Logical Pinpointing

62 02 November 2012 03:33PM

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."

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"

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Comment author: 25 October 2012 03:08:33AM 16 points [-]

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.

Comment author: 01 November 2012 05:06:34PM 9 points [-]

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.")

Comment author: 02 November 2012 10:35:12AM *  5 points [-]

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.)

Comment author: 14 June 2013 03:04:43PM 3 points [-]

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.

Comment author: 04 November 2012 04:36:52AM *  4 points [-]

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.

Comment author: 05 July 2013 07:21:37PM 0 points [-]

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.

Comment author: 02 November 2012 07:10:27PM 7 points [-]

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: Lomegaomega (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 quantifiers? This seems like just the same sort of problem as ruling out non-standard models of arithmetic ("the same sort", not the same, because for the reasons mentioned in (i) it is actually more stringent of a condition.) The point is if you for some reason doubt that we have a categorical grasp of the natural numbers, you are certainly not going to grant that we can enforce a full interpretation of the second order quantifiers. And although it seems intuitively obvious that we have a categorical grasp of the natural numbers, careful consideration of the first incompleteness theorem shows that this is by no means clear.

iii) Given that categoricity results are only up to isomorphism, I don't see how they help you pin down talk of the natural numbers themselves (as opposed to any old omega_sequence). At best, they help you pin down the structure of the natural numbers, but taking this insight into account is easier said than done.

Comment author: 12 December 2013 11:40:27AM 0 points [-]

iii) Given that categoricity results are only up to isomorphism, I don't see how they help you pin down talk of the natural numbers themselves (as opposed to any old omega_sequence). At best, they help you pin down the structure of the natural numbers, but taking this insight into account is easier said than done.

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.

Comment author: 06 November 2012 09:30:13PM 1 point [-]

"Because otherwise you can't talk about numbers as opposed to something else,"

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.

Comment author: 01 November 2012 10:23:19PM 1 point [-]

so AFAIK it's theoretically possible that I'm the first to spell out that idea in exactly that way

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 :-)

Comment author: 01 November 2012 08:50:17PM 11 points [-]

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.

Comment author: 01 November 2012 10:20:40PM 3 points [-]

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"

Comment author: 02 November 2012 09:43:41AM *  16 points [-]

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.

Comment author: 02 November 2012 02:31:47PM 16 points [-]

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

Comment author: 03 November 2012 05:32:04PM 8 points [-]

Done!

Comment author: 02 November 2012 03:40:52PM 3 points [-]

Perfect!

Comment author: 02 November 2012 07:49:52PM *  3 points [-]

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

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.)

Comment author: 01 November 2012 06:54:48PM 11 points [-]

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.

Comment author: 02 November 2012 07:58:10AM 6 points [-]

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.

Comment author: 02 November 2012 09:31:13PM 5 points [-]

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"?

Comment author: 03 November 2012 12:22:33AM 2 points [-]

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?

Comment author: 03 November 2012 07:19:17AM 3 points [-]

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.

Comment author: 03 November 2012 05:36:38PM 3 points [-]

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

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.

Comment author: 04 November 2012 12:28:54AM 2 points [-]

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.

Comment author: 03 November 2012 10:39:08PM *  3 points [-]

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.

Comment author: 04 November 2012 12:40:04AM 2 points [-]

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.

Comment author: 04 November 2012 03:33:27AM *  3 points [-]

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))

Comment author: 04 November 2012 04:16:38AM 3 points [-]

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.

Comment author: 01 November 2012 08:15:31PM *  0 points [-]

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.

Comment author: 01 November 2012 10:17:36PM 1 point [-]

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.

Comment author: [deleted] 02 November 2012 07:05:12PM 1 point [-]

Since numbers are aphysical, how do we relate this to causal relations?

You just begged the question. Eliezer answered you in the OP:

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.

Comment author: 14 November 2012 04:20:18PM 10 points [-]

Thanks for posting this. My intended comments got pretty long, so I converted them to a blog post <a href="http://www.thebigquestions.com/2012/11/14/accounting-for-numbers/">here</a>. 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.

Comment author: 01 November 2012 05:45:09PM 24 points [-]

"- 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.

Comment author: 02 November 2012 02:02:32AM 6 points [-]

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.

Comment author: 06 November 2012 09:34:36PM *  7 points [-]

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.

Comment author: 02 November 2012 09:45:38AM 4 points [-]

I love the elegance of this answer, upvoting.

Comment author: 01 November 2012 05:58:51PM *  4 points [-]

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.

Comment author: 02 November 2012 07:50:26AM 0 points [-]

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?

Comment author: 02 November 2012 09:00:53AM 6 points [-]

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.

Comment author: [deleted] 02 November 2012 06:34:47PM *  2 points [-]

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 the large numbers, no one has observed these quantities. But, for all quantities we have observed so far, mathematical induction has held. We use deduction to apply induction, but that doesn't make the induction any less inductive to begin with. We use the real number system to make predictions in physics. If we have the luxury of making an observation, we should go ahead and update. For companies with limited resources that are trying to develop a useful product to sell to make money, and even more so for Friendly AI (a mistake could end human civilization), it's nice to have a good idea of what an outcome will be before it happens. Bayes' rule provides a systematic way of working with this uncertainty. Maybe, one day, when I put two apples next to two apples on my kitchen table, there will be five (the order in which I move the apples around will affect their quantity), but, if I had to bet one way or the other, I assure you that my money is on this not happening.

Comment author: 04 November 2012 04:53:58PM *  0 points [-]

"- 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."

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.

Comment author: 05 November 2012 05:13:30AM 4 points [-]

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.

Comment author: 04 November 2012 08:49:01PM 2 points [-]

Aren't you just confusing distributions (2d2) and samples ('3') here?

Comment author: 05 November 2012 08:07:03AM 0 points [-]

Thank you, I shall suitably edit my post.

Comment author: 30 December 2012 09:46:58AM *  8 points [-]

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?

Comment author: 30 December 2012 11:56:01AM *  7 points [-]

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.

Comment author: [deleted] 30 December 2012 12:17:07PM 4 points [-]

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.

Comment author: 05 July 2013 08:06:30PM *  0 points [-]

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?

Comment author: [deleted] 13 July 2013 05:54:05PM *  1 point [-]

Still, they could count with beads or rocks, à la the magic sheep-counting bucket.

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.)

Comment author: 30 December 2012 08:08:19PM 3 points [-]

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.

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"?

If I wanted to communicate the idea of numbers to aliens, I don't think I would even talk about logic.

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.

Comment author: 30 December 2012 09:52:41PM 5 points [-]

Is it just coincidence that these nonstandard models don't show up anywhere in the empirical sciences, but real numbers and complex numbers do?

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.

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.

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."

Comment author: 31 December 2012 12:03:30AM *  2 points [-]

How come we never see anything physical that behaves like any of of the non-standard models of first order PA?

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.

Comment author: 30 December 2012 01:14:38PM 1 point [-]

This is basically the theme of the next post in the sequence. :)

Comment author: 23 January 2018 10:33:15AM *  0 points [-]

How come we never see anything physical that behaves like any of of the non-standard models of first order PA?

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.)

Comment author: 04 November 2012 04:33:24AM 7 points [-]

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 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."

This is exactly what I argued and grounded back in this article.

Specifically, that the two premises:

1) rocks behave isomorphically to numbers, and
2) under the axioms of numbers, 2+2 = 4

jointly imply that adding two rocks to two rocks gets four rocks. (See the cute diagram.)

And yet the response on that article (which had an array of other implications and reconciliations) was pretty negative. What gives?

Furthermore, in discussions about this in person, Eliezer_Yudkowsky has (IIRC and I'm pretty sure I do) invoked the "hey, adding two apples to two apples gets four apples" argument to justify the truth of 2+2=4, in direct contradiction of the above point. What gives on that?

Comment author: 02 November 2012 12:19:51PM *  7 points [-]

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 Q(x,x’,y,y’) is true

in first order logic!

The post and comments give some well-known theorems that turn out to rely on such "branching quantifiers", and an encoding of the predicate "there are infinitely many X" which cannot be done in first-order logic.

Comment author: 01 November 2012 11:50:40PM 7 points [-]

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'.

Comment author: 02 November 2012 04:47:35AM 2 points [-]

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

Comment author: 02 November 2012 06:10:52AM 6 points [-]

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

Comment author: 02 November 2012 04:33:22AM 6 points [-]
Comment author: 02 November 2012 10:45:31AM *  5 points [-]

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, non-physical numbery-things. Not only this, but they're a special exception to the theory of reference that's been developed so far, in that you can refer to them without having a causal connection.

The second one (which is similar to what I myself would espouse) doesn't have this problem, because it's just talking about what follows logically from other stuff, but you do then have to explain why we seem to be talking about numbers. And also what people were doing talking about arithmetic before they knew about the Peano axioms. But the real bugbear here is that you then can't really explain logic as part of mathematics. The usual analyisis of logic that we do in maths with the domain, interpretation, etc. can't be the whole deal if we're cashing out the mathematics in terms of logical implication! You've got to say something else about logic.

