Followup to: Causal Reference, Proofs, Implications and Models

The fact that one apple added to one apple invariably gives two apples helps in the teaching of arithmetic, but has no bearing on the truth of the proposition that 1 + 1 = 2.

-- James R. Newman, The World of Mathematics

Previous meditation 1: If we can only meaningfully talk about parts of the universe that can be pinned down by chains of cause and effect, where do we find the fact that 2 + 2 = 4? Or did I just make a meaningless noise, there? Or if you claim that "2 + 2 = 4"isn't meaningful or true, then what alternate property does the sentence "2 + 2 = 4" have which makes it so much more useful than the sentence "2 + 2 = 3"?

Previous meditation 2: It has been claimed that logic and mathematics is the study of which conclusions follow from which premises. But when we say that 2 + 2 = 4, are we really just assuming that? It seems like 2 + 2 = 4 was true well before anyone was around to assume it, that two apples equalled two apples before there was anyone to count them, and that we couldn't make it 5 just by assuming differently.

Speaking conventional English, we'd say the sentence 2 + 2 = 4 is "true", and anyone who put down "false" instead on a math-test would be marked wrong by the schoolteacher (and not without justice).

But what can make such a belief true, what is the belief about, what is the truth-condition of the belief which can make it true or alternatively false? The sentence '2 + 2 = 4' is true if and only if... what?

In the previous post I asserted that the study of logic is the study of which conclusions follow from which premises; and that although this sort of inevitable implication is sometimes called "true", it could more specifically be called "valid", since checking for inevitability seems quite different from comparing a belief to our own universe. And you could claim, accordingly, that "2 + 2 = 4" is 'valid' because it is an inevitable implication of the axioms of Peano Arithmetic.

And yet thinking about 2 + 2 = 4 doesn't really feel that way. Figuring out facts about the natural numbers doesn't feel like the operation of making up assumptions and then deducing conclusions from them. It feels like the numbers are just out there, and the only point of making up the axioms of Peano Arithmetic was to allow mathematicians to talk about them. The Peano axioms might have been convenient for deducing a set of theorems like 2 + 2 = 4, but really all of those theorems were true about numbers to begin with. Just like "The sky is blue" is true about the sky, regardless of whether it follows from any particular assumptions.

So comparison-to-a-standard does seem to be at work, just as with physical truth... and yet this notion of 2 + 2 = 4 seems different from "stuff that makes stuff happen". Numbers don't occupy space or time, they don't arrive in any order of cause and effect, there are no events in numberland.

MeditationWhat are we talking about when we talk about numbers? We can't navigate to them by following causal connections - so how do we get there from here?

...
...
...

"Well," says the mathematical logician, "that's indeed a very important and interesting question - where are the numbers - but first, I have a question for you. What are these 'numbers' that you're talking about? I don't believe I've heard that word before."

Yes you have.

"No, I haven't. I'm not a typical mathematical logician; I was just created five minutes ago for the purposes of this conversation. So I genuinely don't know what numbers are."

But... you know, 0, 1, 2, 3...

"I don't recognize that 0 thingy - what is it? I'm not asking you to give an exact definition, I'm just trying to figure out what the heck you're talking about in the first place."

Um... okay... look, can I start by asking you to just take on faith that there are these thingies called 'numbers' and 0 is one of them?

"Of course! 0 is a number. I'm happy to believe that. Just to check that I understand correctly, that does mean there exists a number, right?"

Um, yes. And then I'll ask you to believe that we can take the successor of any number. So we can talk about the successor of 0, the successor of the successor of 0, and so on. Now 1 is the successor of 0, 2 is the successor of 1, 3 is the successor of 2, and so on indefinitely, because we can take the successor of any number -

"In other words, the successor of any number is also a number."

Exactly.

"And in a simple case - I'm just trying to visualize how things might work - we would have 2 equal to 0."

What? No, why would that be -

"I was visualizing a case where there were two numbers that were the successors of each other, so SS0 = 0. I mean, I could've visualized one number that was the successor of itself, but I didn't want to make things too trivial -"

No! That model you just drew - that's not a model of the numbers.

"Why not? I mean, what property do the numbers have that this model doesn't?"

Because, um... zero is not the successor of any number. Your model has a successor link from 1 to 0, and that's not allowed.

"I see! So we can't have SS0=0. But we could still have SSS0=S0."

What? How -

No! Because -

(consults textbook)

- if two numbers have the same successor, they are the same number, that's why! You can't have 2 and 0 both having 1 as a successor unless they're the same number, and if 2 was the same number as 0, then 1's successor would be 0, and that's not allowed!  Because 0 is not the successor of any number!

"I see. Oh, wow, there's an awful lot of numbers, then. The first chain goes on forever."

It sounds like you're starting to get what I - wait. Hold on. What do you mean, the first chain -

"I mean, you said that there was at least one start of an infinite chain, called 0, but -"

I misspoke. Zero is the only number which is not the successor of any number.

"I see, so any other chains would either have to loop or go on forever in both directions."

Wha?

"You said that zero is the only number which is not the successor of any number, that the successor of every number is a number, and that if two numbers have the same successor they are the same number. So, following those rules, any successor-chains besides the one that start at 0 have to loop or go on forever in both directions -"

There aren't supposed to be any chains besides the one that starts at 0! Argh! And now you're going to ask me how to say that there shouldn't be any other chains, and I'm not a mathematician so I can't figure out exactly how to -

"Hold on! Calm down. I'm a mathematician, after all, so I can help you out. Like I said, I'm not trying to torment you here, just understand what you mean. You're right that it's not trivial to formalize your statement that there's only one successor-chain in the model. In fact, you can't say that at all inside what's called first-order logic. You have to jump to something called second-order logic that has some remarkably different properties (ha ha!) and make the statement there."

What the heck is second-order logic?

"It's the logic of properties! First-order logic lets you quantify over all objects - you can say that all objects are red, or all objects are blue, or 'x: red(x)→¬blue(x)', and so on. Now, that 'red' and 'blue' we were just talking about - those are properties, functions which, applied to any object, yield either 'true' or 'false'. A property divides all objects into two classes, a class inside the property and a complementary class outside the property. So everything in the universe is either blue or not-blue, red or not-red, and so on. And then second-order logic lets you quantify over properties - instead of looking at particular objects and asking whether they're blue or red, we can talk about properties in general - quantify over all possible ways of sorting the objects in the universe into classes. We can say, 'For all properties P', not just, 'For all objects X'."

 

 

 

Okay, but what does that have to do with saying that there's only one chain of successors?

"To say that there's only one chain, you have to make the jump to second-order logic, and say that for all properties P, if P being true of a number implies P being true of the successor of that number, and P is true of 0, then P is true of all numbers."

Um... huh. That does sound reminiscent of something I remember hearing about Peano Arithmetic. But how does that solve the problem with chains of successors?

"Because if you had another separated chain, you could have a property P that was true all along the 0-chain, but false along the separated chain. And then P would be true of 0, true of the successor of any number of which it was true, and not true of all numbers."

I... huh. That's pretty neat, actually. You thought of that pretty fast, for somebody who's never heard of numbers.

"Thank you! I'm an imaginary fictionalized representation of a very fast mathematical reasoner."

