Could you provide an object level counter argument please? A strong one would give me a lot more credence that Luke’s work was not an accurate portrayal of academic philosophy.
(Three would be preferred)
(Object level might look like “philosophers are making useful progress by metric X” or “I expect philosophers’ work to be very useful in area of science a because b” or “doing a PhD in philosophy has lots of value in the world for reasons p, q and r”)
It’s striking how much value there is in academia that I didn’t notice
So look for the value! Don’t write the entire field off, lots of smart people there, probably you are missing something.
But for example quite a few very smart causal inference people are in philosophy. That conference on decision theory MIRI went to in Cambridge was hosted by philosophers. Some philosophers deal with very hard problems that do not map onto empiricism very well, etc.
I think Luke will agree with you on what you say here, though. I remember commenting on one of his posts that was critical of philosophy, saying that his arguments didn’t really apply to the area of philosophy I’m involved in (technical philosophy of science). Luke’s response was essentially, “I agree. I’m not talking about philosophy of science.” I think he’d probably say the same about philosophical work on decision theory and causal inference.
Isn’t that motte/bailey: “philosophy, a diseased discipline” is not a very discriminating title. The best line of his post is this:
I have an efficient filter for skipping past the 95% of philosophy that isn’t useful to me.
And this is definitely ok!
But again, I am not super interested in arguing with people about whether philosophy is worthwhile. I have better things to do. I was only pointing out in response to the OP that I have been harping on LW’s silly anti-academic sentiment for ages, that’s all.
Not sure it’s motte-and-bailey. I do think there are several serious pathologies in large swathes of contemporary philosophy. And I say this not as a dilettante, but a professional philosopher. There are areas of philosophy where these pathological tendencies are being successfully held at bay, and I do think there are promising signs that those areas are growing in influence. But much of mainstream philosophy, especially mainstream metaphysics and epistemology, does suffer from continued adherence to what I consider archaic and unhelpful methodology. And I think that’s what Luke is trying to point out. He does go overboard with his rhetoric, and I think he lacks a feel for the genuine insights of the Western philosophical tradition (as smart and insightful as I think Yudkowsky is, I really find it odd that someone who purports to be reasonably familar with philosophy would cite him as their favorite philosopher). But I think there is a sound point lurking under there, and not merely a banal “motte”-style point.
I was only pointing out in response to the OP that I have been harping on LW’s silly anti-academic sentiment for ages, that’s all.
I absolutely agree with you on the silliness of the anti-academic sentiment.
But much of mainstream philosophy, especially mainstream metaphysics and epistemology, does suffer from continued adherence to what I consider archaic and unhelpful methodology.
Would you mind explaining your perspective? I’m always interested to hear more angles on this, since with my current sample-size being roughly three (Dennett, Railton, Churchland), I tend to think I have an incomplete picture.
Everyone on LW should consider Francis Bacon their patron saint, imo :).
LW seems to have picked the Presbyterian minister Thomas Bayes as its patron saint with the Franciscan friar William of Ockham running a close second :-)
In defense of Luke, when I’ve spent the time to read through philosophy books by strong-naturalist academic philosophers, they’ve often devoted page-counts easily equivalent in length to “Philosophy: a diseased discipline” to carefully, charitably, academically, verbosely tearing non-naturalist philosophy a new asshole. Luke’s post has tended to be a breath of fresh air that I reread after reading any philosophy paper that doesn’t come from a strongly naturalist perspective.
It sincerely worries me that the academics in philosophy who do really excellent work, work that does apply to the real world-that-is-made-of-atoms, work that does map-the-territory, have to spend large amounts of effort just beating down obviously bad beliefs over and over again. You should be able to shoot down a bad idea once, preferably in the peer-review phase, and not have to fight it again and again like a bad zombie.
(Examples of obviously bad ideas: p-zombies, Platonism, Bayesian epistemology (the latter two may require explanation).)
Now, to signal fairness even where I’m blatantly opinionated, plenty of people on LW are indeed irritatingly “men of one idea”, that usually being some variation on AIXI. And in fact, plenty of people on LW hold philosophical opinions I consider obviously bad, like mathematical Platonism.
But the answer to those bad things hasn’t usually been “more philosophy”, as if any philosophy is good philosophy, but instead more naturalism, investing more effort to accommodate conceptual theorizing to the world-that-is-made-of-atoms.
Since significant portions of academic philosophy (for instance, Thomas Nagel) are instead devoted to the view—one that I once expected to be contrarian but which I now find depressingly common—that science and naturalism are wrong, or that they are unjustified, or that they are necessarily incapable of answering some-or-another important question—having one page on a contrarian intellectual-hipsters’ website devoted to ragging on these ought-to-be-contrarian views is a bit of a relief.
If we take Platonism to be the belief that abstract objects (take, for instance, the objects of ZFC set theory) actually exist in a mind-independent way, if not in a particularly well-specified way, then it occurs because people mistake the contents of their mental models of the world for being real objects, simply because those models map the world well and compress sense-data well. In fact, those models often compress most sense-data better than the “more physicalist” truth would: they can be many orders of magnitude smaller (in bits of program devoted to generative or discriminative modelling).
However, just because they’re not “real” doesn’t mean they don’t causally interact with the real world! The point of a map is that it corresponds to the territory, so the point of an abstraction is that it corresponds to regularities in the territory. So naive nominalism isn’t true either: the abstractions and what they abstract over are linked, so you really can’t just move names around willy-nilly. In fact, some abstractions will do better or worse than others at capturing the regularities in sense-data (and in states of the world, of course), so we end up saying that abstractions can exist on a sliding scale from “more Platonic” (those which appear to capture regularities we’ve always seen in all our previous data) to “more nominalist” (those which capture spurious correlations).
Now, for “Bayesian epistemology”, I’m taking the Jaynesian view, which is considered extreme but stated very clearly and precisely, that reasoning consists in assigning probabilities to propositions. People who oppose Bayesianism will usually then raise the Problem of the Prior, and the problem of limited model classes, and so on and so forth. IMHO, the better criticism is simply: propositions are not first-order, actually-existing objects (see above on Platonism)! Consider a proposition to be a set of states some model can be in or not be in, and we can still use Bayesian statistics, including the kinds of complex Bayesian modelling used to model the mind, without endorsing Bayesian philosophy, which would require us to believe in spooky things called “propositions” and “logic”—while also not believing in certain spooky things called “continuous random variables”, which don’t really fit into Cox’s Theorem very well, if I understood Jaynes correctly.
Indeed, even though when you use a Lossy-Correspondence/Compression Theory of Truth, abstract objects become perfectly sensible as descriptions of regularities in concrete objects.
Not really, because most maths is unphysical, ie physics is picking out the physically applicable parts of maths, ie the rest has nothing to correspond to.
If I remember my Lakoff & Núñez correctly, they were arguing that even the most abstract and un-physical-seeming of maths is constructed on foundations that derive from the way we perceive the physical world.
Let me pick up the book again… ah, right. They define two kinds of conceptual metaphor:
Grounding metaphors yield basic, directly grounded ideas. Examples: addition as adding objects to a collection, subtraction as taking objects away from a collection, sets as containers, members of a set as objects in a container. These usually require little instruction.
Linking metaphors yield sophisticated ideas, sometimes called abstract ideas. Examples: numbers as points on a line, geometrical figures as algebraic equations, operations on classes as algebraic operations. These require a significant amount of explicit instruction.