(I think the answer is, loosely, that

1. the "numbers" we talk about are mainly fictional aides to using the system, and
2. the situation of pre-axiom speakers is much like that of English speakers who nonetheless can't explain English grammar.
3. I have no idea what to say about logic! )

I'm curious which of these (or neither) is the correct interpretation of the post, and if it's one of them, what Eliezer's answers are... but perhaps they're coming in another post.

Comment author: [deleted] 03 November 2012 07:10:15PM 6 points [-]

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 puzzle. This asks whether "MU" is a theorem in the MIU system. If it is, then you only need the MIU system to demonstrate this, but if it is not, you need to use reasoning from outside of that system.

As other commenters have already noted, mathematicians are not thinking about ZFC set theory when they prove things (that's not a bad thing; they'd never manage to prove any new results if they had to start from foundations for every proof!). However, mathematicians should be fairly confident that the proofs they create could be reduced down to proofs from the low-level axioms. So Eliezer is definitely right to be worried when a mathematician says "A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.". A proof is a social construct, but it is one, very, very specific kind of social construct. The axioms and inference rules of first-order Peano arithmetic are symbolic representations of our most fundamental notion of what the natural numbers are. The reason for propositional logic, first-order logic, second-order logic, Peano arithmetic, and the scientific method is that humans have little things called "cognitive biases". We are convinced by way too many things that should be utterly unconvincing. To say that a proof is a convincing social construct is...technically...correct (oh how it pains me to say that!)...but that very vague part of what it means for something to be a proof seems to imply that a proof is the utter antithesis of what it was meant for! A mathematical proof should be the most convincing social construct we have, because of how it is constructed.

First-order Peano arithmetic has just a few simple axioms, and a couple simple inference rules, and its symbols have a clear intended interpretation (in terms of the natural numbers (which characterize parts of the web of causality as already explained in the OP)). The truth of a few simple axioms and validity of a couple simple inference rules can be evaluated without our cognitive biases getting in the way. On the other hand, it's probably not a good idea to make "There is a prime number larger than any given natural number." an axiom of a formal system about the natural numbers, because it is not an immediate part of our intuitive understanding of how causal systems that behave according to the rules of the natural numbers behave. We as humans would have to be very, very, confused if a theorem of first-order Peano arithmetic (because we are so sure that its axioms are true and its inference rules are valid) turned out to be the negation of another theorem of Peano arithmetic, but not so confused if the same happened for ZFC set theory, because we do not so readily observe infinite sets in our day-to-day experience. The axioms and inference rules of first-order Peano arithmetic more directly correspond to our physical reality than those of ZFC set theory do (and the axioms and inference rules of the MIU system have nothing to do with our physical reality at all!). If a contradiction in first-order Peano arithmetic were found, though, life would go on. First-order Peano arithmetic does have a lot to do with our physical reality, but not all of it does. It inducts to numbers like 3^^^3 that we will probably never interact with. The ultrafinitists would be shouting "Told you so!"

Now I have said enough to give my direct response to the comment I am replying to. First of all, the dichotomy between "logic" and "mathematics" can be dissolved by referring to "formal systems" instead. A formal system is exactly as entwined with reality as its axioms and inference rules are. In terms of instrumental rationality, the more exotic theorems of ZFC set theory (and MIU) really don't help us, unless we intrinsically enjoy considering the question "What if there were (even though we have no evidence that this is the case) a platonic realm of sets? How would it behave?"

When used as means to an end, the point of a formal system is to correct for our cognitive biases. In other words, the definition of a proof should state that a proof is a "convincing demonstration that should be convincing", to begin with. I suspect Eliezer is so concerned with the Peano axioms because computer programs happen to evidently behave in a very, very mathematical way, and he believes that eventually a computer program will decide the fate of humanity. I share his concerns; I want a mathematical argument that the General Artificial Intelligence that will be created will be Friendly, not anything that might "convince" a few uninformed government officials.

Comment author: 04 November 2012 01:05:17AM *  1 point [-]

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?

First of all, the dichotomy between "logic" and "mathematics" can be dissolved by referring to "formal systems" instead.

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.

Comment author: [deleted] 04 November 2012 05:18:08AM 0 points [-]

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).

Comment author: 04 November 2012 06:52:14AM 0 points [-]

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

First of all, the dichotomy between "logic" and "mathematics" can be dissolved by referring to "formal systems" instead.

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.

Comment author: [deleted] 04 November 2012 03:21:54PM 0 points [-]

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?

Comment author: 02 November 2012 09:50:04PM 2 points [-]

What about "both ways simultaneously, the distinction left ambiguous most of the time because it isn't useful"?

Comment author: 02 November 2012 03:02:06PM *  2 points [-]

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.

Comment author: 02 November 2012 03:17:25PM 7 points [-]

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.

Comment author: 04 November 2012 04:30:24PM 1 point [-]

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.

Comment author: 04 November 2012 09:57:06PM 3 points [-]

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.

Comment author: 04 November 2012 09:54:22PM -1 points [-]

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.

It's useful for reasoning heuristically about conjectures.

Comment author: 05 November 2012 10:28:29AM 0 points [-]

Could I have an example?

Comment author: 02 November 2012 10:30:35PM 1 point [-]

axioms pin down that we're talking about numbers as opposed to something else.

as:

axioms pin down that we're talking about some system that behaves like numbers as opposed to something else.

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.

Comment author: 04 November 2012 01:12:38AM 1 point [-]

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?

Comment author: [deleted] 14 November 2012 08:56:30AM 0 points [-]

Finite subsets of the naturals still behave like naturals.

Comment author: 14 November 2012 12:12:44PM 1 point [-]

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.

Comment author: [deleted] 15 November 2012 12:56:00PM 1 point [-]

True, but the axiom of induction holds, and that is the most useful one.

Comment author: 04 November 2012 02:20:36AM *  0 points [-]

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.

Comment author: 04 November 2012 04:30:49AM 1 point [-]

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.

Comment author: 04 November 2012 04:36:34AM 0 points [-]

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).

Comment author: 04 November 2012 06:46:44AM 1 point [-]

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.

Comment author: 04 November 2012 05:06:04PM 0 points [-]

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 meaningfulness not being dependent on infinitude of universe.

Comment author: [deleted] 14 November 2012 08:55:48AM 0 points [-]

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.

Comment author: 14 November 2012 09:07:57AM 2 points [-]

Did you know that the set of finite strings of a single symbol alphabet also satisfies the Peano Axioms?

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).

Comment author: [deleted] 15 November 2012 01:06:20PM 0 points [-]

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
Comment author: 14 November 2012 02:53:25PM 0 points [-]

I'm not quite sure what you're saying here - that "Numbers" don't exist as such but "the even naturals" do exist?

Comment author: 14 November 2012 03:31:33PM *  0 points [-]

I think s/he is saying there is no Essence of Numberhood beyond satisfaction of the PA's.

Comment author: [deleted] 15 November 2012 12:55:14PM 0 points [-]

Correct.

Comment author: 14 November 2012 04:22:39PM 0 points [-]

Just to be clear, I assume we're talking about the second order Peano axioms here?

Comment author: 02 November 2012 01:31:34PM 1 point [-]

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).

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.

Both of these have some problems. The first one requires you to have weird, non-physical numbery-things.

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.

Not only this, but they're a special exception to the theory of reference that's been developed so far, in that you can refer to them without having a causal connection.

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.

Comment author: 01 November 2012 08:31:24PM 5 points [-]

For thousands of years, mathematicians tried proving the parallel postulate from Euclid's four other postulates, even though there are fairly simple counterexamples which show such a proof to be impossible. I suspect that at least part of the reason for this delay is a failure to appreciate this post's point : that a "straight line", like a "number" has to be defined/specified by a set of axioms, and that a great circle is in fact a "straight line" as far as the first four of Euclid's postulates are concerned.

Comment author: 19 May 2013 03:08:56PM 3 points [-]

That's not correct. Elliptic geometry fails to satisfy some of the other postulates, depending on how they are phrased. I'm not too familiar with the standard ways of making Euclid's postulates rigorous, but if you're looking at Hilbert's axioms instead, then elliptic geometry fails to satisfy O3 (the third order axiom): if three points A, B, C are on a line, then any of the points is between the other two. Possibly some other axioms are violated as well.

Notably, elliptic geometry does not contain any parallel lines, while it is a theorem of neutral geometry that parallel lines do in fact exist.

Hyperbolic geometry was actually necessary to prove the independence of Euclid's fifth postulate, and few would call it a "fairly simple counterexample".

I agree that introducing elliptic geometry (and other simple examples like the Fano plane) earlier on in history would have made the discussion of Euclid's fifth postulate much more coherent much sooner.

Comment author: 01 November 2012 03:50:41PM 5 points [-]

Do we need a process for figuring out which objects are likely to behave like numbers? And as good Bayesians, for figuring out how likely that is?

