Consistent extrapolated beliefs about math?

by cousin_it1 min read4th Sep 201435 comments


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My beliefs about the integers are a little fuzzy. I believe the things that ZFC can prove about the integers, but there seems to be more than that. In particular, I intuitively believe that "my beliefs about the integers are consistent, because the integers exist". That's an uncomfortable situation to be in, because we know that a consistent theory can't assert its own consistency.

Should I conclude that my beliefs about the integers can't be covered by any single formal theory? That's a tempting line of thought, but it reminds me of all these people claiming that the human mind is uncomputable, or that humans will always be smarter than machines. It feels like being on the wrong side of history.

It's also dangerous to believe that "the integers exist" due to my having clear intuitions about them, because humans sometimes make mistakes. Before Russell's paradox, someone could be forgiven for thinking that the objects of naive set theory "exist" because they have clear intuitions about sets, but they would be wrong nonetheless.

Let's explore the other direction instead. What if there was some way to extrapolate my fuzzy beliefs about the integers? In full generality, the outcome of such a process should be a Turing machine that prints sentences about integers which I believe in. Such a machine would encode some effectively generated theory about the integers, which we know cannot assert its own consistency and be consistent at the same time.

So it seems that in the process of extracting my "consistent extrapolated beliefs", something has to give. At some point, my belief in my own consistency has to go, if I want the final result to be consistent.

But if I already know that much about the outcome, it might make sense for me to change my beliefs now, and end up with something like this: "All my beliefs about the integers follow from some specific formal theory that I don't know yet. In particular, I don't believe that my beliefs about the integers are consistent."

I'm not sure if there are gaps in the above reasoning, and I don't know if using probabilistic reflection changes the conclusions any. What do you think?


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In particular, I intuitively believe that "my beliefs about the integers are consistent, because the integers exist". That's an uncomfortable situation to be in, because we know that a consistent theory can't assert its own consistency.

That is true, however you don't appear to be asserting the consistency of your beliefs, you are asserting the consistency of a particular subset of your beliefs which does not contain the assertion of its consistency. This is not in conflict with Gödel's incompleteness theorem which implies that no theory may consistently assert its own consistency. Gödel's incompleteness theorem does not forbid proofs of consistency by more powerful theories: for example there are proofs of the consistency of Peano arithmetic

Yeah, that's a fair point. If I believed the sentence "my beliefs about the integers are consistent", it would be a pretty complicated sentence about the integers, containing an encoding of itself by the diagonal lemma. Maybe you're right that I don't actually believe that, not even intuitively. I just believe a bunch of other sentences, and believe that they are consistent. That would agree with the conclusion of the post, that my beliefs about the integers (both actual and extrapolated) can be covered by some specific formal theory.

I think we can make this more precise. Second-order arithmetic theoretically covers what I believe about the natural numbers. We then need a theory of sets or collections to interpret it. Gaifman, in his discussion of non-standard models, suggests that we go meta instead, and say that whenever we learn a new formula we expand the principle of induction to include this formula. In both cases, the open-ended part comes less from arithmetic than from some larger framework.

So: I don't think my larger 'theory' is necessarily self-consistent. I do think there exists some consistent intermediate theory which includes the consistency of second-order PA, and the uniqueness of its model up to isomorphism.

If we define phi+1 as the union of the statement(s) phi and the assertion of phi's consistency, someone might possibly doubt the consistency of PA+omega, since the concept "omega" need not appear in first-order PA. I find myself hesitating slightly about PA+the Church-Kleene ordinal, although ultimately I think many large cardinals are consistent with(in) set theory. And yes, I think at this point we should start assigning probabilities to mathematical statements.

Marbles exist.

I believe we can apply (at least certain formulations of) integers as elements in models about marbles such as questions about how many marbles fulfill some sharp-line criterion (such as being in a particular urn).

If a formulation of Integers is inconsistent, then that formulation is not a good model for marbles.

