I had a chat with my 10 yo daughter yesterday. We started with discussing philosophy in general and ended up with me breaking it to my daughter, not so gently: We are serfs:

This is a good one to discuss with kids, because it gets to the heart of some of the tricks language can play. The Barber Paradox, like many things thought to be paradoxes, only looks like one because of the time structure ambiguity inherent in language.

In English - and many other
languages I suppose - the habitual form is the same as the present tense: "It rains" can mean either "It rains from time to time" or "It's raining right now." In the so-called Barber Paradox keep in mind that "He shaves himself" is using the habitual form, so there is no contradiction in saying that a man who shaves himself (every morning) also "does not shave himself" tonight (for instance). So we are left with a grammatical ambiguity wherein we can say both "He shaves himself" and "He does not shave himself" with no contradiction, because the former can be taken in the habitual sense and the latter can be taken in the one-time action sense (I don't know the correct grammatical term for this, but I hope the meaning is clear).

That's just a heads up, not the actual explanation of why it's not a paradox. Here's the actual explanation, starting with the statement of the apparent paradox as given in Wikipedia (there may indeed be a version that actually is a paradox, but this is the only one I'm aware of - please point me to an actually paradoxical version if there is one):

It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves.["THE RULE"]

Under this scenario, we can ask the following question: Does the barber shave himself?

Asking this, however, we discover that the situation presented is in fact impossible:

* If the barber does not shave himself, he must abide by the rule and shave himself. * If he does shave himself, according to the rule he will not shave himself.

To see more clearly that there is no paradox, define clearly what "shaves himself" [in the habitual sense] means. Say for example that for a man to "shave himself" means that he shaves every morning. But this is still ambiguous: how many mornings in a row will he have to shave in order for him to satisfy the condition of "shaving every morning"? Yesterday only? Two days? Every morning for the past week week? To make it simple we can say a man "shaves himself" [in the habitual sense] if and only if he shaved himself the previous morning. (There are a number of other reasonable definitions we could choose, but the argument is similar for each of them.)

The barber had to start obeying "the rule" from some time t (could be when he was born, even). Let's say t is 5pm on a given day. Before t, he had either shaved himself or he hadn't. If he had, then in keeping with the rule that he does not shave "men who shave themselves" [in the habitual sense], he does not shave himself [in the one-time action sense] the next morning. Now, since "shaves himself" is defined to mean that he shaved himself the previous morning, there is no logical necessity that he shave himself on that day. The premises don't dictate his actions until the following morning, when he will either shave himself or not.

Come next morning, he will now - by definition - be a man who does not shave himself [in the habitual sense], so he will proceed to shave himself [in the one-time action sense]. Naturally, he then refrains from shaving the morning after that, then resumes, refrains, resumes, etc.

Alternatively, if he hadn't been shaving himself before t, then the next morning he does shave himself [one-time action], the following morning he refrains, shaves, refrains, shaves, refrains, etc.

--

Now all that said, we can make a real "paradox" (actually just an uncompliable rule) by altering the scenario slightly:

Rule: He MUST ALWAYS BE SHAVING all men, and only those men, who aren't shaving themselves (with his octopus arms!)

Question: Is he shaving himself now?

The answer is then, "There is no way to know." The rule is impossible to comply with, so he obviously isn't complying with it now, hence there is no particular reason to expect he is either shaving himself now or not.

If Sweden is free or not kinda depends on your perspective.

I had a t-shirt with a quite that implies tax is theft and a friend from Iran noticed it. He just laughed at the notion of complaining about Swedish taxes and said in his country the government had stolen literally everything and was murdering people in the streets. I had to admit that here freedom it is more of an intellectual exercise then a pressing real concern...

The male barber is the man that, by the end of the day which we are considering, will have been the one and only person whom all the men who have not shaved themselves during that day, will have been shaved by.

Where there is no property there is no justice; a proposition as certain as any demonstration in Euclid

Fools! not to see that what they madly desire would be a calamity to them as no hands but their own could bring

I'd say you are making it to a word game. How about a web page that links to all pages that do not link to themselves?