Their argument is that for any kind of abstract mathematics, if you trace back its origin for long enough, you finally end up at some grounding and linking metaphors that have originally been derived from our understanding of physical reality.
As an example of the technique, they discuss the laws of arithmetic as having been derived from four grounding metaphors: Object Collection (if you put one and one physical objects together, you have a collection of two objects), Object Construction (physical objects are made up of smaller physical objects; used for understanding expressions like “five is made up of two plus three” or “you can factor 28 into 7 times 4″), Measuring Stick (physical distances correspond to numbers; gave birth to irrational numbers, when the Pythagorean theorem was used to prove their existence by assuming that there’s a number that corresponds to the length of the hypotenuse), and Motion Along A Path (used in the sixteenth century to invent the concept of the number line, and the notion of a number as lying between two other numbers).
Now, they argue that these grounding metaphors, each by themselves, are not sufficient to define the laws of arithmetic for negative numbers. Rather you need to combine them into a new metaphor that uses parts of each, and then define your new laws in terms of that newly-constructed metaphor.
Defining negative numbers is straightforward using these metaphors: if you have the concept of a number line, you can define negative numbers as “point-locations on the path on the side opposite the origin from positive numbers”, so e.g. −5 is the point five steps to the left of the origin point, symmetrical to +5 which is five steps to right of the origin point.
Next we can use Motion Along A Path to define addition and subtraction: adding positive numbers is moving towards the right, addition of negative numbers is moving towards the left, subtraction of positive numbers is moving towards the left, and subtraction of negative numbers is moving towards the right. Multiplication by a positive number is also straightforward: if you are multiplying something by n times, you just perform the movement action n times.
But multiplication by a negative number has no meaning in the source domain of motion. You can’t “do something a negative number of times”. A new metaphor must be found, constrained by the fact that it needs to fit the fact that we’ve found 5 (-2) = −10 and that, by the law of commutation (also straightforwardly derivable from the grounding metaphors), (-2) 5 = −10.
Now:
The symmetry between positive and negative numbers motivates a straightforward
metaphor for multiplication by –n: First, do multiplication by the positive
number n and then move (or “rotate” via a mental rotation) to the
symmetrical point—the point on the other side of the line at the same distance
from the origin. This meets all the requirements imposed by all the laws. Thus,
(–2) · 5 = –10, because 2 · 5 = 10 and the symmetrical point of 10 is –10. Similarly,
(–2) · (–5) = 10, because 2 · (–5) = –10 and the symmetrical point of –10 is 10. Moreover,
(–1) · (–1) = 1, because 1 · (–1) = –1 and the symmetrical point of –1 is 1.
The process we have just described is, from a cognitive perspective, another
metaphorical blend. Given the metaphor for multiplication by positive numbers,
and given the metaphors for negative numbers and for addition, we form a
blend in which we have both positive and negative numbers, addition for both,
and multiplication for only positive numbers. To this conceptual blend we add
the new metaphor for multiplication by negative numbers, which is formulated in terms of the blend! That is, to state the new metaphor, we must use
negative numbers as point-locations to the left of the origin,
addition for positive and negative numbers in terms of movement, and
multiplication by positive numbers in terms of repeated addition a positive
number of times, which results in a point-location.
Only then can we formulate the new metaphor for multiplication by negative
numbers using the concept of moving (or rotating) to the symmetrical point-location.
So in other words, we have taken some grounding metaphors and built a new metaphor that blends elements of them, and after having constructed that new metaphor, we use the terms of that combined metaphor to define a new metaphor on top of that.
While this example was in the context of an obviously physically applicable part of maths, their argument is that all of maths is built in this way, starting from physically grounded metaphors which are then extended and linked to build increasingly abstract forms of mathematics… but all of which are still, in the end, constrained by the physical regularities they were originally based on:
The metaphors given so far are called grounding metaphors because they directly
link a domain of sensory-motor experience to a mathematical domain.
But as we shall see in the chapters to come, abstract mathematics goes beyond
direct grounding. The most basic forms of mathematics are directly grounded.
Mathematics then uses other conceptual metaphors and conceptual blends to
link one branch of mathematics to another. By means of linking metaphors,
branches of mathematics that have direct grounding are extended to branches
that have only indirect grounding. The more indirect the grounding in experience,
the more “abstract” the mathematics is. Yet ultimately the entire edifice
of mathematics does appear to have a bodily grounding, and the mechanisms
linking abstract mathematics to that experiential grounding are conceptual
metaphor and conceptual blending.
To take a step back. the discussion is about mathematical Platonism, a theory of mathematical truth which is apparently motivated by the Correspondence theory of truth. That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts, not some special realm of immaterial entities. The relevance of my claim that there are unphysical mathematical truths is that is an argument against the second claim.
Lakoff and Nunez give an account of the origins and nature of mathematical thought that while firmly anti-Platonic doesn’t back a rival theory of mathematical truth, because that is not in fact their area of interest..their interest is in mathematical thinking.
That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts
Who said that? Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
Who said that CToT motivates mathematical Platonism, or who said that CToT is the outstanding theory of mathemtaical truth?
Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
I couldn’t agree more that coherence is the best description of mathematical practice.
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
Whereas the physically inapplicable parts don’t retain real-world correspondence. Correspondence isn’ta n intrinsic, essential part of maths.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
Transfinite mathematics is very interesting and currently has no correspondence to the physical world, at least not in any way that anyone knows about. And you can make the argument that even if there is a correspondence, we will never know about it, because you would have to be sure that actual infinities exist in the physical world, and that would seem pretty hard to confirm.
A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
So it’s abstracting, but then it’s not about the real world anymore.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.
But that’s not at all relevant The existence of unphysical maths is a robust argument against the theory that mathematical truth is true by correspondence to the physical world. The interestingness of such maths is neither here nor there.
while also not believing in certain spooky things called “continuous random variables”, which don’t really fit into Cox’s Theorem very well, if I understood Jaynes correctly.
I found a partial answer to the question I asked in the sibling comment. By chance I happened to need to generate random chords of a circle covering the circle uniformly. In searching on the net for Jaynes’ solution I came across a few fragments of Jaynes’ views on infinity. In short, he insists on always regarding continuous situations as limits of finite ones (e.g as when the binomial distribution tends to the normal), which is unproblematic for all the mathematics he wants to do. That is how the real numbers are traditionally formalised anyway. All of analysis is left unscathed. His wider philosophical objections to such things as Cantor’s transfinite numbers can be ignored, since these play no role in statistics and probability anyway.
I don’t know about the technicalities regarding Cox’s Theorem, but I do notice a substantial number of papers arguing about exactly what hypotheses it requires or does not require, and other papers discussing counterexamples (even to the finite case). The Wikipedia article has a long list of references, and a general search shows more. Has anyone written an up to date review of what Cox-style theorems are known to be sound and how well they suffice to found the mathematics of probability theory? I can google /”Cox’s theorem” review/ but it is difficult for me to judge where the results sit within current understanding, or indeed what the current understanding is.
Has anyone written an up to date review of what Cox-style theorems are known to be sound and how well they suffice to found the mathematics of probability theory?
I don’t know. But I will say this: I am distrustful of a foundation which takes “propositions” to be primitive objects. If the Cox’s Theorem foundation for probability requires that we assume a first-order logic foundation of mathematics in general, in which propositions cannot be considered as instances of some larger class of things (as they can in, for personal favoritism, type theory), then I’m suspicious.
