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diff --git a/build-n-publish.sh b/build-n-publish.sh index 69a8bf0..fabbd29 100755 --- a/build-n-publish.sh +++ b/build-n-publish.sh @@ -54,4 +54,3 @@ echo 'Commit message: ' read message git commit -m "$message" git push -v origin refs/heads/master\:refs/heads/master -# TODO: see if committing and pushing interactively here makes sense. diff --git a/org/An Unpleasant Surprise/An Unpleasant Surprise.org b/org/An Unpleasant Surprise/An Unpleasant Surprise.org new file mode 100644 index 0000000..86876a8 --- /dev/null +++ b/org/An Unpleasant Surprise/An Unpleasant Surprise.org @@ -0,0 +1,138 @@ +#+TITLE:An Unpleasant Surprise +#+DATE: <2024-02-01 Thu 7:27> +#+TAGS: Libertarianism, Philosophy, Philosophy of Mathematics +#+OPTIONS: tex:dvipng + +I'm a libertarian, interested primarily in mathematics, physics, computer science, and economics. I have a recently-acquired pair of degrees in the first two, feel better-read in their foundations than most of my peers, and readily admit the existence of major foundational problems. I myself believe the academy to be irreparably corrupted, which is why I abandoned my former plans of a physics Ph.D. to work in industry for a while, and was going to attempt to lay out a model for meeting the demand for knowledge without force. + +So, that explains my excitement on hearing Steve Patterson's planning to do just that on Tom Woods' show today. I followed the links to his socials, interested to jump on board with this new movement. However, what is offered might well worse than the disease: after reading his quarrels with prevalent philosophy of mathematics, I find a downright muddy thinker that hasn't even bothered to consider the beliefs he attacks on their proponents' terms. I find someone supporting a logical view of mathematical foundations who apparently hasn't even bothered to learn what "logic" or "a contradiction" /is/ except presumably through the leaded-glass distortions of a novelist-wannabe-thinker. + +This has /severe/ consequences for what he is attempting to build: by having a downright terrible understanding of the objects of his critiques, he will be eternally shadow-boxing, ignoring the actual challenges that lead to terrible predictions and sordid manipulations that have come to the political forefront recently. + +Humorously, Patterson brings up Gell-Mann amnesia. For me to trust that his critiques of other disciplines are accurate would be nothing but a further instance of that phenomenon he criticizes. A line-by-line refutation of his article [[https://steve-patterson.com/infinite-things-do-not-exist/]["Infinite Things Do Not Exist"]] will illustrate what I mean (only the parts with some substance included). + +#+begin_quote + +First of all, we have to define our terms. "Infinity" or "infinite" means "without end," "never-completed," or "without boundaries." + +#+end_quote + +Indeed, first of all we do have to define our terms. If we are trying to refute foundations of mathematics, however, we should /actually use the definitions from foundations of mathematics/, so our arguments actually have teeth. + +Infinity, in foundations of mathematics, only applies to sets, or unordered collections. These are the only "things" which comprise the ontology of mathematics; the rest of the taxonomy of mathematical objects are built from or coaxed out of them. A refutation of the general metaphysical concept of infinity, as he has attempted to construct, therefore bears no necessary relation to the term-of-art "infinite" in the context of this ontology. + +Indeed, the mathematical definition is actually meaningfully different: they claim $$\exists X [\exists e (\forall z \lnot (z \in e) \land e \in X) \land \forall y(y \in X \to y \cup \{y\} \in X)]$$ (Zermelo-Fraenkel axiom of infinity, cf. Wikipedia). I.e., there exists a set containing the empty set and all sets buildable by "iterated nesting" starting with the empty set, i.e. there exists a set containing all von Neumann ordinal numbers. + +The claim "$Y$ is infinite" means only that there exists a surjective map from $Y$ to $X$, i.e. one can formally define a correspondence between elements of $Y$ and elements of $X$ that doesn't "miss" any elements of $X$. There is no concept of "without end," "never-completed," or "without boundaries" here without adding more exogeneous structure that allows one to talk about "end," "completion," or "boundary." + +#+begin_quote + +An infinite distance can never be covered—by definition of what we mean by "infinite." There is no "end" to an infinite series—if the series ends, it's finite by definition + +#+end_quote + +Indeed, these are all true statements, but not sound statements—they do not follow from either the definition of infinity he's presented, nor from the definition of infinity he should be refuting. They require a definition of the concept of "distance" in such a way that infinitude applies to it, a corresponding definition of what it means to "cover" something, a definition of "end," a definition of "infinite series," and the introduction of the (comparatively less leap-laden) notation "finite" for the first-order formula "is not infinite." + +#+begin_quote + +Consider the question, "How many positive integers are there?" + +Most people intuitively answer, "There are infinitely many positive integers." Meaning, there isn't some upper-limit on the size of numbers. You can't think of a number that "1" cannot be added