Over at Less Wrong, the estimable Eliezer Yudkowsky attempts to account for the meaning of statements in arithmetic and the ontological status of numbers. I started to post a comment, but it got long enough that I’ve turned my comment into a blog post. I’ve tried to summarize my understanding of Yudkowsky’s position along the way, but of course it’s possible I’ve gotten something wrong.
It’s worth noting that every single point below is something I’ve blogged about before. At the moment I’m too lazy to insert links to all those earlier blog posts, but I might come back and put the links in later. In any event, I think this post stands alone. Because it got long, I’ve inserted section numbers for the convenience of commenters who might want to refer to particular passages.
1. Yudkowsky poses, in essence, the following question:
Yudkowsky phrases the question a little differently. What he actually asks is:
This, I think, threatens to confuse the issue. It’s important to distinguish between the numeral “2”, which is a formal symbol designed to be manipulated according to formal rules, and the noun “two”, which appears to name something, namely a particular number. Because Yudkowsky is asking about meaning and truth, I presume it is the noun, and not the symbol, that he intends to mention. So I’ll stick with my version, and translate his remarks accordingly.
2. Yudkowsky provisionally offers the following answer:
He then provisionally rejects this provisional answer on the grounds (with which I wholeheartedly agree) that “figuring out facts about the natural numbers doesn’t feel like the operation of making up assumptions and then deducing conclusions from them.” He goes on to say: “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.”
He’s certainly right that it feels — to me, and, I am sure to almost everyone who has ever thought much about arithmetic — like the numbers are just “out there”. On the other hand, I’d quibble with Yudkowsky’s assessment of the point of Peano arithmetic. The point isn’t to “allow mathematicians to talk” about numbers; mathematicians from Pythagoras through Dedekind had absolutely no problem talking about numbers in the absence of the Peano axioms. Instead, the point of the Peano axioms was to model what mathematicians do when they’re talking about numbers. Like all good models, the Peano axioms are a simplification that captures important aspects of reality without attempting to reproduce reality in detail.
3. To reach a closer understanding of what numbers are, Yudkowsky imagines trying to explain them to a logician with a full grasp of logic but no grasp of numbers. Here I think Yudkowsky has fooled himself into imagining an impossibility. If you grasp logic, you grasp the idea of a proof. If you grasp the idea of a proof, you grasp the idea of a sequence of logical steps. If you grasp the idea of a sequence of logical steps, you grasp the idea of a sequence. If you grasp the idea of a sequence, you already know a lot about numbers. This is one reason why I believe that any attempt to account for numbers via logic must ultimately be circular.
4. Be that as it may, Yudkowsky goes on to try to explain to his fictional interlocutor what numbers are. He begins by essentially stating the first order Peano axioms: 0 is a number, every number has a successor, no two numbers have the same successor, and so forth. Eventually, he realizes that this approach isn’t taking him quite where he wants to go and makes a bit of a course correction (as we’ll see below). But I think more than a course correction is called for; he’s gone off in entirely the wrong direction. He’s listing the properties of numbers, but not even trying to explain what they are. If I were explaining numbers to a naif, I’d probably start with something like Bertrand Russell’s account of numbers: We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common. Whether or not that works in detail, it’s at least an attempt at a definition, as opposed to a mere list of properties.
5. Yudkowsky, in his fictional conversation with his fictional logician, eventually comes to realize that neither the first order Peano axioms nor any other first-order system can uniquely characterize the natural numbers. This is a consequence of Godel’s Incompleteness Theorem, or even more fundamentally of the Lowenheim-Skolem Theorem. What it means is that no matter what axioms you start with, there are going to be multiple systems that satisfy those axioms; the natural numbers are only one of those systems, so your axioms cannot collectively specify the natural numbers.
6. Yudkowsky solves his problem by passing to second order Peano arithmetic — “second order” meaning that, in addition to using variables to represent numbers, you can also use variables to to represent sets of numbers. He correctly notes that second order Peano arithmetic has a unique model. (I am using the word “model” here in the technical sense of logic, not in the informal social-sciencey sort of way that I used it in point 2 above.) This means that sure enough, there is one and only one system that satisfies all the axioms of second-order arithmetic. And he concludes that:
But this is disastrously wrong for at least two reasons, each of which deserves its own numbered point.
7. Yudkowsky leaps from “the natural numbers can be precisely specified by second order logic” to “the .. study of numbers is equivalent to the logical study of which conclusions follow inevitably from the number-axioms”. This is wrong, wrong, wrong, because second order logic is not logic. Indeed, the whole point of logic is that it is a mechanical system for deriving inferences from assumptions, based on the forms of sentences without any reference to their meanings. (Thus if we assume that all bachelors are unmarried and that Walter is a bachelor, we can infer that Walter is unmarried, without having to know anything at all about who walter is, or what the words “bachelor” and ‘unmarried” mean.) That’s why you’re not allowed to set up an axiom system in which all the true theorems of arithmetic are taken as axioms — there is no mechanical procedure for determining whether a given statement is or is not a true theorem of arithmetic (see Tarski’s theorem on the undefinability of truth) and therefore no mechanical procedure for determining what is or is not an axiom in that system. In second-order Peano arithemetic, we have an analogous problem: The axioms can be identified mechanically, but the rules of inference can’t. A properly programmed computer can examine a first-order proof and tell you if it’s valid or not; that is, it can tell you whether each step does in fact follow logically from some of the previous steps. But no computer can do the same for second-order proofs.