I’m also suspicious of how Cox’s Theorem is supposed to map up to continuous and non-finitary applications of probability—even discrete probability theory, as when dealing with probabilistic programming or the Solomonoff measure. In these circumstances we seem to need the measure-theoretic approach.
Further: if “the extension of classical logic to continuous degrees of plausibility” and “rational propensities to bet” and “measure theory in spaces of normed measure” and “sampling frequencies in randomized conditional simulations of the world” all yield the same mathematical structure, then I think we’re looking at something deeper and more significant than any one of these presentations admits.
In fact, I’d go so far as to say there isn’t really a “Bayesian/Frequentist dichotomy” so much as a “Bayesian-Frequentist Isomorphism”, in the style of the Curry-Howard Isomorphism. Several things we thought were different are actually the same.
I don’t follow your argument re Bayesian epistemology, in fact, I find it not at all obvious. The argument looks like insisting on a different vocabulary while doing the same things, and then calling it statistics rather than epistemology.
while also not believing in certain spooky things called “continuous random variables”, which don’t really fit into Cox’s Theorem very well, if I understood Jaynes correctly.
Can you give a pointer to where he disbelieves in these? He does refer to them apparently unproblematically here and there, e.g. in deducing what a noninformative prior on the chords of a circle should be.
I don’t follow your argument re Bayesian epistemology, in fact, I find it not at all obvious. The argument looks like insisting on a different vocabulary while doing the same things, and then calling it statistics rather than epistemology.
1) Dissolving epistemology to get statistics of various kinds underneath is a good thing, especially since the normal prescription of Bayesian epistemology is, “Oh, just calculate the posterior”, while in Bayesian statistics we usually admit that this is infeasible most of the time and use computational methods to approximate well.
2) The difference between Bayesian statistics and Bayesian epistemology is slight, but the difference between Bayesian statistics and the basic nature of traditional philosophical epistemology that the Bayesian epistemologists were trying to fit Bayesianism into is large.
3) The differences start to become large when you stop using spaces composed of N mutually-exclusive logical propositions arranged into a Boolean algebra. For instance, computational uncertainty and logical omniscience are nasty open questions in Bayesian epistemology, while for an actual statistician it is admitted from the start that models do not yield well-defined answers where computations are infeasible.
Can you give a pointer to where he disbelieves in these?
I can’t, since the precise page number would have to be a location number in my Kindle copy of Jaynes’ book.
If we take Platonism to be the belief that abstract objects (take, for instance, the objects of ZFC set theory) actually exist in a mind-independent way, if not in a particularly well-specified way, then it occurs because people mistake the contents of their mental models of the world for being real objects, simply because those models map the world well and compress sense-data well.
How do you account for the fact that numbers are the same for everyone? Of course, not everyone knows the same things about numbers, but neither does everyone know the same things about Neptune. Nevertheless, the abstract objects of mathematics have the same ineluctability as physical objects. Everyone who looks at Neptune is looking at the same thing, and so is everyone who studies ZFC. These abstract objects can be used to make models of things, but they are not themselves those models.
How do you account for the fact that numbers are the same for everyone?
Two correct maps of the same territory, designed to highlight the same regularities and obscure the same sources of noise, will be either completely the same or, in the noisy case, will approximate each-other.
Just because there’s no Realm of Forms doesn’t mean that numbers can be different for different people without losing their ability to compressively predict regularities in the environment.
What is the territory, that numbers are a map of? I can use them to assemble a map, for example, s=0.5at^2 as a map, or model, of uniformly accelerating bodies, but the components of this are more like the ink and paper used to make a map than they are like a map.
I have a bunch of maps, literal printed maps of various places, and the maps certainly exist as physical objects, alongside the places that they are maps of. They exist independently of me, and independently of whether anyone uses them as a map or as wrapping paper. Likewise, it seems to me, numbers.
Again: they abstract over concrete objects. You get a map that represents lots of territories at the same time by capturing their common regularities and throwing out the details that make them different.
In which case we’re back to the question of why numbers are the same for everyone? You said:
Two correct maps of the same territory, designed to highlight the same regularities and obscure the same sources of noise, will be either completely the same or, in the noisy case, will approximate each-other.
Except you claim there’s no same territory. The aliens in the Andromeda galaxy will have the same numbers as us, as will the sentient truing machines that evolved in a cellular automata. So what common territory are they all looking at?
(Another attempt, this time read the comment before responding.)
Where in the physical word is the common territory between us and the Andromeda aliens? And how about the sentient truing machines that evolved in a cellular automata who aren’t even in the same universe as us?
In which you managed to misspell Turing the same way as in the previous one. ;-)
(As for the actual question, two territories can have some properties in common even if they don’t spatially overlap. In the case of Andromeda there are some non-trivial such properties, i.e. the laws of physics. Mathematics is the study of the properties that all possible territories share, which all are technically tautological but not always immediately obvious to people without infinite computing power.)
So read it again to correct your misunderstanding and then write a response that actually addresses the issues. If you’re just going to ignore my arguments rather than respond to them, I don’t see the point in continuing this conversation.
You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
On the one hand, why? I’m quite happy to say that chess exists. Not everyone will ever see chess, but not everyone will ever see Neptune. Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that’s equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn’t mean that ZFC is inevitable, it means that if you construct two theories both intended to “support” all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward. This is consistent relative to ZFC, and indeed relative to something rather weaker than ZFC, and NFUIC + “all Cantorian sets are strongly Cantorian” (never mind exactly what that means) is equiconsistent with ZFC + some reasonably plausible large-cardinal axioms. OK, fine, so there’s a sense in which NFUIC is ZFC-like, at least as regards consistency strength. But NFUIC’s sets are most definitely not the same as ZFC’s sets. NFUIC has a universal set and ZFC doesn’t; ZFC’s sets are the same sizes as their sets-of-singletons and NFUIC’s often aren’t; NFU has lots and lots of urelements and ZFC has just the single empty set; etc. NFUIC is very unlike ZFC despite these relationships in terms of consistency strength.
[EDITED to add:] Here’s an analogy. You get the same computable functions whether you start with (1) Turing machines, (2) register machines, (3) lambda calculus, or (4) Post production systems. But those are still four very different foundations for computing, they suggest quite different possible hardware realizations and different kinds of notation, they have quite different performance characteristics, etc. (The execution times are admittedly all bounded by polynomials in one another. We could add (5) quantum Turing machines, in which case that would no longer be known to be true.) It’s very interesting that these all turn out to be equivalent in power in some sense, but I wouldn’t call that convergence or suggest that it tells us that (e.g.) lambda-calculus terms have any sort of more exalted metaphysical status than they would if it weren’t for that equivalence.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward.
Yes, ZFC is not quite such an isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward.
Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)
The question is whether the rules of chess have mind-indepedent existence.
Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging? One way is put forward the counterfactual that if the abstract object were different, then the convergence would occur differently. But that seems rather against the spirit of what you are aiming.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
Ask Max Tegmark. :)
I don’t believe in his Level IV multiverse, though. That is, I do draw a distinction between physical and abstract objects.
If you don’t know what your theory is, why undertake to defend it?
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging?
That they are converging is enough. To quote an old saw variously attributed, in mathematics existence is freedom from contradiction.
