I was recently in a conversation on social media that β rather magically, I thought β did not instantly zig into infantile spew. It did, however, take the predicted zag into a stream of superficial (if rather more maturely presented) fatuous nonsense that gave me enough pause to suggest this post.
There is, especially among persons of an especially right-wing political leaning, an inclination to demand simple, clear, and absolute definitions with regard to all terms in general, but especially those with even a patina of political significance. Thus, with the histrionic attacks on that vanishingly small minority of persons who identify as βtransβ (a group that comprises <0.5% the last time I looked it up) the demand by persons of a neo-fascist inclination (and TERFβs; politics makes for strange bedfellows) the self-righteous demand to βdefineβ βwoman.β
(Funny how they never demand a definition of βman.β But then, it is only women whose rights they wish to strip away, after all.)
Back in Aristotleβs day there was yet some hope of justifying a faith in definitions as being foundational to rationality. That said, Diogenes going about Athens with a plucked chicken (a βfeatherles bipedβ) mocking Plato by shouting, βBehold! A man!β might have inspired a measure of humility. But these days, any such sophomoric dependence upon definitions is patently childish, if not downright infantile. Let us call such juvenile insistence the βDictionary Game.β
I am rereading rereading Robert Goldblattβs book, Topoi, though in many respects it seems like Iβm reading it for the first time. When there is enough time and space between myself and some volume or other, that experience of β(re)reading it for the first timeβ is not all that uncommon. It occurred not too long ago with E.P. Thompsonβs The Making of The English Working Class, and Durrellβs Alexandria Quartet. Those works both had in the neighborhood of forty years between the first and the second readings, so I feel less guilty at the sense of surprise and pleasure. With Topoi, my excuses are somewhat more thin, though I can still assert with considerable truth and honesty that thereβs been considerable intellectual development on my part since the first time I tackled the book. I mention this not just to make what amounts to little more than a peculiar Facebook post (some people share pictures of their meal, after all), but to set up a discussion of why a Whiteheadian should pay special attention to that area of abstract algebraic thinking known as Category Theory. Iβll first spend a few words talking about the book itself.
The word βtopoiβ is the plural form of βtopos,β which seems rather more elegant than saying βtoposes.β A topos is a category theoretic structure that is rich in a variety of βniceβ formal characteristics, the details of which Iβll spare you (as that would require an entire book on category theory to explain.) Now, a category (such as might take on the structural features that would further specify it to be a topos) is a mathematical constructions that turns away from βobjectsβ so-called to devote particular attention to functions, transformations, and operations without any special concern for the supposed βwhatβ that is being transformed or operated on. As such, category theory is arguably the purest form of algebraic thinking around. It is scarcely an accident that Leo Corryβs magnificent history of the development of abstract algebra, Modern Algebra and the Rise of Mathematical Structures, ends with the emergence of category theory.
Get your mind out of the gutter, it was nothing like that. I did a presentation at the Personalist Forum conference, which happened to take place fairly close to where I live. (Normally it is at Western Carolina University, but due to scheduling conflicts had to be moved.) This yearβs venue was at the American Institute of Philosophical and Cultural Thought, here in Southern Illinois. The topic is about learning the basic tools needed to genuinely follow Whiteheadβs thought. The title is Learning the βLanguage,β where βlanguageβ is very deliberately scare quoted.
This talk came hard on the heels (as opposed to βheals,β though that too is relevant in an ironic way) of a major surgery Iβd just been through. While complicated enough under the best of circumstances, my procedure proved to be especially difficult. By all estimates, I came through it with flying colors, but I was still quite punchy at the time I made my presentation. I mention this in the talk.
I do word stuff with my mouth.
That being said, it came off quite well. The subject is βclose to my heart,β as it were, and even working from nothing more than an outline I was able to present my case. As I say in the talk, my hope is and remains that the failings of the presentation and the presenter do not mask the fact that there is a legitimate issue and complaint involved in much of existing Whitehead scholarship. Below is the suggested reading list I handed out at the talk, which Iβve expanded a little for this blog post.
As a rule, I despise pictures of myself, and find videos simply unwatchable. I did finally watch this one, and it is less execrable than one might otherwise suppose.