Comment author: 01 November 2012 04:28:10PM *  3 points [-]

Er, yes? I mean it's not like we're born knowing that cars behave like integers and outlet electricity doesn't, since neither of those things existed ancestrally.

Comment author: 02 November 2012 12:44:27PM *  2 points [-]

Wait, what? We may not be born knowing what cars and electricity are, but I would be surprised if we weren't born with an ability (or the capacity to develop an ability) to partition our model of a car-containing section of universe into discrete "car" objects, while not being able to do the same for "electric current" objects.

Comment author: 01 November 2012 04:48:03PM 3 points [-]

I'm pretty sure that we're born knowing cars and carlike objects behave like integers.

Comment author: 02 November 2012 11:56:16PM 4 points [-]

I think our eyes (or visual cortex) knows that certain things (up to 3 or 4 of them) behave like integers since it bothers to count them automatically.

Comment author: 01 November 2012 04:57:58PM 2 points [-]

The ancestral environment included people (who behave like integers over moderate time spans) and water (which doesn't behave like integers)..

The better question would have been "how do people identify objects which behave like integers?".

Comment author: [deleted] 02 November 2012 12:50:38AM 1 point [-]

The better question would have been "how do people identify objects which behave like integers?".

The same way we identify objects which satisfy any other predicate? We determine whether or not something is a cat by comparing it to our knowledge of what cats are like. We determine whether or not something is dangerous by comparing it to our knowledge of what dangerous things are like.

Why do you ask this question specifically of the integers? Is there something special about them?

Comment author: 05 November 2012 03:13:04PM *  4 points [-]

How do you determine whether a physical process "behaves like integers"? The second-order axiom of induction sounds complicated, I cannot easily check that it's satisfied by apples. If you use some sort of Bayesian reasoning to figure out which axioms work on apples, can you describe it in more detail?

Comment author: 13 November 2012 03:48:29PM *  2 points [-]

I don't have an answer to the specific question, only to the class of questions. To approach understanding this, we need to distinguish between reality and what points to reality, i.e, symbols. Our skill as humans is in the manipulation of symbols, as a kind of simulation of reality, with greater or lesser workability for prediction, based in prior observation, of new observations.

"Apples" refers, internally, to a set of responses we created through our experience. We respond to reality as an "apple" or as a "set of apples," only out of our history. It's arbitrary. Counting, and thus "behavior like integers" applies to the simplified, arbitrary constructs we call "apples." Reality is not divided into separate objects, but we have organized our perceptions into named objects.

Examples. If an "apple" is a unique discriminable object, say all apples have had a unique code applied to them, then what can be counted is the codes. Integer behavior is a behavior of codes.

Unique applies can be picked up one at a time, being transferred to one basket or another. However, real apples are not a constant. Apples grow and apples rot. Is a pile of rotten apple an "apple"? Is an apple seed an apple? These are questions with no "true" answer, rather we choose answers. We end up with a binary state for each possible object: "yes, apple," or "no, not apple." We can count these states, they exist in our mind.

If "apple" refers to a variety, we may have Macintosh, Fuji, Golden delicious, etc.

So I have a basket with two apples in it. That is, five pieces of fruit that are Macintosh and three that are Fuji.

I have another basket with two apples in it. That is, one Fuji and one Golden Delicious.

I put them all into one basket. How many apples are in the basket? 2 + 2 = 3.

The question about integer behavior is about how categories have been assembled. If "apple" refers to an individual piece of intact fruit, we can pick it up, move it around, and it remains the same object, it's unique and there is no other the same in the universe, and it belongs to a class of objects that is, again, unique as a class, the class is countable and classes will display integer behavior.

That's as far as I've gotten with this. "Integer behavior" is not a property of reality, per se, but of our perceptions of reality.

Comment author: [deleted] 13 November 2012 03:11:19PM 1 point [-]

Well, it comes from the fact that apples in a bowl is Exclusively just that, as verified by your Bayesian reasoning. There are no other "chains" of successors (shadow apples? I can't even imagine a good metaphor).

So, now you in fact have that bowl of apples narrowed down to {0, S0, SS0, SSS0, ...} which is isomorphic to the natural numbers, so all other natural number properties will be reflected there.

Comment author: 02 November 2012 05:54:01PM *  7 points [-]

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 presents a proof that no mimsy numbers have the Jaberwock property, and then everyone suddenly loses interest in mimsy numbers...

Point being, nothing here definitely justifies thinking that there are numbers, because someone could come along tomorrow and prove ~(2+2=4) and we'd be done talking about "numbers". But I feel really really confident that that won't ever happen and I'm not quite sure how to say whence this confidence. I think this might be similar to your last question, but it seems to dodge RichardKennaway's objection.

Comment author: 03 November 2012 11:36:21AM 2 points [-]

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.")

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

Comment author: 25 October 2012 03:11:59AM 3 points [-]

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?

Comment author: 02 November 2012 01:35:23AM 6 points [-]

(Note: this is my first post. I may be wrong, and if so am curious as to how. Anyway, I figure it's high time that my beliefs stick their neck out. I expect this will hurt, and apologize now should I later respond poorly.)

This may be the answer to a different question, but...

I play lots of role-playing games. Role-playing games are like make-believe; events in them exist in a shared counter-factual space (in the players' imagination). Make-believe has a problem: if two people imagine different things, who is right? (This tends to end with a bunch of kids arguing about whether the fictional T-Rex is alive or dead).

Role-playing games solve this problem by handing authority over various facets of the game to different things. The protagonists are controlled by their respective players, the results of choices by dice and rules, and most of the fictional world by the Game Master.*

So, in a role-playing game, when you ask what is true[RPG], you should direct that question to the appropriate authority. Basically, truth[RPG] is actually canon (in the fandom sense; TV Trope's page is good, but comes with the usual where-did-my-evening-go caveats).

Similarly, if we ask "where did Luke Skywalker go to preschool?", we're asking a question about canon.

That said, even canon needs to be internally consistent. If someone with authority were to claim that Tatooine has no preschools, then we can conclude that Luke Skywalker didn't go to preschool. If an authority claims two inconsistent things, we can conclude that the authority is wrong (namely, in the mathematical sense the canon wouldn't match any possible model).

I've long felt that ideas like morality and liberty are a variety of canon.

Specifically, you can have authorities (a religion or philosopher telling you stuff), and those authorities can be provably wrong (because they said something inconsistent), but these ideas exists in a kind of shared imaginary space. Also, people can disagree with the canon and make up their own ideas.

Now, that space is still informed by reality. Even in fiction, we expect gravity to drop off as the square of distance, and we expect solid objects to be unable to pass through each other.** With ideas, we can state that they are nonsensical (or, at minimum, not useful) if they refers to real things which don't exist. A map of morality is a map of a non-real thing, but morality must interface with reality to be useful, so anywhere the interface doesn't line up with reality, morality (or its map) is wrong.

*This is one possible breakdown. There are many others.

**In most games/stories, anyway. At first glance I'd expect morality to be better bound to reality, but I suppose there's been plenty of people who's moral system boiled down to "don't do anything Ma'at would disapprove of", backed up with concepts like the literal weight of sin (vs. the weight of a feather).

Comment author: 01 November 2012 04:40:28PM 6 points [-]

It so happens that the three "big lies" death mentions are all related to morality/ethics, which is a hard question. But let me take the conversation and change it a bit:

"So we can believe the big ones?"

Yes. Anger. Happiness. Pain. 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 happiness, one molecule of pain.

In this version, the final argument is still correct -- if I take the universe and grind it down to a sieve, I will not be able to say "woo! that carbon atom is an atom of happiness". Since the penultimate question of this meditation was "Is there anything else", at least I can answer that question.

Clearly, we want to talk about happiness for many reasons -- even if we do not value happiness in itself (for ourselves or others), predicting what will make humans happy is useful to know stuff about the world. Therefore, it is useful to find a way that allows us to talk about happiness. Happiness, though, is complicated, so let us put it aside for a minute to ponder something simpler: a solar system. I will simplify here, a solar system is one star and a bunch of planets rotating around it. Though solar systems effect each other through gravity or radiation, most of the effects of the relative motions inside a solar system comes from inside itself, and this pattern repeats itself throughout the galaxy. Much like happiness, being able to talk about solar systems is useful -- though I do not particularly value solar systems in and of themselves, it's useful to have a concept of "a solar system", which describes things with commonalities, and allows me to generalize.

If I grind the universe, I cannot find an atom that is a solar system atom -- grinding the universe down destroys the "solar system" useful pattern. For bounded minds, having these patterns leads to good predictive strength without having to figure out each and every atom in the solar system.

In essence, happiness is no different than solar system -- both are crude words to describe common patterns. It's just that happiness is a feature of minds (mostly human minds, but we talk about how dogs or lizards are happy, sometimes, and it's not surprising -- those minds are related algorithms). I cannot say where every atom is in the case of a human being happy, but some atom configurations are happy humans, and some are not.

So: at the very least, happiness and solar systems are part of the causal network of things. They describe patterns that influence other patterns.