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Vague and empty slogans that might be picked over endlessly, and have been, to no useful purpose. Don’t bother expanding them; it’s a dead end.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Before, you seemed to be pitching in with an opinion on anti-realism versus realism, now you seem to be sayign the whole debate is meaningless. Which is your true belief? It isn’t clear.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Says who? As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Are you a software engineer? Do you believe software engineering has taught you to be clear?
Lisp is worth learning for … the profound enlightenment experience you will have when you finally get it. That experience will make you a better programmer for the rest of your days, even if you never actually use Lisp itself a lot.
As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Well, I have not. Which philosophers would you particularly recommend for this purpose? What in philosophy will assist our most gifted fellow humans in thinking previously impossible thoughts, or is worth learning for the profound enlightenment experience I will have when I finally get it?
Are you a software engineer?
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
Do you believe software engineering has taught you to be clear?
For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts do better. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if the real work was in thinking up a structure and the “stupid implementation” could be left to a few graduate students.
And are either Dijkstra or ESR in a position to directly compare the efficacy of software engineering as a means of clearly expressing philosophical ideas to the efficacy of philosophy as a means of clearly expressing philosophical ideas..ie, do they know anything about philosophy?
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause. It’s just not very significant.
It’s also not news that software engineering teaches you to express software engineering concepts clearly, and it’s not very relevant since the topic is expressing philosophical concepts. Pointing out that someone else is unclear doesn’t make you clear. Pointing about that you can be clear about X doesn’t make you clear about Y.
Which philosophers would you particularly recommend for this purpose?
Getting into conversations where there is mutual commitment to clear communication is the best practice, because you get instant feedback Learning the jargon of philosophy—there are a number of dictionary-style works—is also helpful: after all the jargon is tailored to just the kind of topic you were discussing.
What in philosophy will assist our most gifted fellow humans in thinking previously impossible thought
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
or is worth learning for the profound enlightenment experience I will have when I finally get it?
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause.
I’m not sure it makes sense to label people like E.S. Raymond who are proclaim hacker values as STEM types. Raymond isn’t following the popular science narrative of logical positivism that you find with the typical STEM person.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
Whatever. It’s about the third or fourth change of topic.
Lisp is worth learning for … the profound enlightenment experience you will have when you finally get it. That experience will make you a better programmer for the rest of your days, even if you never actually use Lisp itself a lot.
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
Do you believe software engineering has taught you to be clear?
It has certainly greatly influenced me in that regard. For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts get right. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if she could just think up a structure and leave it to a few graduate students to implement.
(Examples of obviously bad ideas: p-zombies, Platonism, Bayesian epistemology (the latter two may require explanation
I’m not a fan of mathematical Platonism, but physical realists, however hardline, face some very difficult problems regarding the ontologica status of physical law, which make Platonism hard to rule out. (And no, the perenially popular “laws are just descriptions” isn’t a good answer).
P-zombies as a subject worth discussing, or as something that can exist in our univese? But most of the people who discuss PZs don’t think they can exist in our universe. There is some poor quality criticisim of philosophy about as well.
The problems with Bayes are suffcieintly non-obvious to have eluded many or most at LW.
On the one hand, I think that page in specific is actually based on outdated Bayesian methods, and there’s been a lot of good work in Bayesian statistics for complex models and cognitive science in recent years.
On the other hand, I freaking love that website, despite its weirdo Buddhist-philosophical leanings and one or two things it gets Wrong according to my personal high-and-mighty ideologies.
And on the gripping hand, he is very, very right that the way the LW community tends to phrase things in terms of “just Bayes it” is not only a mischaracterization of the wide world of statistics, it’s even an oversimplification of Bayesian statistics as a subfield. Bayes’ Law is just the update/training rule! You also need to discuss marginalization; predictive distributions; maximum-entropy priors, structural simplicity priors, and Bayesian Occam’s Razor, and how those are three different views of Occam’s Razor that have interesting similarities and differences; model selection; the use of Bayesian point-estimates and credible-hypothesis tests for decision-making; equivalent sample sizes; conjugate families; and computational Bayes methods.
Then you’re actually learning and doing Bayesian statistics.
On the miniature nongripping hand, I can’t help but feel that the link between probability, thermodynamics, and information theory means Eliezer and the Jaynesians are probably entirely correct that as a physical fact, real-world event frequencies and movements of information obey Bayes’ Law with respect to the information embodied in the underlying physics, whether or not I can model any of that well or calculate posterior distributions feasibly.
Starting out by expecting a view opposed to your own to be contrarian is a typical form of overconfidence, and not just overconfidence about other people’s opinions.
That seems to be entirely analytic philosophy. My problem is that analytical philosophy is culturally irrelevant. Anthropologists, sociologists, art theorists, and artists talk about continental philosophy, Saussurian (!) linguistics, and psychoanalytic theory. The only things they use from analytic philosophy are arguments like Godel’s incompleteness theorem, Wittgenstein’s later stuff, or Quine’s ontological relativism, that they interpret as saying that analytic philosophy doesn’t work.
Could you provide an object level counter argument please? A strong one would give me a lot more credence that Luke’s work was not an accurate portrayal of academic philosophy.
(Three would be preferred)
(Object level might look like “philosophers are making useful progress by metric X” or “I expect philosophers’ work to be very useful in area of science a because b” or “doing a PhD in philosophy has lots of value in the world for reasons p, q and r”)
I am not very interested in convincing you.
You said:
So look for the value! Don’t write the entire field off, lots of smart people there, probably you are missing something.
But for example quite a few very smart causal inference people are in philosophy. That conference on decision theory MIRI went to in Cambridge was hosted by philosophers. Some philosophers deal with very hard problems that do not map onto empiricism very well, etc.
I think Luke will agree with you on what you say here, though. I remember commenting on one of his posts that was critical of philosophy, saying that his arguments didn’t really apply to the area of philosophy I’m involved in (technical philosophy of science). Luke’s response was essentially, “I agree. I’m not talking about philosophy of science.” I think he’d probably say the same about philosophical work on decision theory and causal inference.
Isn’t that motte/bailey: “philosophy, a diseased discipline” is not a very discriminating title. The best line of his post is this:
And this is definitely ok!
But again, I am not super interested in arguing with people about whether philosophy is worthwhile. I have better things to do. I was only pointing out in response to the OP that I have been harping on LW’s silly anti-academic sentiment for ages, that’s all.
Not sure it’s motte-and-bailey. I do think there are several serious pathologies in large swathes of contemporary philosophy. And I say this not as a dilettante, but a professional philosopher. There are areas of philosophy where these pathological tendencies are being successfully held at bay, and I do think there are promising signs that those areas are growing in influence. But much of mainstream philosophy, especially mainstream metaphysics and epistemology, does suffer from continued adherence to what I consider archaic and unhelpful methodology. And I think that’s what Luke is trying to point out. He does go overboard with his rhetoric, and I think he lacks a feel for the genuine insights of the Western philosophical tradition (as smart and insightful as I think Yudkowsky is, I really find it odd that someone who purports to be reasonably familar with philosophy would cite him as their favorite philosopher). But I think there is a sound point lurking under there, and not merely a banal “motte”-style point.
I absolutely agree with you on the silliness of the anti-academic sentiment.
Would you mind explaining your perspective? I’m always interested to hear more angles on this, since with my current sample-size being roughly three (Dennett, Railton, Churchland), I tend to think I have an incomplete picture.
Everyone on LW should consider Francis Bacon their patron saint, imo :).
