Steve Keen’s Debunking Economics blames debt, not automation.
Essentially, many people currently feel that they are deep in debt, and work to get out of debt. Keen has a ODE model of the macroeconomy that shows various behaviors, including debt-driven crashes.
Felix Martin’s Money goes further and argues that strong anti-inflation stances by central bank regulators strengthen the hold of creditors over debtors, which has made these recent crashes bigger and more painful.
The statements, though contradictory, refer to two different thought experiments.
The two comments, though contradictory, refer to two different thought experiments.
Is it reasonable to take this as evidence that we shouldn’t use expected utility computations, or not only expected utility computations, to guide our decisions?
If I understand the context, the reason we believed an entity, either a human or an AI, ought to use expected utility as a practical decision making strategy, is because it would yield good results (a simple, general architecture for decision making). If there are fully general attacks (muggings) on all entities that use expected utility as a practical decision making strategy, then perhaps we should revise the original hypothesis.
Utility as a theoretical construct is charming, but it does have to pay its way, just like anything else.
P.S. I think the reasoning from “bounded rationality exists” to “non-Bayesian mind changes exist” is good stuff. Perhaps we could call this “on seeing this, I become willing to revise my model” phenomenon something like “surprise”, and distinguish it from merely new information.
Magic Haskeller and Augustsson’s Djinn are provers (or to say it another way, comprehensible as provers, or to say it another way, isomorphic to provers). They attempt to prove the proposition, and if they succeed they output the term corresponding (via the Curry-Howard Isomorphism) to the proof.
I believe they cannot output a term t :: a->b because there is no such term, because ‘anything implies anything else’ is false.
The type constructors that you’re thinking of are Arrow and Int. Forall is another type constructor, for constructing generic polymorphic types. Some types such as “Forall A, Forall B, A → B” are uninhabited. You cannot produce an output of type B in a generic way, even if you are given access to an element of type A.
The type corresponding to a proposition like “all computable functions from the reals to the reals are continuous” looks like a function type consuming some representation of “a computable function” and producing some representation of “that function is continuous”. To represent that, you probably need dependent types—this would be a type constructor that takes an element, not a type, as a parameter. Because not all functions are continuous, the codomain representing “that function is continuous” isn’t in general inhabited. So building an element of that codomain is not necessarily trivial—and the process of doing so amounts to proving the original proposition.
What I’m trying to say is that the types that type theorists use look a lot more like propositions than the types that mundane programmers use.
I think you may be sincerely confused. Would you please reword your question?
If your question is whether someone (either me or the OP) has committed a multiplication error—yes, it’s entirely possible, but multiplication is not the point—the point is anthropic reasoning and whether “I am a Bolzmann brain” is a simple hypothesis.
The arithmetical hierarchy is presuming a background of predicate logic; I was not presuming that. Yes, the type theory that I was gesturing towards would have some similarity to the arithmetical hierarchy.
I was trying to suggest that the answer to “what is a prediction” might look like a type theory of different variants of a prediction. Perhaps a linear hierarchy like the arithmetical hierarchy, yes, perhaps something more complicated. There could be a single starting type “concrete prediction” and a type constructor that, given source type and a destination type, gives the type of statements defining functions that take arguments of the source type and give arguments of the destination type.
The intuitive bridge for me from ultrafinitism to the question “what counts as a prediction?” is that ultrafinitists do happily work with entities like 2^1000 considered as a structure like (“^” “2” “1000″), even if they deny that those structures refer to things that are definitely numbers (of course, they can agree that they are definitely “numbers”, given an appropriately finitist definition of “number”). Maybe extreme teetotalers regarding what counts as a prediction would happily work with things such as computable functions returning predictions, even if they don’t consider them predictions.
Perhaps there is a type theory for predictions, with concrete predictions like “The bus will come at 3 o’clock”, and functions that output concrete predictions like “Every monday, wednesday and friday, the bus will come at 3 o’clock” (consider the statement as a function taking a time and returning a concrete prediction yes or no).
An ultrafinitist would probably not argue with the existence of such a function, even though to someone other than an ultrafinitist, the function looks like it is quantifying over all time. From the ultrafinitist’s point of view, you’re going to apply it to some concrete time, and at that point it’s going to output a some concrete prediction.