Anyway, the next thing I want to talk about is addition. First, suppose that for every x, x + 0 = x.  Next suppose that if x + y = z, then x + Sy = Sz -

"There's no need for that. We're done."

What do you mean, we're done?

"Every number has a successor. If two numbers have the same successor, they are the same number. There's a number 0, which is the only number that is not the successor of any other number. And every property true at 0, and for which P(Sx) is true whenever P(x) is true, is true of all numbers. In combination, those premises narrow down a single model in mathematical space, up to isomorphism. If you show me two models matching these requirements, I can perfectly map the objects and successor relations in them. You can't add any new object to the model, or subtract an object, without violating the axioms you've already given me. It's a uniquely identified mathematical collection, the objects and their structure completely pinned down. Ergo, there's no point in adding any more requirements. Any meaningful statement you can make about these 'numbers', as you've defined them, is already true or already false within that pinpointed model - its truth-value is already semantically implied by the axioms you used to talk about 'numbers' as opposed to something else. If the new axiom is already true, adding it won't change what the previous axioms semantically imply."

Whoa. But don't I have to define the + operation before I can talk about it?

"Not in second-order logic, which can quantify over relations as well as properties. You just say: 'For every relation R that works exactly like addition, the following statement Q is true about that relation.' It would look like, ' relations R: (∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, Sy, z)↔R(Sx, y, z))) → Q)', where Q says whatever you meant to say about +, using the token R. Oh, sure, it's more convenient to add + to the language, but that's a mere convenience - it doesn't change which facts you can prove. Or to say it outside the system: So long as I know what numbers are, you can just explain to me how to add them; that doesn't change which mathematical structure we're already talking about."

...Gosh. I think I see the idea now. It's not that 'axioms' are mathematicians asking for you to just assume some things about numbers that seem obvious but can't be proven. Rather, axioms pin down that we're talking about numbers as opposed to something else.

"Exactly. That's why the mathematical study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms. When you formalize logic into syntax, and prove theorems like '2 + 2 = 4' by syntactically deriving new sentences from the axioms, you can safely infer that 2 + 2 = 4 is semantically implied within the mathematical universe that the axioms pin down. And there's no way to try to 'just study the numbers without assuming any axioms', because those axioms are how you can talk about numbers as opposed to something else. You can't take for granted that just because your mouth makes a sound 'NUM-burz', it's a meaningful sound. The axioms aren't things you're arbitrarily making up, or assuming for convenience-of-proof, about some pre-existent thing called numbers. You need axioms to pin down a mathematical universe before you can talk about it in the first place. The axioms are pinning down what the heck this 'NUM-burz' sound means in the first place - that your mouth is talking about 0, 1, 2, 3, and so on."

Could you also talk about unicorns that way?

"I suppose. Unicorns don't exist in reality - there's nothing in the world that behaves like that - but they could nonetheless be described using a consistent set of axioms, so that it would be valid if not quite true to say that if a unicorn would be attracted to Bob, then Bob must be a virgin. Some people might dispute whether unicorns must be attracted to virgins, but since unicorns aren't real - since we aren't locating them within our universe using a causal reference - they'd just be talking about different models, rather than arguing about the properties of a known, fixed mathematical model. The 'axioms' aren't making questionable guesses about some real physical unicorn, or even a mathematical unicorn-model that's already been pinpointed; they're just fictional premises that make the word 'unicorn' talk about something inside a story."

But when I put two apples into a bowl, and then put in another two apples, I get four apples back out, regardless of anything I assume or don't assume. I don't need any axioms at all to get four apples back out.

"Well, you do need axioms to talk about four, SSSS0, when you say that you got 'four' apples back out. That said, indeed your experienced outcome - what your eyes see - doesn't depend on what axioms you assume. But that's because the apples are behaving like numbers whether you believe in numbers or not!"

The apples are behaving like numbers? What do you mean? I thought numbers were this ethereal mathematical model that got pinpointed by axioms, not by looking at the real world.

"Whenever a part of reality behaves in a way that conforms to the number-axioms - for example, if putting apples into a bowl obeys rules, like no apple spontaneously appearing or vanishing, which yields the high-level behavior of numbers - then all the mathematical theorems we proved valid in the universe of numbers can be imported back into reality. The conclusion isn't absolutely certain, because it's not absolutely certain that nobody will sneak in and steal an apple and change the physical bowl's behavior so that it doesn't match the axioms any more. But so long as the premises are true, the conclusions are true; the conclusion can't fail unless a premise also failed. You get four apples in reality, because those apples behaving numerically isn't something you assume, it's something that's physically true. When two clouds collide and form a bigger cloud, on the other hand, they aren't behaving like integers, whether you assume they are or not."

But if the awesome hidden power of mathematical reasoning is to be imported into parts of reality that behave like math, why not reason about apples in the first place instead of these ethereal 'numbers'?

"Because you can prove once and for all that in any process which behaves like integers, 2 thingies + 2 thingies = 4 thingies. You can store this general fact, and recall the resulting prediction, for many different places inside reality where physical things behave in accordance with the number-axioms. Moreover, so long as we believe that a calculator behaves like numbers, pressing '2 + 2' on a calculator and getting '4' tells us that 2 + 2 = 4 is true of numbers and then to expect four apples in the bowl. It's not like anything fundamentally different from that is going on when we try to add 2 + 2 inside our own brains - all the information we get about these 'logical models' is coming from the observation of physical things that allegedly behave like their axioms, whether it's our neurally-patterned thought processes, or a calculator, or apples in a bowl."

I... think I need to consider this for a while.

"Be my guest! Oh, and if you run out of things to think about from what I've said already -"

Hold on.

"- try pondering this one. Why does 2 + 2 come out the same way each time? Never mind the question of why the laws of physics are stable - why is logic stable? Of course I can't imagine it being any other way, but that's not an explanation."

Are you sure you didn't just degenerate into talking bloody nonsense?

"Of course it's bloody nonsense. If I knew a way to think about the question that wasn't bloody nonsense, I would already know the answer."


Meditation for next time:

Humans need fantasy to be human.

"Tooth fairies? Hogfathers? Little—"

Yes. As practice. You have to start out learning to believe the little lies.

"So we can believe the big ones?"

Yes. Justice. Mercy. Duty. That sort of thing.

"They're not the same at all!"

You think so? Then take the universe and grind it down to the finest powder and sieve it through the finest sieve and then show me one atom of justice, one molecule of mercy.

- Susan and Death, in Hogfather by Terry Pratchett

So far we've talked about two kinds of meaningfulness and two ways that sentences can refer; a way of comparing to physical things found by following pinned-down causal links, and logical reference by comparison to models pinned-down by axioms. Is there anything else that can be meaningfully talked about? Where would you find justice, or mercy?


Mainstream status.

Part of the sequence Highly Advanced Epistemology 101 for Beginners

Next post: "Causal Universes"

Previous post: "Proofs, Implications, and Models"

Logical Pinpointing
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"- try pondering this one. Why does 2 + 2 come out the same way each time? Never mind the question of why the laws of physics are stable - why is logic stable? Of course I can't imagine it being any other way, but that's not an explanation."

Nothing in the process described, of pinpointing the natural numbers, makes any reference to time. That is why it is temporally stable: not because it has an ongoing existence which is mysteriously unaffected by the passage of time, but because time has no connection with it. Whenever you look at it, it's the same, identical thing, not a later, miraculously preserved version of the thing.