to. This conveys the general concept of "infinite." + +#+end_quote + +I don't know how what "most people intuitively answer" in any way influences the precise sense of the term used by experts, any more than what most people mean by "value" or "prefer" influences how Austrian economists are permitted to use it in a technical sense. Certainly, what he argue against cannot be the technical concept if he defines terms as above. + +#+begin_quote + +By the term "actual," I mean "fully-realized," "completed," or "totally encapsulated." + +#+end_quote + +I don't begrudge anyone their definitions, but this is literally just assuming his conclusion, and is totally divorced from any use of the term "actual object" by any person anywhere in history. Additionally, the words he equates "actual" to are inquivalent and have no obvious commonality that I can extract and assign to the new referent he's assigning to the word. By its standard, no emergent objects exist—a waterfall is a fundamentally dynamic entity (its constituent parts are continually changing), and so the waterfall is never fully-realized or completed, because its essential nature is one of change. It is, however, totally (spatially) encapsulated, which illustrates the differences in the "synonyms" he attempts to use here. + +#+begin_quote + +And here we find the elementary error in the conception of an "actual infinite". (sic) I realize my refutation would appear impressive and profound were it complex. + +#+end_quote + +It would appear impressive and profound were it: + +1. actually addressing the belief he claims to refute, +2. using "actual" in any sense other than one he made up for the sole purpose of being able to refute the nonexistent belief he's outlined, +3. actually to contain a valid instantiation of the concept of reasoning it is asserted to employ. + +#+begin_quote + +If it were some difficult, abstract chain of reasoning disproving a century of mathematical thinking—that would surely impress people. But alas, the refutation is not complex. It's outrageously simple. So simple, it is anti-climactic. + +What is never-completed is never completed. + +#+end_quote + +This is simply a false implicit claim about what counts in mathematical or philosophical practice. Look at the corresponding sections on [[https://vixra.org][ViXrA]], and you'll see any number of difficult, abstract chains of reasoning that nevertheless are considered so crankish and obviously errant as to get their authors booted off the extremely lightly moderated [[https://arxiv.org][ArXiV]] from which the service has forked. Difficult, abstract chains of reasoning are incidental instruments to the objectives of the disciplines, not their raisôn d'etré. In fact, simple, crisp, elegant (correct) arguments are considered the crown jewel of these domains. + +Moreover, any time you have any philosophical argument on an issue where there exists substantial disagreement, the barest breath of intellectual humility should cause one to be suspect when it is "too cheap." This arrogance seems to be an all-too-common common vice of the liberty-oriented; I've seen it happen over and over again with midwits on the internet thinking Hoppe's argumentation ethics provides unassailable proof of libertarian ethics despite their not adopting any of the epistemological premises that are essential to the reasoning. + +After this point, he engages in even worse sophistry. I've refuted some highlights. + +#+begin_quote + +What is the curvature of a circle with an infinite radius? + +#+end_quote + +Never in this entire section does he define what "curvature" or "circle" mean, and never gives any reason why the two must necessarily connected. That he doesn't even take the time to consider his implicit priors here is simply baffling to me, and it communicates no actual on-the-ground engagement with mathematics or logic. + +#+begin_quote + +Believe it or not, some mathematicians will say "That is not a contradiction! This just shows the incredible nature of infinities! Paradoxes exist, you've just proved it!" + +#+end_quote + +He gives nothing approximating an argument, much less a proof. The extent of his reasoning is "try to imagine," and then giving a description of what he imagines. There's nothing presented, at all, that would constrain what the reader has imagined. Because, again, he doesn't define what is actually meant by "circle." + +This example has very little to do with any "incredible" nature of infinities[fn:2], and everything to do with the definition of the figures in question, and the limiting process involved. I would advise reading the most minute amount on projective geometry to clarify this example. + +#+begin_quote + +Though calculus can easily be rescued from logical contradictions, set theory cannot. The set theoreticians are absolutely explicit: according to them, some infinities can be fully completed; they have an actual size. + +#+end_quote + +This paragraph almost had me throwing my phone. Not even actual finitists (who do exist, albeit in small numbers, in academia across the globe) are this dishonest about the premises they dislike (they're mostly imanent realists). + +First of all, the axiom of infinity is one you can easily drop, and do finite set theory. This is "rescuing" set theory from "logical contradictions," and I'm sure it's much easier than whatever tortured interpretation of calculus he's come up with. Second of all, he's clearly had no engagement with what set theoreticians