Suggested Readings
Habit of thought:
Alfred North Whitehead, Principles of Mathematics (New York: Henry Holt, 1911.) Free for the download from Project Gutenberg, https://www.gutenberg.org/ebooks/41568
Philip J. Davis and Reuben Hersh, The Mathematical Experience (New York: Harper Paperbacks, 1999.) This book really cannot be praised enough, a book that everyone should read regardless of their interest in Whitehead.
Morris Kline, Mathematics for the Nonmathematician (Mineola: Dover Books, 1985.)
George Polya, Mathematics and Plausible Reasoning, vol. 1 & 2 (Princeton: Princeton University Press, 1990.)
Thomas Tymoczko (Editor), New Directions in the Philosophy of Mathematics (Princeton: Princeton University Press, 1998.) Part of the effort to understand mathematics as inquiry, rather than set theory done badly.
Hermann Weyl, Symmetry (Princeton: Princeton University Press, 1981.) One of those books that earns the label βclassic,β this introduces some of the essential characteristics of group theory without getting into a lot of mathematics.
History:
Edna Kramer, Nature and Growth of Modern Mathematics (Princeton: Princeton University Press, 1981.) For my money, hands down the best general history out there. So of course it is out of print, impossible to find, and insanely expensive.
Morris Kline, Mathematical Thought From Ancient to Modern Times, volumes 1 β 3 (Oxford: Oxford University Press, 1990.) Pretty good, and at least it can be had without mortgaging your first born child.
Leo Corry, Modern Algebra and the Rise of Mathematical Structures (Basel: BirkhΓ€user, 2003.) This is an outstanding book, delving into the origins and problems that led to the emergence of abstract algebra, from the 1820βs to the 1940βs. Whitehead is discussed, but not very closely. Still, the nature of abstract algebra is developed in its historical context to a degree not available anywhere else. By my standards, the book is on the pricey side, but still well worth the investment
Abstract Algebra:
There are plenty of good books out there. The trouble is that it is quite difficult to get your head wrapped around the topic w/o some kind of mentor (i.e., math professor) there to help you out. Keep in mind that math books are the hardest to copy edit, because the editor must be as good at math as the author (which never happens.) So youβll find yourself up against a wall where youβre wondering if you simply donβt understand whatβs being said, or if thereβs a typographical error in the text. I solved the problem by getting an MA at DePaul.
But if you want to give it a go on your own, most any intro book from Dover will do:
Nathan Jacobson, Basic Algebra, vol. I & II (Mineola: Dover Publishing, 2009.) These two volumes are exceptional for their comprehensiveness. I originally acquired these books as first edition hard covers, back when a hard cover cost a little less than a new car. I liked them well enough that when I discovered that Dover had them as eBooks I purchased them again so that Iβd have a copy on my kindle. Be warned, though: the βbasicβ in the title is a tad misleading. These are the books that convinced me I needed to return to graduate school to learn abstract algebra.
There is a certain group of scholars β Iβll name no names β which has taken on such a dominant position in Whitehead scholarship (at least, within the US), that one could arguably characterize their position as βhegemonic.β I have personally met a number of individuals associated with this group, whom Iβll simply call βThe Group,β and freely admit that they are, as individuals, fine, generous, and altogether excellent folks. My complaint here β and I will be complaining rather sharply β is not with any of them as particular persons, but rather with the hegemonic direction in which The Group has taken Whitehead scholarship. That direction is what I am calling βHappy-Fluffy-Touchy-Feely-God-Talkβ (HFTFGT from now on.)
This sentence appears in the last few pages of PR, which are separated from the rest of the volume as Part V. The language and argument of this final, very short βpartβ is fundamentally different from the preceding hundreds-plus pages of text, and this radical difference has led some to wonder just how genuinely integral an element of the rest of the discussion it truly is. In these final, very few pages, Whitehead allows himself to slip into more poetic language, most particularly with the above mentioned one sentence β which Iβll not quote. (If you know, you know, and if you donβt youβll recognize it instantly should you ever read PR to the end.) But members of The Group, and others sympathetic to their program, latched onto that one sentence and ran with it. They ran fast, long, and hard, and are still running. From this we get the HFTFGT of process theology.
And it has swallowed the scholarship whole. So much so that Whiteheadβs triptych of 1919 β 1922 (Enquiry into The Principles of Natural Knowledge, The Concept of Nature, and The Principle of Relativity with Applications), a revolutionary re-evaluation of the entire philosophy of nature, have largely vanished from the canon of Whiteheadβs works that are studied. (Let me reiterate that this is within the US. Chinese scholars, for example, recently celebrated the centennial of those works with no fewer than three separate conferences, one for each book.)
