational aspect of algebraic logic exceptionally plain, in the very first pages and throughout.)
Boole’s logic is, of course, the foundation of boolean algebra, which in turn is the basis of most electronic computers today. (The exceptions are neural-network systems, and even these are often built around a boolean system, only in a highly distributed framework.) What makes Boole’s Investigation so useful is that he is working many ideas out for the first time, and so for the curious autodidact, who might otherwise be less than enthused by mathematical studies, Boole offers a rather more gentle, step-wise introduction to the methods that, with cultivation and habit, can become a full-blown relational approach to thought. This process – the development of relational thinking – is a rather trickier business than simply memorizing a list of informal fallacies, such as were discussed in Thinking About Thinking 1. Many of our habits of language use run contrary to such thinking, even though our languages themselves show (under analysis) no necessary preclusion to relational modes of thought (as emerges from Royce’s little Primer above.)
But one can have a good ear for music, yet will still only become a competent musician through significant study and practice; so too, regardless of one’s native proclivities, the only path into well-developed relational thinking is by working on it in a setting that draws those characteristics out and emphasizes them. In a word, there is no royal road to algebra.