The prevalence of encodings means that we might not be able to “build machines with one or the other”. That is, given that there are basic alternatives A and B and A can encode B and B can encode A, it would take a technologist specializing in hair-splitting to say whether a machine that purportedly is using A is “really” using A at its lowest level or whether it is “actually” using B and only seems to use A via an encoding.
If in the immediate term you want to work with many-sorted first order logic, a reasonable first step would be to implement single-sorted first order logic, and a preprocessor to translate the many-sorted syntax into the single-sorted syntax. Then further development, optimizations and so on, might complicate the story, compacting the preprocessor and single-sorted engine into a single technology that you might reasonably say implements many-sorted logic “natively”.
The prevalence of encodings means that apparently different foundations don’t always make a difference.
It seems to me like this discussion has gotten too far into either/or “winning”. The prevalance of encodings in mathematics and logic (e.g. encoding the concept of pairing in set theory by defining (a, b) to be the set {{a}, {a, b}}, the double negation encoding of classical logic inside intuitionist logic, et cetera) means that the things we point to as “foundations” such as the ZF axioms for set theory are not actually foundational, in the sense of necessary and supporting. ZF is one possible system which is sufficiently strong to encode the things that we want to do. There are many other systems which are also sufficiently strong, and getting multiple perspectives on a topic by starting from different alternative “foundations” is probably a good thing. You do not have to choose just one!
Another way to look at it is—the things that we call foundations of mathematics are like the Planck length—yes, according to our current best theory, that’s the smallest, most basic element. However, that doesn’t mean it is stable; rather, it’s cutting edge (admittedly, Planck length is more at the cutting edge of physics than foundations of mathematics are at a cutting edge of mathematics). The stable things are more like cubits and inches—central, human-scale “theorems” such that if a new alternative foundation of mathematics contradicted them, we would throw away the foundation rather than the theorem. Some candidate “cubit” or “inch” theorems (really, arguments) might be the infinitude of primes, or the irrationality of the square root of 2.
Friedman and Simpson have a program of “Reverse Mathematics”, studying in what way facts that we normally think of as theorems, like the Heine-Borel theorem, pin down things that we normally think of as logic axioms, such as induction schemas. I really think that Simpson’s paper “Partial Realizations of Hilbert’s Program” is insightful and helpful to this discussion: http://www.math.psu.edu/simpson/papers/hilbert.ps
Lakatos’s “Proofs and Refutations” is also helpful, explaining in what sense a flawed and/or informal argument is helpful—Note that Lakatos calls arguments proofs, even when they’re flawed and/or informal, implicitly suggesting that everyone calls their arguments proofs until the flaws are pointed out. I understand he retreated from that philosophical position later in life, but that pre-retreat philosophical position might still be tenable.
Feeling our way into a new formal system is part of our (messy, informal) pebblecraft. Sometimes people propose formal systems starting with their intended semantics (roughly, model theory). Sometimes people propose formal systems starting with introduction and elimination rules (roughly, proof theory). If the latter, people sometimes look for a semantics to go along with the syntax (and vice versa, of course).
For example, lambda calculus started with rules for performing beta reduction. In talking about lambda calculus, people refer to it as “functions from functions to functions”. Mathematicians knew that there was no nontrivial set S with a bijection to the set of all functions from S to S. So something else must be going on. Dana Scott invented domain theory partially to solve this problem. Domains have additional structure, such that some domains D do have bijections with the set of all structure-preserving maps from D to D. http://en.wikipedia.org/wiki/Domain_theory Similarly, modal logics started with just proof theory, and then Saul Kripke invented a semantics for them. http://en.wikipedia.org/wiki/Kripke_semantics
There’s always a syntactical model which you can construct mechanically from the proof theory (at least, that’s my understanding of the downward Lowenheim-Skolem argument) - but Scott and Kripke were not satisfied with that syntactical model, and went looking for something else, more insightful and perspicuous. Adding a “semantic” understanding can increase our (informal) confidence in a formal system—even a formal system that we were already working with. I’m not sure why it helps, but I think it’s part of our pebblecraft that it does help.
Perhaps adding a “semantic” understanding is like another bridge between informal and formal reasoning. These bridges are only partly formal—they’re also partly informal, the concepts and gesturing around the equations and proofs. Having one bridge is sufficient to go onto the Island of Formality, do some stuff, and come off again, but might be more convenient to have two, or three.