What if 2 + 2 varies over something other than time that nonetheless correlates with time in our universe? Suppose 2 + 2 comes out to 4 the first 1 trillion times the operation is performed by humans, and to 5 on the 1 trillion and first time.

I suppose you could raise the same explanation: the definition of 2 + 2 makes no reference to how many times it has been applied. I believe the same can be said for any other reason you may give for why 2 + 2 might cease to equal 4.

Where that is the case, your method of mapping from the reality to arithmetic is not a good model of that process - no more, no less.

1momom2
Seeing as the above response wasn't very upvoted, I'll try to explain in simpler terms. If 2+2 comes out 5 the one-thrillionth-and-first time we compute it, then our calculation does not match numbers. ... which we can tell because? ...and writing this now I realize why the answer was more upvoted, because this is circular reasoning. ':-s Sorry, I have no clue.
8Peterdjones
I couldn't agree more. The timelessness of maths should be read negatively, as indepence of anything else, not as dependence on a timeless realm.
5ArisKatsaris
I love the elegance of this answer, upvoting.
0CCC
I have recently had a thought relevant to the topic; an operation that is not stable. In certain contexts, the operation d is used, where XdY means "take a set of X fair dice, each die having Y sides (numbered 1 to Y), and throw them; add together the numbers on the uppermost faces". Using this definition, 2d2 has value '2' 25% of the time, value '3' 50% of the time, and value '4' 25% of the time. The procedure is always identical, and so there's nothing in the process which makes any reference to time, but the result can differ (though note that 'time' is still not a parameter in that result). If the operation '+' is replaced by the operation 'd' - well, then that is one other way that can be imagined. Edited to add: It has been pointed out that XdY is a constant probability distribution. The unstable operation to which I refer is the operation of taking a single random integer sample, in a fair manner, from that distribution.
5hirvinen
The random is not in the dice, it is in the throw, and that procedure is never identical. Also, XdY is a distribution, always the same, and the dice are just a relatively fair way of picking a sample.
2JoachimSchipper
Aren't you just confusing distributions (2d2) and samples ('3') here?
0CCC
Thank you, I shall suitably edit my post.
0Eliezer Yudkowsky
But the question isn't, "Why don't they change over time," but rather, "why are they the same on each occasion". It makes no reference to occasion? Sure, but even so, why doesn't 2 + 2 = a random number each time? Why is the same identical thing the same?

I'm not sure what the etiquette is of responding to retracted comments, but I'll have a go at this one.

Why is the same identical thing the same?

That's what I mean when I say they are identical. It's not another, separate thing, existing on a separate occasion, distinct from the first but standing in the relation of identity to it. In mathematics, you can step into the same river twice. Even aliens in distant galaxies step into the same river.

However, there is something else involved with the stability, which exists in time, and which is capable of being imperfectly stable: oneself. 2+2=4 is immutable, but my judgement that 2+2 equals 4 is mutable, because I change over time. If it seems impossible to become confused about 2+2=4, just think of degenerative brain diseases. Or being asleep and dreaming that 2+2 made 5.

2[anonymous]
So the question becomes, "If "2+2" is just another way of saying "4", what is the point of having two expressions for it?" My answer: As humans, we often desire to split a group of large, distinct objects into smaller groups of large, distinct objects, or to put two smaller groups of large, distinct, objects, together. So, when we say "2 + 2 = 4", what we are really expressing is that a group of 4 objects can be transformed into a group of 2 objects and another group of 2 objects, by moving the objects apart (and vice versa). Sharing resources with fellow humans is fundamental to human interaction. The reason I say, "large, distinct objects" is that the rules of addition do not hold for everything. For example, when you add "1" particle of matter to "1" particle of antimatter, you get "0" particles of both matter and antimatter. Numbers, and, yes, even logic, only exist fundamentally in the mind. They are good descriptions that correspond to reality. The soundness theorem for logic (which is not provable in the same logic it is describing) is what really begins to hint at logic's correspondence to the real world. The soundness theorem relies on the fact that all of the axioms are true and that inference rules are truth-preserving. The Peano axioms and logic are useful because, given the commonly known meaning we assign to the symbols of those systems, the axioms do properly describe our observations of reality and the inference rules do lead to conclusions that continue to correspond to our observations of reality (in (one of) the correct domain(s), groups of large, distinct, objects). We observe that quantity is preserved regardless of grouping; this is the associative property (here's another way of looking at it). The mathematical proof of the soundness theorem is useless for convincing the hard skeptic, because it uses mathematical induction itself! The principle of mathematical induction is called such because it was formulated inductively. When it comes to

Mainstream status:

The presentation of the natural numbers is meant to be standard, including the (well-known and proven) idea that it requires second-order logic to pin them down. There's some further controversy about second-order logic which will be discussed in a later post.

I've seen some (old) arguments about the meaning of axiomatizing which did not resolve in the answer, "Because otherwise you can't talk about numbers as opposed to something else," so AFAIK it's theoretically possible that I'm the first to spell out that idea in exactly that way, but it's an obvious-enough idea and there's been enough debate by philosophically inclined mathematicians that I would be genuinely surprised to find this was the case.

On the other hand, I've surely never seen a general account of meaningfulness which puts logical pinpointing alongside causal link-tracing to delineate two different kinds of correspondence within correspondence theories of truth. To whatever extent any of this is a standard position, it's not nearly widely-known enough or explicitly taught in those terms to general mathematicians outside model theory and mathematical logic, just like the standard position on "proof". Nor does any of it appear in the S. E. P. entry on meaning.

[-]Benya170

Very nice post!

Bug: Higher-order logic (a standard term) means "infinite-order logic" (not a standard term), not "logic of order greater 1" (also not a standard term). (For whatever reason, neither the Wikipedia nor the SEP entry seem to come out and say this, but every reference I can remember used the terms like that, and the usage in SEP seems to imply it too, e.g. "This second-order expressibility of the power-set operation permits the simulation of higher-order logic within second order.")

[-]Dan123110

A few points:

i) you don't actually need to jump directly to second order logic in to get a categorical axiomatization of the natural numbers. There are several weaker ways to do the job: L_omega_omega (which allows infinitary conjunctions), adding a primitive finiteness operator, adding a primitive ancestral operator, allowing the omega rule (i.e. from the infinitely many premises P(0), P(1), ... P(n), ... infer AnP(n)). Second order logic is more powerful than these in that it gives a quasi categorical axiomatization of the universe of sets (i.e. of any two models of ZFC_2, they are either isomorphic or one is isomorphic to an initial segment of the other).

ii) although there is a minority view to the contrary, it's typically thought that going second order doesn't help with determinateness worries (i.e. roughly what you are talking about with regard to "pinning down" the natural numbers). The point here is that going second order only works if you interpret the second order quantifiers "fully", i.e. as ranging over the whole power set of the domain rather than some proper subset of it. But the problem is: how can we rule out non-full interpretations of the quan... (read more)