are /actually saying/: they have a specific definition of "size" called "cardinality," and confusing it for other characterizations of size that coincide on fintie sets are where most "paradoxes" about infinite things arise. If he had actually engaged with set theory, he would both: + +1. know that cardinality has a specific meaning distinct from colloquial size, and +2. that it is defined literally just by taking a representative of an equivalence class under bijection, i.e. the "size" of a countably infinite set is /by definition/ just the natural number set (just like the "size" of a set of 3 objects is the number 3, which is by definition a particular set with 3 objects). This does not even implicitly or abstractly carry with it any notion of "completing" an infinite process. + +#+begin_quote + +Now re-examine the concept of infinity. "Infinite" means "never-ending," "incomplete," "always-bigger-than." + +But "always-bigger-than"is another way of saying "Not merely A. /More than A/." + +In other words, the very term "infinite" is an /explicit denial of identity/ + +Therefore, an "infinite thing" is "a thing which is itself, /and more-than-itself at the same time/." An outright contradiction. + +#+end_quote + +This passage is where I stopped reading, as it became clear that he /has not even had any interaction with logic, philosophical or otherwise/. Because this simply is not what "contradiction" means. + +First off, he's done the thing again where he's stated three inequivalent words as if they were synonyms in order to define a concept in a way that is entirely private so that he can delude himself into believing he's made an argument. An infinite set is not (strictly) bigger than itself. In the case of the integers, its /elements/ always have larger elements. These are completely different orders. These are different classes of objects. I can't possibly imagine how anyone could make this most basic of object-type errors and expect to be taken seriously in their opinions on philosophy of mathematics, much less found an /institute/ on the basis of said errors. An infinite set, indeed, is what it is. It simply /is not/ more than itself; no one since Cantor other than he has ever attempted to abuse concepts to assert so. + +Moreover, the final nail in the coffin, in order to assert that something is a contradiction, he must demonstrate that the premises entail both $A$ and $\lnot A$. He does not even attempt to give an argument why "is" and "more than" are such that "thing" and "more than thing" are necesarily different. He merely /asserts/ it. This is the opposite of an argument. + +The root of his misunderstanding appears to be the mistaken belief that the "things mathematics are about" are in some way procedural in nature, and that when a mathematician writes down something that would seem to encode a nonterminating process, his only option for reasoning about that object is to somehow complete the nonterminating process. In fact, he can just...use the properties that define the object he has notated, without ever actually completing a supertask. To prove that every natural number is either odd or even, I need not check every natural number. Nevertheless, my deductions will hold for all (infinitely many) natural numbers. Mathematics is not computation, and computation is not mathematics. Specifications and algorithms are distinct. + +I would expect anyone to actually take the time to /learn/ something about the things he criticizes before going out and trying to solicit people's email addresses and (presumably) money. His ends are too noble for him to tarnish the entire concept of stateless research by abject ignorance about the subjects he purports to improve. He doesn't even have to believe in infinite sets; there are a lot of finitists who do great work and whom I respect. But those finitists necessarily know the first thing about logic, because they actually perform mathematics, and as such, don't attempt to make the claim that infinite sets are logically contradictory. They simply claim that they don't exist for non-logical reasons, and therefore infinite mathematics is valid deduction from false premises, i.e. it isn't "about" anything useful. If infinite sets were /actually/ logically contradictory, especially via a 4-step deduction, software capable of automated proof generation would have found the contradictory sentences 50 years ago. + +His general point stands, though. People writing introductory statistics textbooks for life and social scientists have completely /bastardized/ the (mostly fine, in my opinion) foundations of statistics. The epistemological vices that flow from the "null ritual" (see Gerd Gigernzer's work) cause an eternal 95%-confidence lake of fire in which the non-physical sciences have been locked for a century. He is, however, too bold in extrapolating this observation. It's not a flaw native to higher levels of the Comte's hierarchy and gets passed down that causes this—it's a genuinely original mutation, adapted to these disciplines' students' mathematical and philosophical ineptitude, had I to guess. [fn:1] Hopefully, Patterson can do some self-reflection, improve his methodology, and produce a system that produces genuine knowledge and can actually out-compete the corrupt, statist hellhole we're all running from. + +* Footnotes +[fn:2] I find their nature rather mundane, actually, and consider the popular mathematics communicators that try to make them seem "mysterious" to