Tag Archives: Logic

Complexity – It Ain’t Simple (part 2 of 2)

30 Monday Aug 2021

Posted by in Complexity, Logic, Mathematics, Philosophy of Logic

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The guiding motto in the life of every natural philosopher should be, Seek simplicity and distrust it.”

– Alfred North Whitehead, The Concept of Nature (end of chapter VII.)

Ultimately, the only way we know how to measure the complexity of some process or phenomenon – beyond excruciatingly vague and unhelpful statements like, “this is really complicated” – is by measuring how hard it is to solve the mathematical equations used to characterize the problem. All the rest, even when palpably, indisputably true, is just hand-waving. Sometimes hand-waving makes us feel better, because we need to burn off the energy pent up in our frustration. But it never really tells us anything. On the other hand, we really do have some effective means of measuring how hard it is to solve some mathematical equation or other, and we’ve refined such measures significantly over the past fifty years because such measures tell us a great deal about what we can and cannot do with our beloved computers (which includes all of your portable and handheld devices, in case you weren’t sure.)

Some problems simply cannot be solved. This even despite the fact that the problems in question seem perfectly reasonable ones that are well and clearly formulated. (Actually, being well formulated makes it easier to demonstrate when a problem cannot be solved.) Some problems can be solved, albeit with certain qualifications, while still others are “simply” and demonstrably solvable.i However, saying that a problem is “solvable” – even in the pure and “simple” sense (notice how I keep scare-quoting that word) – doesn’t mean that it can be solved in any useful or practical sense. If the actual computation of a solution ultimately demands more time &/or computer memory space than exists or is possible within the physical universe, then it is unclear how we mere mortals benefit from this theoretical solvability.ii It is these latter considerations that bring us into the realm of computational complexity.

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Complexity – It Ain’t Simple (part 1 of 2)

24 Tuesday Aug 2021

Posted by in Logic, Process Philosophy, Whitehead

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Some sixty-one years ago, the American philosopher Willard Van Orman Quine wrote a famous essay, “On Simple Theories of a Complex World.” Actually, referring to this as a “famous essay” is a tad redundant, since Quine is one of those people who only ever wrote famous essays. But setting that observation (bordering on sour grapes) aside, Quine goes on to observe the difficulty in saying just what does qualify as simplicity. He further observes the legitimate psychological and formal reasons while theory builders so ardently crave simple theories: the simpler the theory, the more readily it can be employed in our various cognitive activities. Of course, too simple a theory leaves us with no purchase on the world what-so-ever. “God willed it” is about as simple a theory as you can come up with, but it is also as singularly useless a theory as anyone could ever imagine; it provides absolutely no insight, a complete absence of predictive power, and only an illusion of emotional comfort for those readily distracted by vacuous hand waving.

A “Rube Goldberg” machine.

Quine was writing more than a decade before the emergence of computational complexity as a sub-field of abstract Computer Science, in which upper and lower bounds for kinds of complexity (and thus, conversely, forms of simplicity) was even formulated. But we do now have a variety of ways to address Quine’s concerns about how to characterize complexity and simplicity. I’ll say more about this in a moment. What I want to start with a more controversial proposition: Namely, Quine got it backwards. In a very real sense, it is the world that is fundamentally simple and our theories that are complex.

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The Infinite

22 Thursday Jul 2021

Posted by in Logic, Mathematics, Ontology

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I don’t anticipate any explicitly Whiteheadian considerations this time around, but all my thoughts are informed by my Whitehead scholarship, so you never know. What I want to talk about here is the idea of infinity. I say “idea,” rather than “concept,” because even within the relatively constrained bounds of formal mathematics infinity is not one thing. Outside of the bounds of mathematics matters are significantly worse, little or since no effort is made to constrain such talks, or even render it potentially intelligible, with formally legitimate techniques.

Speaking of “outside the bounds,” the ancient Greek word for the infinite is “apeiron” (ἄπειρον), which translates as “unlimited” – the “a” being the negation (“un”) and “peiron” meaning limited or bounded. Clever as they were, the Greeks lacked our additional 2,300 years of mathematical study, so the idea that one can have something that is infinite (unbounded) – for example, the length of the perimeter of a geometrical figure – i.e. a perimeter that exceeds any possible length, measurable either in practice or the ideal, that is nevertheless bounded by an easily measured finite figure (a circle, for example) would never have occurred to them.i But the figure above, the Koch snowflake, is precisely such a figure. (Details can be found HERE. As is my wont, I skip the technical details which will take up more text than this blog post.)