3VAuroch
Generally, things being identical up to isomorphism is considered to make them the same thing in all senses that matter. If something has all the same properties as the natural numbers, in every respect and every particular, then that's no different from merely changing the names. This is a pretty basic mathematical concept, and that you aren't familiar with it makes me question the rest of this comment as well.
5bryjnar
I think philosophers who think that the categoricity of second-order Peano arithmetic allows us to refer to the natural numbers uniquely tend to also reject the causal theory of reference, precisely because the causal theory of reference is usually put as requiring all reference to be causally guided. Among those, lots of people more-or-less think that references can be fixed by some kinds of description, and I think logical descriptions of this kind would be pretty uncontroversial. OTOH, for some reason everyone in philosophy of maths is allergic to second-order logic (blame Quine), so the categoricity argument doesn't always hold water. For some discussion, there's a section in the SEP entry on Philosophy of Mathematics. (To give one of the reasons why people don't like SOL: to interpret it fully you seem to need set theory. Properties basically behave like sets, and so you can make SOL statements that are valid iff the Continuum Hypothesis is true, for example. It seems wrong that logic should depend on set theory in this way.)
7Polymath
This is a facepalm "Duh" moment, I hear this criticism all the time but it does not mean that "logic" depends on "set theory". There is a confusion here between what can be STATED and what can be KNOWN. The criticism only has any force if you think that all "logical truths" ought to be recognizable so that they can be effectively enumerated. But the critics don't mind that for any effective enumeration of theorems of arithmetic, there are true statements about integers that won't be included -- we can't KNOW all the true facts about integers, so the criticism of second-order logic boils down to saying that you don't like using the word "logic" to be applied to any system powerful enough to EXPRESS quantified statements about the integers, but only to systems weak enough that all their consequences can be enumerated. This demand is unreasonable. Even if logic is only about "correct reasoning", the usual framework given by SOL does not presume any dubious principles of reasoning and ZF proves its consistency. The existence of propositions which are not deductively settled by that framework but which can be given mathematical interpretations means nothing more than that our repertoire of "techniques of correct reasoning", which has grown over the centuries, isn't necessarily finalized.
3Luke_A_Somers
The Abstract Algebra course I took presented it in this fashion. I have a hard time seeing how you could even have abstract algebra without this notion.
3SilasBarta
What about Steven Landsburg's frequent crowing on the Platonicity of math and how numbers are real because we can "directly perceive them"? How does this relate to it? EDIT: Well, he replies here.
1Kenny
I was wondering what he thought about this! While I greatly sympathize with the "Platonicity of math", I can't shake the idea that my reasoning about numbers isn't any kind of direct perception, but just reasoning about an in-memory representation of a model that is ultimately based on all the other systems that behave like numbers. I find the arguments about how not all true statements regarding the natural numbers can be inferred via first-order logic tedious. It doesn't seem like our understanding of the natural numbers is particularly impoverished because of it.
2pozorvlak
I remember explaining the Axiom of Choice in this way to a fellow undergraduate on my integration theory course in late 2000. But of course it never occurred to me to write it down, so you only have my word for this :-)
0[anonymous]
This post definitely deserves a lot of credit.
-1MBlume
If memory serves, Hofstadter uses roughly this explanation in GEB.
1DavidS
This is pretty close to how I remember the discussion in GEB. He has a good discussion of non-Euclidean geometry. He emphasizes that originally the negation of Parallel Postulate was viewed as absurd, but that now we can understand that the non-Euclidean axioms are perfectly reasonable statements which describe something other than plane geometry we are used to. Later he has a bit of a discussion of what a model of PA + NOT(CON(PA)) would look like. I remember finding it pretty confusing, and I didn't really know what he was getting at until I red some actual logic theory textbooks. But he did get across the idea that the axioms would still describe something, but that something would be larger and stranger than the integers we think we know.
0Peterdjones
??? IRC, Hofstadter is a firm formalist, and I don't see how that square with EYs apparent Correspondence Theory. At least i don't see the point in correspondence if hat is being corresponded to is itself generated by axioms.

Thanks for posting this. My intended comments got pretty long, so I converted them to a blog post here. The gist is that I don't think you've solved the problem, partly because second order logic is not logic (as explained in my post) and partly because you are relying on a theorem (that second order Peano arithmetic has a unique model) which relies on set theory, so you have "solved" the problem of what it means for numbers to be "out there" only by reducing it to the question of what it means for sets to be "out there", which is, if anything, a greater mystery.

So this is where (one of the inspirations for) Eliezer's meta-ethics comes from! :)

A quick refresher from a former comment:

Cognitivism: Yes, moral propositions have truth-value, but not all people are talking about the same facts when they use words like "should", thus creating the illusion of disagreement.

... and now from this post:

Some people might dispute whether unicorns must be attracted to virgins, but since unicorns aren't real - since we aren't locating them within our universe using a causal reference - they'd just be talking about different models, rather than arguing about the properties of a known, fixed mathematical model.

(This little realization also holds a key to resolving the last meditation, I suppose.)

I've heard people say the meta-ethics sequence was more or less a failure since not that many people really understood it, but if these last posts were taken as a perequisite reading, it would be at least a bit easier to understand where Eliezer's coming from.

3Nick_Tarleton
Agreed, and disappointed that this comment was downvoted.
[-]MBlume130

This is a really good post.

If I can bother your mathematical logician for just a moment...

Hey, are you conscious in the sense of being aware of your own awareness?

Also, now that Eliezer can't ethically deinstantiate you, I've got a few more questions =)

You've given a not-isomorphic-to-numbers model for all the prefixes of the axioms. That said, I'm still not clear on why we need the second-to-last axiom ("Zero is the only number which is not the successor of any number.") -- once you've got the final axiom (recursion), I can't seem to visualize any not-isomorphic-to-numbers models.

Also, how does one go about proving that a particular set of axioms has all its models isomorphic? The fact that I can't think of any alternatives is (obviously, given the above) not quite sufficient.

Oh, and I remember this story somebody on LW told, there were these numbers people talked about called...um, I'm just gonna call them mimsy numbers, and one day this mathematician comes to a seminar on mimsy numbers and presents a proof that all mimsy numbers have the Jaberwock property, and all the mathematicians nod and declare it a very fine finding, and then the next week, he comes back, and pre... (read more)

1Viliam_Bur
I guess it is not necessary. It was just an illustration of a "quick fix", which was later shown to be insufficient.

You just say: 'For every relation R that works exactly like addition, the following statement S is true about that relation.' It would look like, '∀ relations R: (∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz))) → S)', where S says whatever you meant to say about +, using the token R.

The expression '(∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz)))' is true for addition, but also for many other relations, such as a '∀x∀y∀z: R(x, y, z)' relation.

5moshez
I'm not sure that adding the conjunction (R(x,y,z)&R(x,y,w)->z=w) would have made things clearer...I thought it was obvious the hypothetical mathematician was just explaining what kind of steps you need to "taboo addition"

Yes, the educational goal of that paragraph is to "taboo addition". Nonetheless, the tabooing should be done correctly. If it is too difficult to do, then it is Eliezer's problem for choosing a difficult example to illustrate a concept.

This may sound like nitpicking, but this website has a goal is to teach people rationality skills, as opposed to "guessing the teacher's password". The article spends five screens explaining why details are so important when defining the concept of a "number", and the reader is supposed to understand it. So it's unfortunate if that explanation is followed by another example, which accidentally gets the similar details wrong. My objections against the wrong formula are very similar to the in-story mathematician's objections to the definitions of "number"; the definition is too wide.