be downright enemies of the people—it manufactures confusion such as his. + +[fn:1] This isn't to say that those higher levels don't have problems, of course. The foundations of mathematics work wonderfully, but are unwieldly to use in practice. Ideally, we'd work in proof assistants, so that papers would contain unassailably correct deductions (contingent on the correctness of the verifier's software stack); doing this in a Hilbert-style deductive system and brute ZFC would be agony. Casting mathematical foundations in more ergonomic, computable terms like homotopy type theory is an active area of research. Various philosophical interpretations of physics have severe problems—heck, the standard model, the best-confirmed prediction about the external world mankind has ever made /must be wrong/ (it implies that some interactions at higher energy have infinite yield). We simply have neither the experiments nor sufficiently sophisticated mathematics (specifically: a set-theoretic model for the path-integral) to fix it yet. However, I have seen absolutely no evidence that these deficiencies cause problems for the lower, non-social sciences. 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A basic contemplation on cases suggests an omission: what might have direction but no magnitude? + +They're called infinitesimals. And mathematicians have been pretending to not believe in them for 2000 years. + +* Nilsquares, Rigid Rods, and Microstraightness (Macrogayness?) + +Calculus, fundamentally, is about continuous change. And the very nature of "continuous" itself has always been turgid: only the hallucinogenic stimulant ZFC has been able to, very recently, convince people that continua consist of individual points. + +Arguably, the seeds of calculus lie in the work of the ancient Greek mathematicians Archimedes, Antiphon, and Bryson. They, following Aristotle's deflation of Zeno's various paradoxes of motion, viewed continua as successions of overlapping instants, each of which was /not/ a dimensionless quantity but retained the central /directionality/ of the continuum as a whole. Archimedes routinely viewed volumes as sums of (lots of) /indivisibles/, plane-like sections, in order to derive expressions for those volumes in terms of linear dimensions. The other two viewed curves as sums of (lots of) discrete, short segments0—rigid rods just long enough to have a slope but no appreciable length. + +Newton viewed time-dependent quantities (fluents) as having rates of change (fluxions) at each instant computable by evaluating the quantity perturbed by an infinitely small quantity and ignoring "doubly infinitely small" powers of that quantity in the result. Leibniz similarly employed infinitesimally small quantities, but in a more geometric context TODO: Leibniz's visual proof of the FTC + + Euler's textbook on calculus, which is remarkably similar in order and content to modern presentations, has the following "proof" of Newton's nilsquare property of $dx$, quite humorous to the modern eye [fn:2]: algebra tells us that if we can divide by zero then $\frac{0}{0} = a$ for any $a ∈ \mathbb{R}$. If we let $dx = 0$, then this rearranges to $a \cdot dx = 0$; he concieved of this as two different ways of writing zero save for the fact one is permitted to apply all usual rules of algebra to these terms, and so $\frac{dx}{dx} = 1$. This allowed him to say + $$1 = 1 + 0 = 1 + dx = \frac{dx + (dx)^2}{dx} \Leftrightarrow dx = dx + (dx)^2 \Leftrightarrow (dx)^2 = 0.$$ + + Weirstrass's limit approach to analysis, and the set-theoretic constructions of the continuum courtesy Cauchy and Dedekind, claim to save us from the absurdities apparent here (some of which Berkeley had pointed out contemporary to Newton). However, these approaches still pervade pedagogy and application. Anyone taking calculus will learn the "washer method," useful in computing volumes. Every calculus book has a picture of a little tangent line segment, with $dx$ its $x$ component and $dy$ its $y$ component. Everyone thinks in infinitesimals before limits. And anyone with knowledge of physics, even from memes, will know that infinitesimal derivations are the rule rather than the exception! A course in classical mechanics wouldn't be complete unless the professor a) doesn't explain what a variation is, b) assumes you know properties like $(\delta q)^2 = 0$ and the multivariable chain rule for variations anyway, and c) invokes the fundamental lemma of the calculus of variations as if it's the most unobjectionable of axioms. In statistical mechanics, the basic principles relating state variables are still stated in infinitesimal form. And for engineers, limits have long since exited the mind; it's $dx$ (secretly, $\Delta x$) all the way down . + + What we observe is a vast rift between theory and practice, between formalism and intuition. And that should strike any mathematician as an opportunity. + +* Abstract Nonsense to the Rescue + +Two approaches evolved, roughly coeval, to close this gap. The first is a logical method, which views infinitesimals as reciprocals of transfinite cardinals in a field of "hyperreal" numbers; it is provably syntax sugar for limits [fn:3]. It requires such terrifying phrases as "nonstandard models of Peano arithmetic" and "ultrafilter" to understand and apply, but successfully resurrects infinitesimals, and was used to prove some longstanding problems in analysis [fn:4]. The second fell from the clouds of category theory, is much more