That's just an impossibility, which you can call a "paradox" because it seems as if it should be possible - but only for that reason. In principle, it's no different than saying, "How about a square circle?"

The male barber is the man that, by the end of the day which we are considering, will have been the one and only person whom all the men who have not shaved themselves during that day, will have been shaved by.

Right, so there can be no such man. If by "paradox" we simply mean that this result is surprising, because the idea of a barber shaving everyone who doesn't shave themselves seems "reasonable," then I agree - it's surprising for a while. But it's not surprising once you realize that there can be no such man.

Right, so there can be no such man. If by "paradox" we simply mean that this result is surprising, because the idea of a barber shaving everyone who doesn't shave themselves seems "reasonable," then I agree - it's surprising for a while. But it's not surprising once you realize that there can be no such man.

By that definition there can be no paradoxes, can there? I tend to agree. Once there's a paradox, check your premises. Which is what happened when Russel's Paradox was formulated. It had big impact on mathematic and logic science. Set theory had before then allowed for sets to be able to include themselves. That premise had to be checked! Far from "simply" a word game, I would say.

I'm trying to get some reddit attention to my blog post. Now whenever I say anything libertarian-ish on reddit I always seem to attract people who are somehow scared of liberty. If you've got some spare time I'd appreciate some help =)

By that definition there can be no paradoxes, can there? I tend to agree. Once there's a paradox, check your premises.

Exactly.

PEZ:

Which is what happened when Russel's Paradox was formulated. It had big impact on mathematic and logic science. Set theory had before then allowed for sets to be able to include themselves.

I've been looking for more on Russel's Paradox, because the Wikipedia article on it seems silly. The idea of a set that contains itself never made sense in the first place. And the notion of a "set of all non-squares" immediately raises the question, "What is the universe of discourse!?" Not covered in the article it seems.

The article also mentions this:

In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:

"The reason why a function cannot be its own argument is that the
sign for a function already contains the prototype of its argument, and
it cannot contain itself. For let us suppose that the function F(fx)
could be its own argument: in that case there would be a proposition
'F(F(fx))', in which the outer function F and the inner function F must
have different meanings, since the inner one has the form O(f(x)) and
the outer one has the form Y(O(fx)). Only the letter 'F' is common to
the two functions, but the letter by itself signifies nothing. This
immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou)
. Ou = Fu'. That disposes of Russell's paradox." (Tractatus Logico-Philosophicus, 3.333)

But does not say if or why Wittgenstein is wrong.

PEZ:

Far from "simply" a word game, I would say.

The Wikipedia version is just wordplay. It was only after I wrote that long post that I realized it could be salvaged by a different interpretation. I guess I should have looked in the Stanford encyclopedia of philosophy instead.

I love reddit. Threw in a comment just for kicks, but it seems you've got them covered.

It's wonderful to read your comment. I've read it (reddit, get it?) three times now. =)

Thanks for letting me know you think I've got it covered. I was going to ask how I'm doing, being all new to libertarianism as I am. Though, I can see from reading your comment (I really did read it three times) that I still have a lot to learn.

Ideas are the ultimate power. Power is not being able to kill someone. Power is being able to kill someone and then convince everyone that it was OK. Power is not about being able to steal from someone, but about stealing from someone and then convincing everyone that it was OK. Government is one big sales pitch.

"A stable liberal democracy...a place were people do not starve to death, political violence is almost unheard of..."

Not against each other. They might declare it, but they do not fight it. Wikipedia is your friend.

Undisturbed? Tax evaders are jailed or killed. Actually the same holds for major law breakers in general.

Don't take part in the economy and a just system can not tax you.
Then, today's system is not completely just in this regard. We do have
"fastighetsskatten" (which I consider HIGHLY immoral). But there are
many "nolltaxerare".

Save some money, buy a small plot of land in the middle of nowhere
and perform no trade with people unless you want to pay tax. This could
be done in a collective also (I think you only need to pay taxes then
when trading with the outside world). I believe it could be achieved in
Sweden, but I see no reason to do it. Buying one's "freedom" like this
is in the reach of anyone with time on their hands. But without trading
with those who pay taxes, life will be hard. Some people probably live
like this as we speech. (I heard of an old man that only now and then
went in to town to trade eggs for milk, but I don't know if it's true)

I agree with what you say about "ownership" not being an absolute right, because nothing is an absolute right.