Even those works of Whiteheadβs that do receive some attention receive it only selectively. Thus part IV of PR, for example, is often skipped over and ignored with students sometimes being told to ignore it because it is βirrelevant.β One might, alternatively, point out that part IV is the beating heart of Whiteheadβs entire relational system, where he presents his mature mereotopology, his non-metrical theory of curvature (βflat lociβ), his subtle theory of physical connectedness and causality (βstrainsβ), his completed theory on the internalization of relatedness as the flipside to the theory of the externalization of relatedness found in part III, etc. But part IV also involves a lot of logical and mathematical thinking βstuff,β and so one can just skip over that because it doesnβt feed into HFTFGT. A more cautious reader might suspect that what this rather demonstrates is that it is HFTFGT that is flopping around looking for relevance. But such cautious readers are not being invited into the club, and their professors are not encouraging their students to adopt such cautious approaches.
It is partly as a result of this narrow and eminently disputable presentation of Whiteheadβs philosophy that many outside the field who might otherwise profit from engaging with Whiteheadβs ideas (especially persons in the sciences), explicitly reject the notion out of hand. Because, after all, Whitehead is βnothing more thanβ a lot of HFTFGT. And people βjust know this to be the caseβ because they are constantly and loudly reminded of this βfactβ by those experts who are only interested in HFTFGT.
(Of course, persons in the physical sciences tend to reject any suggestion of engaging in philosophy because it is, after all, philosophy. They often do this as they explicitly engage in philosophical discourse; and do so badly.)
Such a reductionist caricature of Whiteheadβs thought is, of course, the worst sort grotesquely fatuous twaddle imaginable. Let me repeat, Whitehead wholly re-imagines Nature in a relationally robust and holistic framework that is original, insightful, and logically rigorous. But consider in comparison what your grasp of Christianity might be were it the case that all you ever heard about it came from the neo-fascist Christian Dominionist fundamentalists. Your idea of Christ would look more like Adolf Hitler. (By the bye, in contrast to the neo-fascists, the advocates of HFTFGT promote a vastly more Christ-inspired vision of God and the gospels that is genuinely loving and caring for ALL of creation.) And so it becomes increasingly difficult to even suggest to people who are not already heavily, even exclusively, invested in HFTFGT to cast even a casual eye on Whiteheadβs work.
Which brings us to the matter of how a vine can kill a tree.
There is a method of killing a tree called βgirdling.β A tree grows out as well as up. But if something is tightly bound around the outside of the trunk (it is βgirdledβ) the tree can no longer grow outwards. And it is these outer portions that carry the nutrients up the trunk to the rest of the tree. So the effect is like a garrote.
A vine is capable of girdling a tree. There is no malevolence involved, no ill or predatory intent; but the effect is the same. This is what βThe Groupβ is doing, I would argue, to the larger tree of Whitehead scholarship. (One of the ironies here is that they themselves are being girdled by the neo-fascist Christian Dominionist fundamentalists, who deny that liberal β never mind process β theology even qualifies as Christianity, or as anything other than the work of the Devil, even though this form of βdevilryβ is demonstrably truer to the Gospels. But just try to find someone who is not already an expert in the field who is even aware of the existence of process theology.)
I donβt want the HFTHGT people to go away, but I would like to see a serious effort on their part to acknowledge that their project emerges from a vanishingly small corner of Whiteheadβs work. I donβt want to chop down the vine, but I would like the vine to stop strangling the tree. This would include exercising some genuine circumspection about what they attribute to Whitehead, as opposed to what they themselves rather freely speculate about, far beyond anything he β in his meticulous, mathematically rigorous and disciplined way β ever pretended to entertain.
Yes, I have been away from this blog for a long time. No, I am not going to talk about that.
Iβve been thinking a great deal about the connections (possible and otherwise) between various aspects of theoretical computer science, and reasoning in general and empirical science in particular. When I talk about βtheoretical computer scienceβ, I definitely do not mean applied problems such as the rendered graphics in an FPSRPG (and that shot most assuredly DID hit, you cheating bastards!) No, I mean the mathematical and logical puzzles associated with what it is possible to compute, in the absolute limit of possibility, and what (among that collection of puzzles) can be reasonably computed given the physical and temporal constraints of the universe.