convenient to apply, has nice geometric semantics, explains why classical infinitesimal theories ran into problems, and also solves an analogous formalism-intuition gap in differential geometry. + +Smooth infinitesimal analysis, this new hotness is named. And the rules are straightforward to state [fn:5]: + +1. Reasoning about continua is constructive. +2. We have a continuum $R$ forming a field (the elements are thought of as (possibly infinitesimal) segments on the continuum). +3. There is a strict order $<$ on elements of $R$ that is compatible with $+$ and $\cdot$; all elements of $R$ are strictly between 0 and 1. +4. Square roots are computable. +5. (Microstraightness) For any function $C$ and any point $P$ on it, there is a (small) nondegenerate segment of $C$—a microsegment—around $P$ which is straight, that is, $C$ is /microstraight/ around $P$. +6. (Integration) For any + + + +* Footnotes + +[fn:5] Extracted from John Bell's /Primer of Smooth Infinitesimal analysis/. +[fn:4] Bernstein and Robinson 1966 made a ginormous leap on the invariant subspace problem, proving it for the case of polynomially compact operators. +[fn:3] Robinson 1960. +[fn:2] This was introduced to me by Michael Penn, who cites A. Ferzola 1994. +[fn:1] Not a quote; I made this up. But he basically said that. diff --git a/staging/building/building.org b/staging/building/building.org new file mode 100644 index 0000000..d7fff91 --- /dev/null +++ b/staging/building/building.org @@ -0,0 +1,63 @@ +#+TITLE: The 4 Ways to Build Things +#+DATE: <2023-11-30 Thu 14:35> +#+TAGS: Engineering, Construction, Philosophy + +Construction is the combination of stable primitive material objects into more complicated and interesting stable material objects. This should be uncontroversial enough. However, what is surprising is that there seems to be precisely 4 kinds of combination it is possible to use to combine them: blocking, sticking, bonding, joining, fastening, and resting. Furthermore, it is possible to define these concepts rigorously, and prove that they are the only possible means, subject to a relatively unobjectionable axiomatization of what "material object" means. + +* The Setup + +We assume that any given solid object can be effectively modeled by a collection of classical particles, each constrained to be no more than some specific distance $\epsilon$ from its neighbor (but possibly more constrained as well). This can be viewed as a graph or network, where the vertices represent particles and the edges are lines between each pair of constrained particles. These edges are, of course, of length less than $\epsilon$. Furthermore, we assume that these objects are solid, i.e. that the particles from two different objects are repulsive at short ranges and dense enough that the interiors of the objects never intersect. In order for "interior" to make sense, we also assume that the constraints modeled as such a graph has edges that form a polytope; the interior of this polytope is the interior of the object. This is sufficiently general to reproduce qualitative features of phenomena like flexibility and elasticity, but not general enough to reproduce phenomena like thermal expansion, fractures, or absorption. Last, we assume that it is only the constraints above and external fields that can affect the motion of the particles in any situation. + +TODO: diagrams. + +This is a reasonably good approximation of homogeneous molecules interacting via bonds that don't break, as atoms are particle-like on the scale of ordinary constructions and are attracted by forces that are short-range repulsive, medium-range attractive, and long-range made irrelevant by kinetic energy (TODO: Lennard-Jones potential diagram). + +Given any such object moving under a free Lagrangian and its internal constraints (a system $L$), we want to investigate and classify the resulting systems $L'$ that result from making modifications to this scenario. Specifically, we claim that all $L'$ that constitute systems of this type (particle systems constrained by "bonds" and "solidity") can be realized by some combination of the following 5 particularly natural operations applied to $L$ (restricted): + +1. adding particles, +2. removing particles, +3. adding inter-particle constraints, +4. removing inter-particle constraints, and +5. introducing an external field that affects all the particles. + +Additionally, we can define a "distance" between two objects by the minimum number of operation applications it is necessary to perform on one of them so that some accessible state of the result is an accessible state of the other object. This raises interesting algorithmic questions about the easiest way to realize a particular object. + +We are also interested whether operations that more closely imitate practically possible construction methods share the universality property: it's rare that it's possible to add or remove particles or constraints in the interior of an object, so if the operations are restricted to particles on the surface of the object, can we still build everything? In a similarly practical vein, we explore the transformations thought of as the combination of separate objects into one or the separation of one object into multiple, and transformations that decrease or increase the number of degrees of freedom + + + + + +Since all (material) constructions consist of finitely many combinations, it suffices to restrict attention to a single combination of two such objects. To