Mercy is easier than justice and duty. Mercy is a specific configuration of atoms behaving a human in a specific way -- even though the human feels they are entitled to cause another human hurt ("feeling entitled" is a set of specific human-mind-configurations, regardless of whether "entitlement" actually exists), but does not do so (for specific reasons, etc. etc.). In short, mercy describes specific patterns of atoms, and is part of causal networks.

Duty and justice -- I admit that I'm not sure what my reductionist metaethics are, and so it's not obvious what they mean in the causal network.

Comment author: 01 November 2012 11:32:38PM *  7 points [-]

We could make it even easier :P

You say that a tiger has stripes, but I looked at some tiger atoms and didn't see any stripes.

The harder question is what is a valid way of figuring out the important properties of the system.

Comment author: 01 November 2012 05:22:39PM 4 points [-]

The statement that the world is just is a lie. There exist possible worlds that are just - for instance, these worlds would not have children kidnapped and forced to kill - and ours is not one of them.

Thus, justice is a meaningful concept. Justice is a concept defined in terms of the world (pinned-down causal links) and also irreducibly normative statements. Normative statements do not refer to "the world". They are useful because we can logically deduce imperatives from them. "If X is just, then do X." is correct, that is:

Do the right thing.

Comment author: 01 November 2012 09:32:18PM 2 points [-]

I am not entirely sure how you arrived at the conclusion that justice is a meaningful concept. I am also unclear on how you know the statement "If X is just, then do X" is correct. Could you elaborate further?

In general, I don't think it is a sufficient test for the meaningfulness of a property to say "I can imagine a universe which has/lacks this property, unlike our universe, therefore it is meaningful."

Comment author: 02 November 2012 07:22:06PM 1 point [-]

I did not intend to explain how i arrived at this conclusion. I'm just stating my answer to the question.

Do you think the statement "If X is just, then do X" is wrong?

Comment author: 03 November 2012 05:40:11PM 0 points [-]

Like army1987 notes, it is an instruction and not a statement. Considering that, I think "if X is just, then do X" is a good imperative to live by, assuming some good definition of justice. I don't think I would describe it as "wrong" or "correct" at this point.

Comment author: 04 November 2012 05:11:44PM 0 points [-]

OK. Exactly what you call it is unimportant.

What matters is that it gives justice meaning.

Comment author: 02 November 2012 08:47:42PM 0 points [-]

It may be incomplete. Do you have a place for Mercy?

Comment author: 04 November 2012 05:15:00PM 2 points [-]

The reason I'm not making distinctions among different moral words, though such distinctions exist in language, is that it seems the only new problem created by these moral words is understanding morality. Once you understand right and wrong, just and unjust can be defined just like you define regular words, even if something can be just but immoral.

Comment author: [deleted] 03 November 2012 03:24:48PM 1 point [-]

the statement "If X is just, then do X"

That's an instruction, not a statement.

Comment author: 02 November 2012 03:24:54AM 3 points [-]

They exist in the same sense that numbers exist, or that meaningful existence exists, or that meaningfulness exists.

Once you grind the universe into powder, none of those things exists anymore.

Comment author: 01 November 2012 06:09:39PM *  3 points [-]

I was going to say that yes, I think there is another kind of thing that can be meaningfully talked about, and "justice" and "mercy" and "duty" have something to do with that sort of thing, but a more prototypical example would be "This court has jurisdiction". Especially if many experts were of the opinion that it didn't, but the judge disagreed, but the superior court reversed her, and now the supreme court has decided to hear the case.

But then I realized that there was something different about that kind of "truth": I would not want an AI to assign a probability to the proposition The court did, in fact, have jurisdiction (nor to, oh, It is the duty of any elected official to tell the public if they learn about a case of corruption, say). I think social constructions can technically be meaningfully talked about among humans, and they are important as hell if you want to understand human communication and behavior, but I guess on reflection I think that the fact that I would want an AI to reason in terms of more basic facts is a hint that if we are discussing epistemology, if we're discussing what sorts of thingies we can know about and how we can know about them, rather than discussing particular properties of the particularly interesting thingies called humans, then it might be best to say that "The judge wrote in her decision that the court had jurisdiction" is a meaningful statement in the sense under consideration, but "The court had jurisdiction" is not.

Comment author: 03 November 2012 01:16:07PM 2 points [-]

I would find them under the category of patterns.

A neural network is very good at recognising patterns; and human brains run on a neural network architecture. Given a few examples of what a word does or does not mean, we can quickly recognise the pattern and fit it into our vocabulary. (Apparently, this can be used in language classes; the teacher will point to a variety of objects, indicating whether they are or are not vrugte, for example; and it won't take that many examples before the student understands that vrugte means fruit but not vegetables).

Justice and mercy are not patterns of objects, but rather patterns of action. The man killed his enemy, but has a wife and children to support; sending him to Death Row might be just, but letting him have some way of earning money while imprisoned might be merciful. Similarly, happy, sad, and angry are emotional patterns; a person acts in this way when happy, and acts in that way when sad.

Comment author: 02 November 2012 05:38:32PM 2 points [-]

Justice, mercy, duty, etc are found by comparison to logical models pinned down by axioms. Getting the axioms right is damn tough, but if we have a decent set we should be able to say "If Alex kills Bob under circumstances X, this is unjust." We can say this the same way that we can say "Two apples plus two apples is four apples." I can't find an atom of addition in the universe, and this doesn't make me reject addition.

Also, the widespread convergence of theories of justice on some issues (eg. Rape is unjust.) suggests that theories of justice are attempting to use their axioms to pin down something that is already there. Moral philosophers are more likely to say "My axioms are leading me to conclude rape is a moral duty, where did I mess up?" than "My axioms are leading me to conclude rape is a moral duty, therefore it is." This also suggests they are pinning down something real with axioms. If it was otherwise, we would expect the second conclusion.

Comment author: 02 November 2012 08:58:15PM *  1 point [-]

"theories of justice are attempting to use their axioms to pin down something that is already there"

So in other words, duty, justice, mercy--morality words--are basically logical transformations that transform the state of the universe (or a particular circumstance) into an ought statement.

Just as we derive valid conlcusions from premises using logical statements, we derive moral obligations from premises using moral statements.

The term 'utility funcion' seems less novel now (novel as in, a departure from traditional ethics).

Comment author: 04 November 2012 05:16:37PM 1 point [-]

This is my view.

Comment author: 04 November 2012 08:00:01PM 0 points [-]

Not quite. They don't go all the way to completing an ought statement, as this doesn't solve the Is/Ought dichotomy. They are logical transformations that make applying our values to the universe much easier.

"X is unjust" doesn't quite create an ought statement of "Don't do X". If I place value on justice, that statement helps me evaluate X. I may decide that some other consideration trumps justice. I may decide to steal bread to feed my starving family, even if I view the theft as unjust.

Comment author: [deleted] 01 November 2012 05:11:58PM 2 points [-]

I've thought about this for a while, and I feel like you can replace "Fantasy" and "Lies" with "Patterns" in that dialogue, and have it make sense, and it also appears to be an answer to your questions. That being said, it also feels like a sort of a cached thought, even though I've thought about it for a while. However, I can't think of a better way to express it and all of the other thoughts I had appeared to be significantly lower caliber and less clear.

Comment author: 01 November 2012 09:17:08PM 0 points [-]

In people's brains, and in papers written by philosophy students.

Comment author: 03 November 2012 01:02:35AM 1 point [-]

"Justice" and "mercy" can be found by looking at people, and in particular how people treat each other. They're physical things, although they're really complicated kinds of physical things.

Comment author: 06 November 2012 09:53:24PM 6 points [-]

In particular, the kind of thing that is destroyed when you grind it down into powder.

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. Cars. Chairs. Bicycles. 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 car, one molecule of bicycle.

Susan and Death, in Hogfather by Terry Pratchett

Same thing.

Comment author: 01 November 2012 07:58:15PM 0 points [-]

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?

You find them inside counterfactual statements about the reactions of an implied hypothetical representative human, judging under under implied hypothetical circumstances in which they have access to all relevant knowledge. There is clearly justice if a wide variety of these hypothetical humans agree that there is, under a wide variety of these hypothetical circumstances; there is clearly not justice if they agree that there is not. If the hypothetical people disagree with each other, then the definition fails.

Talking about things like justice, mercy and duty is meaningful, but the meanings are intermediated by big, complex webs of abstractions which humans keep in their brains, and the algorithms people use to manipulate those webs. They're unambiguous only to the extent to which people successfully keep those webs in sync with each other. In practice, our abstractions mainly work by combining bags of weak classifiers and feature-weighted similarity to positive and negative examples. This works better for cases that are similar to the training set, worse for cases that are novel and weird, and better for simpler abstractions and abstractions built on simpler constituents.

Comment author: 01 November 2012 08:02:39PM *  0 points [-]

Why couldn't the hypothetical omniscient people inside the veil of ignorance decide that justice doesn't exist? Or if they could, how does that paragraph go towards answering the meditation? What distinguishes them from the hypothetical death who looks through everything in the universe to try to find mercy? Aren't you begging the question here?