Computability: What can or cannot be computed, period. For example, can you write a program that will test all other programs to see if they run. Absolutely not! Take the program itself, flip a few relations, and then feed that to itself and you will force it into an infinite loop that it cannot solve. Due to the logician Alonzo Church, this is known as the βHalting Problem.β One of the favorite ways of demonstrating a problem is unsolvable is by proving its solution would also solve the Halting Problem.
Getting to this place where I can write what is obvious.
It is not an accident that within a matter of a couple of days, this βSupremeβ court has viciously curtailed human rights while indefensibly expanding gun rights. But take a look at the picture below. Look at it carefully. This is not a person.
This is a variant on the AR-15 assault rifle. Infantile purists will delaminate if you say “AR” stands for “assault rifle.”
But if you are a woman, if you are non-binary, LGBTQIA+, it basically has more rights before this court than you do. In other words, before this court, you are not a person. Add BIPOC to the previous list, because unless you are white and male, if you go parading around with an assault weapon you are unlikely to be allowed to survive, never mind pass unharrassed, by our massively militarized Law Enforcement.
With the savagely ideological βSupremeβ Court prepared to erase the rights of actual human beings on on no other account than that they are women (and consequently, donβt count), it seemed like a timely moment to set down my other projects and cast an eye upon the subject of abortion. Now, Whitehead himself never addressed the topic, so no pretense can be made to declare what his thoughts on the subject might have been. We can say, however, that his personal conclusions, were he to have any, are not really relevant here, as we want to develop a view of the subject within the context of process metaphysics, and not any one scholars individual declaration. That being said, it must also be added that other ways of working out conclusions other than those offered here will also be possible within the stated domain.
First off, what is a βpersonβ? We should immediately drop any thought of conflating βpersonβ with βhuman being.β βProperly developedβ human beings seem clearly to be persons, but not all persons will be human beings, developed or otherwise. Non-terrestrial intelligences, for you science fiction enthusiasts, are clearly persons without being human. But many would argue that terrestrial non-human animals are also persons, deserving of our care and ethical considerations. These (humans) would be those variously involved in animal rights activism and concerns. It is a tricky subject that Iβll not pursue here, though I admit to being a little troubled by my failure to embrace vegetarianism. Iβm sure youβll have noticed by now that Iβve not tackled the previous scare-quoted qualifier βproperly developed.β I promise, we will get back to that.
But more needs to be said about βperson.β A person is an agent, and an agent is something capable of intentional activities, behaviors, and/or stances. There is a philosophical school known as βPersonalismβ that takes this as a metaphysical βprimitive,β which is to say, first premise. There is what we might call the βliteβ version, that argues persons are metaphysically primary because there can be no interpretation of the world without intentional agents actually interpreting the world. As stated, this position is very hard to dispute, since any attempt to do so cheats by presupposing an interpreter in the form of a βGodβs eye view on the world,β while pretending to be βobjective.β But that βGodβs eyeβ is an interpreter, an intentional agent. Then there is the βHeavyβ version of personalism that says everything is a person (in some sense.) An electron is βinterpretingβ itβs world via itβs electromagnetic field. This is a trickier position, but one that deserves serious treatment, regardless of oneβs final conclusions. But the subtleties are beyond the scope of this current essay (or pretty much any essay of only 1450 words.)
The guiding motto in the life of every natural philosopher should be, Seek simplicity and distrust it.β
β 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 β beyond excruciatingly vague and unhelpful statements like, βthis is really complicatedβ β 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 weβve 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 werenβt 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 βsimplyβ and demonstrably solvable.i However, saying that a problem is βsolvableβ β even in the pure and βsimpleβ sense (notice how I keep scare-quoting that word) β doesnβt 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.
Some sixty-one years ago, the American philosopher Willard Van Orman Quine wrote a famous essay, βOn Simple Theories of a Complex World.β Actually, referring to this as a βfamous essayβ 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. βGod willed itβ 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 Quineβs concerns about how to characterize complexity and simplicity. Iβll 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.
I donβt 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 βidea,β rather than βconcept,β 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 βoutside the bounds,β the ancient Greek word for the infinite is βapeironβ (αΌΟΡιΟΞΏΞ½), which translates as βunlimitedβ β the βaβ being the negation (βunβ) and βpeironβ 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) β for example, the length of the perimeter of a geometrical figure β 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.)
THE QUANTUM of EXPLANATION 