make "combination" more precise, we consider (WLOG) the first object having a free Lagrangian, save for its internal constraints, and then consider augmentations of this initial system that maintain the first object + +we meanperforming some sort of alteration to the case of free objects far away, such as bring them close together to touch, altering the internal geometry of the objects, or adding an external field into consideration, that reduces the number of degrees of freedom of the objects. + +By "sticking," we mean the introduction of new constraints of the same flavor between the exterior particles of two objects, without introducing additional particles. + +By "bonding," we mean the introduction of new constraints of the same flavor between the exterior particles of two objects, via the inclusion of additional mediator particles. + +By "joining," we mean using the geometry of two objects themselves to interlock them together, using their solidity to constrain them without the introduction of any third object. + +By "fastening," we mean introducing a third object whose geometry interlocks the two objects together using its solidity. + +By "resting," we mean exploiting an external field to hold the objects against each other. + +* The Exhaustiveness Argument. + +In the general case, the two objects are far away from each other. They move completely freely, at least locally; this preclueds any possibility of their interiors intersecting or of their particles being connected by chains of constitutive constraints. + +To impose additional restrictions, the objects must be close (in the sense that it is possible for infinitesimal motions of the particles consistent with the internal constraints to violate some of the constraints between the two bodies, i.e. either to break a relational constraint chain or to make their interiors intersect). + +This happens in four cases: either there are additional constraints between the particles of the objects or there are not, or there are additional particles introduced or there are not. + +By assumption, the only things that affect the motion of the particles are their internal constraints, their solidity, and external fields. + +* Practical Examples + +- Wood +- Metal +- Rope +- Fabric +- Glass +- Plastic +- Ceramics diff --git a/staging/building/ltximg/org-ltximg_503065b6e650ba4cb74c1cf694a4d537bee10c92.png b/staging/building/ltximg/org-ltximg_503065b6e650ba4cb74c1cf694a4d537bee10c92.png Binary files differnew file mode 100644 index 0000000..3afd4ee --- /dev/null +++ b/staging/building/ltximg/org-ltximg_503065b6e650ba4cb74c1cf694a4d537bee10c92.png diff --git a/staging/hypercomputer/hypercomputer.org b/staging/hypercomputer/hypercomputer.org new file mode 100644 index 0000000..0597535 --- /dev/null +++ b/staging/hypercomputer/hypercomputer.org @@ -0,0 +1,9 @@ +#+TITLE: CSRNGs Make Our CPUs Hypercomputers +#+DATE: <2023-11-09 Thu 17:24> +#+TAGS: Computer Science, Church, Turing, Computability + +* People Consistently Misinterpret The Church-Turing Thesis + +There's an excellent [[https://plato.stanford.edu/entries/church-turing/][SEP article]] on what Turing's original claim was, and how rampant and severe inaccurate statements of it are. The gist: he claims that any procedure that could be carried out, in principle, by a human computer with pencil and paper could be carried out by the namesake machine. /Not/, as many people claim, that any procedure any machine could compute could also be computed by a Turing machine. + +The distance between these theses contains machines that exploit physical processes that produce non-computable results, such as a digit stream of [[https://mathworld.wolfram.com/ChaitinsConstant.html][Chaitin's constant]] or solving the [[https://en.wikipedia.org/wiki/Word_problem_for_groups][word problem for groups]]. Most existing proposals are pretty outlandish. diff --git a/staging/modularity/modularity.org b/staging/modularity/modularity.org new file mode 100644 index 0000000..1648260 --- /dev/null +++ b/staging/modularity/modularity.org @@ -0,0 +1,30 @@ +#+TITLE: Modularity is Sheafy +#+DATE: <2023-11-22 Wed 11:39> +#+TAGS: Software, Philosophy, Engineering + +In my article Software Equals Hardware, I argue that electrical engineers and computer architects have solved the problem of modularity/code-at-scale. However, the discussion of precisely what "modular" means there was, as usual, underspecified to point of error. + +On reflection, I feel like a system is modular if it promotes, enhances, or forces things such that some "local" (or small-scale) properties are sufficient to ensure some "global" (or large-scale properties). For example, writing code in a conventional module system carries a set of local restrictions (all code is declared in a module, public functionality in a module needs to be exported, library code needs to be imported, etc.) that together promote a desirable property of the whole codebase (eliminated or much decreased risk of unintentional name collisions). So also for avoiding mutable global state (can't write or alter global variables, but now you can make assumptions about linking), for lexical scope (TODO), and so on. As this looks awful similar to certain structural patterns in mathematics, it led to the formalization: + +Modular systems are sheafified. + +* The Abstract Nonsense + +Of course, unless you're an algebraic geometer, algebraic topologist, or category theorist, this formalization isn't terribly intuitive (and