So the study of second-order consequences is not logic at all; to tease out all the second-order consequences of your second-order axioms, you need to confront not just the forms of sentences but their meanings. In other words, you have to understand meanings before you can carry out the operation of inference. But Yudkowsky is trying to derive meaning from the operation of inference, which won’t work because in second-order logic, meaning comes first.
8. Even putting all that aside, Yudkowsky is relying on a theorem when he says that second-order Peano arithmetic has a unique model. That theorem requires a substantial dose of set theory. So in order to avoid taking numbers as primitive objects, he’s effectively resorted to taking sets as primitive objects. But why is it any more satisfying to take set theory as “given” than to take numbers as “given”? Indeed, the formal study of numbers precedes the formal study of sets by millennia, which suggests that numbers are a more natural starting point than sets are. Whether or not you buy that argument, it’s important to recognize that Yudkowsky has “solved” the problem of accounting for numbers only by reducing it to the problem of accounting for sets — except that he hasn’t even done that, because his reduction relies on pretending that second order logic is logic.
9. All of which leaves us with the problem of accounting for numbers, and for the meaning of statements like “two plus two equals four”. To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary. I’ve given an account in The Big Questions of how the existence of numbers can account for the existence of the physical universe; I think it would be very difficult to go in the opposite direction (though I’ve seen some pretty good attempts). Therefore, accepting numbers as primary and accounting for the universe as a necessary consequence of numbers seems to me to be the ontologically parsimonious thing to do, and I like parsimony.
10. Needless to say, point 9 is not a proof. But I know of no alternative story that strikes me as even remotely plausible. Moreover, the alternative stories all seem to go wrong in pretty much the same ways; for example, every single point above is one I’ve blogged about before in other contexts, but here they are, all being relevant again.
Absolutely love the number theory. Your blog posts, along with GEB are the only sources I’ve found on this subject that blend just the right amount of rigor and accessibility so that I can understand what’s going on without being a math major, while at the same time don’t have to feel like it’s being dumbed down too much. I will check out Less Wrong as well.
I accept your Platonism, but Y’s conclusion is still true. If he is wrong, and there is no such equivalence, then what is the counterexample? Is there some mathematical knowledge of integers that is nonlogical? Or is there some logical conclusion that is not mathematical? Where does Y go wrong with his conclusion?
Nice post.
On point 3, here’s how a logician could exist without an understanding of sequences. When they are presented with axioms and rules of inference, they absorb them into part of their consciousness. They allow these axioms to drift around in their mind, and occasionally realise that a rule of inference applies to certain statements, and derive a new statement, which joins the cloud of axioms they are ruminating on. If you ask the logician if a certain statement is true, they ruminate for a while in a completely nonsequential way, until they suddenly realise that the statement we are asking about has been derived. (Of course, if our query is about a false statement, this could take forever).
Now, you and I are aware that for the proven statement, there is a sequential logical proof. The logician could even prove this for us. We could even, via a series of clever question, get the proof out of him and write it down. However, the logician remains always unaware of the sequence of steps taken to derive any particular statement.
Steve, I have a few questions about your views:
1. You claim that we can not only reason about natural numbers, we can also perceive them directly. If that’s really the case, then why do most people not share your belief that we have this extra-sensory perception? Would’nt someone know if they were hearing or deaf, for instance? Also, does your “ESP” have any observable consequences? Are there any facts about natural numbers that perception can tell us but thought without this perception cannot?
2. What exactly is the basis for your distinction between natural numbers and real numbers? As I told you in a previous thread, the second-order theory of real numbers has a unique model upto isomorphism, just like second-order Peano arithmetic, and natural numbers are definable by quantifying over sets of real numbers, just as real numbers are definable using sets of natural numbers. So why isn’t the viewpoint that we all have the same idea of real numbers but different notions of natural numbers equally valid as your view that we all have the same idea of natural numbers but different notions of real numbers?
3. It’s true that logical proofs consist of sequences of statements, but do you need to grasp the idea of a proof to do a proof? Do you need to understand what a sequence is in order for your thought process to be sequential?
So there’s no such thing as a proof-checker (as in a piece of software) for second-order proofs?
So how do mathematicians check other mathematicians’ proofs? Sorry if this is a dumb question.
Steven, you mention “A properly programmed computer can examine a first-order proof and tell you if it’s valid or not… But no computer can do the same for second-order proofs. ”
Would you say that a human can validate a second-order proof? Surely humans are not magical and are carrying out a sequence of small steps that could be programmed if they were known, yes?
CC:
So how do mathematicians check other mathematicians’ proofs?