This and previous articles in this series emphasize attaching meaning to sequences of symbols via discussion and gesturing toward models. That strategy doesn’t seem compatible with your article regarding sheep and pebbles. Isn’t there a way to connect sequences of symbols to sheep (and food and similar real-world consequences) directly via a discipline of “symbolcraft”?
I thought pebblecraft was an excellent description of how finitists and formalists think about confusing concepts like uncountability or actual infinity: Writing down ”… is uncountably infinite …” is part of our pebblecraft, and we have some confidence that when we get to the bottom line of the formal argument, we’ll be able to interpret the results “in the usual way”. The fact that someone (perhaps an alien) might watch our pebblecraft and construe us (in medias res) to be referring to (abstract, mathematical) things other than the (abstract, mathematical) things that we usually construe ourselves to be referring to doesn’t stop the pebblecraft magic from working.
Mathematics and logic are part of a strategy that I’ll call “formalization”. Informal speech leans on (human) biological capabilities. We communicate ideas, including ideas like “natural number” and “set” using informal speech, which does not depend on definitions. Informal speech is not quite pointing and grunting, but pointing and grunting is perhaps a useful cartoon of it. If I point and grunt to a dead leaf, that does not necessarily pin down any particular concept such as “dead leaves”. It could just as well indicate the concept “things which are dry on top and wet on bottom”—including armpits. There’s a Talmudic story about a debate that might be relevant here. Nevertheless (by shared biology) informal communication is possible.
In executing the strategy of formalization, we do some informal argumentation to justify and/or attach meaning to certain symbols and/or premises and/or rules. Then we do a chain of formal manipulations. Then we do some informal argumentation to interpret the result.
Model theory is a particular strategy of justifying and/or attaching meaning to things. It consists of discussing things called “models”, possibly using counterfactuals or possible worlds as intuition boosters. Then it defines the meaning of some strings of symbols first by reference to particular models, and then “validity” by reference to all possible models, and argues that certain rules for manipulating (e.g. simplifying) strings of symbols are “validity-preserving”.
Model theory is compelling, perhaps because it seems to offer “thicker” foundations. But you do not have to go through model theory in order to do the strategy of formalization. You can justify and/or attach meaning to your formal symbols and/or rules directly. A simple example is if you write down sequences of symbols, explain how to “pronounce” them, and then say “I take these axioms to be self-evident.”.
One problem with model theory from my perspective is that it puts too much in the metatheory (the informal argumentation around the formal symbol-manipulation). Set theory seems to me like something that should be in the formal (even, machine-checkable?) part, not in the metatheory. It’s certainly possible to have two layers of formality, which in principle I have no objection to, but model theoretical arguments often seem to neglect the (informal) work to justify and attach meaning to the outer layer. Furthermore, making the formal part more complicated has costs.
I think this paper by Simpson: Partial Realizations of Hilbert’s Program might be illuminating.
I’m concerned that you’re pushing second order logic too hard, using a false fork—such and so cannot be done in first order logic therefore second-order logic. “Second order” logic is a particular thing—for example it is a logic based on model theory. http://en.wikipedia.org/wiki/Second-order_logic#History_and_disputed_value
There are lots of alternative directions to go when you go beyond the general consensus of first-order logic. Freek Wiedijk’s paper “Is ZF a hack?” is a great tour of alternative foundations of mathematics—first order logic and the Zermelo-Frankel axioms for set theory turns out to be the smallest by many measures, but the others are generally higher order in one way or another. http://www.cs.ru.nl/~freek/zfc-etc/zfc-etc.pdf
“First order logic can’t talk about finiteness vs. infiniteness.” is sortof true in that first-order logic in a fixed language is necessarily local—it can step from object to object (along relation links) but the number of steps bounded by the size of the first-order formula. However, a standard move is to change the language, introducing a new sort of object “sets” and some axioms regarding relations between the old objects and the new objects such as inclusion, and then you can talk about properties of these new objects such as finiteness and infiniteness. Admittedly this standard move is informal or metatheoretic; we’re saying “this theory is the same as that theory except for these new objects and axioms”.
This is what the Zermelo-Frankel axioms do, and the vast majority of mathematics can be done within ZF or ZFC. Almost any time you encounter “finiteness” and “infiniteness” in a branch of mathematics other than logic, they are formalizable as first-order properties of first-order entities called sets.
