Let’s get (a little) mathematical. If you’re still reading, good for you!
I spend a fair amount of time reading various logic texts. Most of that time, these days, is spent on texts that are shared with a “Creative Commons” license, and are thus freely downloadable. This is for two reasons: first, I am deeply offended by contemporary text book prices. For example, Hurley’s logic book (you can look that up on your own) is around $100.00 for the more recent editions. Not as bad as Calculus text books, but certainly extreme when one considers that the material presented can be had for free from other sources. So, despite the overwhelming improbability of it ever occurring, I can’t stop myself from thinking about the scam inherent in textbook pricing, and thinking how I, as a would-be teacher, might better serve my students w/o bankrupting them.
The second reason is that I just really like the subject, and want to keep my nose in the books on this subject at all times. Like playing the cello, if you stop practicing, you lose whatever mastery you may once have possessed. (The cello analogy is in reference to the great Pablo Casals and the possibly apocryphal response(s) he gave to why he always practiced so diligently.) Since I am otherwise utterly penurious, my choice of texts to “practice” with are limited to what I can download for free. With respect to topics within mathematics, including formal logic, the range of materials is actually enormous, and the quality exceptionally good. One of these books is the Open Logic Text by the Open Logic Group (“OL”), licensed under Creative Commons international attribution 4.0. (I believe I have fulfilled my legal obligations in the forgoing; full .PDF HERE.) I very much approve of this text, and almost anyone but me would never have even the slightest critique to offer regarding its exceptionally comprehensive coverage of the topic in a readily understandable fashion. But I do have one criticism, one that pretty much no one but a Whiteheadian would ever think to make. And that is about their too sanguine opening about the centrality of sets, and their uncritical acceptance of an intransigently object structured thinking.
Here are a few words from OL that set the stage for us:
A set is a collection of objects, considered independently of the way it is specified, of the order of the objects in the set, or of their multiplicity. The objects making up the set are called elements or members of the set. If a is an element of a set X, we write a X (otherwise, a X). (OL, pg. 19, emphasis added for clarity.)
Notice that what is primary here are the objects; how, why, and even that they are gathered together is treated as a matter of no importance. In this view, objects owe their reality to nothing else, and stand forth as irreducible kernel’s of absolute self-identity. This claim is supposed to be innocent of any philosophical (much less, metaphysical) content, a supposition that is wildly at odds with the facts.
For example, there may be adequate philosophical reasons for treating the symbols “” and “” as metaphysically innocent for the purposes of the specific mathematical inquiry at hand, but that does not make them innocent simpliciter. The theory of objects and self-identity that is built into orthodox set theory is an exceptionally aggressive theory of external relations. External relations basically claim that self-identity is the first, the primary fact-of-facts, that is given at the outset of everything else, and all other forms of relatedness are external to that self-identity and, often enough, wholly parasitic upon that ultimate, absolute, fact-of-facts. Notice also that the phrasing in the above was not casual or accidental: they are philosophical reasons for justifying a specific mathematical inquiry.
Nothing prevents us from inverting this order of things. A more mereological approachi might be as follows: We begin with a (mereological) whole, with or without minimal units (called “atoms”). “Elements” progressively emerge as derivative abstractions from the relations of how other mereological wholes overlap with our initial choice. These progressive abstractive refinements (which can look very much like an unlabeled Venn diagram) may or may not ultimately terminate in selecting mereological atoms, even if such atoms (as considered in some particular schema of inquiry) are or are not taken to exist. Thus, what set theory and its metaphysics of absolute, atomic, self-identity takes as the most indisputable fact of reality, this position treats as a high abstraction, a result of extreme synthesis of compounded relational realities rather than an immediate given. The Polish logician Lesniewski, co-inventor of contemporary mereology with Alfred North Whitehead (both of whom worked in complete ignorance of the other’s efforts) developed his mereological theses specifically as an alternative to set theory as the foundation of mathematics.ii Rather than treating self-identity as primitive and all relations as external to it, this position treats self-identity as an ultimate, constituted result of all the relations that are internal to it. One can agree or disagree with these positions in their various flavors, but one cannot pretend that one position is somehow less of a metaphysical commitment than the other.
Far away from the abstractions of mathematical reasoning, there are solid reasons for rejecting the thesis of self-identity as primitive, and external relatedness as basic, in the arena of human experience. Every bit of evidence from paleontology, anthropology, sociology, psychology, history and – data-driven! – economics, shows quite conclusively that human identity is something that is built from the relational structures in which it is embedded, and not an irreducible fact-of-facts with which we must begin, groundless libertarian ideology to the contrary notwithstanding. Indeed, libertarian wishful thinking is the same manifestation of a baseless metaphysics of external relatedness at the human level that set theory manifests in abstract mathematics.
This is a peculiar turn for our inquiry to have taken. We started in mathematics, found ourselves in metaphysics, wound up in human personal and social structures, and were handed an analogy between the two points that form our start and end. The fascination with externality that we find in set theory is undoubtedly less pernicious than the effect that assumption has in social matters, but any habit of disregarding one’s metaphysical commitments – being a habit – is never innocuous, since such behaviors inevitably carry over into other modes of action and inquiry.
One of the things that make Whitehead’s philosophy worthy of study is that he took a much more subtle path to the issues regarding internal and external forms of relatedness than just naively accepting one while rejecting the other. Unlike Lesniewski, Whitehead was motivated by his knowledge of contemporary physics and his sensitivity to the realities of human experience. The idea of taking an infinitesimal point (a set-theoretic “element” of the real number line) as primitive, to say nothing as exhibiting any possible physical or experiential reality, struck him as pure nonsense, and so he resorted to a more internally related mereological notion coupled with a theory of abstraction to reconstruct such ideas on lines that were more empirically and logically justifiable.
And being neither simple nor naive, Whitehead never tried to eliminate or reduce either external nor internal relations the one to the other. Rather, he argued for a path that would unite the two approaches. But that unification was not as (externally related) individual things, but as (internally connected) processes of development. Thus, rather than external and internal relations as substantive objects in their own right, his metaphysics is one of the externalizing and internalizing of relatedness. After all, his magnum opus is entitled Process and Reality for a reason. And here we can see some hint of why he never joined other logicians in their lathered embrace of set theory. Perhaps we can learn from this.
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iFull disclosure, I wrote the article on Cosmology in this volume.
iiThis program has proven exceedingly difficult, and makes a pragmatic argument in favor of set theory.