They think about the meanings of the statements being made and ask themselves whether they make sense.
CC: Mathematicians do not check other mathematicians’ 2nd-order proofs. As Steve says, 2nd-order logic is not logic. Proofs need to be reducible to 1st-order logic, where they can be checked using the rules of logic.
I found Russell’s statement odd:
‘We say that two sets of objects are “equinumerous” if they can be placed in one-one correspondence with each other; a “number” is that which all sets equinumerous to a given set have in common.’
Are the concepts of “two sets” and “one-one correspondence” meaningful without defining the natural numbers first?
Justin:
Are the concepts of “two sets” and “one-one correspondence” meaningful without defining the natural numbers first?
Yes! Read Russell on this; he makes it all very clear.
Some people may not recall that Landsburg acknowledged that his argument about the inherent appeal of the natural numbers reflects a form of extra-sensory perception. See The Big Questions, Chap. 7, “On What There Obviously Is”:
I suspect the answer to the two other questions — why people deny that the appeal of natural numbers reflects ESP, and whether we perceive things about natural numbers that cannot be derived from “pure reason” – are related. If I conclude that we can derive all our thoughts about numbers from pure reason (or reason combined with experience), then I need not rely on the ESP thesis. Landsburg and others argue that we can’t derive the natural numbers as a unique model of any set of axioms, yet we still somehow perceive natural numbers as unique. These facts suggest that our appreciation of natural numbers is not merely a function of our reasoning. And if the appreciation is not generated within ourselves, it must be generated beyond ourselves; we must be perceiving, not generating, numbers.
I gotta say, the idea grows on me with each exposition.
nobody.really:
we can’t derive the natural numbers as a unique model of any set of axioms, yet we still somehow perceive natural numbers as unique. These facts suggest that our appreciation of natural numbers is not merely a function of our reasoning. And if the appreciation is not generated within ourselves, it must be generated beyond ourselves; we must be perceiving, not generating, numbers.
Well put.
“we can’t derive the natural numbers as a unique model of any set of axioms”
Well put perhaps but not quite true. We can if we allow second order logic can’t we? Yes, I do see you want to disallow that Steve. But it does mean this conclusion depends on that dismissal. That’s important. You are arguing from human intuition, and indeed claiming for it great powers, but also demanding that human intuition spurn second-order logic.
@justin: As an intuitive aid, think of simple machines or computers. Does a computer need to be able to count to tell which pile of apples is bigger? No. Just toss out an apple at a time from each pile. Stop when one pile is gone, and look at the other. So you can see the notion of more than is simpler than anything with counting.
Not only does this machine tell you which is greater, but also, if the machine actually destroys the apples, it stimulates the economy.
Roger wrote: “CC: Mathematicians do not check other mathematicians’ 2nd-order proofs. As Steve says, 2nd-order logic is not logic. Proofs need to be reducible to 1st-order logic, where they can be checked using the rules of logic.”
So if a proof can’t be converted to FOL, then it’s no good? I’m genuinely asking.
CC: You are confusing two meanings of the word “proof”. Technically, a proof is a sequence of statements, each of which is either an axiom or follows from earlier statements via valid rules of inference. Informally, a proof is a clear, convincing, incontrovertible argument. For the most part, mathematicians deal in the latter, not the former. Often, what makes an argument clear and convincing is the conviction that it could, given enough time and effort, be translated into a proof in the technical sense. But this is certainly not always the case. Sometimes the clear convincing arguments come first and only then (if at all) does someone go back and ask what axioms would be necessary to derive these conclusions formally. Historically, people made clear convincing arguments long before anyone had conceived the notion of a formal system.
Historically, Aristotle made clear convincing arguments in 350BC, and Euclid had a formal system for doing geometry and arithmetic proofs in 300BC. So the gap was not that long in recorded history.
@Roger: Informal proofs of the Pythagorean theorem date back to ancient Egypt.
Euclid lived in Egypt, and is thought to have relied on earlier mathematicians. Did you mean more ancient Egypt? This 1991 letter says the theorem was Hellenic, and that claims of an older African origin are false.
SL: So can any “valid” convincing proof be translated into FOL?
Counter-proposal: the natural numbers are not “out there”, they are in fact “in here”. That is to say, they are an emergent phenomenon of the peculiar way in which the human brain processes both sensory and self-generated inputs.
…which is not to say that they’re not real in any useful sense: we can make predictions about the physical world based on them which are reliable enough to be useful. (Similarly, it’s pretty clear that our perception of time is entirely idiosyncratic, but general relativity still works.) But there’s no need to retreat to Plato’s cave to explain their “intuitive” nature: they’re intuitive because they reflect the (so far still largely unmapped) deep structures of our own brain.
As a weak justification for this view, I offer the condition of acalculia, a post-stroke (or other brain damage) phenomenon in which basic arithmatic abilities in previously baseline adults become impaired, sometimes to the point of being unable to say which of two integers are “larger”.
(A strong justification will have to wait until we make contact with a species outside our own genome who nonetheless have a mathematics: updates as they are warranted.)
