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ec429 comments on How to Convince Me That 2 + 2 = 3 - Less Wrong

53 Post author: Eliezer_Yudkowsky 27 September 2007 11:00PM

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Comment author: ec429 14 September 2011 04:48:15AM *  14 points [-]

For a while this confused me, because I incorrectly identified what part of Eliezer's argument I thought was wrong.

Suppose I were to make all those observations suggesting that combining two objects with two objects produced three objects. I would not conclude that 2+2=3, rather I would conclude that objects were not modelled by Peano Arithmetic. (This much has been said by other commenters). But then I only 'know' Peano Arithmetic through the (physical) operation of my own brain.

Here's how to convince me that 2+2=3. Suppose I look at the proof from (peano axioms) to (2+2=4), and suddenly notice that an inference has been made that doesn't follow from the inference rules (say, I notice that the proof says a + (b⁺) = (a+b)⁺ and I know full well that the correct rule is (a⁺)+(b⁺)=(a+b)⁺). I correct this 'error' and follow through to the end of the proof, and conclude the result 2+2=3. What do I do? I consider that this observation is more likely if 2+2=3 than if 2+2=4, and so I update on that. It's still more likely that 2+2=4, because it's more likely that I've made an error this time than that everyone who's analysed that proof before has made an error (or rather, that I have not heard of anyone else spotting this error). But clearly there is something to update on, so my prior probability that 2+2=3 is not zero. However, I also maintain that if in fact the proof of 2+2=4 is correct, then it remains correct whether or not I am convinced of it, whether or not I exist, and even whether or not physical reality exists. So it is a priori true, but my knowledge of it is not a priori knowledge (because the latter does not exist).

I think this is what Eliezer was trying to say with "Unconditional facts are not the same as unconditional beliefs.", but this seems to be glossed over and almost lost within the confusion about earplugs. The article's failure to distinguish between a mathematical theory and a mathematical model (map and territory, possibly?) came very close to obscuring the actual point. This article does not explain how to convince Eliezer that 2+2=3, it explains how to convince Eliezer that PA does not model earplugs - and since the latter is not an a priori truth, it is much less interesting that knowledge of it is not a priori either.

Comment author: komponisto 14 September 2011 05:07:40AM 1 point [-]

The article's failure to distinguish between a mathematical theory and a mathematical model (map and territory, possibly?)

Exactly. This is one of Eliezer's few genuine philosophical mistakes, one which, four years later, he's still making.

Comment author: Eliezer_Yudkowsky 14 September 2011 05:41:38AM 1 point [-]

I know very well the difference between a collection of axioms and a collection of models of which those axioms are true, thank you.

A lot of people seem to have trouble imagining what it means to consider the hypothesis that SS0+SS0 = SSS0 is true in all models of arithmetic, for purposes of deriving predictions which distinguish it from what we should see given the alternative hypothesis that SS0+SS0=SSSS0 is true in all models of arithmetic, thereby allowing internal or external experience to advise you on which of these alternative hypotheses is true.

Comment author: [deleted] 14 September 2011 05:53:52AM 1 point [-]

Reading your essay I wondered whether it would have been more effective if you had chosen bigger numbers than 2, 2, and 3. e.g. "How to convince me that 67+41 = 112."

Comment author: ec429 14 September 2011 06:05:19AM 7 points [-]

That would have been a damn nuisance, because throughout the rest of this comment thread we'd have been writing unhelpfully long strings of Ss. ;)

Comment author: [deleted] 14 September 2011 06:13:55AM 1 point [-]

I was proud of this comment and I comfort myself with your explanation for why it got the response it did.

Comment author: ec429 14 September 2011 06:03:20AM 2 points [-]

I, at least, was not suggesting that you don't know the difference, merely that your article failed to take account of the difference and was therefore confusing and initially unconvincing to me because I was taking account of that difference.

However (and it took me too damn long to realise this; I can't wait for Logic and Set Theory this coming year), I wasn't talking about "models" in the sense that pebbles are a Model of the Theory PA. I was talking in the sense that PA is a model of the behaviour observed in pebbles. If PA fails to model pebbles, that doesn't mean PA is wrong, it just means that pebbles don't follow PA. If a Model of PA exists in which SS0+SS0 = SSS0, then the Theory PA materially cannot prove that SS0+SS0 ≠ SSS0, and if such a proof has been constructed from the axiomata of the Theory then either the proof is in error (exists a step not justified by the inference rules), or the combination of axiomata and inference rules contains a contradiction (which can be rephrased as "under these inference rules, the Theory is not consistent"), or the claimed Model is not in fact a Model at all (in which case one of the axiomata does not, in fact, apply to it).