Your suggestion: '∀x∀y∀z∀w: R(x, 0, x) ∧ (R(x, y, z)↔R(x, Sy, Sz)) ∧ ((R(x, y, z)∧R(x, y, w))→z=w)'

My alternative: '∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, y, z)↔R(x, Sy, Sz)) ∧ (R(x, y, z)↔R(Sx, y, Sz))'.

Both seem correct, and anyone knows a shorter (or a more legible) way to express it, please contribute.

[-]Kindly200

Shorter (but not necessarily more legible): ∀x∀y∀z: (R(x, 0, z)↔(x=z)) ∧ (R(x, Sy, z)↔R(Sx, y, z)).

5Viliam_Bur
Perfect!
5Benya
The version in the article now, ∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)↔R(x, Sy, Sz)), is better than before, but it leaves open the possibility that R(0,0,7) as well as R(0,0,0). One more possibility is: "Not in second-order logic, which can quantify over functions as well as properties. (...) It would look like, '∀ functions f: ((∀x∀y: f(x, 0) = x ∧ f(x, Sy) = Sf(x, y)) → Q)' (...)" (I guess I'm not entirely in favor of this version -- ETA: compared to Kindly's fix -- because quantifying over relations surely seems like a smaller step from quantifying over properties than does quantifying over functions, if you're new to this, but still thought it might be worth pointing out in a comment.)

Your idea of pinning down the natural numbers using second order logic is interesting, but I don't think that it really solves the problem. In particular, it shouldn't be enough to convince a formalist that the two of you are talking about the same natural numbers.

Even in second order PA, there will still be statements that are independent of the axioms, like "there doesn't exist a number corresponding to a Godel encoding of a proof that 0=S0 under the axioms of second order PA". Thus unless you are assuming full semantics (i.e. that for any collection of numbers there is a corresponding property), there should be distinct models of second order PA for which the veracity of the above statement differs.

Thus it seems to me that all you have done with your appeal to second order logic is to change my questions about "what is a number?" into questions about "what is a property?" In any case, I'm still not totally convinced that it is possible to pin down The Natural Numbers exactly.

5Eliezer Yudkowsky
I'm assuming full semantics for second-order logic (for any collection of numbers there is a corresponding property being quantified over) so the axioms have a semantic model provably unique up to isomorphism, there are no nonstandard models, the Completeness Theorem does not hold and some truths (like Godel's G) are semantically entailed without being syntactically entailed, etc.
6dankane
OK then. As soon as you can explain to me exactly what you mean when you say "for any collection of numbers there is a corresponding property being quantified over", I will be satisfied. In particular, what do you mean when you say "any collection"?
3Eugine_Nier
If you're already fine with the alternating quantifiers of first-order logic, I don't see why allowing branching quantifiers would cause a problem. I could describe second order logic in terms of branching quantifiers.
2dankane
Huh. That's interesting. Are you saying that you can actually pin down The Natural Numbers exactly using some "first order logic with branching quantifiers"? If so, I would be interested in seeing it.
2Eugine_Nier
Sure: It is not the case that: there exists a z such that for every x and x’, there exists a y depending only on x and a y’ depending only on x’ such that Q(x,x’,y,y’,z) is true where Q(x,x’,y,y’,z) is ((x=x' ) → (y=y' )) ∧ ((Sx=x' ) → (y=y' )) ∧ ((x=0) → (y=0)) ∧ ((x=z) → (y=1))
3dankane
Cool. I agree that this is potentially less problematic than the second order logic approach. But it does still manage to encode the idea of a function in it implicitly when it talks about "y depending only on x", it essentially requires that y is a function of x, and if it's unclear exactly which functions are allowed, you will have problems. I guess first order logic has this problem to some degree, but with alternating quantifiers, the functions that you might need to define seem closer to the type that should necessarily exist.
1Eliezer Yudkowsky
Are you claiming that this term is ambiguous? In what specially favored set theory, in what specially favored collection of allowed models, is it ambiguous? Maybe the model of set theory I use has only one set of allowable 'collections of numbers' in which case the term isn't ambiguous. Now you could claim that other possible models exist, I'd just like to know in what mathematical language you're claiming these other models exist. How do you assert the ambiguity of second-order logic without using second-order logic to frame the surrounding set theory in which it is ambiguous?
3dankane
I'm not entirely sure what you're getting at here. If we start restricting properties to only cut out sets of numbers rather than arbitrary collections, then we've already given up on full semantics. If we take this leap, then it is a theorem of set theory that all set-theoretic models of the of the natural numbers are isomorphic. On the other hand, since not all statements about the integers can be either proven or disproven with the axioms of set theory, there must be different models of set theory which have different models of the integers within them (in fact, I can build these two models within a larger set theory). On the other hand, if we continue to use full semantics, I'm not sure how you clarify to be what you mean when you say "a property exists for every collection of numbers". Telling me that I should already know what a collection is doesn't seem much more reasonable than telling me that I should already know what a natural number is.
3Eliezer Yudkowsky
Doesn't the proof of the Completeness Theorem / Compactness Theorem incidentally invoke second-order logic itself? (In the very quiet way that e.g. any assumption that the standard integers even exist invokes second-order logic.) I'm not sure but I would expect it to, since otherwise the notion of a "consistent" theory is entirely dependent on which models your set theory says exist and which proofs your integer theory says exist. Perhaps my favorite model of set theory has only one model of set theory, so I think that only one model exists. Can you prove to me that there are other models without invoking second-order logic implicitly or explicitly in any called-on lemma? Keep in mind that all mathematicians speak second-order logic as English, so checking that all proofs are first-order doesn't seem easy.
2dankane
I am admittedly in a little out of my depth here, so the following could reasonably be wrong, but I believe that the Compactness Theorem can be proved within first order set theory. Given a consistent theory, I can use the axiom of choice to extend it to a maximal consistent set of statements (i.e. so that for every P either P or (not P) is in my set). Then for every statement that I have of the form "there exists x such that P(x)", I introduce an element x to my model and add P(x) to my list of true statements. I then re-extend to a maximal set of statements, and add new variables as necessary, until I cannot do this any longer. What I am left with is a model for my theory. I don't think I invoked second order logic anywhere here. In particular, what I did amounts to a construction within set theory. I suppose it is the case that some set theories will have no models of set theory (because they prove that set theory is inconsistent), while others will contain infinitely many. My intuition on the matter is that if you can state what you are trying to say without second order logic, you should be able to prove it without second order logic. You need second order logic to even make sense of the idea of the standard natural numbers. The Compactness Theorem can be stated in first order set theory, so I expect the proof to be formalizable within first order set theory.
0[anonymous]
I'm not entirely sure what you're getting at here. If we start restricting properties to only cut out sets of numbers rather than arbitrary collections, then we've already given up on full semantics. If we take this leap, then it is a theorem of set theory that all set-theoretic models of the of the natural numbers are isomorphic. On the other hand, since not all statements about the integers can be either proven or disproven with the axioms of set theory, there must be different models of set theory which have different models of the integers within them (in fact if you give me an inaccessible cardinal, I build these two models within a larger set theory). On the other hand, if we continue to use full semantics, I'm not sure how you clarify to be what you mean when you say "a property exists for every collection of numbers". Telling me that I should already know what a collection is doesn't seem much more reasonable than telling me that I should already know what a natural number is.
0chaosmosis
I think this is his way of connecting numbers to the previous posts. If "a property" is defined as a causal relation, which all properties are, then I think this makes sense. It doesn't provide some sort of ultimate metaphysical justification for numbers or properties or anything, but it clarifies connections between the two and such a justification isn't really possible anyways.
0dankane
I don't think that I understand what you mean here. How can these properties represent causal relations? They are things that are satisfied by some numbers and not by others. Since numbers are aphysical, how do we relate this to causal relations. On the other hand, even with a satisfactory answer to the above question, how do we know that "being in the first chain" is actually a property, since otherwise we still can't show that there is only one chain.
2[anonymous]
You just begged the question. Eliezer answered you in the OP:
-2chaosmosis
I can't think of an example, but I'm thinking that if a property existed then it would be a causal relation. A property wouldn't represent a causal relation, it would be one. I wasn't thinking mathematically but instead in terms of a more commonplace understanding of properties as things like red and yellow and blue. The argument made by the simple idea of truth might be a way to get us from physical states (which are causal relations) to numbers. If you believe that counting sheep is a valid operation, then quantifying color also seems fine. The reason I spoke in terms of causal relations is because I believe understanding qualities as causal relations between things allows us to deduce properties about things through a combination of Salmonoff Induction and the method described in this post. Are you questioning the idea that numbers or properties are a quality about objects? If so, what are they? I'm feeling confused though. If the definition of property used here doesn't connect to or means something completely different than facts about objects, then I'm way off base. I might also be off base for other reasons. Not sure.
0dankane
I am questioning the idea that numbers (at least the things that this post refers to as numbers) are a quality about objects. Numbers, as they are described here, are an abstract logical construction.