Behold!
@Roger: I don’t know if the ancient Egyptians proved the general theorem. They knew it for 3-4-5 and 12-13-1. I have seen discussions of Mesopotamian and Indian proofs too. All were less formal than Euclid’s.
CC:
So can any “valid” convincing proof be translated into FOL?
The answer to the question you asked is: Sure. Not only can any convincing proof be translated into FOL, so can any unconvincing proof. All you have to do is take your conclusion as an axiom.
The answer to the question you probably meant to ask is: No first order system suffices to prove all of the true theorems of arithmetic.
Ken, they had examples, not proofs. If the Indians or anyone else even had the concept of a proof before the Greeks, I’d like to see it.
Steve, I don’t think that you answered CC’s question. He wants to translate a proof, not look for a new one.
CC, the answer is yes, all proofs can be translated to first order logic, with certain minor technical exceptions. For example, some statements are nonfirstorderizability. There are work-arounds for such difficulties.
Doctor Memory,
Various physical traumas cause blindness. This in no way invalidates the existence of color or varying frequencies occurring in the light spectrum. Light very much exists independent of humans, much less the human mind. As do their frequencies. Color is how humans perceive these wavelengths.
To better understand that the integers are “out there” rather than “in here”, consider the periodic table. The elements in this chart have their characteristics based on the integer number of protons, electrons, and neutrons. In fact, when the periodic table was first arranged, many of the entries in that table were empty, but based solely on the integer number of protons, it was known ahead of time what elements would go where and what types of basic properties they would have.
Or consider the Fibonacci sequence. The patterns formed by the Fibonacci sequence, which is based on the arithmetic of the integers, occurs all over the place, without any need for human perception. Both of these things, as well as a number of other things, come about due to integers and their properties, without the existence of a human brain to perceive them. The notion of ‘1’ is how humans perceive the first two elements of the Fibonacci sequence and ‘2’ is the third.
Ken: valid points all, but if integers had the same properties (or even particularly similar) as photons, we wouldn’t be having this discussion in the first place.
Or to put it another way: I can have a box with a single photon in it, so long as I don’t care about knowing its velocity. How many integers are in that box?
How many integers are in that box?
I’m not really sure what you are asking. Your question reads like asking how many gravities are in the box.
Gravity exists whether or not it’s measured. It isn’t a thing to be touched or counted. To understand it, humans have developed a theory for it. When we find an observation that doesn’t fit the theory for gravity, we don’t say gravity doesn’t exist. We say our theory is wrong.
Similarly, the integers exist whether or not we think about them. We develop a theory to describe them and label zero zero and one one. The Peano axioms theorize the structure of the integers. If we found that after some sequence of statements starting from the axioms that 2+2=5, no one will say the integers don’t exist. Everyone will say the axioms are wrong.
Since we know many of the basic properties of the integers, like 2+2=5, without having to resort to axioms, i.e., human perception, when our human perceptions turn out to yield some thing contradictory about the integers, we say our perceptions (axioms) are wrong. In other words, the integers are not a product of our perception. They don’t reside in our brains. They are “out there”.
Perhaps the following is an example of a second-order proof that can’t be translated into a first-order proof, and perhaps it will even help people understand the distinction.
Suppose we have an axiomatic system, S, for talking about the integers. Suppose S is consistent (even though we may not know that).
We’ll prove : “There are true statements (about the integers) that can’t be proved in S. Here’s an axample…”
First, I follow Godel’s proof to construct a certain mathematical statement. This is purely first-order logic.
Then I point out that this statement, P, means “P has no proof in
S”. This is where the second order logic begins.
If P isn’t true, then “P does not have no proof in S”, that is, “P has a proof in S”, which means S is able to prove false statements. It’s inconsistent.
Therefore, P has no proof, and is true. We just proved it.
The inference is valid, but is clearly outside S. What’s more, we’ve relied on what P “means”, and on the concept of “truth”. The rule of inference we’ve used would be impossible to pin down in any first order logical system.
Mike H, P does not have a proof in S, but it still has 1st-order proofs outside S.
There is a scene in the new Lincoln movie where Abe makes an argument based on an axiom from Euclid in a “2000 year old book”. So even movie goers are expected to know that formal systems have been around for a while.
Roger: when you say P does have a first order proof outside S, is that the same as Steve saying ” Sure…all you have to do is take your conclusion as an axiom.”? Honestly, I find this stuff interesting, and examples like Mike H’s very useful.
@Harold 31: It may be be, it need not be.
Steve posted a cute new proof of Godels’ Incompleteness theorem some months ago. Worth hunting up. But for some purposes you really need to understand the techniques Godel used in his proof. He built a machine and then ground out his theorem. The machine itself is important.
The machine is a rigorous and precise way that statements in arithmetic can make assertions about other statements in arithmetic. So if you specify a formal system with all it’s rules his machine can produce a statement X that makes assertions about satemetns A, B, C, D, etc. This is how he was able to make precise the Epimenides paradox. That paradox is this sentence: “This sentence is false.”