if you are, it isn't terribly formal). + +To those schools of wizards is known the concept of a "presheaf:" intuitively, a structure that has some notion of "local" and "global." Obviously, our programs have such a notion; computer scientists tend to call it "scope." By linguistic inference, these wizards also talk about "sheaf;" this means a presheaf that has some way to coherently glue together local structure into global structure. In computer science, we can identify this with composition (which is universally recognized as desirable, even essential to modularity). However, for a given presheaf, there are generally many different sheaves over it (many different ways to compose programs that agree with a given notion of scope). + +The wizards save us with a delightfully-named ritual: /sheafification/. There is a way to, for every presheaf on a suitable structure (called a "site"), produce the corresponding sheaf that is, in some sense, the most general (is "left-adjoint to the inclusion functor of sheaves into presheaves," whatever that means). Therefore, if we design programs such that our composition agrees with the most general composition our notion of scope induces, we can expect to get the most general programs. + +Note: the sheafification still allows for freedom in two dimensions: the choice of what the underlying structure is, and the choice of what locality means. We must appeal to additional philosophical and practical concerns to get a unique definition of "modular," but it cannot be overstated how valuable the restriction of the debate to those two choices is. + +* Its Application + +It would be really nice if we could use this definition to actually /discover/ modular abstractions, or to /prove/ that a given abstraction is maximally modular. The former I'll leave to the reader; to bolster this definition, here are several cases in which the received wisdom on modular design aligns with its predictions. + +** Module Systems + +The problem here is: we've written a bunch of code libraries, and we want to have a way to call functions from the libraries + +** Scoping Rules +** diff --git a/staging/reverent/reverent.org b/staging/reverent/reverent.org new file mode 100644 index 0000000..6fb3001 --- /dev/null +++ b/staging/reverent/reverent.org @@ -0,0 +1,63 @@ +#+TITLE: Reverent Constructions +#+DATE: <2023-12-02 Sat 04:23> +#+TAGS: Aesthetics, Art, Construction, Philosophy + +It seems like everyone hates the way things are. + +Material things, that is (though Lord knows it's otherwise true); common attitudes identify a chronic decline in their quality, both for function and form. Things are, the story goes, not designed to be /durable/ or /fixable/ anymore, but designed to break and need authorized dealer service, just to milk every last cent from you. They're designed to lock you in and lock you out, to prevent interoperability and ecosystem exit, to drive mindless consumption and enrich their unscrupulous inventors. They're also not designed to be /beautiful/ or /interesting/. All art, so it goes, is headed the way of (post)modernist absurdist brutalist pragmatism (choose your least favorite couple adjectives). Everything that wasn't made /back then/ is derivative, overly academic, or downright offensive. + +Regardless of the truth of this narrative, from the things it despises we may extract an interesting question: what unites the things they /value?/ + +As in, what are the features in a material object that would satisfy its believers? It's certainly /not/ a mere revivalist impulse: at least some of its supporters will point to historical objects /across/ historical cultures and movements, and its supporters are certainly culturally diverse. Nor is it nostalgia: most champion objects were made centuries before those doing the championing. + +I believe they want reverent constructions. + +* The Basic Idea + +Reverent constructed objects are designed to draw attention to the divine. What the divine is, of course, varies across cultures and individuals. I argue for my own perspective, left open-ended enough that I hope it becomes fairly general. I believe the divine is expressed: + +- in mathematics, through classifications, uniqueness results, canonical forms, universal properties, the infinite(simal), and the idea of zero; +- in science, through universal structure as derived from modern cosmology, fundamental nondeterminism, complex emergent dynamic phenomena, emergent order, remarkable material properties, the anticipatory nature of living systems, and the ubiquitous unseen; +- in history and anthropology, through ancient orthography, scripts, symbolism, and iconography; +- in philosophy, through consistency, subsidiarity and hierarchy, adherance to principle, voluntary exchange, and private property; +- in aesthetic principles themselves, through fitness for purpose (with particularly general functional properties valorized), the nature of things (in the medieval sense of the processes and principles that sustain existence), the heavens (i.e. upward directions), and coherence. + +Parts of objects are classified as /instrumental/ and /final/ according to how sensitive the function of the object is to their modification. + +* Deduced Principles +** Function +These functional properties are prized: + +- modularity, +- repairability, +- auditability and ease of access to functional components, +- durability, +- configurability, +- efficient use of space, +- resiliance to disaster, +- bootstrappability, +- safety. + +In