Comment author: 11 November 2012 12:21:37AM *  6 points [-]

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.

Comment author: 12 November 2012 08:09:39PM 1 point [-]

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.

Agreed, and disappointed that this comment was downvoted.

Comment author: 01 November 2012 05:47:43PM 8 points [-]

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

Comment author: 01 November 2012 05:58:21PM 12 points [-]

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.

Comment author: 01 November 2012 06:10:00PM 6 points [-]

It's a pretty hard balance to strike that's probably different for everyone, between incomprehensibility and boringness.

Comment author: [deleted] 01 November 2012 11:36:25PM 6 points [-]

I already more-or-less knew most of the stuff in the previous posts in this sequences and still didn't find them boring.

Comment author: 01 November 2012 11:58:18PM 3 points [-]

Agree. When I first read The Simple Truth, I thought Eliezer was endorsing pragmatism over correspondence.

Comment author: 02 November 2012 12:03:56AM 3 points [-]

I'm still wondering what The Simple Truth is about. My best guess is that it is a critique of instrawmantalism.

Comment author: [deleted] 02 November 2012 03:54:00AM 3 points [-]

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.

Comment author: 18 December 2012 03:35:43AM 2 points [-]

Awesome, I was looking for a good explanation of the Peano axioms!

About six months ago I had a series of arguments with my housemate, who's been doing a philosophy degree at a Catholic university. He argued that I should leave the door open for some way other than observation to gather knowledge, because we had things like maths giving us knowledge in this other way, which meant we couldn't assume we'd come up with some other other way to discover, say, ethical or aesthetic truths.

I couldn't convince him that all we could do in ethics was reason from axioms, because he didn't understand that maths was just reasoning from axioms --- and I didn't actually understand the Peano axioms, so I couldn't explain them.

So, thanks for the post.

Comment author: 03 November 2012 01:05:51PM 2 points [-]

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.

My short answer is "because we live in a causal universe".

To expand on that:

Logic is a process that has been specifically designed to be stable. Any process that has gone through a design specifically intended to make it stable, and refined for stability over generations, is going to have a higher probability of being stable. Logic, in short, is more likely than anything else in the universe to be stable.

So then the question is not why logic specifically is stable - that is by design - but rather whether it is possible for anything in the universe to be stable. And there is one thing that does appear to be stable; that if you have the same cause, then you will have the same effect. That the universe is (at least mostly) causal. It is that causality that gives logic its stability, as far as I can see.

Comment author: 03 November 2012 09:34:29AM *  2 points [-]

"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."

But the axiom schema of induction does not completely exclude nonstandard numbers. Sure if I prove some property P for P(0) and for all n, P(n) => P(n+1) then for all n, P(n); then I have excluded the possibility of some nonstandard number "n" for which not P(n) but there are some properties which cannot be proved true or false in Peano Arithmetic and therefore whose truth hood can be altered by the presence of nonstandard numbers.

Can you give me a property P which is true along the zero-chain but necessarily false along a separated chain that is infinitely long in both directions? I do not believe this is possible but I may be mistaken.

Comment author: 03 November 2012 11:03:28PM 1 point [-]

But the axiom schema of induction does not completely exclude

Eliezer isn't using an axiom schema, he's using an axiom of second order logic.

Comment author: 03 November 2012 11:57:49PM *  1 point [-]

I don't see what the difference is... They look very similar to me.

At some point you have to translate it into a (possibly infinite) set of first-order axioms or you wont be able to perform first-order resolution anyway.

Comment author: 04 November 2012 03:18:08AM 0 points [-]

What's wrong with second order resolution?

Comment author: 03 November 2012 05:49:09PM *  1 point [-]

Not sure if I understand the point of your argument.

Are you saying that in reality every property P has actually three outcomes: true, false, undecidable? And that those always decidable, like e.g. "P(n) <-> (n = 2)" cannot be true for all natural numbers, while those which can be true for all natural numbers, but mostly false otherwise, are always undecidable for... some other values?

Can you give me a property P which is true along the zero-chain but necessarily false along a separated chain that is infinitely long in both directions? I do not believe this is possible but I may be mistaken.

I don't know.

Let's suppose that for any specific value V in the separated chain it is possible to make such property PV. For example "PV(x) <-> (x <> V)". And let's suppose that it is not possible to make one such property for all values in all separated chains, except by saying something like "P(x) <-> there is no such PV which would be true for all numbers in the first chain and false for x".

What would that prove? Would it contradict the article? How specifically?

Comment author: 03 November 2012 06:29:50PM 1 point [-]

Are you saying that in reality every property P has actually three outcomes: true, false, undecidable?

By Godel's incompleteness theorem yes, unless your theory of arithmetic has a non-recursively enumerable set of axioms or is inconsistent.

And that those always decidable, like e.g. "P(n) <-> (n = 2)" cannot be true for all natural numbers, while those which can be true for all natural numbers, but mostly false otherwise, are always undecidable for... some other values?

I'm having trouble understanding this sentence but I think I know what you are asking about.

There are some properties P(x) which are true for every x in the 0 chain, however, Peano Arithmetic does not include all these P(x) as theorems. If PA doesn't include P(x) as a theorem, then it is independent of PA whether there exist nonstandard elements for which P(x) is false.

Let's suppose that for any specific value V in the separated chain it is possible to make such property PV. What would that prove? Would it contradict the article? How specifically?

I think this is what I am saying I believe to be impossible. You can't just say "V is in the separated chain". V is a constant symbol. The model can assign constants to whatever object in the domain of discourse it wants to unless you add axioms forbidding it.

Honestly I am becoming confused. I'm going to take a break and think about all this for a bit.

Comment author: 03 November 2012 09:59:14AM *  1 point [-]

If our axiom set T is independent of a property P about numbers then by definition there is nothing inconsistent about the theory T1 = "T and P" and also nothing inconsistent about the theory T2= "T and not P".

To say that they are not inconsistent is to say that they are satisfiable, that they have possible models. As T1 and T2 are inconsistent with each other, their models are different.

The single zero-based chain of numbers without nonstandard numbers is a single model. Therefore, if there exists a property about numbers that is independent of any theory of arithmetic, that theory of arithmetic does not logically exclude the possibility of nonstandard elements.

By Godel's incompleteness theorems, a theory must have statements that are independent from it unless it is either inconsistent or has a non-recursively-enumerable theorem set.

Each instance of the axiom schema of induction can be constructed from a property. The set of properties is recursively enumerable, therefore the set of instances of the axiom schema of induction is recursively enumerable.

Every theorem of Peano Arithmetic must use a finite number of axioms in its proof. We can enumerate the theorems of Peano Arithmetic by adding increasingly larger subsets of the infinite set of instances of the axiom schema of induction to our axiom set.

Since the theory of Peano Arithmetic has a recursively enumerable set of theorems it is either inconsistent or is independent of some property and thus allows for the existence of nonstandard elements.

Comment author: 02 November 2012 10:11:13AM 2 points [-]

I love this post, and will be recommending it.

Speaking as a non-mathematician I think I would have tried to express 'there's only one chain' by saying something like 'all numbers can be reached by a finite amount of repetititions of considering the successor of a number you've already considered, starting from zero'.

Comment author: 02 November 2012 12:48:49PM 4 points [-]

We can try to write that down as "For all x, there is an n such that x = S(S(...S(0)...)) repeated n times."

The two problems that we run into here are: first, that repeating S n times isn't something we know how to do in first-order logic: we have to say that there exists a sequence of repetitions, which requires quantifying over a set. Second, it's not clear what sort of thing "n" is. It's a number, obviously, but we haven't pinned down what we think numbers are yet, and this statement becomes awkward if n is an element of some other chain that we're trying to say doesn't exist.

Comment author: 05 May 2015 10:29:06AM *  0 points [-]

repeating S n times isn't something we know how to do in first-order logic

Why not? Repeating S n times is just addition, and addition is defined in the peano first order logic axioms. I just took these from my textbook:

∀y.plus(0,y,y)

∀x.∀y.∀z.(plus(x,y,z) ⇒ plus(s(x),y,s(z)))

∀x.∀y.∀z.∀w.(plus(x,y,z) ∧ ¬same(z,w) ⇒ ¬plus(x,y,w))

I've also seen addition defined recursively somehow, so each step it subtracted 1 from the second number and added 1 to the first number, until the second number was equal to zero. Something like this:

∀x.∀y.∀z.∀w.(plus(x,y,z) ⇒ plus(s(x),w,z) ∧ same(s(w),y))

From this you could define subtraction in a similar way, and then state that all numbers subtracted from themselves, must equal 0. This would rule out nonstandard numbers.

Comment author: 05 May 2015 11:15:13AM 3 points [-]

From this you could define subtraction in a similar way, and then state that all numbers subtracted from themselves, must equal 0. This would rule out nonstandard numbers.