Play with it for a bit.
The trick is that in the Epimenides sentence “this” is very squishy. Not quite precsiely defined. The world does not end because twist as you might you cannot a true paradox out of it because of the squishiness of “this”. Godel’s machine makes that problem go away.
I realize this is a bit loose and abstract but if you keep a hold of this central idea the discussions make more sense: Godel found a way to make mathematics talk about itself without ambiguity.
A good book is Godel’s Proof by Nagel and Newman.
@Doctor Memory: I don’t know if your idea makes sense but at least it’s empirically minded!
My view, perhaps silly, but I will put it forward for consideration. Two plus two is four because whenever I take two of anything and add two more of the same thing I get four of those things, regardless of what the thing is. Thus, I can drop the qualification “thing” and simply say two plus two is four.
It is a true statement, but an empirical and provisional truth. Perhaps a day will come when some “thing” is discovered that does not have this property. Unlikely, perhaps. Even in the quantum world, with its manifest weirdness, two plus two is four. But if that day comes, two plus two will no longer equal four without qualification.
Neil: That’s fine for “two plus two is four”. Now what about “Every number is the sum of four squares”?
OK, I understand. Steve thinks that natural numbers are real. I agree. He thinks that they are complex, though. I agree with this statement as well.
@35: Will that still be true when “2 + 2 = 4” fails?
Steve: Isn’t that a theorem?
@MR: Steve doesn’t just think they are real. He thinks they are reality.
(And yes I got the R, C joke)
Quote Ken: Your question reads like asking how many gravities are in the box.
Phrased that way, it would certainly be silly. But I can ask how we might calculate the gravitational force between two particles in a box, or between a photon and the box itself. IOW, I can ask how “much” gravity is “in” the box for suitably tortured definitions of “much” and “in”. Still can’t ask how many or how much integers are in there though. :)
(In case it’s not obvious: I’m not exactly married to the “deep mental structures” line, I just find it slightly less unconvincing than any other explanation so far. Still waiting on some E.T.s to drop by and settle the issue, really.)
But Doctor Memory, how many ETs fit in a box?
Doctor Memory,
Let me see if I understand you correctly. I’m still not sure I understand what it is you are asking, so please tell me if I’m wrong. You’re saying imagine a universe with a single object in it, right? Then asking if the laws of arithmetic apply because there isn’t anymore than one thing, claiming that two cannot exist, because only a single thing exists. Is that right?
First, this implies that in this universe, there cannot be an infinite number of integers because the number of objects in this universe is finite. I think that it is clear that this is very false.
Secondly, if anyone/anything, whatever observed this universe, it would be noted that it only had a single, “one”, object .
Or are you trying to claim this: the observer is required to note that there is one object for the integer one to exist? And that without an observer noting it, “one” wouldn’t exist? Because if that’s your argument, then I could ask: Does that single object exist if no one observes it? Does a falling tree make a sound if no one is there to hear it? The answers to these four questions are no, no, yes, and yes.
Otherwise, you are claiming the philosophy of solipsism. Do you admit that something can exist outside of your own perception? I claim yes. It sounds like you say yes as well, by considering a universe with only a single object, which presumably is not your own mind. You seem to be saying a mind is not needed to acknowledge the existence of a universe with a single, “one”, object. While it’s true term “one” is a human invention to describe this situation, as all language is is a series of symbols to represent something.
But “one” exists, even if it is not named, the same as your imagined single object universe. Can you explain to me how a single object universe can exist, without the need for a mind to observe/perceive it, but 2+2=4 cannot?
@Ken: Imagine one particle that can move back and forth in time and weave all that is.
Doctor Memory (and perhaps others):
Here’s what I totally don’t get:
You seem more willing to believe that a rock is “out there” rather than “in here” than to believe that the natural numbers are “out there” rather than “in here”. I have never understood why people find the former easier to swallow than the latter.
You might observe that rocks occupy space and time while numbers don’t, but what does that have to do with the “in here/out there” distinction? Isn’t it at least equally plausible that the things that occupy space and time are precisely the things that are “in here”? After all, space and time are largely mental constructs, so why shouldn’t their contents be?
Steve, here is the distinction I see between rocks and numbers. You can find out information about a rock using sensory observation, including information that you do not gleam from thinking about rocks. On the other hand, all the information about numbers that we have seem to arise from thought alone. In other words, even if we accept your belief that we perceive numbers directly, we do not find any information from this perception that we do not already gleam merely from thinking about numbers. So that’s one reason to be skeptical that numbers are really “out there” rather than just a construct of human thought.
You can find out information about a rock using sensory observation, including information that you do not gleam from thinking about rocks. On the other hand, all the information about numbers that we have seem to arise from thought alone
This sounds like an argument for Steve’s point, rather than the one you were trying to make.
After all, you basically said :
* to find out about rocks, we need sensations and thought.
* to find out about numbers, we need thought.
Seems to me that if I accept this, I should place more confidence in numbers. After all, both thought and sensation is fallible, so your ideas about rocks are built on a shakier foundation. No?