How come we never see anything physical that behaves like any of of the non-standard models of first order PA? Given that's the case, it seems like we can communicate the idea of numbers to other humans or even aliens by saying "the only model of first order PA that ever shows up in reality", so we don't need second order logic (or the other logical ideas mentioned in the comments) just to talk about the natural numbers?

9Qiaochu_Yuan
The natural numbers are supposed to be what you get if you start counting from 0. If you start counting from 0 in a nonstandard model of PA you can't get to any of the nonstandard bits, but first-order logic just isn't expressive enough to allow you to talk about "the set of all things that I get if I start counting from 0." This is what allows nonstandard models to exist, but they exist only in a somewhat delicate mathematical sense and there's no reason that you should expect any physical phenomenon corresponding to them. If I wanted to communicate the idea of numbers to aliens, I don't think I would even talk about logic. I would just start counting with whatever was available, e.g. if I had two rocks to smash together I'd smash the rocks together once, then twice, etc. If the aliens don't get it by the time I've smashed the rocks together, say, ten times, then they're either so bad at induction or so unfamiliar with counting that we probably can't meaningfully communicate with them anyway.
5A1987dM
The Pirahã are unfamiliar with counting and we still can kind-of meaningfully communicate with them. I agree with the rest of the comment, though.
0Kenny
I was ready to reply "bullshit", but I guess if their language doesn't have any cardinal or ordinal number terms ... Still, they could count with beads or rocks, à la the magic sheep-counting bucket. It's understandable why they wouldn't really need counting given their lifestyle. But I wonder what they do (or did) when a neighboring tribe attacks or encroaches on their territory? Their language apparently does have words for 'small amount' and 'large amount', but how would they decide how many warriors to send to meet an opposing band?
1A1987dM
Here's a decent argument that they probably don't have words for numbers because they don't count, rather than the other way round, contra pop-Whorfianism. (Otherwise I guess they'd just borrow the words for numbers from Portuguese or something, as they probably did with personal pronouns from Tupi.)
3Wei Dai
Is it just coincidence that these nonstandard models don't show up anywhere in the empirical sciences, but real numbers and complex numbers do? I'm wondering if there is some sort of deeper reason... Maybe you were hinting at something by "delicate"? Good point. I guess I was trying to make the point that Eliezer seems a bit obsessed with logical pinpointing (aka categoricity) in this post. ("You need axioms to pin down a mathematical universe before you can talk about it in the first place.") Before we achieved categoricity, we already knew what mathematical structure we wanted to talk about, and afterwards, it's still useful to add more axioms if we want to prove more theorems.
8Qiaochu_Yuan
The process by which the concepts "natural / real / complex numbers" vs. "nonstandard models of PA" were generated is very different. In the first case, mathematicians were trying to model various aspects of the world around them (e.g. counting and physics). In the second case, mathematicians were trying to pinpoint something else they already understood and ended up not quite getting it because of logical subtleties. I'm not sure how to explain what I mean by "delicate." It roughly means "unlikely to have been independently invented by alien mathematicians." In order for alien mathematicians to independently invent the notion of a nonstandard model of PA, they would have to have independently decided that writing down the first-order Peano axioms is a good idea, and I just don't find this all that likely. On the other hand, there are various routes alien mathematicians might take towards independently inventing the complex numbers, such as figuring out quantum mechanics. I guess Eliezer's intended response here is something like "but when you want to explain to an AI what you mean by the natural numbers, you can't just say The Things You Use To Count With, You Know, Those."
4Shmi
Umm... wouldn't they be considered "standard" in this case? I.e. matching some real-world experience? Let's imagine a counterfactual world in which some of our "standard" models appear non-standard. For example, in a purely discrete world (like the one consisting solely of causal chains, as EY once suggested), continuity would be a non-standard object invented by mathematicians. What makes continuity "standard" in our world is, disappointingly, our limited visual acuity. Another example: in a world simulated on a 32-bit integer machine, natural numbers would be considered non-standard, given how all actual numbers wrap around after 2^32-1. Exercise for the reader: imagine a world where a certain non-standard model of first order PA would be viewed as standard.
1Eliezer Yudkowsky
This is basically the theme of the next post in the sequence. :)
0cousin_it
Qiaochu's answer: because PA isn't unique. There are other (stronger/weaker) axiomatizations of natural numbers that would lead to other nonstandard models. I don't think that answer works, because we don't see nonstandard models of these other theories either. wedrifid's answer: because PA was designed to talk about natural numbers, not other things in reality that humans can tell apart from natural numbers. My answer: because PA was designed to talk about natural numbers, and we provably did a good job. PA has many models, but only one computable model. Since reality seems to be computable, we don't expect to see nonstandard models of PA in reality. (Though that leaves the mystery of whether/why reality is computable.)

First post in this sequence that lives up to the standard of the old classics. Love it.

Yeah, but I've found the previous posts much more useful for coming up with clear explanations aimed at non-LWers, and I presume they'd make a better introduction to some of the core LW epistemic rationality than just throwing "The Simple Truth" at them.

7Armok_GoB
It's a pretty hard balance to strike that's probably different for everyone, between incomprehensibility and boringness.
8A1987dM
I already more-or-less knew most of the stuff in the previous posts in this sequences and still didn't find them boring.
2lukeprog
Agree. When I first read The Simple Truth, I thought Eliezer was endorsing pragmatism over correspondence.
5Shmi
I'm still wondering what The Simple Truth is about. My best guess is that it is a critique of instrawmantalism.
3[anonymous]
In my opinion, Causal Diagrams and Causal Models is far superior to Timeless Causality. I am not saying that there is anything wrong with "Timeless Causality", or any of Eliezer's old posts, but this sequence goes into enough depth of explanation that even someone who has not read the older sequences on Less Wrong would have a good chance of understanding it.
[-]Klao90

You just say: 'For every relation R that works exactly like addition, the following statement S is true about that relation.' It would look like, '∀ relations R: (∀x∀y∀z: R(x, 0, x) ∧ (R(x, y, z)→R(x, Sy, Sz))) → S)', where S says whatever you meant to say about +, using the token R.