If you taboo one-predicate ‘matter’, please specialize the two-place predicate (X matters to Y) to Y = “the OP’s subsequent use of this article”, and use the resulting one-place predicate.
I am not worried about apparent circularity. Once I internalized the Lowenheim-Skolem argument that first-order theories have countable “non-standard” models, then model theory dissolved for me. The syntactical / formalist view of semantics, that what mathematicians are doing is manipulating finite strings of symbols, is always a perfectly good model, in the model theoretic sense. If you want to understand what the mathematician is doing, you may look at what they’re doing, rather than taking them at their word and trying to boggle your imagination with images of bigness. Does dissolving model theory matter?
There’s plenty of encodings in mathematics—for example, using first-order predicate logic and the ZFC axioms to talk about second-order logic, or putting classical logic inside of intuitionistic logic with the double negation translation. Does the prevalence of encodings (analogous to the Turing Tarpit) matter?
Formal arguments, to be used in the real world, occur as the middle of an informal sandwich—first there’s an informal argument that the premises are appropriate or reasonable, and third there’s an informal argument interpreting the conclusion. I understand the formal part of this post, but I don’t understand the informal parts at all. Nonstandard (particularly countable) models are everywhere and unavoidable (analogously Godel showed that true but unprovable statements are everywhere and unavoidable). Against that background, the formal success of second-order logic in exiling nonstandard models of arithmetic doesn’t seem (to me) a good starting point for any third argument.
Does it matter if you don’t have formal rules for what you’re doing with models?
Do you expect what you’re doing with models to be formalizable in ZFC?
Does it matter if ZFC is a first-order theory?
It may not be possible to draw a sharp line between things that exist from the things that do not exist. Surely there are problematic referents (“the smallest triple of numbers in lexicographic order such that a^3+b^3=c^3”, “the historical jesus”, “the smallest pair of numbers in lexicographic order such that a^3+24=c^2”, “shakespeare’s firstborn child”) that need considerable working with before ascertaining that they exist or do not exist. Given that difficulty, it seems like we work with existence explicitly, as a theory; it’s not “baked in” to human reasoning.
Guy Steele wrote a talk called “Growing a Language”, where one of his points is that building hooks (such as functions) into the language definition to allow the programmer to grow the language is more important than building something that is often useful, say, complex numbers or a rich collection of string manipulation primitives. Maybe talking about the structure of “theories of X” would be valuable. Perhaps all theories have examples (including counterexamples as a specific kind of example) and rules (including definitions as a specific kind of example) - thats the kind of thing that I’m suggesting might be more like a hook.
One model for time travel might be a two dimensional piece of paper with a path or paths drawn wiggling around on it. If you scan a “current moment” line across the plane, then you see points dancing. If a line and its wiggles are approximately perpendicular to the line of the current moment, then the dancing is local and perhaps physical. Time travel would be sigmoid line, first a “spontaneous” creation of a pair of points, then the cancellation of one (“reversed”) point with the original point.
An alternative story is of a line next to a loop—spontaneous creation of two “virtual” particles, one reversed, followed by those two cancelling.
Would J.K. Rowling’s book be causal if we add to the lore that Time Turners are well understood to cause (unasked) “virtual bearers” very like their bearers? The virtual bearers could be reified by the real bearer using the time turner, or if they are not reified, they will “cancel” themselves by using their own, virtual, time turner.
I think this addition changes the time turner from a global / acausal constraint on possible universes to a local / causal constraint, but I could very well be mistaken. Note that the reversed person is presumably invisible, but persuasive—perhaps they are a disjunctive geas of some sort.
If humans are bad at mental arithmetic, but good at, say, not dying—doesn’t that suggest that, as a practical matter, humans should try to rephrase mathematical questions into questions about danger?
E.g. Imagine stepping into a field crisscrossed by dangerous laser beams in a prime-numbers manner to get something valuable. I think someone who had a realistic fear of the laser beams, and a realistic understanding of the benefit of that valuable thing would slow down and/or stop stepping out into suspicious spots.
Quantifying is ONE technique, and it’s been used very effectively in recent centuries—but those successes were inside a laboratory / factory / automation structure, not in an individual-rationality context.