LW seems to have picked the Presbyterian minister Thomas Bayes as its patron saint with the Franciscan friar William of Ockham running a close second :-)
If I had to pick one, I’d go with Laplace.
In defense of Luke, when I’ve spent the time to read through philosophy books by strong-naturalist academic philosophers, they’ve often devoted page-counts easily equivalent in length to “Philosophy: a diseased discipline” to carefully, charitably, academically, verbosely tearing non-naturalist philosophy a new asshole. Luke’s post has tended to be a breath of fresh air that I reread after reading any philosophy paper that doesn’t come from a strongly naturalist perspective.
It sincerely worries me that the academics in philosophy who do really excellent work, work that does apply to the real world-that-is-made-of-atoms, work that does map-the-territory, have to spend large amounts of effort just beating down obviously bad beliefs over and over again. You should be able to shoot down a bad idea once, preferably in the peer-review phase, and not have to fight it again and again like a bad zombie.
(Examples of obviously bad ideas: p-zombies, Platonism, Bayesian epistemology (the latter two may require explanation).)
Now, to signal fairness even where I’m blatantly opinionated, plenty of people on LW are indeed irritatingly “men of one idea”, that usually being some variation on AIXI. And in fact, plenty of people on LW hold philosophical opinions I consider obviously bad, like mathematical Platonism.
But the answer to those bad things hasn’t usually been “more philosophy”, as if any philosophy is good philosophy, but instead more naturalism, investing more effort to accommodate conceptual theorizing to the world-that-is-made-of-atoms.
Since significant portions of academic philosophy (for instance, Thomas Nagel) are instead devoted to the view—one that I once expected to be contrarian but which I now find depressingly common—that science and naturalism are wrong, or that they are unjustified, or that they are necessarily incapable of answering some-or-another important question—having one page on a contrarian intellectual-hipsters’ website devoted to ragging on these ought-to-be-contrarian views is a bit of a relief.
That word, “obviously”, I don’t think it means what you think it means :-)
Could you provide that explanation?
Sure.
If we take Platonism to be the belief that abstract objects (take, for instance, the objects of ZFC set theory) actually exist in a mind-independent way, if not in a particularly well-specified way, then it occurs because people mistake the contents of their mental models of the world for being real objects, simply because those models map the world well and compress sense-data well. In fact, those models often compress most sense-data better than the “more physicalist” truth would: they can be many orders of magnitude smaller (in bits of program devoted to generative or discriminative modelling).
However, just because they’re not “real” doesn’t mean they don’t causally interact with the real world! The point of a map is that it corresponds to the territory, so the point of an abstraction is that it corresponds to regularities in the territory. So naive nominalism isn’t true either: the abstractions and what they abstract over are linked, so you really can’t just move names around willy-nilly. In fact, some abstractions will do better or worse than others at capturing the regularities in sense-data (and in states of the world, of course), so we end up saying that abstractions can exist on a sliding scale from “more Platonic” (those which appear to capture regularities we’ve always seen in all our previous data) to “more nominalist” (those which capture spurious correlations).
Now, for “Bayesian epistemology”, I’m taking the Jaynesian view, which is considered extreme but stated very clearly and precisely, that reasoning consists in assigning probabilities to propositions. People who oppose Bayesianism will usually then raise the Problem of the Prior, and the problem of limited model classes, and so on and so forth. IMHO, the better criticism is simply: propositions are not first-order, actually-existing objects (see above on Platonism)! Consider a proposition to be a set of states some model can be in or not be in, and we can still use Bayesian statistics, including the kinds of complex Bayesian modelling used to model the mind, without endorsing Bayesian philosophy, which would require us to believe in spooky things called “propositions” and “logic”—while also not believing in certain spooky things called “continuous random variables”, which don’t really fit into Cox’s Theorem very well, if I understood Jaynes correctly.
The motivation actually seems to be the Correspondence Theory of Truth..that is mentioned several timesin subsequent comments.
Indeed, even though when you use a Lossy-Correspondence/Compression Theory of Truth, abstract objects become perfectly sensible as descriptions of regularities in concrete objects.
Not really, because most maths is unphysical, ie physics is picking out the physically applicable parts of maths, ie the rest has nothing to correspond to.
If I remember my Lakoff & Núñez correctly, they were arguing that even the most abstract and un-physical-seeming of maths is constructed on foundations that derive from the way we perceive the physical world.
Let me pick up the book again… ah, right. They define two kinds of conceptual metaphor:
Their argument is that for any kind of abstract mathematics, if you trace back its origin for long enough, you finally end up at some grounding and linking metaphors that have originally been derived from our understanding of physical reality.
As an example of the technique, they discuss the laws of arithmetic as having been derived from four grounding metaphors: Object Collection (if you put one and one physical objects together, you have a collection of two objects), Object Construction (physical objects are made up of smaller physical objects; used for understanding expressions like “five is made up of two plus three” or “you can factor 28 into 7 times 4″), Measuring Stick (physical distances correspond to numbers; gave birth to irrational numbers, when the Pythagorean theorem was used to prove their existence by assuming that there’s a number that corresponds to the length of the hypotenuse), and Motion Along A Path (used in the sixteenth century to invent the concept of the number line, and the notion of a number as lying between two other numbers).
Now, they argue that these grounding metaphors, each by themselves, are not sufficient to define the laws of arithmetic for negative numbers. Rather you need to combine them into a new metaphor that uses parts of each, and then define your new laws in terms of that newly-constructed metaphor.
Defining negative numbers is straightforward using these metaphors: if you have the concept of a number line, you can define negative numbers as “point-locations on the path on the side opposite the origin from positive numbers”, so e.g. −5 is the point five steps to the left of the origin point, symmetrical to +5 which is five steps to right of the origin point.
Next we can use Motion Along A Path to define addition and subtraction: adding positive numbers is moving towards the right, addition of negative numbers is moving towards the left, subtraction of positive numbers is moving towards the left, and subtraction of negative numbers is moving towards the right. Multiplication by a positive number is also straightforward: if you are multiplying something by n times, you just perform the movement action n times.
But multiplication by a negative number has no meaning in the source domain of motion. You can’t “do something a negative number of times”. A new metaphor must be found, constrained by the fact that it needs to fit the fact that we’ve found 5 (-2) = −10 and that, by the law of commutation (also straightforwardly derivable from the grounding metaphors), (-2) 5 = −10.
Now:
So in other words, we have taken some grounding metaphors and built a new metaphor that blends elements of them, and after having constructed that new metaphor, we use the terms of that combined metaphor to define a new metaphor on top of that.
While this example was in the context of an obviously physically applicable part of maths, their argument is that all of maths is built in this way, starting from physically grounded metaphors which are then extended and linked to build increasingly abstract forms of mathematics… but all of which are still, in the end, constrained by the physical regularities they were originally based on:
To take a step back. the discussion is about mathematical Platonism, a theory of mathematical truth which is apparently motivated by the Correspondence theory of truth. That is being rivaled by another theory, also motivated by CToT, wherein the truth-makers of mathematical statements are physical facts, not some special realm of immaterial entities. The relevance of my claim that there are unphysical mathematical truths is that is an argument against the second claim.
Lakoff and Nunez give an account of the origins and nature of mathematical thought that while firmly anti-Platonic doesn’t back a rival theory of mathematical truth, because that is not in fact their area of interest..their interest is in mathematical thinking.