Personally, I think libertarianism loses most of its charm when you
remove the concept of inalienable rights. I don't claim that you can
find an invisible "justice" particle or the like. I just claim that
certain kinds of behavior would make me dangerous to the perpetrator
and that I'm also willing to conspire with others to stop such
behavior. I'm a quite benign individual. But if I run across some
serious human rights abuses in my daily life, I hope I get the idea to
contact the police before I do something stupid :-)

To me, libertarianism is just the opinion that people
can manage matters and disputes amongst themselves without central
monopoly control.

With present day social/technological development we can only choose
between the misery of private or public monopolies/oligopolies in
certain sectors. Sure, I believe that it would not be the end of the
world if we privatized the justice and police systems. I only have a
problem with all the hellish trouble it would create. (reasoning given
below)

The real question is why we even need central planning in the first place.

We will need it to avoid an humanitarian catastrophe until the world
becomes in our minds a small village. To not create an humanitarian
catastrophe it would require:

Tools for a minimally schooled individual to make sense out of
terabytes of disintegrated and sometimes false statistics gathered from
hundreds of million of sources.

Tools for said individual to produce and share such statistics with the world. Even when the powers that be do not agree.

Tools for said individual and his/her relevant peers to reach a
somewhat wide consensus about what is to be done. Querying compliance
of people claiming to be part of consensus is also necessary.

Said tools needs to be robust against sabotage.

The manufacture and design knowledge of such tools need to be widely spread.

They must be cheap

Must be designable and manufactureable in a small village.

They must not rely on an underlying infrastructure in danger of monopoly or oligopoly formation (today's internet is)

With such tools, mankind could get stuff done without central
planning or even involuntary taxation. But today, it would be like
introducing nation-state and liberal democracy in a pre-railroad and
pre-printing-press society.

Even your idea about forcing certain people to donate to the poor is possible without a central government

If you want to perform genocide, large scale organized rapes and
other mischief, never put it in writing! When some of this "mischief"
was unofficially organized in some military units during the balkan
wars, soldiers complied. But when they started to write "rape duty"
planning lists for the unit, people started to complain. If it is one
thing that I do not trust self organized systems to, it is violence.
The consensus making and constitution of a liberal democracy is one
hell of a safety check. Not almighty, but useful.

Even if I condone the use of force in certain occasions, it is not something I take lightly.

All a democratic monopoly on force can do is bring the most power-hungry to the top (aye but isn't that plain to see!

I've spent some time helping politicians. Most of them are not the
power hungry type (I did not find one single such person). But they did
very often suffer from the delusion that they were the most competent
and intelligent person in the room :-) All the rules and media
surveillance in a modern democracy is kind of annoying for power hungry
bastards.

bringing out the worst in people and the businesses at
the top who are more incentivized to use the state as a bludgeon
against their competitors than to actually compete.

This is the hallmark of the democratic system failing. Some
societies fail more (US) some less (Sweden) (both are fine democracies,
the US constitution is just a little bit too old). But to be honest, a
good tactician will always use his/her environment to best effect. The
state is the environment in this case.

Common law traditions show that central government is superfluous.

No, it shows how true common law failed above village level.

Common law can only be as bad as the people in the society are; government can be much, much worse.

You don't need a state for genocide, but a stable liberal democracy is an excellent protection from such madness.

Google John Hasnas The Obviousness of Anarchy to see the case made in detail.

I did, and hence my delay in answering.

It was an interesting read (found some pdf by google), but I think
he underestimate how much humans differ. As long as people are willing
to die and kill for their beliefs, shit will happen in a pluralistic
justice system. Certain kind of international disputes is actually hell
if the parties do not intend to do business in the future (custody
disputes are a good example).