PS – in the oil and gas industry, trusting your intuition about numbers gets the good stuff out safely. Trusting your intuition about rocks leads to this. (Ok, that’s a gross oversimplification, I agree. Scratch it.)
Keshav Srinivsan: I find this a thoroughly unsatisfying answer. Why would you trust your sensory observations any more than you trust your extra-sensory perceptions?
Mike H: Your answer to Keshav Srinivasan is even better than mine!
Steve, at least my sensory observations tell me what I do not know from my own thoughts. Whether the new information they’re conveying to me is correct information about reality, at least it’s new. If I do have extra-sensory perception of numbers, then this perception seems to give me absolutely no information other then what is already there in my own thoughts. So does that suggest that my extra-sensory perceptions come from my own mind, whereas my sensory observation come from outside my mind.
Another issue is that the supposed extra-sensory perception is a bit too accurate. I have observed things about rocks that turned out to be wrong later on. In other words, assuming rocks are real, there is a difference between the apparent properties of rocks and the real properties of rocks. But is there any perception about numbers (as opposed to thought about numbers) which you had but you later found out was not accurate? Is there ever a disparity between the properties you perceive numbers to have and the properties that numbers actually have?
“at least my sensory observations tell me what I do not know from my own thoughts”
but only if you think about them.
“If I do have extra-sensory perception of numbers, then this perception seems to give me absolutely no information other then what is already there in my own thoughts”
You are assuming without evidence that your own thoughts get all their information from your senses. Why not assume (or allow the possibility) that we have some information already built in at the start?
Steven, I hope to get to this next week (currently in a crunch mode) but a couple of quick questions before I reply:
1) Do you hold that humans are uncomputable?
2) Had you already read the prior post in the series, http://lesswrong.com/lw/f43/proofs_implications_and_models/ ?
Eliezer:
1) I do not hold that humans are uncomputable.
2) I have not read the prior post in the series, but will do so ASAP.
3) I am dashing out the door for a trip to Boston; otherwise I’d expand a bit on both of the above.
3a) Said trip to Boston might also slightly delay any response to your response.
4) Welcome to The Big Questions!
Keshav Srinivasan:
But is there any perception about numbers (as opposed to thought about numbers) which you had but you later found out was not accurate? Is there ever a disparity between the properties you perceive numbers to have and the properties that numbers actually have?
Happens all the time! I convince myself something is true, I try to prove it, I fail, eventually I find a counterexample, at the end of the day I understand numbers a little better than I did before.
Mike H –
* to find out about rocks, we need sensations and thought.
* to find out about numbers, we need thought.
Seems to me that if I accept this, I should place more confidence in numbers. After all, both thought and sensation is fallible, so your ideas about rocks are built on a shakier foundation”
You’re saying we have more evidence for the existence of rocks so we should be less confident in their existence? This is right up Eliezer’s alley.
Steve, that wasn’t what I was asking. There may well be false beliefs about numbers that you’ve convinced yourself of purely by thinking, but has your extra-sensory perception of numbers ever led you astray?
What is “out there” and what is “in here”?
I don’t think the question can be answered empirically. A solipsist thinks everthing is “in here”. Most of us would say he is as crazy as a loon, but we cannot prove to him that he is wrong, because from his perspective we are “in here”. Those who like the many world interpretation of quantum mechanics think that anything we can imagine, like unicorns, are out there somewhere as long as no natural law is violated.
My philosophical stance is to ask what do I think would NOT be “out there” had not a sufficient level of intelligence (human or otherwise) arisen? I would say poetry, music, art, basketball, theory, perfect geometrical forms, numbers,…. They exist because we exist.
SL:”After all, space and time are largely mental constructs, so why shouldn’t their contents be?”
Equally, aren’t mental contents the result of underlying physical process? Can you cite any thoughts unconnected to brains or computers?
@Martin-2 #54 I was replying to Keshav #45, who said as the basis for an objection to Steve’s argument (and I paraphrase) that if “our knowledge of rocks comes from thinking about our sensations, but our knowledge of numbers comes just by thinking” then rocks are more real. I was pointing out that his conclusion seemed illogical, for the reasons I mentioned. I didn’t think empiricism was being discussed.
Please don’t take my paraphrase of Keshav’s phrasing a hypothetical premise of an objection to Steve’s argument as my opinion. It probably shouldn’t be taken as Keshav’s or Steve’s either without a number of pinches of rock salt.
Mike H – Ahh “thinking about our sensations” not “thinking and through sensations”. That’s why you can compound the error.
Neil:
You wrote:
Two plus two is four because whenever I take two of anything and add two more of the same thing I get four of those things, regardless of what the thing is.
I wrote:
That’s fine for “two plus two is four”. Now what about “Every number is the sum of four squares”?
You wrote:
Isn’t that a theorem?
Yes, but that’s not what’s relevant here. The relevant question is: In your approach, what does this statement mean?
You want to claim that “two plus two is four” is a empirical truth about “things”. Is “Every number is a sum of four squares” an empirical truth about “things”? Or do you need a whole new theory of meaning for universally quantified statements like this?