If a formulation of integers does not allow you to say things about marbles that you want to be able to say, then that formulation is not powerful enough.

Generally there's a belief that there an infinitive amount of integers but we don't believe that there something like an infinitive amount of marbles.

Okay. If I want to know something about marbles and someone begins invoking infinities, that raises a yellow flag for me.

If I want to know about the asymptotic behavior of an algorithm, then no such flag is raised.

I think you meant to say "We don't believe that there are infinite different amounts of marbles", since the amount of marbles in an infinite amount of marbles isn't even described by an integer.

So if I want to describe filling a vase with a marble being cut in half when a n unsplit would fall off I have to come up with a more powerfull formulation of integers that allows fractions? That can't be it because fractions are no longer a model of integers.

No, you just broke your model by doing something not covered by it.

But it covers only what it is able to say. Thus any attempt to be more expressive breaks it.

edit: actually the theory works just fine. It isn't even broken but it is a different theory. If I would had said that this was a theory of "amounts" this would have been clearly progress that should be welcomed. But what if in my pretheoretic sense I equivocate "integers" and "amounts" (as could be assumed if I can't fraction). Thus when wanting a better theory it's ambigous whether I want or don't want it to cover that kind of scenario.

Exactly. If it acts like integers, then use integers. The above example, you tried to use integers despite the underlying phenomenon not acting like integers. That broke it.

The question is more about your beliefs about marbles, and what would happen if you tried to extrapolate those beliefs all the way. My argument is that some parts of Platonism wouldn't survive such extrapolation.

Do you conflate the mathematical meaning of "existence" with the colloquial one? As others mentioned, what happens if you replace the colloquial "exist" with something like "is accurate" or "is useful"? After all, integers are a mathematical model, not a physical object, and models are ultimately judged by their usefulness and accuracy. For example, the objects in the naive set theory may not exist in the mathematical sense, but they still appear in this model (of limited but non-zero usefulness and accuracy).

My post is dealing with the question whether human beliefs about the integers can be covered by some specific formal theory, or all formal theories are inadequate for that purpose. I've seen many people argue for the latter point of view, in the post I try to argue for the former.

Ah, I see. I misunderstood your point. You don't worry about your current beliefs, you worry about the extrapolated ones. How do you see this represented? An algorithm, which, when fed a theorem, outputs a proof/disproof/proof of undecidability? Or something that spits all possible true statements, unprompted, never shutting up?

Yeah. Judging from the comments, it looks like most people misunderstood the point, though the point is right there in the title :-( I think the result of extrapolation should be an algorithm that spits out all true statements, or an algorithm that checks proofs. These are equivalent, and known as "effectively generated" or "recursively axiomatizable" theories. It can't be an algorithm that answers provable/disprovable/independent, because that's impossible even for PA, and I'd like to think that I believe in more than just PA.

What do you mean with "exist"? What would it mean if integers don't exist?

I don't know. All occurrences of "exist" in my post are inside quotes, referring to intuitions that I don't know how to explain. For what it's worth, these intuitions are similar to what's described in Landsburg's post.

if you don't mean any ontological import you might want to phare it as "integers is system that works". The negation would be things that dont' work. For example a triangle with a right corner with angles summing over 180 degrees. But even then you have to specify the background assumptions as those kinds of triangles actually work out. Usually a mind defaults to a euclidean mindset while the applied concepts could apply to non-euclidean context too.

That integers exist could mean a number of things. Like that x + y = y + x for every x and y that is a value. However the logic of non-commuting values has been figured out. Therefore that sentence would be false. There are things that don't fall under this rule meaning this is not an universal rule. Having some assumtion that you need not have violated in your life doesn't mean such a violation would be impossible. The only way back would be to explicitly declare the delineation of that context ie the various properties needed. But then "integers exists" becomes just "assuming integers integers is all there is" which isn't very surprising or would need any explanation.