I should probably write down what I think I know about the epistemic status of mathematics and why I think I know it, because I'm pretty sure I disagree quite strongly with you (and my prior probability of me being right and you being wrong is rather low).

Comment author: Oscar_Cunningham 14 September 2011 09:15:02AM 9 points [-]

Scientists and mathematicians use the word "model" in exactly opposite ways. This is occasionally confusing.

Comment author: komponisto 14 September 2011 06:05:56AM *  1 point [-]

I know very well the difference between a collection of axioms and a collection of models of which those axioms are true, thank you.

Then why do you persist in saying things like "I don't believe in [Axiom X]/[Mathematical Object Y]"? If this distinction that you are so aptly able to rehearse were truly integrated into your understanding, it wouldn't occur to you to discuss whether you have "seen" a particular cardinal number.

I understand the point you wanted to make in this post, and it's a valid one. All the same, it's extremely easy to slip from empiricism to Platonism when discussing mathematics, and parts of this post can indeed be read as betraying that slip (to which you have explicitly fallen victim on other occasions, the most recent being the thread I linked to).

Comment author: Eliezer_Yudkowsky 14 September 2011 06:26:53AM *  10 points [-]

I don't think people really understood what I was talking about in that thread. I would have to write a sequence about

  • the difference between first-order and second-order logic
  • why the Lowenheim-Skolem theorems show that you can talk about integers or reals in higher-order logic but not first-order logic
  • why third-order logic isn't qualitatively different from second-order logic in the same way that second-order logic is qualitatively above first-order logic
  • the generalization of Solomonoff induction to anthropic reasoning about agents resembling yourself who appear embedded in models of second-order theories, with more compact axiom sets being more probable a priori
  • how that addresses some points Wei Dai has made about hypercomputation not being conceivable to agents using Solomonoff induction on computable Cartesian environments, as well as formalizing some of the questions we argue about in anthropic theory
  • why seeing apparently infinite time and apparently continuous space suggests, to an agent using second-order anthropic induction, that we might be living within a model of axioms that imply infinity and continuity
  • why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction, or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for, on the surface of things, than we have evidence favoring the physical existability of models of infinity and continuity, or the mathematical sensibility of talking about the integers or real numbers.
Comment author: [deleted] 14 September 2011 06:33:15AM 3 points [-]

Lowenheim-Skolem, maybe?

But that does not imply that you can't talk about integers or reals in first order logic. And indeed you can talk about integers and real numbers using first-order logic, people do so all the time.

Comment author: Eliezer_Yudkowsky 14 September 2011 06:39:19AM 1 point [-]

Only in the same sense that you can talk about kittens by saying "Those furry things!" There'll always be some ambiguity over whether you're talking about kittens or lions, even though kittens are in fact furry and have all the properties that you can deduce to hold true of furry things.

Comment author: [deleted] 14 September 2011 07:02:33AM 3 points [-]

Not in the same sense at all. All of the numbers that you have ever physically encountered were nameable, definable, computable. Moreover they came to you with algorithms for verifying that one of them was equal to another.

Comment author: Vladimir_Nesov 14 September 2011 10:00:39AM *  2 points [-]

Yes, and that's OK. I suspect you can't do qualitatively better than that (viz ambient set-theoretic universe for second-order logic), but it's still possible (necessary?) to work under this apparent lack of absolute control over what it is you are dealing with. Even though (first order) PA doesn't know what "integers" are, it's still true that the statements it believes valid are true for "integers", it's useful that way (just as AIs or humans are useful for making the world better). It is a device that perceives some of the properties of the object we study, but not all, not enough to rebuild it completely. (Other devices can form similarly imperfect pictures of the object of study and its relationship with the device perceiving it, or of themselves perceiving this process, or of the object of study being affected by behavior of some of these devices.)

Likewise, we may fail to account for all worlds that we might be affecting by our decisions, but we mostly care about (or maybe rather have non-negligible consequentialist control over) "real world" (or worlds), whatever this is, and it's true that our conclusions capture some truth about this "real world", even if it's genuinely impossible for us to ever know completely what it is. (We of course "know" plenty more than was ever understood, and it's a big question how to communicate to a FAI what we do know.)

Comment author: CronoDAS 14 September 2011 06:45:51AM 1 point [-]

In other words, a first uncountable ordinal may be perfectly good math, but it's not physics?