I believe that the human brain is so plastic and complex that the
difference between individuals can be larger than what you would expect
from between species. Two individual's sense of justice or lack thereof
can be utterly irreconcilable. Their way of experiencing the world can
be as different as night and day. In a situation like this, there would
need to exist a will for collaboration between them for true justice to
be made in a pluralistic court system. When people lived all their
lives in the same village, the village elders were often enough.
Sometimes things can be solved unofficially today also (companies in
the same business, between neighbors, etc) but seldom a murder and the
like. (then it is and was families that want/ed to remain on good
footing with each other)

So many points I don't have time to respond right now, but this is a nice debate I welcome anyone else to join.

First, the Barber Paradox is weaker than Russell's paradox itself, since one can answer it by saying that the barber is lying. On Russell's paradox, we have to remember (with Quine) that the statement - x is a paradox - is made relative to some deductive system. What is a deductive system? It's an attempt to model, with axioms and rules of inference, real reasoning that goes on in the real world. In particular, in setting up our metamathematical system, mathematicians (at least those who worry about this sort of thing, rather than just having at it) want to model, in a formal system (as formal as can be, anyway) what common sense tells us works in mathematics. Frege, in particular, is concerned to build all of the actually-existing mathematical system from logic - that is, he wants math without mathematical axioms. So we get the basic logical principles, the rules of inference which Frege claims model the way we actually think (or the way the world is - I'm not clear from Frege which way he sees it, but from our perspective here it doesn't matter.) One of these is basic logical principle 5. Russell then creates a paradox, not relative to real reasoning, or to the real world, but relative to Frege's system, in particular basic logical principle 5. The point of the paradox, as is acknowledged by everyone involved or interested - Russell, Frege, later work in set theory and foundations - is that, actually, basic logical principle 5 is not reflective of how reason works. Russell suggests a very limited form of comprehension, namely the theory of types. Later, Zermelo and Frankel suggested an axiomatic approach to foundations, in particular, they want to demand that a set come into existence not through comprehension, but only when specified to exist by the axioms. The result, ZFC, is believed by most people in the field to correctly model the way mathematicians actually view the world. Godel attempted to build another system, but it was shown that the two have equivalent theories. Currently, mathematicians are divided on the question of certain undecided questions in ZFC, such as large cardinals, CH, and, if not CH, then some questions as to the behavior of the witnesses (Martin's Axiom.) Interestingly, choice also yields many conclusions referred to as paradoxes, yet they don't have the structure of Russell's paradox, in that they fail to be self-contradictory. The best known is Banach Tarski, but this is a claim about a topological object, not a basketball or glass globe (as it is commonly stated) and hence cannot really be said to contradict physical evidence. Most mathematicians, then, accept choice. (I don't).

One of these is basic logical principle 5. Russell then creates a paradox, not relative to real reasoning, or to the real world, but relative to Frege's system, in particular basic logical principle 5. The point of the paradox, as is acknowledged by everyone involved or interested - Russell, Frege, later work in set theory and foundations - is that, actually, basic logical principle 5 is not reflective of how reason works.

Thank you so much. I've been looking for someone knowledgeable about this. What is basic logical principle 5?

So many points I don't have time to respond right now, but this is a nice debate I welcome anyone else to join.

Indeed. Seems my fellow countryman and I disagree on what we think would happen should more libertarianism-ish principles be applied. Also I would like to see him reason more pro-central planning and control than against the lack thereof. No time for it now though. I'll return later today.

First, a little background. Frege's logicist program is quite different from what is done in analytic philosophy today, and from what is done in 'foundations' today. In particular, Frege opposes the use of mathematical axioms. Instead, he wants to say mathematics (although his book on the subject covered only arithmetic, he indicated a plan for future books to cover all of existing mathematics) is just fancy logic. All you need to do, he says, is recognize the principles of logic (giving basic logical principles) and mathematics will, in a sense, pop out. Basic logical principle 5 is supposed to be the basic rule of thought (again, or of reality, I'm not sure what he held here) that justifies the use of sets in mathematics (there was not yet a distinction between classes and sets, that is part of ZF). It is phrased as : the value-range of a function f is the same as that of a function g iff for every x, f(x)=g(x). This implies that, in philosophical terms, for every sense there is an extension, and an object is in the extension of the characteristic predicate iff it has the sense of the predicate. Today, we recognize this as having the same theory as an unlimited axiom of comprehension - that is, for any description, you can form the set of objects meeting that description. It is from here that Russell derives Russell's paradox by describing 'things which are not elements of themselves.'