As long as you aren't clear what you mean with "exist" the rest of the argument is build on quicksand.

How so? The argument is about disbelieving intuitions about "existence" when they seem to contradict well-known math results.

If you start to reject naive assumptions about 'existence' the straightforward way is to do ontology and get a concept of 'existence' that's not naive.

In logical positivism there the attitude that you don't need to do any ontology, but when it comes to issues like this that just isn't helpful.

Landsburg writes:

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.

But the set of all possible minds is so vast that the fact that numbers feel real to us feels to my mind as extremely weak evidence that numbers are real.

I think (it's not my post) that it's supposed to be evidence that numbers are real because the set of all possible minds is vast. Because there are so many possible minds, it's unlikely that a mind chosen randomly from that set would have similar intuitions about numbers to mine. It's even more unlikely that a third mind would also have those intuitions. Yet, for some reason, this vastly unlikely thing happens anyway. This implies that there is some reason which is responsible for all the minds feeling the same way. One such reason would be "the intuition is correct, and the minds have correctly figured it out".

(Other possible reasons could be "all the minds are biased in the same way" or some other reason unrelated to truth of the idea, but nobody's saying it's proof, it's just evidence.)

Though the number of possible minds is vast, I think the likelihood of two minds sharing an intuitive concept of number is high, because minds (or perhaps I should say consciousnesses) process information sequentially. Perhaps it is akin to the shared perception of rhythm, which is not limited to human minds. I suppose you have already seen it, but this video is amazing: the dancing cockattoo.

I will state my beliefs regarding the integers: they are practically useful in guiding myself through the world. All other philosophical statements are complicated and often murky, so I will simply resolve myself as uncertain about "the nature of integers' existence" until such time that I have studied the topic more deeply or it becomes of practical importance.

Patrick Hayes did a similar thing for physics. Dennett calls it sophisticated naïve physics, a special case of axiomatic anthropology.

Patrick Hayes, the artificial intelligence researcher, once set out on a project to axiomatize the naïve (or folk) physics of liquids. The idea was to provide a robot with the propositions it would need to use as its core beliefs if it was going to interact with people (who rely on folk physics every day). It proved to be more challenging than he had anticipated, and he wrote an interesting paper about the project, "The Naïve Physics Manifesto" (Hayes, 1978). In the naïve physics of liquids, everything that strikes naïve folks as counterintuitive is, of course, ruled out: siphons are "impossible" and so are pipettes, but you can mop up liquid with a fluffy towel, and pull water out of a well with a suction pump. A robot equipped with such a store of "knowledge" would be as surprised by a siphon as most of us were when first we saw one in action. Hayes's project was what I would call sophisticated naïve physics because he was under no illusions; he knew the theory he was trying to axiomatize was false, however useful in daily life. This was an exercise in what might be called axiomatic anthropology: you treat what the folks say - and agree about - as your axioms or theorems, and try to render the data set consistent, resolving any contradictions encountered. And, of course, he didn't bother rounding up any actual informants; he figured that he knew the naïve physics of liquids as well as any normal person did, so he used himself as his sole informant: axiomatic auto-anthropology.

If you are truly concerned with this, why not subscribe to the Gerhard Goentz line of argumentation? Transfinite induction makes good sense to me.

we know that a consistent theory can't assert its own consistency.

Godel is only interested in countably axiomatizable theories of mathematics (theories that can be constructed from countable sets of axioms). I would argue his conclusions only apply to some well-formed axiomatic theories.

I think you're imagining an exit that isn't there. For example, ZFC can formalize Gentzen's proof of consistency of PA, and supports quite a bit of transfinite induction. Yet it still has to obey Gödel's theorem, like any other recursively axiomatizable theory.

(Also, all sets of axioms are countable, because they are subsets of the set of all finite strings, which is countable. I assume you meant to say something else, but I can't guess what.)