each case, an object is monotonically better improved by improving each variable, /ceteris paribus/. + +** Shape +While there's no specific prejudice against straight lines, they're best used to emphasize symmetries of more interesting objects. Generally, shapes that are of interest will be accompanied by some mathematical characterization of interest, e.g. an algebraic variety, a catenary, or a transcendental function. Excellence in craft can be judged by the degree of coherence between the physical curve and the abstract curve. Constructions should draw the eye upwards to an apex or beyond, emphasizing the heavens. +** Color +Mirroring cosmology, the instrumental parts that support and background the object's function should be muted, dark, and cool. The final parts that are of primary interest and see the most activity can be more lively, lighter, and warmer. To accomplish coherence among means, colors should generally be solid, with patterns only in light contrast and geometrically interesting. However, objects that are materially distinct and are near or touching by accident or for practical concerns are better in different colors than otherwise. +** Texture and Finish +For natural materials, +** Material +Coherence of means implies that, ideally, only analogous materials are mixed. Wood should go with wood, stone with stone, metal with metal, plastic with plastic, etc. All else equal, materials of more similar relevent aesthetic qualities should be used together than otherwise. Using more natural materials for objects that help sustain existence most directly creates a pleasing analogy. +** Mode of Assembly +** Embellishment +Can be extremely intricate, just not distracting. +** Scripts +** Fonts +* Some Concrete Ideas +- Windows that are cloud chambers, making the unseen cosmic rays that surround us visible. +- Dark-colored bookshelves with brightly-colored, mismatched book bindings. +- Senary numerals +- Woods like cherry and walnut for structural purposes; white oak for highlights +- Equally well, things can have a predefined place or the order can be emergent. +- +* External Criticism diff --git a/staging/software=hardware/software=hardware.org b/staging/software=hardware/software=hardware.org new file mode 100644 index 0000000..ef75cfa --- /dev/null +++ b/staging/software=hardware/software=hardware.org @@ -0,0 +1,58 @@ + +#+TITLE: Software Equals Hardware +#+DATE: <2023-11-19 Sun 06:36> +#+TAGS: Software, Computer Architecture, Electronics, Philosophy, Engineering + +80% of the debate about software engineering best-practice concerns the means to ensure so-called "modular" code. The other 20% is about a supposed dichotomy between developer comfort and program performance. + +I think the reason that we disagree about these concepts is that +1. there is a division between computer architects and computer programmers, and +2. those divided have decided they work on fundamentally different problems. + +* The Hard Problem of Reprogrammability + +Why do we need software at all? + +In the evenly-rotating economy, every algorithm would run on its own application-specific computer (not necessarily electrical; not necessarily digital). We, by contrast, have uncertainty about what algorithms we want to run. Ergo, we need reasonably generic computers that we can reconfigure to run whatever algorithms we decide to use. This is a Hard Problem™. + +Historically, computer architects have + +We have historically decided that all data and programs we want to run can be encoded as natural numbers, and noticed that natural number operations can be implemented by register machines, and noticed that register machines can be implemented by Boolean functions, and that Boolean functions can be implemented by electrical semiconductor circuits, to great success. + +The original motivation for this is Gödel proof numbering. + +This is why characters are encoded; this is why one-dimensional syntaxes are far more convenient. + +* The Asymptotic Illuminati +That there isn't a canonical way to implement a given algorithm poses problems for the theoretical analysis of procedures. The conventional solution is wanting for many practical purposes (it at least needs to be complemented by discussion of constants). +** Their Propaganda +Algorithmics in books. +** What They Don't Want You To Know +Algorithmics in practice. +** The Red-Pill Approach +Analyzing algorithms by dependently typing over the machines they execute on, comparing machines by algorithmic performance, etc + +* Hardware Mythologies +Computer architects have decided that programmers should be kept out of their turf. +** Behind the ABI Veil +Modern (RISC) computer architecture, especially the von Neumann bottleneck. +** Our Capricious Gods +*** The L Trinity +Cache locality, especially practical problems. +*** Ghosts of Dead Programs +Branch prediction +** Dead Cults +High-level language architectures and reconfigurable computing. +Noticing what von Neumann machines do well: extreme transistor density, and highly branching code. +** The Neo-Pagans +Modern accelerators +*** Seizing the Means of Pipelining +CISC cache-manipulation instructions +*** Holors Above All +Modern GPU acceleration + +** Dialectic Synthesis +Reconfigurable register machines (a von Neumann/Harvard processor and an FPGA, where the programmer can create new machine instructions that are implemented in the FPGA). + +* Software Mythologies +** The Messiah: GHC Optimization Directives |