That will not rule out nonstandard models of the first-order Peano axioms. If a subtraction predicate is defined by:

∀x. sub(x,0,x)

∀x.∀y.∀z. sub(x,y,z) ⇒ sub(s(x),s(y),z)

then you don't need to add that all numbers subtracted from themselves, must equal 0. ∀x.sub(x,x,0) is already a theorem, which can be proved almost immediately from those axioms and the first-order induction schema. Being a theorem, it is true in all models. Every nonstandard element of a nonstandard model, subtracted from itself, gives 0.

It may seem odd that a statement proved by induction is necessarily true even of those elements of a non-standard model that, in our mental picture of them, cannot be reached by counting upwards from zero, but the induction axiom scheme explicitly says just that: if P(0) and ∀x.(P(x) ⇒ P(s(x))) then ∀x.P(x). The conclusion is not limited to standard values of x, because the language cannot distinguish standard from non-standard values.

Comment author: 07 May 2015 09:54:05AM 0 points [-]

If you already have an axiom of induction then you've already ruled out nonstandard numbers and this isn't an issue. I was trying to show that without the second order logic axiom of induction, you can rule out nonstandard numbers.

The recursive subtract predicate will never reach zero on a nonstandard number, therefore it can not be true that n-n=0.

Comment author: 07 May 2015 10:57:14AM 0 points [-]

If you already have an axiom of induction then you've already ruled out nonstandard numbers and this isn't an issue. I was trying to show that without the second order logic axiom of induction, you can rule out nonstandard numbers.

Without second-order logic, you cannot rule out nonstandard numbers. As Epictetus mentioned, the Lowenheim-Skolem Theorem implies that if there is a model of first-order Peano arithmetic, there are models of all infinite cardinalities.

You have to distinguish the axioms from the meanings one intuitively attaches to them. We have an intuitive idea of the natural numbers, and the Peano axioms (including the induction schema) seem to be true of them. However, ZFC set theory (for example) provably contains models of those axioms other than the natural numbers of our intuition.

The induction schema seems to formalise our notion that every natural number is reachable by counting up from zero. But look more closely and you can intuitively read it like this: if you can prove that P is true of every number you can reach by counting, then P is true of every number (even those you can't reach by counting, if there are any).

The predicate "is a standard number" would be a counterexample to that, but the induction schema is asserted only for formulas P expressible in the language of Peano arithmetic. Given the existence of non-standard models, the fact that "is a standard number" does not satisfy the induction schema demonstrates that it is not definable in the language.

The subtraction predicate provably satisfies ∀n. n-n = 0. Therefore every model of the Peano axioms satisfies that -- it would not be a model if it did not.

(Technical remark: I should not have added "sub" as a new symbol, which creates a different language, an extension of Peano arithmetic. Instead, "sub(x,y,z) should be introduced as a metalinguistic abbreviation for y+z=x, which is a formula of Peano arithmetic. One can still prove ∀x. sub(x,x,0), and without even using induction. Expanding the abbreviation gives x+0 = x, which is one of the axioms, e.g. as listed here.)

Comment author: 05 May 2015 11:40:14AM 1 point [-]

I refer you to the Lowenheim-Skolem Theorem:

Every (countable) first-order theory that has an infinite model, has a model of size k for every infinite cardinal k. You cannot use first-order logic to exclude non-standard numbers unless you want to abandon infinite models altogether.

Comment author: 05 May 2015 02:40:28PM 0 points [-]

Repeating S n times is not addition: addition is the thing defined by those axioms, no more, and no less. You can prove the statements:

∀x. plus(x, 1, S(x))

∀x. plus(x, 2, S(S(x)))

∀x. plus(x, 3, S(S(S(x))))

and so on, but you can't write "∀x. plus(x, n, S(S(...n...S(x))))" because that doesn't make any sense. Neither can you prove "For every x, x+n is reached from x by applying S to x some number of times" because we don't have a way to say that formally.

From outside the Peano Axioms, where we have our own notion of "number", we can say that "Adding N to x is the same as taking the successor of x N times", where N is a real-honest-to-god-natural-number. But even from the outside of the Peano Axioms, we cannot convince the Peano Axioms that there is no number called "pi". If pi happens to exist in our model, then all the values ..., pi-2, pi-1, pi, pi+1, pi+2, ... exist, and together they can be used to satisfy any theorem about the natural numbers you concoct. (For instance, sub(pi, pi, 0) is a true statement about subtraction, so the statement "∀x. sub(x, x, 0)" can be proven but does not rule out pi.)

Comment author: 07 May 2015 10:08:59AM 0 points [-]

"For every x, x+n is reached from x by applying S to x some number of times" because we don't have a way to say that formally.

But that's what I'm trying to say. To say n number of times, you start with n and subtract 1 each time until it equals zero. So for addition, 2+3 is equal to 3+2, is equal to 4+1, is equal to 5+0. For subtraction you do the opposite and subtract one from the left number too each time.

If no number of subtract 1's cause it to equal 0, then it can't be a number.

Comment author: 07 May 2015 02:53:17PM 0 points [-]

I know that's what you're trying to say because I would like to be able to say that, too. But here's the problems we run into.

1. Try writing down "For all x, some number of subtract 1's cause it to equal 0". We can write the "∀x. ∃y. F(x,y) = 0" but in place of F(x,y) we want "y iterations of subtract 1's from x". This is not something we could write down in first-order logic.

2. We could write down sub(x,y,0) (in your notation) in place of F(x,y)=0 on the grounds that it ought to mean the same thing as "y iterations of subtract 1's from x cause it to equal 0". Unfortunately, it doesn't actually mean that because even in the model where pi is a number, the resulting axiom "∀x. ∃y. sub(x,y,0)" is true. If x=pi, we just set y=pi as well.

The best you can do is to add an infinitely long axiom "x=0 or x = S(0) or x = S(S(0)) or x = S(S(S(0))) or ..."

Comment author: 08 May 2015 04:32:28AM 0 points [-]

I think I'm starting to get it. That there is no property that a natural number could be defined as having, that a infinite chain couldn't also satisfy in theory.

That's really disappointing. I took a course on logic and the most inspiring moment was when the professor wrote down the axioms of peano arithmitic. They are more or less formalizations of all the stuff we learned about numbers in grade school. It was cool that you could just write down what you are talking about formally and use pure logic to prove any theorem with them. It's sad that it's so limited you can't even express numbers.

Comment author: 01 November 2012 05:43:49PM *  1 point [-]

I love this inquiry.

Numbers do not appear in reality, other than "mental reality." 2+2=4 does not appear outside of the mind. Here is why:

To know that I have two objects, I must apply a process to my perception of reality. I must recognize the objects as distinct, I must categorize them as "the same" in some way. And then I apply another process, "counting." That is applied to my collected identifications, not to reality itself, which can just as easily be seen as unitary, or sliced up in a practically infinite number of ways.

Number, then, is a product of brain activity, and the observed properties of numbers are properties of brain process. Some examples.

I put two apples in a bowl. I put two more apples in the bowl. How many apples are now in the bowl?

We may easily say "four," because most of the time this prediction holds. However, it's a mixing bowl, used as a blender, and what I have now is a bowl of applesauce. How many apples are in the bowl? I can't count them! I put four apples in, and none come out! Or some smaller number than four. Or a greater number (If I add some earth, air, fire, and water, and wait a little while....)

Apples are complex objects. How about it's two deuterium molecules? (Two deuterons each, with two electrons, electronically bound.) How about the bowl is very small, confining the molecules, reducing their freedom of movement, and their relative momentum is, transiently, close to zero?

How many deuterons? Initially, four, but ... it's been calculated that after a couple of femtoseconds, there are none, there is one excited atom of Beryllium-8, which promptly decays into two helium nuclei and a lot of energy. In theory. It's only been calculated, it's not been proven, it merely is a possible explanation for certain observed phenomena. Heh!

The point here: the identity of an object, the definition of "one," is arbitrary, a tool, a device for organizing our experience of reality. What if it's two red apples and two green apples? They don't taste the same and they don't look the same, at least not entirely the same. What we are counting is the identified object, "apple." Not what exists in reality. Reality exists, not "apples," except in our experience, largely as a product of language.

The properties of numbers, so universally recognized, follow from the tools we evolved for predicting behavior, they are certainly not absolutes in themselves.

Hah! "Certainly." That, with "believe" is a word that sets off alarms.

Comment author: 01 November 2012 05:58:22PM -1 points [-]

The fact that one apple added to one apple invariably gives two apples....

It's almost a tautology. What we have is an iterated identification. There are two objects that are named "apple," they are identical in identification, but separate and distinct. This appears in time. I'm counting my identifications. The universality of 1+1 = 2 is a product of a single brain design. For an elephant, the same "problem" might be "food plus food equals food."

Comment author: 01 November 2012 08:19:02PM *  1 point [-]

Basically, you're saying that for an elephant, apples behave like clouds, because the elephant has a concept of apple that is like our concept of cloud. (I hope real elephants aren't this dumb). I like this a lot, it clarifies what I felt was missing from the cloud analogy.