In ZF, it is believed that we capture the intuitions and things Frege wanted without the paradox by the subset axiom - for any set S and a predicate, you can form the subset of S whose elements are st they make the predicate true. This is sometimes referred to as the axiom of limited comprehension, and is what allows the distinction between sets (objects that may be worked with mathematically) and classes (which are simply descriptions without extensions.)

This is sometimes referred to as the axiom of limited comprehension, and is what allows the distinction between sets (objects that may be worked with mathematically) and classes (which are simply descriptions without extensions.)

Thanks again! The Wikipedia article refers to a set such as "the set of all non-squares," but I'm immediately thinking, "What is the universe of discourse?!" Is the difference between a set and a class basically that the universe of discourse (or whatever the analog may be in math) is delineated or defined, while for classes it is not? It seems that if the universe of discourse if clearly defined a set could not contain itself in the first place. This has been bothering me for several weeks.

Well, those are logically possible. What I was pointing out is that there are some things we can say in language that appear normal and well-formed, but actually are incoherent or logically impossible. (Interestingly, if you write a visual representation of the statement, "There is a website that links to every site that does not link to itself," you get an infinite series of frames that looks like what you see when you face two mirrors at each other. Same with other paradoxes like, "This sentence is false" and "If this sentence is true, Santa Claus exists.")

Sorry, I was completely unclear. This question have nothing to do with Russel's paradox or even paradoxes in general. The square rectangle, or circular oval, is a semi-classic object oriented modeling problem (in programming).

That would make your answer even more interesting. =) Implementing a class hierarchy some programmers try to think in concepts of "contracts". Each subclass has to promise not to break any contracts of it's superclass. Let's say you're developing a drawing program with some vector shapes. When you add the square shape it's easy to think like: "A square is a Rectangle which is a Shape". Rectangle sets up a contract that it should have four sides and right-angles. It also makes sense to add to the contract that when the length of one side is changed, the length of its opposite side is also changed, but not the two other sides. A square, naturally has to violate that contract...

hanks again! The Wikipedia article refers to a set such as "the set of all non-squares," but I'm immediately thinking, "What is the universe of discourse?!" Is the difference between a set and a class basically that the universe of discourse (or whatever the analog may be in math) is delineated or defined, while for classes it is not? It seems that if the universe of discourse if clearly defined a set could not contain itself in the first place. This has been bothering me for several weeks.

The approach that says that by defining the universe, you can simply outlaw self-membership, is actually the approach that Russell took to Russell's paradox. Like Frege, Russell was in the logicist project, and hence wanted to preserve as much of Frege as possible. It is Russell that Godel addressed in his famous theorem, although it turned out to apply to formalism a la Hilbert as well. There have been two problems with Russell's theory of types, the first being Godel's theorem just mentioned, the other (more fatal as a mathematical approach) is that is outlaws much of actually existing mathematics.

In axiomatic mathematics, on the other hand, such as ZF, the universe of discourse is simply those things that come into existence through the axioms. It turns out, though, that axiom sets have multiple models, hence we don't know certain things about the universe (put Platonically. For a formalist, they just describe multiple universes.) For the purpose of making a set, though, it doesn't matter what the universe of discourse is. Some sets come from axioms, and the universe is irrelevant. Others come from other sets, and all those operations are absolute. It doesn't matter what universe you're working in if you say A is the subset of B st...all that matters is what is in B. The set of all non-squares only really makes sense out of the naturals, or the integers - out the reals, what do you mean by square?

A class, though, is in mathematics treated as non-existent. As math is actually done, proper classes (sets are also classes, but existent ones) are formed by taking all or most of the universe. For instance, every ordinal is a set - but the group of all the ordinals, while it can be described {x:x is an ordinal} is not specified by any axiom to exist. Thus, we treat it as simply a predicate, and do not assume that the collection it implies actually exists. This doesn't exactly solve philosophical problems, it does let us do math without them getting in our way.