Who said that? Actual formal systems run on a coherence theory of truth: if the theory is consistent (and I do mean consistent according to a meta-system, so Goedel and Loeb aren’t involved right now), then it’s a theory. It may also be a totally uninteresting theory, or a very interesting theory. The truth-maker for a mathematical statement is just whether it has a model (and if you really wanted to, you could probably compile that into something about computation via the Curry-Howard Correspondence and some amount of Turing oracles). But the mere truth of a statement within a formal system is not the interesting thing about the statement!
Who said that CToT motivates mathematical Platonism, or who said that CToT is the outstanding theory of mathemtaical truth?
I couldn’t agree more that coherence is the best description of mathematical practice.
This one.
Or rather, who claimed that the truth-makers of mathematical statements are physical facts?
Insofar as logic consists in information-preserving operations, the non-physically-applicable parts of math still correspond to the real world, in that they preserve the information about the real world which was put into formulating/locating the starting formal system in the first place.
This is what makes mathematics so wondrously powerful: formality = determinism, and determinism = likelihood functions of 0 or 1. So when doing mathematics, you get whole formal systems where the theorems are always at least as true as the axioms. As long as any part of the system corresponds to the real world (and many parts of it do) and the whole system remains deterministic, then the whole system compresses information about the real world.
Whereas the physically inapplicable parts don’t retain real-world correspondence. Correspondence isn’ta n intrinsic, essential part of maths.
Sure, you can come up with a formal system that bears no correspondence to the real world whatsoever. Mathematicians just won’t consider it very interesting most of the time.
They call it “pure mathematics”.
Transfinite mathematics is very interesting and currently has no correspondence to the physical world, at least not in any way that anyone knows about. And you can make the argument that even if there is a correspondence, we will never know about it, because you would have to be sure that actual infinities exist in the physical world, and that would seem pretty hard to confirm.
What do cardinals correspond to?
I suppose it’s about the language, not about the model. Still.
Screw it. I’ll just go do a PhD thesis on how abstraction works, and then anyone who wants to actually understand can read that.
A lot of pure math takes the form of: “let’s take something in the real world, like ‘notion of containment in a bag’ and run off with it.” So it’s abstracting, but then it’s not about the real world anymore. There are no cardinals in the real world, but there are bags.
Yes it is. It still consists in information from the real world. The precise structure was chosen out of an infinite space of possible structures based precisely on its ability to generalize scenarios from “real life”.
Consider, for instance, real numbers and continuity. The real world is not infinitely divisible—we know this now! But at one time, when these mathematical theories were formulated, that was a working hypothesis, and in fact, people could not divide things small enough to actually find where they became discrete. So continuity, as a mathematical construct, started out trying to describe the world, and was later found to have more interesting implications even when it was also found to be physically wrong.
Ok, so you would say inaccessible cardinals, versions of the continuum hypothesis etc. are “about the real world.”
At this point, I am bowing out of this conversation as we aren’t using words in the same way.
I think you’re engaging in deepities here. It is clearly true that all of mathematics historically descends from thoughts about the real world. It is clearly false that all of mathematics is directly about the real world. Using the same words for both claims, “mathematics is about the real world”, is the deepity.
That is news to me. Physicists, even fundamental physicists, still talk about differential geometry and Hilbert spaces and so on. There are speculations about an underlying discrete structure on the Planck scale or below, but did anyone refound physics on that basis yet? Stephen Wolfram made some gestures in that direction in his magnum opus; but I read a physicist writing a review of it saying that Wolfram’s idea of explaining quantum entanglement that way was already known not to work.
I’m actually only using it for the former.
You may be thinking of Scott Aaronson’s review of “A new kind of science”.
I think it’s at least arguable that there are plenty of cardinals in the real world (e.g., three) even though there very likely aren’t the “large cardinals” that set theorists like to speculate about.
(Of course there are also cardinals and cardinals.)
What was the point of writing that? Do you think I was talking about “3”?
I (1) was genuinely unsure whether you were asserting that numbers (even small positive integers) are too abstract to “live” in the real world—a reasonable assertion, I think, though I thought it probably wasn’t your position—and (2) thought it was amusing even if you weren’t.
But that’s not at all relevant The existence of unphysical maths is a robust argument against the theory that mathematical truth is true by correspondence to the physical world. The interestingness of such maths is neither here nor there.
I found a partial answer to the question I asked in the sibling comment. By chance I happened to need to generate random chords of a circle covering the circle uniformly. In searching on the net for Jaynes’ solution I came across a few fragments of Jaynes’ views on infinity. In short, he insists on always regarding continuous situations as limits of finite ones (e.g as when the binomial distribution tends to the normal), which is unproblematic for all the mathematics he wants to do. That is how the real numbers are traditionally formalised anyway. All of analysis is left unscathed. His wider philosophical objections to such things as Cantor’s transfinite numbers can be ignored, since these play no role in statistics and probability anyway.
I don’t know about the technicalities regarding Cox’s Theorem, but I do notice a substantial number of papers arguing about exactly what hypotheses it requires or does not require, and other papers discussing counterexamples (even to the finite case). The Wikipedia article has a long list of references, and a general search shows more. Has anyone written an up to date review of what Cox-style theorems are known to be sound and how well they suffice to found the mathematics of probability theory? I can google /”Cox’s theorem” review/ but it is difficult for me to judge where the results sit within current understanding, or indeed what the current understanding is.
I don’t know. But I will say this: I am distrustful of a foundation which takes “propositions” to be primitive objects. If the Cox’s Theorem foundation for probability requires that we assume a first-order logic foundation of mathematics in general, in which propositions cannot be considered as instances of some larger class of things (as they can in, for personal favoritism, type theory), then I’m suspicious.
I’m also suspicious of how Cox’s Theorem is supposed to map up to continuous and non-finitary applications of probability—even discrete probability theory, as when dealing with probabilistic programming or the Solomonoff measure. In these circumstances we seem to need the measure-theoretic approach.
Further: if “the extension of classical logic to continuous degrees of plausibility” and “rational propensities to bet” and “measure theory in spaces of normed measure” and “sampling frequencies in randomized conditional simulations of the world” all yield the same mathematical structure, then I think we’re looking at something deeper and more significant than any one of these presentations admits.
In fact, I’d go so far as to say there isn’t really a “Bayesian/Frequentist dichotomy” so much as a “Bayesian-Frequentist Isomorphism”, in the style of the Curry-Howard Isomorphism. Several things we thought were different are actually the same.
I don’t follow your argument re Bayesian epistemology, in fact, I find it not at all obvious. The argument looks like insisting on a different vocabulary while doing the same things, and then calling it statistics rather than epistemology.
Can you give a pointer to where he disbelieves in these? He does refer to them apparently unproblematically here and there, e.g. in deducing what a noninformative prior on the chords of a circle should be.
1) Dissolving epistemology to get statistics of various kinds underneath is a good thing, especially since the normal prescription of Bayesian epistemology is, “Oh, just calculate the posterior”, while in Bayesian statistics we usually admit that this is infeasible most of the time and use computational methods to approximate well.
2) The difference between Bayesian statistics and Bayesian epistemology is slight, but the difference between Bayesian statistics and the basic nature of traditional philosophical epistemology that the Bayesian epistemologists were trying to fit Bayesianism into is large.
3) The differences start to become large when you stop using spaces composed of N mutually-exclusive logical propositions arranged into a Boolean algebra. For instance, computational uncertainty and logical omniscience are nasty open questions in Bayesian epistemology, while for an actual statistician it is admitted from the start that models do not yield well-defined answers where computations are infeasible.