Comment author: Eliezer_Yudkowsky 14 September 2011 06:53:06AM 0 points [-]

I don't believe it's good math until it becomes possible to talk about the first uncountable ordinal, in the way that you can talk about the integers. Any first-order theory of the integers, like first-order PA, will have some models containing supernatural numbers, but there are many different sorts of models of supernatural numbers, you couldn't talk about the supernaturals the way you can talk about 3 or the natural numbers. My skepticism about "the first uncountable ordinal" is that there would not exist any canonicalizable mathematical object - nothing you could ever pin down uniquely - that would ever contain the first uncountable ordinal inside it, because of the indefinitely extensible character of well-ordering. This is a sort of skepticism of Platonic existence - when that which you thought you wanted to talk about can never be pinned down even in second-order logic, nor in any other language which does not permit of paradox.

Comment author: Sniffnoy 14 September 2011 07:00:38AM 6 points [-]

You seem to keep forgetting that the whole notion of "second-order logic" does not make sense without some ambient set theory. (Unless I am greatly misunderstanding how second-order logic works?) And if you have that, then you can pin down the natural numbers (and the first uncountable ordinal) in first-order terms in this larger theory.

Comment author: Eliezer_Yudkowsky 14 September 2011 07:18:37AM 0 points [-]

Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense. It's just that the ambient models in the second-order theory include collections of possible predicates of any objects that get predicates attached, or if you prefer, people who speak in second-order logic think that it makes as much sense to say "all possible collections that include some objects and exclude others, but still include and exclude only individual objects" as "all objects".

Comment author: [deleted] 14 September 2011 07:24:36AM 6 points [-]

Only to the same degree that first-order logic requires an ambient group of models (not necessarily sets) to make sense.

Well, it makes sense to me without any models. I can compute, prove theorems, verify proofs of theorems and so on happily without ever producing a "model" for the natural numbers in toto, whatever that could mean.

Comment author: CronoDAS 14 September 2011 07:55:29AM 0 points [-]

Hmmm...

::goes and learns some more math from Wikipedia::

Okay... I now know what an ordinal number actually is. And I'm trying to make more sense out of your comment...

So, re-reading this:

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for

So if I understand you correctly, you don't trust anything that can't be defined up to isomorphism in second-order logic, and "the set of all countable ordinals" is one of those things?

(I never learned second order logic in college...)

Comment author: komponisto 14 September 2011 07:37:31AM 6 points [-]

Everything sounded perfectly good until the last bullet:

why believing that things like a first uncountable ordinal can contain reality-fluid in the same way as the wavefunction

ERROR: CATEGORY. "Wavefunction" is not a mathematical term, it is a physical term. It's a name you give to a mathematical object when it is being used to model the physical world in a particular way, in the specific context of that modeling-task. The actual mathematical object being used as the wavefunction has a mathematical existence totally apart from its physical application, and that mathematical existence is of the exact same nature as that of the first uncountable ordinal; the (mathematical) wavefunction does not gain any "ontological bonus points" for its role in physics.

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness

Pinning down a single model up to isomorphism might be a nice property for a set of axioms to have, but it is not "reality-conferring": there are two groups of order 4 up to isomorphism, while there is only one of order 3; yet that does not make "group of order 3" a "more real" mathematical object than "group of order 4".

Comment author: ata 14 September 2011 10:15:57AM *  6 points [-]

I would like very very much to read that sequence. Might it be written at some point?

Comment author: ec429 14 September 2011 06:34:25AM 4 points [-]

Hmm, funny you should treat "I don't believe in [Mathematical Object Y]" as Platonism. I generally characterise my 'syntacticism' (wh. I intend to explain more fully when I understand it the hell myself) as a "Platonic Formalism"; it is promiscuously inclusive of Mathematical Objects. If you can formulate a set of behaviours (inference rules) for it, then it has an existing Form - and that Form is the formalism (or... syntax) that encapsulates its behaviour. So in a sense, uncountable cardinals don't exist - but the theory of uncountable cardinals does exist; similarly, the theory of finite cardinals exists but the number '2' doesn't.

This is of course bass-ackwards from a map-territory perspective; I am claiming that the map exists and the territory is just something we naïvely suppose ought to exist. After all, a map of non-existent territory is observationally equivalent to a map of manifest reality; unless you can observe the actual territory you can't distinguish the two. Taking as assumption that the observe() function always returns an object Map, the idea that there is a territory gets Occamed out.