The amazing thing about Gödel's theorem is how general it is. I mentioned that in the post. Any Turing machine that prints theorems (regardless of the internal mechanism) must obey Gödelian limitations, as must any Turing machine that receives proofs and checks them for correctness by any method. The only way to escape these limits is by hypercomputation, but I wouldn't hold my breath.

I don't doubt that just about anything can be formalized in ZFC or some extension of it. I am aware that a Turing machine can print any recursively axiomatizable theory.

all sets of axioms are countable, because they are subsets of the set of all finite strings

The set of all finite strings is clearly order-able. Anything constructed from subsets of this set is countable in that it has cardinality aleph_1 or less (even if it contains the set).

I read this book on something called language theory (I think it's now called "formal language theory"), an attempt to apply the idea that all mathematics is represented in the language of finite strings. According to the text as I remember it, the set of all finite strings is equivalent in size to the set of all the statements that can be made in closed languages.

My question is, treating math as an open language, is it possible to axiomatize in a semantically meaningful way, consistent with the bulk of constructive mathematics? I believe the answer is yes, but I would genuinely like to hear your thoughts on the subject.

The reason I think this question is worth asking for three reasons. 1) from a purely structuralist/historical perspective, new concepts enter math all the time and they often challenge the consistency of some portion if not all of mathematics. True they are explained in terms of old concepts, but from a purely observational point of view, the language of math behaves much more like an open language than a closed one. 2) I believe all theories have axioms whether overtly stated or hidden deep within nomenclature. If any set of axioms is both incomplete and inconsistent, then the only way of evaluating competing theories is to compare them. But we can play the stronger weaker logic game all day without knowing if we're forming a closed loop. From that point of view, it becomes even more important to consider the possibility of a theory that explains why some theories work for some things and not other. So I close my eyes and try to imagine the parameters of a theory that is complete and consistent. I think Godel is right -- so it has to have uncountably many axioms otherwise paradox. 3) This is the part I don't know how to explain in mathematical terms. Which axioms to use ... I mean if Zorn's Lemma and the axiom of choice are the same thing, then the axioms we see must be as much a consequence of the language as they are a reflection of whatever is the core of mathematics. When I read a textbook in number theory, I'm seeing the axioms of algebra transformed to fit a different way of thinking of numbers. The concepts are conserved, but the form they take is just a mask and I know that there are questions we don't know how to answer yet. But there is a general pattern that all branches of mathematics follow -- all try to eliminate the extraneous and unnecessary, to streamline axioms to fit the demands of the language ... If we are to devise a self consistent theory of sets, the first axiom (after the definition of a set, of addition, of inequality, of the null-set, of infinity) would be the axiom of incompleteness. After all, if the list of axioms never terminates, the Turing machine can't halt. :-)

4) I don't like the idea of questions that cannot be answered or at least outlawed for the sake of sanity.

With that in mind, I think it's okay to have unanswered questions about integers.

Sorry, I don't understand what you're talking about. Can you give an example of a theory with uncountably many axioms?

(A): There exists a function f:R->R

and the axioms, for all r in R:

(A_r): f(r)=0

(The graph of f is just the x-axis.)

This might be expressible with a finite axiomatisation (e.g. by building functions and arithmetic in ZFC), and indeed I've given a finite schema, but I'm not sure it's 'fair' to ask for an example of a theory that cannot be compressed beyond uncountably many axioms; that would be a hypertask, right? I think that's what Joshua's getting at in the sibling to this comment.

Example infers more than one representation could exist, which for an object this large would be absurd.

It sounds like mathematical platonism which appeals to some like (seemingly) Roger Penrose, but it seems connected to other networks of concepts at least in part and I do not think it should be taken as a given. Perhaps in the future we can model when such belief systems will arise given access to other information based on the history of the individual in questiin, but we are not quite there yet. For further reference one could examine the examples of computationally model led social behavior in a topic someone created here regarding the reverse engineering of belief systems

puts on asshole face We could also do this thing with the other thing that this guy suggested.

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