I can’t, since the precise page number would have to be a location number in my Kindle copy of Jaynes’ book.
A brief quote will do, enough words to find them in my copy.
How do you account for the fact that numbers are the same for everyone? Of course, not everyone knows the same things about numbers, but neither does everyone know the same things about Neptune. Nevertheless, the abstract objects of mathematics have the same ineluctability as physical objects. Everyone who looks at Neptune is looking at the same thing, and so is everyone who studies ZFC. These abstract objects can be used to make models of things, but they are not themselves those models.
Two correct maps of the same territory, designed to highlight the same regularities and obscure the same sources of noise, will be either completely the same or, in the noisy case, will approximate each-other.
Just because there’s no Realm of Forms doesn’t mean that numbers can be different for different people without losing their ability to compressively predict regularities in the environment.
What is the territory, that numbers are a map of? I can use them to assemble a map, for example, s=0.5at^2 as a map, or model, of uniformly accelerating bodies, but the components of this are more like the ink and paper used to make a map than they are like a map.
I have a bunch of maps, literal printed maps of various places, and the maps certainly exist as physical objects, alongside the places that they are maps of. They exist independently of me, and independently of whether anyone uses them as a map or as wrapping paper. Likewise, it seems to me, numbers.
If there is no Realm of Forms, what territory are you referring to?
The ordinary physical universe, presumably.
As TheAncientGeek said, the ordinary physical universe. “Abstract” objects abstract over concrete objects.
And where is the ordinary physical universe do these abstractions live?
Again: they abstract over concrete objects. You get a map that represents lots of territories at the same time by capturing their common regularities and throwing out the details that make them different.
So do you claim these abstractions actually exist?
The abstract maps exist. The abstract territory does not.
In which case we’re back to the question of why numbers are the same for everyone? You said:
Except you claim there’s no same territory. The aliens in the Andromeda galaxy will have the same numbers as us, as will the sentient truing machines that evolved in a cellular automata. So what common territory are they all looking at?
The physical world: see here.
(Another attempt, this time read the comment before responding.)
Where in the physical word is the common territory between us and the Andromeda aliens? And how about the sentient truing machines that evolved in a cellular automata who aren’t even in the same universe as us?
In which you managed to misspell Turing the same way as in the previous one. ;-)
(As for the actual question, two territories can have some properties in common even if they don’t spatially overlap. In the case of Andromeda there are some non-trivial such properties, i.e. the laws of physics. Mathematics is the study of the properties that all possible territories share, which all are technically tautological but not always immediately obvious to people without infinite computing power.)
How about you try reading the comment your responding to next time. It’s even extremely short so you don’t have much of an excuse.
Instead of assuming that I didn’t, you could also have assumed that I just misunderstood it.
So read it again to correct your misunderstanding and then write a response that actually addresses the issues. If you’re just going to ignore my arguments rather than respond to them, I don’t see the point in continuing this conversation.
Your previous comment already made me uninterested in continuing the conversation.
You could equally say that everyone who looks at the rules of chess sees the same thing. In order to show some inevitability to ZFC, you have to show that unconnected parties arriving at it independently.
On the one hand, why? I’m quite happy to say that chess exists. Not everyone will ever see chess, but not everyone will ever see Neptune. Among all the games that could be played, chess is but one grain of sand on the beach. But the grain of sand exists regardless of whether anyone sees it.
On the other hand, there has been, I believe, a substantial tendency for people devising alternative axioms for the concepts of sets to come up with things equiconsistent to ZFC or to subsets of ZFC, and with fairly direct translations between them. Compare also the concept of computability, where there is a very strong tendency for different ways to answer the question “what is computation?” to come up with equivalent definitions.
It is (I think) true that if you try to come up with an alternative foundation for mathematics you are likely to get something that’s equivalent to some subset of ZFC perhaps augmented with some kind of large cardinal axiom. But that doesn’t mean that ZFC is inevitable, it means that if you construct two theories both intended to “support” all of mathematics without too much extravagance, you can often more or less implement one inside the other.
But that doesn’t mean that ZFC specifically has any particular inevitability. Consider, e.g., NFU + Infinity + Choice (as used e.g. in Randall Holmes’s book “Elementary set theory with a universal set”) which I’ll call NFUIC henceforward. This is consistent relative to ZFC, and indeed relative to something rather weaker than ZFC, and NFUIC + “all Cantorian sets are strongly Cantorian” (never mind exactly what that means) is equiconsistent with ZFC + some reasonably plausible large-cardinal axioms. OK, fine, so there’s a sense in which NFUIC is ZFC-like, at least as regards consistency strength. But NFUIC’s sets are most definitely not the same as ZFC’s sets. NFUIC has a universal set and ZFC doesn’t; ZFC’s sets are the same sizes as their sets-of-singletons and NFUIC’s often aren’t; NFU has lots and lots of urelements and ZFC has just the single empty set; etc. NFUIC is very unlike ZFC despite these relationships in terms of consistency strength.
[EDITED to add:] Here’s an analogy. You get the same computable functions whether you start with (1) Turing machines, (2) register machines, (3) lambda calculus, or (4) Post production systems. But those are still four very different foundations for computing, they suggest quite different possible hardware realizations and different kinds of notation, they have quite different performance characteristics, etc. (The execution times are admittedly all bounded by polynomials in one another. We could add (5) quantum Turing machines, in which case that would no longer be known to be true.) It’s very interesting that these all turn out to be equivalent in power in some sense, but I wouldn’t call that convergence or suggest that it tells us that (e.g.) lambda-calculus terms have any sort of more exalted metaphysical status than they would if it weren’t for that equivalence.
Yes, ZFC is not quite such an isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infintiely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s system NF was the only significant system of set theory that was not known to be equiconsistent with ZFC,
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Yes, ZFC may be not quite such a starkly isolated landmark of thinginess as computability is, which is why I said “a strong tendency”. And anyway, these alternative formalisations of set theory mostly have translations back and forth. Even ZFA (which has sets-within-sets-within-etc infinitely deep) can be modelled in ZFC. It’s not a subject I’ve followed for a long time, but back when I did, Quine’s NF was the only significant system of set theory for which this had not been done. I don’t know if progress has been made on that since.
(ETA: I found this review of NF from 2011. Its consistency was still open then.)
As for computable functions, yes, the different ways of getting at the class have different properties, but that just makes them different roads leading to the same Rome.
Randall Holmes says he has a proof of the consistency of NF relative to ZFC (and in fact something weaker, I think). He’s said this for a while, he’s published a few versions of his proof (mostly different in presentation in the interests of clarity, rather than patching bugs), and I think the general feeling is that he probably does have a proof but it hasn’t yet been thoroughly checked by others. (Who may be holding off because he’s still changing his mind about the best way of writing it down.)
The question is whether the rules of chess have mind-indepedent existence.
Where are these grains, ie the rules of every possible game? Are they in our universe, or some heavenly library of babel?
so how do we cash out the idea that these things are converging on an abstract object , rather than just converging? One way is put forward the counterfactual that if the abstract object were different, then the convergence would occur differently. But that seems rather against the spirit of what you are aiming.
Ask Max Tegmark. :)
I don’t believe in his Level IV multiverse, though. That is, I do draw a distinction between physical and abstract objects.
That they are converging is enough. To quote an old saw variously attributed, in mathematics existence is freedom from contradiction.