I would change the statement to be something other than 'S', say 'Q', as S is already used for 'successor'.

4tim
I agree that the use of S here was confusing. Also, there is one too many right parens.

Requesting feedback:

"Whenever a part of reality behaves in a way that conforms to the number-axioms - for example, if putting apples into a bowl obeys rules, like no apple spontaneously appearing or vanishing, which yields the high-level behavior of numbers - then all the mathematical theorems we proved valid in the universe of numbers can be imported back into reality. The conclusion isn't absolutely certain, because it's not absolutely certain that nobody will sneak in and steal an apple and change the physical bowl's behavior so that it doesn't m

... (read more)

Terry Tao's 2007 post on nonfirstorderizability and branching quantifiers gives an interesting view of the boundary between first- and second-order logic. Key quote:

Moving on to a more complicated example, if Q(x,x’,y,y’) is a quaternary relation on four objects x,x’,y,y’, then we can express the statement

For every x and x’, there exists a y depending only on x and a y’ depending on x and x’ such that Q(x,x’,y,y’) is true

...but it seems that one cannot express

For every x and x’, there exists a y depending only on x and a y’ depending only on x’ such that

... (read more)

I'm a little confused as to which of two positions this is advocating:

  1. Numbers are real, serious things, but the way that we pick them out is by having a categorical set of axioms. They're interesting to talk about because lots of things in the world behave like them (to some degree).

  2. Mathematical talk is actually talk about what follows from certain axioms. This is interesting to talk about because lots of things obey the axioms and so exhibit the theorems (to some degree).

Both of these have some problems. The first one requires you to have weird, no... (read more)

7[anonymous]
I'm not sure exactly what Eliezer intends, but I'll put in my two cents: A proof is simply a game of symbol manipulation. You start with some symbols, say '(', ')', '¬', '→', '↔', '∀', '∃', 'P', 'Q', 'R', 'x', 'y', and 'z'. Call these symbols the alphabet. Some sequences of symbols are called well-formed formulas, or wffs for short. There are rules to tell what sequences of symbols are wffs, these are called a grammar. Some wffs are called axioms. There is another important symbol that is not one of the symbols you chose - this is the '⊢' symbol. A declaration is the '⊢' symbol followed by a wff. A legal declaration is either the '⊢' symbol followed by an axiom or the result of an inference rule. An inference rule is a rule that declares that a declaration of a certain form is legal, given that certain declarations of other forms are legal. A famous inference rule called modus ponens is part of a formal system called first-order logic. This rule says: "If '⊢ P' and '⊢ (P → Q)' (where P and Q are replaced with some wffs) are valid declarations, then '⊢ Q' is also a valid declaration." By the way, a formal system is just a specific alphabet, grammar, set of axioms, and set of inference rules. You also might like to note that if '⊢ P' (where P is replaced with some wff) is a valid declaration, then we also call P a theorem. So now we know something: In a formal system, all axioms are theorems. The second thing to note is that a formal system does not necessarily have anything to do with even propositional logic (let alone first- or second-order logic!). Consider the MIU system (open link in WordPad, on Windows), for example. It has four inference rules for just messing around with the order of the letters, 'M', 'I', and 'U'! That doesn't have to do with the real world or even math, does it? The third thing to note is that, though a formal system can tell us what wffs are theorems, it cannot (directly) tell us what wffs are not theorems. And hence we have the MU puzz
1bryjnar
A few things: 1. I don't think we disagree about the social construct thing: see my other comment where I'm talking about that. 2. It sounds like you pretty much come down in favour of the second position that I articulated above, just with a formalist twist. Mathematical talk is about what follows from the axioms; obviously only certain sets of axioms are worth investigating, as they're the ones that actually line up with systems in the world. I agree so far, but you think that there is no notion of logic beyond the syntactic? Aren't you just dropping the distrinction between syntax and semantics here? One of the big points of the last few posts has been that we're interested in the semantic implications, and the formal systems are a (sound) syntactic means of reaching true conclusions. From your post it sounds like you're a pretty serious formalist, though, so that may not be a big deal to you.
0[anonymous]
Definitely position two. I would describe first-order logic as "a formal encapsulation of humanity's most fundamental notions of how the world works". If it were shown to be inconsistent, then I could still fall back to something like intuitionistic logic, but from that point on I'd be pretty skeptical about how much I could really know about the world, beyond that which is completely obvious (gravity, etc.). What did I say that implied that I "think that there is no notion of logic beyond the syntactic"? I think of "logic" and "proof" as completely syntactic processes, but the premises and conclusions of a proof have to have semantic meaning; otherwise, why would we care so much about proving anything? I may have implied something that I didn't believe, or I may have inconsistent beliefs regarding math and logic, so I'd actually appreciate it if you pointed out where I contradicted what I just said in this comment (if I did).
0bryjnar
Looking back, it's hard to say what gave me that impression. I think I was mostly just confused as to why you were spending quite so much time going over the syntax stuff ;) And made me think that you though that all logical/mathematical talk was just talk of formal systems. That can't be true if you've got some semantic story going on: then the syntax is important, but mainly as a way to reach semantic truths. And the semantics don't have to mention formal systems at all. If you think that the semantics of logic/mathematics is really about syntax, then that's what I'd think of as a "formalist" position.
2[anonymous]
Oh, I think I may understand your confusion, now. I don't think of mathematics and logic as equals! I am more confident in first-order logic than I am in, say, ZFC set theory (though I am extremely confident in both). However, formal system-space is much larger than the few formal systems we use today; I wanted to emphasize that. Logic and set theory were selected for because they were useful, not because they are the only possible formal ways of thinking out there. In other words, I was trying to right the wrong question, why do mathematics and logic transcend the rest of reality?
5khafra
In contrast with my esteemed colleague RichardKennaway, I think it's mostly #2. Before the Peano axioms, people talking about numbers might have been talking about any of a large class of things which discrete objects in the real world mostly model. It was hard to make progress in math past a certain level until someone pointed out axiomatically exactly which things-that-discrete-objects-in-the-real-world-mostly-model it would be most productive to talk about. Concordantly, the situation of pre-axiom speakers is much like that of people from Scotland trying to talk to people from the American South and people from Boston, when none of them knows the rules of their grammar. Edit: Or, to be more precise, it's like two scots speakers as fluent as Kawoomba talking about whether a solitary, fallen tree made a "sound," without defining what they mean by sound.

Aye, right. Yer bum's oot the windae, laddie. Ye dinna need tae been lairnin a wee Scots tae unnerstan, it's gaein be awricht! Ane leid is enough.