In axiomatic mathematics, on the other hand, such as ZF, the universe of discourse is simply those things that come into existence through the axioms.

That's quite a relief, as that's how I wanted to think about it.

JAlanKatz:

For the purpose of making a set, though, it doesn't matter what the universe of discourse is. Some sets come from axioms, and the universe is irrelevant.

Right, just definitions and axioms.

JAlanKatz:

As math is actually done, proper classes (sets are also classes, but existent ones) are formed by taking all or most of the universe. For instance, every ordinal is a set - but the group of all the ordinals, while it can be described {x:x is an ordinal} is not specified by any axiom to exist. Thus, we treat it as simply a predicate, and do not assume that the collection it implies actually exists.

Great, no need to bother with that then. But about Gödel now, Wikipedia says in one of its articles about Truth,

Wikipedia:

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system.

This implies that "true," since Gödel, means something other than "provable from the axioms." Yet as I understand, the Gödel sentence G in a consistent formal theory T is

G = "G is not provable in T"

Isn't the idea that G is true but not provable? If so, what is the new notion of true?

This implies that "true," since Gödel, means something other than "provable from the axioms." Yet as I understand, the Gödel sentence G in a consistent formal theory T is

G = "G is not provable in T"

Isn't the idea that G is true but not provable? If so, what is the new notion of true?

Correct, although the = there needs explanation. In any case, here's the idea. The deductive system used in math can be proven to be both consistent and complete, so if you have a group of axioms and they imply something, then you can prove that thing. What Godel showed to be incomplete is an system of axioms for mathematics - in particular, any useful axiom system. Godel himself was a Platonist, and so he described things this way - there is a real mathematical universe, somewhere, where math happens. We attempt to create an axiomatic system such that, through deduction, we can prove all and only those things that are true in that universe. Here's an easy way - just make every true statement into an axiom. That works, but we then have an undecidable axiom system. On the other hand, if you want a useful axiom system, it won't do what you say. Either it will prove everything, or it will fail to prove some true fact about mathematics.

So what does a non-Platonist make of this? It's harder to describe. From my point of view, axioms define structures (the same as the formalist notion) and we construct axioms to make a mathematical structure model a structure that exists somewhere else - either a thought pattern, or a physical system, or whatever. We want it to model it precisely, and Godel shows that it cannot - there will be facts about the real system that cannot be proven with the axioms, or else the axioms will prove false things about the system. I call my position an Aristotelian position, to distinguish it from the Platonists.

What of a pure formalism? I have no idea what a pure formalist makes of Godel's theorem. That's probably why there are so few of them since Godel.

if you have a group of axioms and they imply something, then you can prove that thing.

Aren't those two the same thing? (Logical implication and provability)

JAlanKatz:

we can prove all and only those things that are true in that universe.

But what does "true" mean if not "provable"? OK, from a Platonist perspective I see, but what does a Platonist think of as truth?

JAlanKatz:

From my point of view, axioms define structures (the same as the formalist notion) and we construct axioms to make a mathematical structure model a structure that exists somewhere else - either a thought pattern, or a physical system, or whatever.

Yeah, usually right. (I could make a haphazard mathematical system just for fun, or in hopes it might someday be useful.)

JAlanKatz:

We want it to model it precisely, and Godel shows that it cannot - there will be facts about the real system that cannot be proven with the axioms, or else the axioms will prove false things about the system.

Now this is interesting. I suppose the examples would be things similar to the halting problem?

My apprehension about Gödel's theorem has been, where is truth being defined, and isn't truth a norm of assertion? If (1) truth is a norm of assertion (i.e., a statement automatically asserts its own truth by its very fact of being a statement rather than just a free-floating utterance), and if (2) what is true is defined to be simply that which follows from the axioms, then

G = "G is not provable in T"

G = "G is true, and G is not provable in T" [by (1)]

G = "G is provable in T, and G is not provable in T" [by (2)]

Which seems like just a normal contradiction. So either (1) doesn't hold, (2) doesn't hold, or the theorem doesn't hold. Since the theorem has been widely accepted, I can only assume there is broad agreement that either (1) or (2) doesn't hold. Yet what I've looked at seems to indicate there is much debate on such issues. That's what's weird.