Having it explicitly stated is helpful. It leads to the insight that at bottom, outside of directly useful concepts and into pure ontology/epistemology, there are no isolated individual integers. There is only relative magnitude on a broad continuum. This makes approaching QM much simpler.

Comment author: 02 November 2012 01:07:31AM 3 points [-]

Mmmm. This is all projected onto elephants, but maybe something like what you say. I was just pointing to a possible alternate processing mode. An elephant might well recognize quantity, but probably not through counting, which requires language. Quantity might be recognized directly, by visual comparison, for example. Bigger pile/smaller pile. More attraction vs. less attraction, therefore movement toward bigger pile. Or smell.

Comment author: [deleted] 02 November 2012 01:33:35AM 1 point [-]

Would you argue, then, that aliens or AIs might not discover the fact that 1 + 1 = 2, or even consider it a fact at all?

Comment author: 07 November 2012 04:54:06PM -2 points [-]

EY is talking from a position of faith that infinite model theory and second-order logic are good and reasonable things.

It is possible to instead start from a position of doubt that the infinite model theory and second order logic are good and reasonable things (based on my memory of having studied in college whether model theory and second order logic can be formalized within Zermelo-Frankel set theory, and what the first-order-ness of Zermelo-Frankel has to do with it.).

We might be fine with a proof-theoretic approach, which starts with the same ideas "zero is a number", "the successor of a number is a different number", but then goes to a proof-theoretic rule of induction something like "I'd be happy to say 'All numbers have such-and-such property' if there were a proof that zero has that property and another also proof that if a number has that property, then its successor also has that property."

We don't need to talk about models at all - in particular we don't need to talk about infinite models.

Second-order arithmetic is sufficient to get what EY wants (a nice pretty model universe) but I have two objections. First it is too strong - often the first sufficient hammer that you find in mathematics is rarely the one you should end up using. Second, the goal of a nice pretty model universe presumes a stance of faith in (infinite) model theory, but the infinite model theory is not formalized. If you do formalize it then your formalization will have alternative "undesired" interpretations (by Lowenheim-Skolem).

Comment author: [deleted] 08 November 2012 09:54:21PM *  2 points [-]

EY is talking from a position of faith that infinite model theory and second-order logic are good and reasonable things.

I think this is a fallacy of gray. Mathematicians have been using infinite model theory and second-order logic for a while, now; this is strong evidence that they are good and reasonable.

Edit: Link formatting, sorry. I wish there was a way to preview comments before submitting....

Comment author: 09 November 2012 09:06:01PM 1 point [-]

Second-order logic is not part of standard, mainstream mathematics. It is part of a field that you might call mathematical logic or "foundations of mathematics". Foundations of a building are relevant to the strength of a building, so the name implies that foundations of mathematics are relevant to the strength of mainstream mathematics. A more accurate analogy would be the relationship between physics and philosophy of physics - discoveries in epistemology and philosophy of science are more often driven by physics than the other way around, and the field "philosophy of physics" is a backwater by comparison.

As is probably evident, I think the good, solid mathematical logic is intuitionist and constructive and higher-order and based on proof theory first and model theory only second. It is easy to analogize from their names to a straight line between first-order, second-order, and higher-order logic, but in fact they're not in a straight line at all. First-order logic is mainstream mathematics, second-order logic is mathematical logic flavored with faith in the reality of infinite models and set theory, and higher-order logic is mathematical logic that is (usually) constructive and proof-theoretic and built with an awareness of computer science.

Comment author: 09 November 2012 09:25:29PM 3 points [-]

Comment author: 10 November 2012 01:18:56AM 2 points [-]

To an extent, but I think it's obvious that most mathematicians couldn't care less whether or not their theorems are expressible in second-order logic.

Comment author: 10 November 2012 01:46:15AM 3 points [-]

Yes, because most mathematicians just take SOL at face value. If you believe in SOL and use the corresponding English language in your proofs - i.e., you assume there's only one field of real numbers and you can talk about it - then of course it doesn't matter to you whether or not your theorem happens to require SOL taken at face value, just like it doesn't matter to you whether your proof uses ~~P->P as a logical axiom. Only those who distrust SOL would try to avoid proofs that use it. That most mathematicians don't care is precisely how we know that disbelief in SOL is not a mainstream value. :)

Comment author: 10 November 2012 06:38:00PM 5 points [-]

The standard story is that everything mathematicians prove is to be interpreted as a statement in the language of ZFC, with ZFC itself being interpreted in first-order logic. (With a side-order of angsting about how to talk about e.g. "all" vector spaces, since there isn't a set containing all of them -- IMO there are various good ways of resolving this, but the standard story considers it a problem; certainly in so far as SOL provides an answer to these concerns at all, it's not "the one" answer that everybody is obviously implicitly using.) So when they say that there's only one field of real numbers, this is supposed to mean that you can formalize the field axioms as a ZFC predicate about sets, and then prove in ZFC that between any two sets satisfying this predicate, there is an isomorphism. The fact that the semantics of first-order logic don't pin down a unique model of ZFC isn't seen as conflicting with this; the mathematician's statement that there is only one complete ordered field (up to isomorphism) is supposed to desugar to a formal statement of ZFC, or more precisely to the meta-assertion that this formal statement can be proven from the ZFC axioms. Mathematical practice seems to me more in line with this story than with yours, e.g. mathematicians find nothing strange about introducing the reals through axioms and then talk about a "neighbourhood basis" as something that assigns to each real number a set of sets of real numbers -- you'd need fourth-order logic if you wanted to talk about neighbourhood bases as objects without having some kind of set theory in the background. And people who don't seem to care a fig about logic will use Zorn's lemma when they want to prove something that uses choice, which seems quite rooted in set theory.

Now I do think that mathematicians think of the objects they're discussing as more "real" than the standard story wants them to, and just using SOL instead of FOL as the semantics in which we interpret the ZFC axioms would be a good way to, um, tell a better story -- I really like your post and it has convinced me of the usefulness of SOL -- but I think if we're simply trying to describe how mathematicians really think about what they're doing, it's fairer to say that they're just taking set theory at face value -- not thinking of set theory as something that has axioms that you formalize in some logic, but seeing it as as fundamental as logic itself, more or less.

Comment author: 10 November 2012 10:59:33PM 3 points [-]

Um, I think when an ordinary mathematician says that there's only one complete ordered field up to isomorphism, they do not mean, "In any given model of ZFC, of which there are many, there's only one ordered field complete with respect to the predicates for which sets exist in that model." We could ask some normal mathematicians what they mean to test this. We could also prove the isomorphism using logic that talked about all predicates, and ask them if they thought that was a fair proof (without calling attention to the quantification over predicates).

Taking set theory at face value is taking SOL at face value - SOL is often seen as importing set theory into logic, which is why mathematicians who care about it all are sometimes suspicious of it.

Comment author: 11 November 2012 12:56:42AM 2 points [-]

Um, I think when an ordinary mathematician says that there's only one complete ordered field up to isomorphism, they do not mean, "In any given model of ZFC, of which there are many, there's only one ordered field complete with respect to the predicates for which sets exist in that model." We could ask some normal mathematicians what they mean to test this.

The standard story, as I understand it, is claiming that models don't even enter into it; the ordinary mathematician is supposed to be saying only that the corresponding statement can be proven in ZFC. Of course, that story is actually told by logicians, not by people who learned about models in their one logic course and then promptly forgot about them after the exam. As I said, I don't agree with the standard story as a fair characterization of what mathematicians are doing who don't care about logic. (Though I do think it's a coherent story about what the informal mathematical English is supposed to mean.)

Taking set theory at face value is taking SOL at face value - SOL is often seen as importing set theory into logic, which is why mathematicians who care about it all are sometimes suspicious of it.

Is it a fair-rephrasing of your point that what normal mathematicians do requires the same order of ontological commitment as the standard (non-Henkin) semantics of SOL, since if you take SOL as primitive and interpret the ZFC axioms in it, that will give you the correct powerset of the reals, and if you take set theory as primitive and formalize the semantics of SOL in it, you will get the correct collection of standard models? 'Cause I agree with that (and I see the value of SOL as a particularly simple way of making that ontological commitment, compared to say ZFC). My point was that mathematical English maps much more directly to ZFC than it does to SOL (there's still coding to be done, but much less of it when you start from ZFC than when you start from SOL); e.g., you earlier said that "[o]nly those who distrust SOL would try to avoid proofs that use it", and you can't really use ontological commitments in proofs, what you can actually use is notions like "for all properties of real numbers", and many notions people actually use are ones more directly present in ZFC than SOL, like my example of quantifying over the neighbourhood bases (mappings from reals to sets of sets of reals).

Comment author: 10 November 2012 03:56:12PM -1 points [-]

I agree with this statement - and yet you did not contradict my statement that second order logic is also not part of mainstream mathematics.

