I’m a little confused as to which of two positions this is advocating:
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).
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
the “numbers” we talk about are mainly fictional aides to using the system, and
the situation of pre-axiom speakers is much like that of English speakers who nonetheless can’t explain English grammar.
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.
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.
I don’t think we disagree about the social construct thing: see my other comment where I’m talking about that.
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.
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).
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.
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?
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.
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.
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.
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.
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.
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).
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
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?
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.
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.
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).
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.
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.
I’m a little confused as to which of two positions this is advocating:
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).
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.
I’m a little confused as to which of two positions this is advocating:
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).
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
the “numbers” we talk about are mainly fictional aides to using the system, and
the situation of pre-axiom speakers is much like that of English speakers who nonetheless can’t explain English grammar.
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.
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.
A few things:
I don’t think we disagree about the social construct thing: see my other comment where I’m talking about that.
It sounds like you pretty much come down in favour of the second position that I articulated above, just with a formalist twist. Mathematical talk is about what follows from the axioms; obviously only certain sets of axioms are worth investigating, as they’re the ones that actually line up with systems in the world. I agree so far, but you think that there is no notion of logic beyond the syntactic?
Aren’t you just dropping the distrinction between syntax and semantics here? One of the big points of the last few posts has been that we’re interested in the semantic implications, and the formal systems are a (sound) syntactic means of reaching true conclusions. From your post it sounds like you’re a pretty serious formalist, though, so that may not be a big deal to you.
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).
Looking back, it’s hard to say what gave me that impression. I think I was mostly just confused as to why you were spending quite so much time going over the syntax stuff ;) And
made me think that you though that all logical/mathematical talk was just talk of formal systems. That can’t be true if you’ve got some semantic story going on: then the syntax is important, but mainly as a way to reach semantic truths. And the semantics don’t have to mention formal systems at all. If you think that the semantics of logic/mathematics is really about syntax, then that’s what I’d think of as a “formalist” position.
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?
In contrast with my esteemed colleague RichardKennaway, I think it’s mostly #2. Before the Peano axioms, people talking about numbers might have been talking about any of a large class of things which discrete objects in the real world mostly model. It was hard to make progress in math past a certain level until someone pointed out axiomatically exactly which things-that-discrete-objects-in-the-real-world-mostly-model it would be most productive to talk about.
Concordantly, the situation of pre-axiom speakers is much like that of people from Scotland trying to talk to people from the American South and people from Boston, when none of them knows the rules of their grammar. Edit: Or, to be more precise, it’s like two scots speakers as fluent as Kawoomba talking about whether a solitary, fallen tree made a “sound,” without defining what they mean by sound.
Aye, right. Yer bum’s oot the windae, laddie. Ye dinna need tae been lairnin a wee Scots tae unnerstan, it’s gaein be awricht! Ane leid is enough.
What about “both ways simultaneously, the distinction left ambiguous most of the time because it isn’t useful”?
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.
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.
It’s useful for reasoning heuristically about conjectures.
Could I have an example?
I would read this:
as:
Lots of things in both real and imagined worlds behave like numbers. It’s most convenient to pick one of them and call them “The Numbers” but this is really just for the sake of convenience and doesn’t necessarily give them elevated philosophical status. That would be my position anyway.
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.
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).
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
I’m not quite sure what you’re saying here—that “Numbers” don’t exist as such but “the even naturals” do exist?
I think s/he is saying there is no Essence of Numberhood beyond satisfaction of the PA’s.
Correct.
Just to be clear, I assume we’re talking about the second order Peano axioms here?
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?
Finite subsets of the naturals still behave like naturals.
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.
True, but the axiom of induction holds, and that is the most useful one.
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.
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.
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).
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.
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.
I read it as (1), with a side order of (2). Mathematical talk is also about what follows from certain axioms. The axioms nail it down so that mathematicians can be sure what other mathematicians are talking about.
Not weird, non-physical numbery-things, just non-physical numbery-things. If they seem weird, maybe it’s because we only noticed them a few thousand years ago.
No more than a magnetic field is a special exception to the theory of elasticity. It’s just a phenomenon that is not described by that theory.
But EY insists that maths does come under correspondence/reference!
“to delineate two different kinds of correspondence within correspondence theories of truth.”″