N.b. I think you would be hard pressed to find a mathematician who believes that “Every number is a sum of four squares” is an empirical truth.
Steve, a bit confused about second order logic. In the linked post answering Coupon Clipper’s question, you said that a second order logic proof CAN be checked by a computer, its just that not all rules of inference can be listed in advance. But in this post you seem to be saying that a second order proof CANNOT be checked by a computer — which is correct?
Steve:
You are too kind to give my ramblings credence, but for the record and for what it is worth, here is what I think.
I think the four-suare theorem is a true statement which follows logically from previously accepted statements, some of which in turn may be themselves theorems. Ultimately, going down the pyramid of turtles, you reach some fundamentals, such as the addition and subtraction of natural numbers, and perhaps operations on sets and groups, which were induced from particular empirical cases (like adding and subtracting rocks). Someone realized that that the operations were independent of the material substrate, and a universal mental concept was created.
In short, I think (to paraphrase Kronecker), “The natural numbers (and perhaps some other stuff) were induced from empiricism, and all the rest is the work of man.”
nivedita:
You can specify some rules of inference, and a computer can check a proof that uses those rules of inference. It can’t check a proof that uses rules of inference you haven’t specified.
Steve, I’m definitely familiar with the result you’re talking about, but the proofs I’ve seen typically start with the assumption that you have a proposed formal system of second-order logic, with effectively specified axioms and rules of inference, and then prove that such a system cannot be both sound and complete withe respect to standard semantics. But you seem to be alluding to a slightly different proof, where you start with a set of axioms that are effectively specified, and then prove that the set of rules of inference required to make the system sound and complete cannot be effectively specified.
Of course the two proofs are equivalent, but I have a question about your preferred proof: is it possible to give good a NON-effective description of the rules of inference required? One such description, of course, would just be “all the valid consequences of the axioms”. But can you give a non-trivial non-effective description of the rules?
Keshav Srinivasan:
But can you give a non-trivial non-effective description of the rules?
I expect that anything we’d recognize as a “description” would be recursive, so no.
Steve, I didn’t mean a description we could actually use. I just meant any description, no matter how useless, that is even slightly less trivial than saying “all the valid consequences of the axioms”. To give an example of a non-trivial non-recursive description, the truths of first-order arithmetic can be described as “theorems of first-order Peano arithmetic plus the omega rule.”
Let me put my question in more formal terms: what is the least powerful oracle that you need to attach to a Turing machine in order to decide what is and isn’t a rule of inference of second-order logic?
Keshav Srinivasan:
what is the least powerful oracle that you need to attach to a Turing machine in order to decide what is and isn’t a rule of inference of second-order logic?
Interesting question. Offhand, I don’t know.
66 and 67: What would an answer to this question even look like?
Martin-2: I gave an example of what answer to such a question would look like. There is no recursive description of the truths of arithmetic, yet it is possible to give a good nonrecursive description of them: just add the omega rule (which is non-recursive) to first-order Peano arithmetic, and then the consequences of that system would be precisely the truths of arithmetic.
In case you’re not familiar with it, the omega rule states that if for each n it is possible to prove P(n), then you can conclude that for all n, P(n). Of course, you couldn’t make a computer program implement this rule, because it would take infinitely long to make a proof of P(n) for all n, so the program would never terminate and thus you would never be able to conclude that for all n, P(n).
Steve, slightly off-topic, but could you explain why you view the natural numbers and the real numbers on unequal terms? It seems like any claim you can make about the status of the natural numbers, I can make an analogous claim about the real numbers. You say that we all have the same idea of the natural numbers, because 2nd-order Peano arithmetic has a unique model upto isomorphism, but you can make the exact same statement about the second-order theory of real numbers. You say the real numbers are defined using a set-theoretic construction using the natural numbers, but you can instead define the natural numbers in terms of set theory and real numbers. So who is to say which one is more basic, and which has a stronger claim at some unique ontological or epistemological status?
I want to question Steve’s ideas about intuition.
Steve’s argument is basically this:
we can perceive and reason about the natural numbers.
We don’t do this on the basis of a number of axioms, which wouldn’t be adequate anyway even in theory.
No, we must be intuiting a thing, the natural numbers.
My question is how does Steve know I am intuiting the same things?
All we ever exchange about these are statements, formal at some level.
Even names and digits require conventional understanding.
Steve cannot present 446637839123 directly to my mind, he needs an intermediary language.
And when we make decisions about them we use statements and rules not number objects directly most of the time.
Steve doesn’t just grasp 2455327792884633417773499902 and know from intimate contact it is or is not divisble by 25526783.
He uses symbolic and linguistic intermediaries.
So maybe it’s just *those* which exist in Steve’s mind, and mine.
@Keshav 70: You can have different sets of reals though right, and different cardinalities? Same N whether the continuum hypothesis is true or false (granting it can be either for the moment). So while a certain body of theorems might be the same the underlying *object* (in Steve’s view) would be different.