3Armok_GoB
What about "both ways simultaneously, the distinction left ambiguous most of the time because it isn't useful"?
0Peterdjones
EY seems to be taken with the resemblance between a causal diagram and the abstract structure of axioms, inferences and theorems in mathematcal logic. But there are differences: with causality, our evidence is the latest causal output, the leaf nodes. We have to trace back to the Big Bang from them.However, in maths we start from axioms, and cannot get directly to the theorems or leaf nodes. We could see this process as exploring a pre-existing territory, but it is hard to see what this adds, since the axioms and rules of inference are sufficient for truth, and it is hard to see, in EY's presentation how literally he takes the idea.
2Eliezer Yudkowsky
Er, no, causal models and logical implications seem to me very different in how they propagate modularly. Unifying the two is going to be troublesome.
-4Eugine_Nier
It's useful for reasoning heuristically about conjectures.
0Peterdjones
Could I have an example?
0Giles
I would read this: as: Lots of things in both real and imagined worlds behave like numbers. It's most convenient to pick one of them and call them "The Numbers" but this is really just for the sake of convenience and doesn't necessarily give them elevated philosophical status. That would be my position anyway.
0[anonymous]
The Peano Arithmetic talks about the Successor function, and jazz. Did you know that the set of finite strings of a single symbol alphabet also satisfies the Peano Axioms? Did you know that in ZFC, defining the set all sets containing only other members of the parent set with lower cardinality, and then saying {} is a member obeys the Peano Axioms? Did you know that saying you have a Commutative Monoid with right division, that multiplication with something other than identity always yields a new element and that the set {1} is productive, obey the Peano Axioms? Did you know the even naturals obey the Peano Axioms? Did you know any fully ordered set with infimum, but no supremum obey the Axioms? There is no such thing as "Numbers," only things satisfying the Peano Axioms.
1ArisKatsaris
Surely the set of finite strings in an alphabet of no-matter-how-many-symbols satisfies the Peano axioms? e.g. using the English alphabet (with A=0, B=S(A), C=S(B)....AA=S(Z), AB=S(AA), etc would make a base-26 system).
0[anonymous]
Single symbol alphabet is more interesting, (empty string = 0, sucessor function = append another symbol) the system you describe is more succinctly described using a concatenation operator: * 0 = 0, 1 = S0, 2 = S1 ... 9 = S8. * For All b in {0,1,2,3,4,5,6,7,8,9}, a in N: ab = a x S9 + b From these definitions we get, example-wise: * 10 = 1 x S9 + 0 = SSSSSSSSSS0
0Giles
I'm not quite sure what you're saying here - that "Numbers" don't exist as such but "the even naturals" do exist?
0Peterdjones
I think s/he is saying there is no Essence of Numberhood beyond satisfaction of the PA's.
0[anonymous]
Correct.
0Giles
Just to be clear, I assume we're talking about the second order Peano axioms here?
0bryjnar
We don't know whether the universe is finite or not. If it is finite, then there is nothing in it that fully models the natural numbers. Would we then have to say that the numbers did not exist? If the system that we're referring to isn't some physical thing, what is it?
0[anonymous]
Finite subsets of the naturals still behave like naturals.
0bryjnar
Not precisely. In many ways, yes, but for example they don't model the axiom of PA that says that every number has a successor.
0[anonymous]
True, but the axiom of induction holds, and that is the most useful one.
0Giles
I've realised that I'm slightly more confused on this topic than I thought. As non-logically omniscient beings, we need to keep track of hypothetical universes which are not just physically different from our own, but which don't make sense - i.e. they contain logical contradictions that we haven't noticed yet. For example, let T be a Turing machine where we haven't yet established whether or not T halts. Then one of the following is true but we don't know which one: * (a) The universe is infinite and T halts * (b) The universe is infinite and T does not halt * (c) The universe is finite and T halts * (d) The universe is finite and T does not halt If we then discover that T halts, we not only assign zero probability to (b) and (d), we strike them off the list entirely. (At least that's how I imagine it, I haven't yet heard anyone describe approaches to logical uncertainty). But it feels like there should also be (e) - "the universe is finite and the question of whether or not T halts is meaningless". If we were to discover that we lived in (e) then all infinite universes would have to be struck off our list of meaningful hypothetical universes, since we are viewing hypothetical universes as mathematical objects. But it's hard to imagine what would constitute evidence for (or against) (e). So after 5 minutes of pondering, that more or less maps out my current state of confusion.
0bryjnar
I think you're confused if you think the finitude of the universe matters in answering the mathematical question of whether T halts. Answering that question may be of interest for then figuring out whether certain things in our universe that behave like Turning machines behave in certain ways, but the mathematical question is independent. Your confusion is that you think there need to be objects of some kind that correspond to mathematical structures that we talk about. Then you've got to figure out what they are, and that seems to be tricky however you cut it.
0Giles
I agree that the finitude of the universe doesn't matter in answering the mathematical question of whether T halts. I was pondering whether the finitude of the universe had some bearing on whether the question of T halting is necessarily meaningful (in an infinite universe it surely is meaningful, in a finite universe it very likely is but not so obviously so).
0bryjnar
Surely if the infinitude of the universe doesn't affect that statement's truth, it can't affect that statement's meaningfulness? Seems pretty obvious to me that the meaning is the same in a finite and an infinite universe: you're talking about the mathematical concept of a Turing machine in both cases.
0Giles
Conditional on the statement being meaningful, infinitude of the universe doesn't affect the statement's truth. If the meaningfulness is in question then I'm confused so wouldn't assign very high or low probabilities to anything. Essentially: * I have a very strong intuition that there is a unique (up to isomorphism) mathematical structure called the "non-negative integers" * I have a weaker intuition that statements in second-order logic have a unique meaningful interpretation * I have a strong intuition that model semantics of first-order logic is meaningful * I have a very strong intuition that the universe is real in some sense It's possible that my intuition might be wrong though. I can picture the integers in my mind but my picture isn't completely accurate - they basically come out as a line of dots with a "going on forever" concept at the end. I can carry on pulling dots out of the "going on forever", but I can't ever pull all of them out because there isn't room in my mind. Any attempt to capture the integers in first-order logic will permit nonstandard models. From the vantage point of ZF set theory there is a single "standard" model, but I'm not sure this helps - there are just nonstandard models of set theory instead. Similarly I'm not sure second-order logic helps as you pretty much need set theory to define its semantics. So if I'm questioning everything it seems I should at least be open to the idea of there being no single model of the integers which can be said to be "right" in a non-arbitrary way. I'd want to question first order logic too, but it's hard to come up with a weaker (or different) system that's both rigorous and actually useful for anything. I've realized one thing though (based on this conversation) - if the universe is infinite, defining the integers in terms of the real world isn't obviously the right thing to do, as the real world may be following one of the nonstandard models of the integers. Updating in favor of meaning
0Richard_Kennaway
I read it as (1), with a side order of (2). Mathematical talk is also about what follows from certain axioms. The axioms nail it down so that mathematicians can be sure what other mathematicians are talking about. Not weird, non-physical numbery-things, just non-physical numbery-things. If they seem weird, maybe it's because we only noticed them a few thousand years ago. No more than a magnetic field is a special exception to the theory of elasticity. It's just a phenomenon that is not described by that theory.
2Peterdjones
But EY insists that maths does come under correspondence/reference! "to delineate two different kinds of correspondence within correspondence theories of truth.""

I think it's worth mentioning explicitly that the second-order axiom introduced is induction.