There is a good reason why I should want to do something so ontologically bizarre: by removing referents, and semantics, and manifest reality; by retaining only syntax, and rejecting the suggestion that one logic really "models" another, we finally solve the problems of Gödel (I'm a mathematician, not a philosopher, so I'm allowed to invoke Gödel without losing automatically) and the infinite descent when we say "first order logic is consistent because second-order logic proves it so, and we can believe second-order logic because third-order logic proves it consistent, and...". When all you are doing is playing symbol games stripped of any semantics, "P ∧ ¬P" is just a string, and who cares if you can derive it from your axiomata? It only stops being a string when you apply your symbol games to what you unknowingly label as "manifest reality", when you (essentially) claim that symbol game A (Peano arithmetic) models symbol game B (that part of the Physics game that deals with the objects you've identified as pebbles).

Platonism is not a means of excluding a mathematical object because it's not one of the Forms; it is a means of allowing any mathematical object to have a Form whether you like it or not. I don't believe in God, but there still exists a Form for a mathematical object that looks a lot like "a universe in which God exists". It's just that conceiving of a possible world only makes an arrow from your world to its, not an arrow in the reverse direction, hence why "A perfect God would have the quality of existence" is such a laughable non-starter :) What if someone broke out of a hypothetical situation in your room right now?

Comment author: [deleted] 17 September 2011 02:06:53AM *  0 points [-]

(I'm a mathematician, not a philosopher, so I'm allowed to invoke Gödel without losing automatically)

As a fellow mathematician, I want to point out that it doesn't mean you win automatically, either. Just look at Voevodsky's recent FOM talk at the IAS.

Comment author: ec429 17 September 2011 04:59:19PM 0 points [-]

Well, of course I don't win automatically. It's just that there's a kind of Godwin's Law of philosophy, whereby the first to invoke Gödel loses by default.

Comment author: Andreas_Giger 14 September 2011 06:20:58AM *  0 points [-]

A lot of people seem to have trouble imagining what it means to consider the hypothesis that SS0+SS0 = SSS0 is true in all models of arithmetic

Maybe I'm misinterpreting you, but could you explain how any non-symmetric equation can possibly be true in all models of arithmetic?

Comment author: Eliezer_Yudkowsky 14 September 2011 06:46:09AM 2 points [-]

SS0 isn't a free variable like "x", it is, in any given model of arithmetic, the unique object related by the successor relation to the unique object related by the successor relation to the unique object which is not related by the successor relation to any object, which is how mathematicians say "Two".

Comment author: ec429 14 September 2011 06:59:07AM 2 points [-]

Although as a mathmo myself I should point out that, to save time, we usually pronounce it "Two". :)

Comment author: Andreas_Giger 14 September 2011 07:10:50AM *  1 point [-]

I am quite familiar with TNT. However either you are talking about models of arithmetic based on peano axioms, in which case e.g. SS0 + SS0 = SSS0 simply cannot be true, for it contradicts these axioms and if both the peano axioms and said equation were true, you wouldn't have a model of arithmetics; or (what I'm assuming) you are actually talking about non-peano arithmetics, in which case there is no compelling reason why any equation of this kind should generally be true anyway.

On another note, it seems that bayesianism is heavily based on peano arithmetic, so refuting peano arithmetic by means of bayesianism seems like refuting bayesianism rather than refuting peano arithmetic, at least to me.

Comment author: Tyrrell_McAllister 14 September 2011 08:03:35PM *  7 points [-]

Maybe I'm misinterpreting you, but could you explain how any non-symmetric equation can possibly be true in all models of arithmetic?

The purpose of the article is only to describe some subjective experiences that would cause you to conclude that SS0+SS0 = SSS0 is true in all models of arithmetic. But Eliezer can only describe certain properties that those subjective experiences would have. He can't make you have the experiences themselves.

So, for example, he could say that one such experience would conform to the following description: "You count up all the S's on one side of the equation, and you count up all the S's on the other side of the equation, and you find yourself getting the same answer again and again. You show the equation to other people, and they get the same answer again and again. You build a computer from scratch to count the S's on both sides, and it says that there are the same number again and again."

Such a description gives some features of an experience. The description provides a test that you could apply to any given experience and answer the question "Does this experience satisfy this description or not?" But the description is not like one in a novel, which, ideally, would induce you to have the experience, at least in your imagination. That is a separate and additional task beyond what this post set out to accomplish.

Comment author: Andreas_Giger 14 September 2011 09:55:33PM *  1 point [-]

Yes, I am aware of that. However, I don't think two pebbles on the table plus another two pebbles on the table resulting in three pebbles on the table could cause anyone sane to conclude that SS0 + SS0 = SSS0 is true in all models of arithmetic. In order to be convinced of that, you would have to assign "PA doesn't apply to pebbles" a lower prior probability than "PA is wrong".