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Strains, Planes, and Flat Loci

22 Saturday May 2021

Posted by in Emergence, Logic, measurement, Mereology, Metaphysics, Process Philosophy

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A running joke that Dr. Auxier and I incorporated into our booki was the phrase, “skip to page 337.” The pagination reference is to the Free Press edition of the corrected version of Whitehead’s monumental work of metaphysics, Process and Reality (“PR” hereafter.) Page 337 of PR is the start of the fifth part of the work, his rather poetic discussion on “God,” beyond the more concrete arguments of the preceding 337 pages. By “concrete” it should be understood that Whitehead’s “God” is not some religion inspiring big daddy in the sky that you go to church to beg candy from. Uneducated rumors to the contrary not withstanding, Whitehead never invented words. But at many points in his tome on “speculative philosophy” (his preferred term for what others call “metaphysics”) he needed to identify an “omega point” which served as the entirely impersonal foundation for the rational structure of the world as well as the “font of creativity.” He called this “God.” Were he inclined to use non-English words, a better choice might have been the Greek “arché” (αρχη). But Whitehead was Whitehead, and that was never going to happen, and so it did not.

Setting aside for the moment the question of “God,” there are some important issues in the material that the people skipping over to pg. 337 are, in fact, skipping over, in their stampeding rush to gin up a “Whiteheadian” theology. There are two things I want to talk about that are left all but untouched in the secondary literature on Whitehead, one of which is interesting and the other is downright revolutionary. These things appear in the pages that many scholars ignore when the skip to pg. 337. They are what Whitehead called “strains” and “flat loci.” I’ll address these in order. But first I’ll devote a paragraph to the work on natural philosophy that Whitehead developed in the years preceding PR.

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The Implicit “All”

12 Tuesday Nov 2019

Posted by in Critical Thinking, Logic

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One of the fundamental units in logical analysis is that function/operatori lovingly known as “the quantifier.” Most logic texts content themselves with just two: “all” and “some,” formally symbolized as “” and “” respectively. Thus, to say that, “All X is p,” one is asserting that every (or any) instance of X is also an instance of p, or is characterized by p, etc. Similarly, when someone says only that, “Some X is p,” the claim is made that, if one looks hard enough, one will find at least one instance where X is p. There are ways of precisifying (one of those $5.00 words philosophers love to use) the above statements, but there is hardly any need to do so here. It suffices to have a general idea. Two points I’ll mention in passing. First, in most formal contexts (substructural logics are an example of an exception), “all” and “some” are defined as being interchangeable using “not”: thus, “not-All X is not-p” is taken to mean “Some X is p,” and conversely. Secondly, these are not the only quantifiers possible: “many” and “most” are also examples. But these last two are difficult to formalize (to say the least) and by a polite convention among logicians they are generally ignored wholesale.Implicit All

As the title of this post states, I wish to talk about what I am calling the implicit “all”; uses of the “all” quantifier in which that quantifier is functioning but not explicitly stated. This happens quite often, in point of fact, and is not problematical in itself. Where problems do arise is when that usage is not merely implicit, but actively denied as a means of evading the consequences of what someone has actively stated or written. When this happens, we are faced not merely with a logical error, but an overt act of dishonesty. The dishonesty becomes not merely overt but blatant when, even after the implicit “all” is pointed out, the individual continues to deny it. Continue reading →

Objects and Relations

19 Monday Nov 2018

Posted by in Logic, Process Philosophy, Relationalism

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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.Venn diagram

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. Continue reading →

Self-Identity

27 Tuesday Feb 2018

Posted by in Logic, Mathematics, Metaphysics, Ontology, Process Philosophy, Relationalism

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I was not an especially “outward looking” or alert youth, working rather to shut the world out rather than invest painful consideration into something that was already almost unbearably painful. But occasionally my habits of thinking would turn themselves outward, to chew on a puzzle that had managed to break through my protective shell and demand my attention. This happened twice that I can recall in high school: the first time, after an especially depressing episode I realized I needed to make a study of reading people – perhaps, more importantly, I realized that I could learn this, and I began picking up clues effectively and rapidly. The second, and first genuinely philosophical moment, was when I “discovered” the “problem of evil” as it related to the born-again Christianity I’d been emotionally bullied into accepting by various members of my family – personal responsibility is a joke, of course, in any world dominated by an omniscient and omnipotent creator god. This began my “angry atheist” phase, which went on for another decade (until I’d actually read a substantial bit of philosophy.)Acropolis1