Aren't those two the same thing? (Logical implication and provability)

By logical implication, we mean "Whenever A is true in a structure, then B is true in a structure." By provable, we mean "can write a formal deduction whose first line is A, every line is an axiom or the result of applying laws of inference to previous lines, and whose last line is B." We can certainly construct a deductive calculus that is not complete - just take out all the laws of inference.

AJ:

But what does "true" mean if not "provable"? OK, from a Platonist perspective I see, but what does a Platonist think of as truth?

A Platonist (in math) thinks that a sentence is true just in case the proposition expressed by that sentence (using modern terminology to describe Platonic thought) is a statement that has no counterexamples in the "universe of mathematics" - which is among the forms. Modern Platonists tend not to mention forms, though.

AJ:

Yeah, usually right. (I could make a haphazard mathematical system just for fun, or in hopes it might someday be useful.)

Sure, you could, and attempts have been made to do so - but it turns out that any such attempt tends to model a system of thought, since it's hard to break out of the way your mind works. Now, some systems - set theory, for instance - were developed to model a system of thought, but turned out later to have physical applications as well (Dr. Long might have something to say on this.)

The fact is, though, that whatever you were hoping to model, it turns out your system has other models that are not similar to it. This is the source of the Skolem paradox.

AJ:

Now this is interesting. I suppose the examples would be things similar to the halting problem?

The halting problem is a corollary of Godel's theorem in recursion, so yes. It seems to most of us (some would disagree) that before you run the program, it is true or false that it will halt in a finite amount of time. Yet the answer is not computable. This is just Godel's theorem translated into the language of recursion. How would someone disagree? You could hold that anything not computable is indeterminate - until we run it, there is no answer. On the other hand, a formalist can simply deny that theory of recursion has much at all to do with computers running actual programs.

AJ:

My apprehension about Gödel's theorem has been, where is truth being defined, and isn't truth a norm of assertion? If (1) truth is a norm of assertion (i.e., a statement automatically asserts its own truth by its very fact of being a statement rather than just a free-floating utterance), and if (2) what is true is defined to be simply that which follows from the axioms, then

G = "G is not provable in T"

G = "G is true, and G is not provable in T" [by (1)]

G = "G is provable in T, and G is not provable in T" [by (2)]

Which seems like just a normal contradiction. So either (1) doesn't hold, (2) doesn't hold, or the theorem doesn't hold. Since the theorem has been widely accepted, I can only assume there is broad agreement that either (1) or (2) doesn't hold. Yet what I've looked at seems to indicate there is much debate on such issues. That's what's weird.

Even if a statement automatically asserts its own truth - I'd say asserts the truth of its proposition, but whatever - the sentence can be wrong about that, too. Consider the sentence "5 has no successor in the natural numbers." It asserts its own truth, but it's wrong about that. Next, it simply isn't true that to assert your own truth is to assert your own provability. 2 is incorrect. Finally, even if I followed your argument, it would turn out that G is a contradiction, hence a false statement - but then it's provable.

By logical implication, we mean "Whenever A is true in a structure, then B is true in a structure." By provable, we mean "can write a formal deduction whose first line is A, every line is an axiom or the result of applying laws of inference to previous lines, and whose last line is B." We can certainly construct a deductive calculus that is not complete - just take out all the laws of inference.

It seems to me that there are "laws of inference" that are inherent, or simply are, by virtue of how we define the terms. Given A -> B and B -> C, I don't think it makes sense to state that without a special law of inference we cannot conclude that A -> C. (Is that what you're saying?) Of course that's fine in the case where we want to model a special kind of "logic" that only shares the common word but isn't strictly related to logical reasoning.

JAlanKatz:

A Platonist (in math) thinks that a sentence is true just in case the proposition expressed by that sentence (using modern terminology to describe Platonic thought) is a statement that has no counterexamples in the "universe of mathematics" - which is among the forms.