The statement "PA applies to pebbles" (or anything else for that matter) doesn't follow of the peano axioms in any way and is therefore not part of peano arithmetic. So what if peano arithmetic doesn't apply to pebbles, there are other arithmetics that don't either, and that doesn't make them any wrong. You're using them everyday in situations where they do apply.

A mathematical theory doesn't consist of beliefs that are based on evidence; it's an axiomatic system. There is no way any real-life situation could convince me that PA is false. Saying "SS0 + SS0 = SSS0 is true in all models of arithmetic" sounds like "0 = S0" or "garble asdf qwerty sputz" to me. It just doesn't make any sense.

Mathematics has nothing to do with experience, only to what extent mathematics applies to reality does.

Comment author: Tyrrell_McAllister 14 September 2011 10:08:03PM 1 point [-]

Mathematics has nothing to do with experience.

That you have certain mathematical beliefs has a lot to do with the experiences that you have had. This applies in particular to your beliefs about what the theorems of PA are.

Comment author: Andreas_Giger 14 September 2011 10:11:55PM *  1 point [-]

Sorry, I edited the statement in question right before you posted that because I anticipated a similar reaction. However, you're still wrong. It has only to do with my beliefs to what extent peano arithmetic applies to reality, which is something completely different.

Edit: Ok, you're probably not wrong, but it rather seems we are talking about different things when we say "mathematical beliefs". Whether peano arithmetic applies to reality is not a mathematical belief for me.

Comment author: Andreas_Giger 14 September 2011 10:17:14PM 0 points [-]

And another thing: It might be possible that if peano arithmetic didn't apply to reality I wouldn't have any beliefs about peano arithmetic because I might not even think of it. However there is no way I could establish the peano axioms and then believe that SS0 + SS0 = SSS0 is true within peano arithmetic. It's just not possible.

Comment author: Tyrrell_McAllister 14 September 2011 10:18:39PM *  1 point [-]

Consider the experiences that you have had while reading and thinking about proofs within PA. (The experience of devising and confirming a proof is just a particular kind of experience, after all.) Are you saying that the contents of those experiences have had nothing to do with the beliefs that you have formed about what the theorems of PA are?

Suppose that those experiences had been systematically different in a certain way. Say that you consistently made a certain kind of mistake while confirming PA proofs, so that certain proofs seemed to be valid to you that don't seem valid to you in reality. Would you not have arrived at different beliefs about what the theorems of PA are?

That is the sense in which your beliefs about what the theorems of PA are depend on your experiences.

Comment author: Andreas_Giger 14 September 2011 10:23:47PM 0 points [-]

I'm not sure I 100% understand what you're saying, but the question "which beliefs will I end up with if logical reasoning itself is flawed" is of little interest to me.

Comment author: [deleted] 14 September 2011 06:08:07AM *  2 points [-]

Maybe "earplugs do not model PA," not the other way around? (Edit: just saw this excellent clarification.)

Number-handling is an older science than Peano arithmetic, and especially older than model theory. The numbers 2 and 3 would "exist" even if PA were shown to have no models. At least, the notation 2 and 3 would still be relevant to things that really exist.

It is very easily verified that 2 + 2 does not equal 3, but not effortlessly verified. It takes a positive amount of effort to verify it, and there is a positive amount risk of having made a mistake while doing so.

Comment author: hairyfigment 15 September 2011 06:36:57PM 10 points [-]

You're over-thinking this. Take a look at this real-world example of a "neurological fault":

Now I knew where I was. Soon I would come to interchange 27 with its two ramps, A and B. B led away from my destination and A directly into it. It had always struck me as strange that one reached 27B before 27A. I recalled drawing that on a map to give to someone who was going to visit me. My breathing has returned to normal and my panic had disappeared. I come up to the first sign for the interchange.

"27A"

I could hardly breathe. That was not possible. 27A was after 27B. I knew that. I considered for a moment the possibility that on the previous night, shortly after I drove on this very highway, construction workers had descended en masse on the interchanges and somehow moved them. That seemed far more possible than that my clear (and detailed) memory could be so wrong. 27A looked exactly as I remembered, except that now I could see 27B clearly in the distance and in the past I had to turn my head to see it.

I exited on the ramp that I knew wasn’t there twenty-four hours previously to find myself on a well-remembered road. And soon I was home.

Now imagine that happening on a massive scale. Say that right after reading this comment you experience evidence, like that which the OP describes, going against your memories of what happens when you put one pair of objects next to another pair. (This includes "mental confirmation that XXX - XX = XX", though not a formal proof in PA.) Would that make you doubt your memories of what PA says? Would you want to check the proof that it says (2+2=4) in case your current memory is lying about that as well?