If you don’t know what your theory is, why undertake to defend it?
Vague again. Realists think mathematical existence, is a real, literal existence for which non-contradiuction is a criterion, whereas antirealists think mathematical existence is a mere metaphor with no more content than non-contradiction.
Perhaps I should finish with my usual comment that studying philosophy is useful because it allows you to articulate your theories, or, failing that, to notice when there are no clear concepts behind your words.
Vague and empty slogans that might be picked over endlessly, and have been, to no useful purpose. Don’t bother expanding them; it’s a dead end.
Much of the source material has failed to learn that lesson, and is useful in this regard primarily as negative examples. Some philosophers even say so. As a discipline for forcing you to discover the fallibility of subjectively clear and distinct ideas, and to arrive at actually clear ideas that demonstrably work, there are better fields of study, for those who are able to learn. Software design, for example.
Before, you seemed to be pitching in with an opinion on anti-realism versus realism, now you seem to be sayign the whole debate is meaningless. Which is your true belief? It isn’t clear.
Says who? As a software engineer, and philosopher (and scientist), I have found philosophy to be the best training for expressing abstract ideas clearly.
Are you a software engineer? Do you believe software engineering has taught you to be clear?
These two, for example, from opposite ends of the academic/professional spectrum:
E.W. Dijkstra, “The Humble Programmer”
E.S. Raymond, “How To Become A Hacker”
Well, I have not. Which philosophers would you particularly recommend for this purpose? What in philosophy will assist our most gifted fellow humans in thinking previously impossible thoughts, or is worth learning for the profound enlightenment experience I will have when I finally get it?
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts do better. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if the real work was in thinking up a structure and the “stupid implementation” could be left to a few graduate students.
And are either Dijkstra or ESR in a position to directly compare the efficacy of software engineering as a means of clearly expressing philosophical ideas to the efficacy of philosophy as a means of clearly expressing philosophical ideas..ie, do they know anything about philosophy?
It’s not news that when two or more STEM type are gathered together they will recite the Mantra Against the Philosophers, in the expectation of reaping agreement and maybe even applause. It’s just not very significant.
It’s also not news that software engineering teaches you to express software engineering concepts clearly, and it’s not very relevant since the topic is expressing philosophical concepts. Pointing out that someone else is unclear doesn’t make you clear. Pointing about that you can be clear about X doesn’t make you clear about Y.
Getting into conversations where there is mutual commitment to clear communication is the best practice, because you get instant feedback Learning the jargon of philosophy—there are a number of dictionary-style works—is also helpful: after all the jargon is tailored to just the kind of topic you were discussing.
That’s a different topic, but it happens...philosophers have been criticised for entertaining ideas that are too weird, among other things.
That’s a different topic again.
Weirdness is not the same thing as previously impossible thought. In the strongest form of “impossible thought” you will not even understand the claim being made enough that it registers with you.
I’m not sure it makes sense to label people like E.S. Raymond who are proclaim hacker values as STEM types. Raymond isn’t following the popular science narrative of logical positivism that you find with the typical STEM person.
Whatever. It’s about the third or fourth change of topic.
Well, I have not. Which philosophers would you particularly recommend?
These two, for example, from opposite ends of the academic/professional spectrum:
E.W. Dijkstra, “The Humble Programmer”
E.S. Raymond, “How To Become A Hacker”
Software design and implementation has been a large part of my job and my recreations for all of my adult years. I have never taken a course of study in the subject.
It has certainly greatly influenced me in that regard. For example, I was familiar with the fallacy of suggestively named tokens long before reading Eliezer wrote of it on LessWrong, the fallacy of taking the subjective feeling that a task is simple for its actual simplicity, and probably various other things that are now just a part of my mental furniture.
While the lessons are there to be learned, that does not mean that everyone will learn them. I have rolled my eyes many times over what I would call junk XML languages, where the creators have done no more than write down English names for every concept they can think of in some domain of discourse, sprinkle pointy brackets over them, write a DTD, and believe they’ve achieved something. They have not. In the field of procedural humanoid animation, in which I have worked, there have been many attempts to generate animation from a human-written script specifying the movements, but, well, it would take too long to say what I think is wrong with most of them that my own efforts get right. I once heard a distinguished researcher in the field even say “I am not interested in stupid implementation”, as if she could just think up a structure and leave it to a few graduate students to implement.
llyershpitser, no matter how hard you try you will never get this message across on LW. This kind of message is unfortunately too hard ingrained.
I’m not a fan of mathematical Platonism, but physical realists, however hardline, face some very difficult problems regarding the ontologica status of physical law, which make Platonism hard to rule out. (And no, the perenially popular “laws are just descriptions” isn’t a good answer).
P-zombies as a subject worth discussing, or as something that can exist in our univese? But most of the people who discuss PZs don’t think they can exist in our universe. There is some poor quality criticisim of philosophy about as well.
The problems with Bayes are suffcieintly non-obvious to have eluded many or most at LW.
On the one hand, I think that page in specific is actually based on outdated Bayesian methods, and there’s been a lot of good work in Bayesian statistics for complex models and cognitive science in recent years.
On the other hand, I freaking love that website, despite its weirdo Buddhist-philosophical leanings and one or two things it gets Wrong according to my personal high-and-mighty ideologies.
And on the gripping hand, he is very, very right that the way the LW community tends to phrase things in terms of “just Bayes it” is not only a mischaracterization of the wide world of statistics, it’s even an oversimplification of Bayesian statistics as a subfield. Bayes’ Law is just the update/training rule! You also need to discuss marginalization; predictive distributions; maximum-entropy priors, structural simplicity priors, and Bayesian Occam’s Razor, and how those are three different views of Occam’s Razor that have interesting similarities and differences; model selection; the use of Bayesian point-estimates and credible-hypothesis tests for decision-making; equivalent sample sizes; conjugate families; and computational Bayes methods.
Then you’re actually learning and doing Bayesian statistics.
On the miniature nongripping hand, I can’t help but feel that the link between probability, thermodynamics, and information theory means Eliezer and the Jaynesians are probably entirely correct that as a physical fact, real-world event frequencies and movements of information obey Bayes’ Law with respect to the information embodied in the underlying physics, whether or not I can model any of that well or calculate posterior distributions feasibly.
Starting out by expecting a view opposed to your own to be contrarian is a typical form of overconfidence, and not just overconfidence about other people’s opinions.
Sometimes, yes. However, I rather expect that naturalism should be the consensus.
He could have saved himself some trouble by writing “Philosophy: a Partly Diseased Disciplien” or “Philosophy: a Bit of a Curate’s Egg”.
How about:
a link to the article by Luke that you’re talking about
the names of some good current philosophy journals
I think the article Ilya has in mind is this one: Philosophy, a diseased discipline.
I can help with the second request:
The British Journal for the Philosophy of Science
That seems to be entirely analytic philosophy. My problem is that analytical philosophy is culturally irrelevant. Anthropologists, sociologists, art theorists, and artists talk about continental philosophy, Saussurian (!) linguistics, and psychoanalytic theory. The only things they use from analytic philosophy are arguments like Godel’s incompleteness theorem, Wittgenstein’s later stuff, or Quine’s ontological relativism, that they interpret as saying that analytic philosophy doesn’t work.
Analytic philosophers will find the strength to carry on, somehow.
Good for analytic philosphy, but my real concern is with literature. Literature today is captive to bad philosophy. Poetry, even more explicitly so.