I’m not convinced there is an underlying object myself, hecne my skolem-like attack on what the contents of minds really are, but in Steve’s terms there is an asymmetry.
Ken B: You have to compare apples to apples. It is certainly true that there are different models, of different cardinalities, for the first-order theory of real numbers (called the theory of real-closed fields), just as there are different models, of different order-types, for first-order theory of real numbers. Now the result you mentioned, that N is the same no matter what, is an expression of the result that there is only one model (upto isomorphism) for *second-order* Peano arithmetic. There is also a similar result that there is only one model upto isomorphism for the second-order theory of real numbers. See my comment here:
http://www.thebigquestions.com/2012/09/13/simple-as-abc/#comment-63796
Also, just as the real numbers are definable in second-order Peano arithmetic, the natural numbers are definable in the second-order theory of real numbers. So at least as far as these results go, the natural numbers and the real numbers are on equal footing.
Sorry, I meant to say “just as there are different models, of different order-types, for first-order Peano arithmetic.”
Kesahv: Yes, if by ‘you’ you mean Steve. :) I am actually one of the more strident critics of Steve’s metaphysics here. I have even advanced the argument that in a strong sense R is unique as is C, given N. I think you put it better though!
Ken B: I think I would take Steve’s side concerning the argument you’ve advanced. If we define real numbers in terms of sets of natural numbers, then there is some ambiguity as to what sets of natural numbers there ARE, even keeping the natural numbers themselves fixed, so there would be a corresponding ambiguity concerning the real number system. Similarly, if we define the natural numbers in terms of sets of real numbers as I did in the comment I linked to, then there’s an ambiguity concerning what sets of real numbers there are, and thus an ambiguity concerning the natural number system.
“… second order logic is not logic. Indeed, the whole point of logic is that it is a mechanical system for deriving inferences from assumptions, based on the forms of sentences without any reference to their meanings… In second-order Peano arithemetic, we have an analogous problem: The axioms can be identified mechanically, but the rules of inference can’t.”
I never studied second order logic, but I looked up Wikipedia (always questionable, but anyway). Quote: “no deductive system for second-order formulas that simultaneously satisfies these three desired attributes:
(Soundness) ….
(Completeness) …
(Effectiveness) There is a proof-checking algorithm that can correctly decide whether a given sequence of symbols is a valid proof or not.”
hmmmm… it says all 3 cannot be true, but not that ‘effectiveness’, per se, is impossible.
Mike H: ”
Martin-2: “You can find out information about a rock using sensory
observation, including information that you do not gleam from thinking
about rocks. On the other hand, all the information about numbers that
we have seem to arise from thought alone.”
This sounds like an argument for Steve’s point, rather than the one you were trying to make.
After all, you basically said :
* to find out about rocks, we need sensations and thought.
* to find out about numbers, we need thought.
Seems to me that if I accept this, I should place more confidence in numbers. After all, both thought and sensation is fallible, so your ideas about rocks are built on a shakier foundation.”
Not bad.
Let’s try another approach: Darwin.
I exist (and I’ll presume you do). Why? I, and my honorable ancestors, were able to survive in a hostile universe… through reliable sensory perception. A necessary condition, no? Those who couldn’t tell a lion from a pussycat… well, you know the story.
Following your logic, motor vehicle operation is safer with eyes CLOSED, since one is exposed only to the vagaries of ‘mental construct’, rather than the added fallibility of sensory input.
hmmmm….
Paul T:
hmmmm… it says all 3 cannot be true, but not that ‘effectiveness’, per se, is impossible
Are you suggesting that second order Peano arithmetic might not be sound? And if that’s what you *are* suggesting, then surely we’ve dispensed with second order PA as a foundation for arithmetic.
And as for completeness, this is of course the whole point. Remember the context: The claim is being made that we can interpret the notion of “arithmetic truth” in terms of “consequences of a second-order system”.
PT: “it says all 3 cannot be true, but not that ‘effectiveness’, per se, is impossible”
SL: “Are you suggesting that second order Peano arithmetic might not be sound? And if that’s what you *are* suggesting, then surely we’ve dispensed with second order PA as a foundation for arithmetic.”
I have no horse in this race. I’m not here to debate, but maybe learn something.
An unsound logic doesn’t seem too promising. However, if Peano arithmetic IS unsound, that might come in handy, for challenging my monthly Visa bill –
SL: “And as for completeness, this is of course the whole point. Remember the context: The claim is being made that we can interpret the notion of “arithmetic truth” in terms of “consequences of a second-order system”.”
I looked at the Peano axioms long ago. I never knew there was an isuue regarding first vs. second order systems. I’m surprised to learn that no proof procedure exists for second order logic.
The Wikipedia article claims the 3 conditions cannot all be satisfied. So, maybe 2 out of 3? If arithmetic is incomplete, per Godel, then implicitly there COULD be such a proof procedure, contra your claim. So something’s inconsistent here.
Steve (52) – “I do not hold that humans are uncomputable”
(O) – “the estimable Eliezer Yudkowsky”
Dissed!