You know, it never looks good when the proprietor of a highly contentious web site hides behind his commentators. It tends to destroy the forum as an appropriate setting for serious intellectual discussions. So I trust that Mark has merely been busy, or better yet, that he recognizes the futility of trying to defend his indefensible behavior.
In any case, I’ll go ahead and pave the way to a final resolution of the situation by dispelling any remaining doubt that Mark is incompetent to evaluate the essay he’s been attacking here. Fortunately, an analysis of the first “substantive” paragraph of his critique will be sufficient for that purpose.
Here’s the paragraph:
“Right from the start, we can see the beginnings of how he’s going to use a supposedly set-theoretic notion, in a very peculiar way. I don’t know anyone who seriously thinks that the universe is a set. Sets are a tool that we use to construct abstract models that describe things. The universe isn’t a set; it’s the universe. And yet a huge part of his argument is, ultimately, based on “disproving” the idea that the universe is a set, based on silly word-games.”
Let’s have a look the above paragraph sentence by sentence.
Sentence 1: “Right from the start, we can see the beginnings of how he’s going to use a supposedly set-theoretic notion, in a very peculiar way.”
I don’t know what this means; it’s “geek” to me. It’s probably an error, but in the spirit of evaluative competence, I’ll withhold judgment.
Sentence 2: “I don’t know anyone who seriously thinks that the universe is a set.”
Error 1: That’s vanishingly unlikely. Materialists think that the universe is a set of material objects (e.g., atoms and subatomic particles in various combinations) on which all else can be secondarily defined and/or causally supervened. Any assertion by Mark that he doesn’t know at least one person subscribing to such a viewpoint is simply incredible, especially given the atheist-materialist circles in which he runs. (Mark describes himself as a “religious, theistic, reconstructionist Jew,” but that’s beside the point; merely that he attended a modern university is enough to tell us that he has rubbed elbows with many atheistic materialists.)
But materialism is almost beside the point; all we need here is the scientific method. With its unrelenting emphasis on observation of, and experimentation on, material objects including the measurement devices thereby affected, the scientific method demands that everything in science be related to observables and the objects to which they are attached, which, being individually discernable, qualify as elements of sets (with all appropriate distinctions applied; e.g., sets of physical objects or events are countable, while sets of points in a continuum are uncountable).
In search of counterexamples, one may be tempted to point to such things as time and process, “empty space”, various kinds of potential, forces, fields, waves, energy, causality, the spacetime manifold, quantum wave functions, “laws of nature”, “the mathematical structure of physical reality,” and so on as “non-material components of the universe”, but these are predicates whose physical relevance utterly depends on observation of the material content of the universe. To cut them loose from the elements of observational sets would be to deprive them of observational content and empty them of all physical meaning.
Sentence 3: “Sets are a tool that we use to construct abstract models that describe things. “
Error 2: More accurately, the concept “set” is a formal entity into which real content may be mapped by description or definition. To preclude content is to sever the mapping and render the “tool” descriptively useless.
Everything discernable (directly perceptible) within the physical universe, including the universe itself (as a coherent singleton), can be directly mapped into the set concept; only thusly are secondary concepts endowed with physical content. One ends up with sets, and elements of sets, to which various otherwise-empty concepts are attached. Unfortunately, in standard theory, this attachment is reminiscent of sessile mollusks which have glued themselves to foreign bodies, and this is a problem for set theory as a descriptive language for the universe (or as a foundational language of the mathematical formalisms applied to the universe by science), as it is subject to a crippling form of dualism which separates functions from the sets they relate. But while set concept is obviously necessary - these other concepts are rendered physically meaningless without it - this in no way implies its sufficiency on any scale of reference.
Sentence 4: “The universe isn’t a set; it’s the universe.”
Error 3: This is an instance of logical negation amounting to an absolute distinction between “set” and “the universe”. It asserts the nonexistence of structural overlap between “universe” and “set” on all levels of reference, thus precluding shared structure.
Let’s take a closer look. Mark isn’t just saying
4a. “The universe is *in part* a set, but ultimately *more than* just a set (of objects, events, etc.)”;
he’s saying
4b. “The universe is *not* a set, period.”
These statements are mathematically distinct. Mark’s statement, 4b, implies that the universe is nowhere a set, i.e., that neither it nor any of its contents can be mapped into a collection or aggregation of objects, elements, points, or any other discernable entities as content. But this is completely absurd.
Indeed, if the “set” concept is free of physical content, then this precludes the use of any measurement device for observation or experimentation, and in fact, reference to anything that is observationally discernable and quantifiable in number, as this would provide physical content for the “set” concept. Whoops, no more science!
Obviously, the universe IS a (structured) set, but not MERELY a set in the context of any established version of set theory. Its description requires a more powerful mathematical language incorporating the “set” concept within a more capacious formal entity (which, of course, was largely the point of my little essay, which was written back before “everybody knew” that standard set theory could not be rehabilitated as a foundational language). Hello, CTMU, and hello, SCSPL!
In short, the author of Sentence 4 (i.e., Mark) is either mathematically illiterate, or he’s trying a bit clumsily to agree with me in all essential respects, but doesn’t quite know it due to the depth of his own incomprehension.
Sentence 5: “And yet a huge part of his argument is, ultimately, based on ‘disproving’ the idea that the universe is a set, based on silly word-games.”
Error 4: This statement consists of two parts:
5a: “His argument is based on ‘disproving’ the idea that the universe is a set” (I’ll be charitable and assume that Mark knows what proof actually entails in mathematics);
5b: “This attempted disproof, and the argument based on it, are silly word games.”
Quibbles aside, statement 5a is close to accurate; I do, after all, maintain that the universe is not merely a set, but something with greater expressive capacity (properly including that inherent in the set concept itself). However, statement 5b amounts to an accusatory misconstruction of the writer’s personal incomprehension, for which there is no excuse.
And that’s just one little paragraph. Its completely erroneous character conclusively establishes that Mark’s critique fails value criteria 1-3 enumerated above, and that Mark himself fails all three adjoining competency criteria … which, somewhat to his credit, he freely admits.
Summary: Explaining the errors made by Mark at the very beginning of his critique requires more space than is occupied by Mark’s statements themselves. Mark actually generates errors at roughly the same rate, and arguably faster than the rate, at which he writes about the “errors” of others!
Even though this may not seem like serious business to some readers, it certainly is. If Mark does not desist in his nonsense, it may well turn out to be something he regrets for the rest of his life. This is not because he is merely wrong; we all live and learn. It is because Mark often lacks any clue regarding the wrong turns he has taken, and in order to distract himself from his frustration, habitually lashes out at the sources of his confusion like a vindictive child. Any failure of comprehension precipitates him into a fit of pique, at which point he disastrously (for him) attempts to damage the understanding and the reputations of others without just cause.
I’m sure it would be a relief for all concerned if this were the end of my participation here. So I hope that’s the case…all the more so because if it is not, then one way or another, things will only go further downhill for Mark.