The third “break through” (second genuinely philosophical one) happened when I was in the army. I was stationed some 18 kliks from what was (at the time) the East German border, in the Central German highlands, as an electronics tech in an Improved HAWK anti-aircraft missile battery. Every year, each such unit chose a squad of people to be sent to NAMFI, Crete, to spend a few days training, culminating in firing a live bird at a drone target. As it happened (then, at least), the entire trip involved several days both before and after the actual training which were free time for the troops to explore the island or, as several of us chose to do, take the ferry from Souda bay to the Piraeus and Athens. So it came to pass that I climbed the steps up the hill of the Acropolis. Except, that’s not quite right. Nobody actually walked on those steps, and it wasn’t out of respect for their antiquity. Continue reading →

Making Sense

12 Monday Feb 2018

Posted by in General Philosophy, Inquiry, Logic, Metaphysics, Philosophy of Science, Process Philosophy, Whitehead

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Whitehead set out to make sense of things. After witnessing all of his attempts to point out how Einstein’s general theory of relativity failed to make the sense it claimed to make (and still fails to do so, but the model centrists won’t permit empirical evidence to get in the way of their clever mathematics), he arguably decided that he needed to step back from epistemology and philosophy of science, to present a more logically primary argument, in the metaphysical form of his “philosophy of organism.” Whitehead centered his argument on what I and Randy Auxier named “the quantum of explanation,” a logical (rather than ontological) center, around which Whitehead constructed his subtle and complex system of making sense. It has been suggested that Whitehead’s magnum opus, Process and Reality, is one of the five most difficulty works in the Western philosophical canon. I’m not inclined to argue with such a sentiment, since the most that could be credibly argued is that it might be knocked back to sixth place. For my part, I’m not sure what work could manage that feat.No Sense

One of the points that Randy and I tried to emphasize was that the process of “making sense” was itself a rather complex process, in which the most active word in the proceeding is process: this is not an object you hold, but an activity you engage in. So despite my habitual focus upon contemporary science &/or concerns, this is actually as classic an issue as you can find in the Western philosophical canon. (And I just don’t have the expertise to speak with even casual ignorance about the Eastern canon, a source of inestimable insight and subtlety. I am, however, inclined – ignorant as I am – to suspect that what I have to say here can find its analogs in that tradition.) Continue reading →

The World is a Circle

15 Wednesday Feb 2017

Posted by in Authoritarians, Critical Thinking, Logic

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The title is an ironic gesture to a disturbingly cheerful (some, like me, might say saccharin) tune by Bacharach and David, but my intention is to talk about what is less happily categorized as circular reasoning. This is one of those fallacies that has been recognized for so long that the medievals gave it a Latin name: petitio principii. It is also one of those painful failures of basic reasoning that goes beyond the narrow confines of formal logic, or introductory critical thinking classes. This is one of those monsters of bad thinking that empower authoritarian minded individuals and their enablers to unshamefacedly spout about “alternative facts” and other infantile drivel. You see, the problem with a circle, as well as with a mind that reasons in one, is that the circle is closed; inquiry, on the other hand, is (by necessity) open and ongoing.disturbing-1

I’ve talked before (several times, in fact) about what Altemeyer describes as the “compartmentalization” that occurs in authoritarian belief and ideology. One can scarcely dignify this latter as “thinking,” regardless of the degree of sophisticated cleverness employed in maintaining those compartments as air tight against all facts and logic. Authoritarian thinkers, following Hamlet’s example, keep their minds, bounded in a nutshell and count themselves kings of infinite space, were it not that they have bad dreams. (Of course, Hamlet was being ironic, and mocking his interlocutors, something the Mango Mussolini’s enthusiasts entirely fail to grasp.) The thing is, these people choose to be bounded by a nutshell, all the while imagining themselves in princely command of infinite space. Meanwhile, their bad dreams (which are the trailings of reality, dogging them despite their dogmatism) are the sources of their willing embrace of Trumpian neo-fascism. Because the nutshell – the “nut house” – in which they have bound their minds is a tightly enclosed circle that permits no entry from reality. Continue reading →