A topologist might care about manifolds or homeomorphisms - they do not care about foundations of mathematics - and it is not the case that only one foundation of mathematics can support topology. The weaker foundation is preferable.

Comment author: 10 November 2012 04:41:04PM 2 points [-]

The last sentence is not obvious at all. The goal of mathematics is not to be correct a lot. The goal of mathematics is to promote human understanding. Strong axioms help with that by simplifying reasoning.

Comment author: 10 November 2012 05:37:00PM *  1 point [-]

If you assume A and derive B you have not proven B but rather A implies B. If you can instead assume a weaker axiom Aprime, and still derive B, then you have proven Aprime implies B, which is stronger because it will be applicable in more circumstances.

Comment author: 11 November 2012 07:29:56AM 0 points [-]

In what "circumstances" are manifolds and homeomorphisms useful?

Comment author: 11 November 2012 03:37:04PM 0 points [-]

If you were writing software for something intended to traverse the Interplanetary transfer network then you would probably use charts and atlases and transition functions, and you would study (symplectic) manifolds and homeomorphisms in order to understand those more-applied concepts.

If an otherwise useful theorem assumes that the manifold is orientable, then you need to show that your practical manifold is orientable before you can use it - and if it turns out not to be orientable, then you can't use it at all. If instead you had an analogous theorem that applied to all manifolds, then you could use it immediately.

Comment author: 11 November 2012 05:20:47PM 2 points [-]

There's a difference between assuming that a manifold is orientable and assuming something about set theory. The phase space is, of course, only approximately a manifold. On a very small level it's - well, something we're not very sure of. But all the math you'll be doing is an approximation of reality.

So some big macroscopic feature like orientability would be a problem to assume. Orientability corresponds to something in physical reality, and something that clearly matters for your calculation.

The axiom of choice or whatever set-theoretic assumption corresponds to nothing in physical reality. It doesn't matter if the theorems you are using are right for the situation, because they are obviously all wrong, because they are about symplectic dynamics on a manifold, and physics isn't actually symplectic dynamics on a manifold! The only thing that matters is how easily you can find a good-enough approximation to reality. More foundational assumptions make this easier, and do not impede one's approximation of reality.

Note that physicists frequently make arguments that are just plain unambiguously wrong from a mathematical perspective.

Comment author: 08 November 2012 10:17:42PM 0 points [-]

Well, it's strong evidence that mathematicians find these things useful for publishing papers.

Comment author: 16 July 2015 09:58:48PM 1 point [-]

"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."

Ok NOW I finally get the whole Peano arithmetic thing. ...Took me long enough. Thanks kindly, unusually-fast-thinking mathematician!

Comment author: 14 April 2013 03:49:17PM *  1 point [-]

The boundary between physical causality and logical or mathematical implication doesn’t always seem to be clearcut. Take two examples.

(1) The product of two and an integer is an even integer. So if I double an integer I will find that the result is even. The first statement is clearly a timeless mathematical implication. But by recasting the equation as a procedure I introduce both an implied separation in time between action and outcome, and an implied physical embodiment that could be subject to error or interruption. Thus the truth of the second formulation strictly depends on both a mathematical fact and physical facts.

(2) The endpoint of a physical process is causally related to the initial conditions by the physical laws governing the process. The sensitivity of the endpoint to the initial conditions is a quite separate physical fact, but requires no new physical laws: it is a mathematical implication of the physical laws already noted. Again, the relationship depends on both physical and mathematical truths.

Is there a recognized name for such hybrid cases? They could perhaps be described as “quasi-causal” relationships.

Comment author: 08 December 2012 12:50:23AM 0 points [-]

x + Sy = Sz -

That looks a bit odd.

Comment author: 08 December 2012 12:58:05AM 2 points [-]

I think the idea is that one speaker got cut off by the other after having said "x+Sy=Sz".

Comment author: [deleted] 08 December 2012 01:47:49AM *  2 points [-]

If this were Wikipedia, someone would write a rant about the importance of using typographically correct characters for the hyphen, the minus sign, the en dash, and the em dash ( - − – and — respectively).

Comment author: 08 December 2012 05:31:00AM 0 points [-]

Yeah, I understood that after about 10 seconds of confusion, which seems unnecessary.

Comment author: 21 November 2012 12:01:08AM 0 points [-]

I'm new here, so watch your toes...

As has been mentioned or alluded to, the underlying premise may well be flawed. By considerable extrapolation, I infer that the unstated intent is to find a reliable method for comprehending mathematics, starting with natural numbers, such that an algorithm can be created that consistently arrives at the most rational answer, or set of answers, to any problem.

Everyone reading this has had more than a little training in mathematics. Permit me to digress to ensure everyone recalls a few facts that may not be sufficiently appreciated. Our general education is the only substantive difference between Homo Sapiens today and Homo Sapiens 200,000 years ago.

With each generation the early education of our offspring includes increasingly sophisticated concepts. These are internalized as reliable, even if the underlying reasons have been treated very lightly or not at all. Our ability to use and record abstract symbols appeared at about the same time as farming. The concept that "1" stood for a single object and "2" represented the concept of two objects was establish along with a host of other conceptual constructs. Through the ensuing millennia we now have an advanced symbology that enables us to contemplate very complex problems.

The digression is to point out that very complex concepts, such as human logic, require a complex symbology. I struggle with understanding how contemplating a simple artificially constrained problem about natural numbers helps me to understand how to think rationally or advance the state of the art. The example and human rationality are two very different classes of problem. Hopefully someone can enlighten me.

There are some very interesting base alternatives that seem to me to be better suited to a discussion of human rationality. Examining the shape of the Pareto front generated by PIBEA (Prospect Indicator Based Evolutionary Algorithm for Multiobjective Optimization Problems) runs for various real-world variables would facilitate discussions around how each of us weights each variable and what conditional variables change the weight (e.g., urgency).

Again, I intend no offense. I am seeking understanding. Bear in mind that my background is in application of advanced algorithms in real-world situations.

Comment author: 11 November 2012 12:28:41AM 0 points [-]

Due to all this talk about logic I've decided to take a little closer look at Goedel's theorems and related issues, and found this nice LW post that did a really good job dispelling confusion about completeness, incompleteness, SOL semantics etc.: Completeness, incompleteness, and what it all means: first versus second order logic

If there's anything else along these lines to be found here on LW - or for that matter, anywhere, I'm all ears.

Comment author: 04 November 2012 02:38:33AM *  0 points [-]

a

Comment author: 02 November 2012 03:02:47AM *  0 points [-]

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

The property "is the only number which is not the successor of any number" manifestly is false for every Sx.

There is a number ' (spoken "prime"). The sucessor of ' is '. ' and ' are the same number.

There is a number A. Every property which is true of 0, and for which P(Sx) is true whenever P(x) is true, is true of A. The successor of A is B. The successor of B is C. The successor of C is A.

Both of these can be eliminated by adding a property P1: EDIT for correctness: It is true of a number y that if Sx=y, then y≠x; It is further true of the number Sy that if Sx=y, Sy≠x. &etc But P1 was not required in your description of numbers.

There is also an infinite series, ... -3,-2,-1,o,1,2,3... which also shares all of the properties zero for which P(Sx) is true whenever P(x) is true.

I can't easily find a way to exclude any of the infinite chains using the axioms described here.

Comment author: 01 November 2012 11:23:08PM *  0 points [-]

"Why does 2+2 come out the same way each time?"

Thoughts that seem relevant:

1. Addition is well defined, that is if x=x' and y=y' then x+y = x'+y'. Not every computable transformation has this property. Consider the non-well-defined function <+> on fractions given by a/b <+> c/d = (a+c)/(b+d) We know that 3/9 = 1/3 and 2/5 = 4/10 but 7/19 != 3/8.

2. We have the Church-Rosser Theorem http://en.wikipedia.org/wiki/Church%E2%80%93Rosser_theorem as a sort of guarantee (in the lambda calculus) that if I compute one way and you compute another, then we can eventually reach common ground.

3. If we consider "a logic" to be a set of rules for manipulaing strings, then we can come up with some axioms for classical logic that characterize it uniquely. That is to say, we can logically pinpoint classical logic (say, with the axioms of boolean algebra) just like we can we can logically pinpoint the natural numbers (with the peano axioms).

Comment author: [deleted] 02 November 2012 01:25:06AM 2 points [-]

I'd say that your "non-well-defined function on fractions" isn't actually a function on fractions at all; it's a function on fractional expressions that fails to define a function on fractions.

Comment author: 02 November 2012 05:42:52AM 1 point [-]

Fair enough. We could have "number expressions" which denote the same number, like "ssss0", "4", "2+2", "2*2". Then the question of well-definedness is whether our method of computing addition gives the same result for each of these different number expressions.

Comment author: 01 November 2012 07:25:20PM 0 points [-]

Because you can prove once and for all that in any process which behaves like integers, 2 thingies + 2 thingies = 4 thingies.

I expected at this point the mathematician to spell out the connection to the earlier discussion of defining addition abstractly - "for every relation R that works exactly like addition..."