If this is in debate, then how do you state the following, which seems to imply that truth is something other than provability (but what?)?

JAlanKatz:

Next, it simply isn't true that to assert your own truth is to assert your own provability. 2 is incorrect.

Even kids understand that "You're not not fat" just means "You're fat" without a proof, and a proof could add nothing to their understanding or certainty. Adults understand more, and seasoned mathematicians even more. If you told a hyper-intelligent being the ZF axioms and certain definitions, and then later told him an advanced theorem based on those axioms and theorems, he might say, "You already told me that." So it seems from point of view of a sufficiently intelligent creature, to speak of a theorem as "provable" would be as irrelevant as telling a kid it's provable that "not not fat" means "fat." In other words, it seems it's only our human limitations that make us use a word like "provable" instead of simply "follows from premises."

JAlanKatz:

This is the source of the Skolem paradox.

Uh oh, another paradox. I'm addicted to these things lately. Will have to check it out.

JAlanKatz:

This is just Godel's theorem translated into the language of recursion. How would someone disagree? You could hold that anything not computable is indeterminate - until we run it, there is no answer.

I'll have to think about that one for a while.

JAlanKatz:

Consider the sentence "5 has no successor in the natural numbers." It asserts its own truth, but it's wrong about that.

Yeah, so like "A and ~A" it's a false statement (I'd say that since it's a contradiction, it doesn't follow from any non-contradictory premises).

JAlanKatz:

Next, it simply isn't true that to assert your own truth is to assert your own provability. 2 is incorrect.

In Human Action, Mises wrote something to the effect that all mathematical theorems were already contained in the axioms. For reasons I mentioned above (the point of view of a hyper-intelligent being), it seems that provable just means "follows from premises." So - and this is really what I'm hoping to figure out - if "true" doesn't mean "follows from premises" then what does it mean? I know there are many theories of truth, but if that debate were at play Gödel's theorem wouldn't be so widely accepted.

JAlanKatz:

Finally, even if I followed your argument, it would turn out that G is a contradiction, hence a false statement - but then it's provable.

If "provable" just means "follows from the premises," then we have

G = "G follows from the premises of T, and G does not follow from the premises of T"

So G is a contradiction and hence false, meaning - I propose - that G does not follow from the premises (since a contradiction can't follow from non-contradictory premises/axioms).

It seems to me that there are "laws of inference" that are inherent, or simply are, by virtue of how we define the terms. Given A -> B and B -> C, I don't think it makes sense to state that without a special law of inference we cannot conclude that A -> C. (Is that what you're saying?) Of course that's fine in the case where we want to model a special kind of "logic" that only shares the common word but isn't strictly related to logical reasoning.

Well, this is precisely the point of coming up with deductions. When we describe a formal deduction, we are talking about something that can be carried out mechanically. Picture having a computer do it - which is precisely why there are so many applications to recursion theory, by the way. (In fact, my professor wrote a text where he presents the entire proof of Godel's theorem in the language of recursion, rather than logic.) Now, one would hope that the rules of inference we program into the computer match up well with what really happens. But the computer doesn't know or care about meaning - I can just as easily tell the computer that if it sees A->B on one line, and A on another, then it may conclude -B on a third, as I can anything useful. So if I did that, I'd be able to prove false things, and unable to prove true things. This would be (for our purposes) a poor deductive system, since one can't mess up logical implication in the same way. The universe follows certain rules, and it doesn't care what we program. So if it is the case that A implies B, and A happens, B will happen, even if we've proved something different. So showing that a certain deductive system is complete and consistent is precisely showing that the rules selected to characterize the system do the same things as the rules operative in the real world. This is not obvious at all times, especially the completeness. Frege, after all, came up with a rule that he believed did model how things happen in the real world which was inconsistent (basic logical principle 5.)

So, once we've designed a good deductive system, we will be assured that whenever some situation models the axiom set S, and if T can be proven from S, that the same situation models T. This is completeness and consistency of the deductive system.

Godel's incompleteness proof, on the other hand, is about completeness of a set of axioms. Essentially, he wants to ask if it is possible to come up with a set of axioms which completely characterize the particular universe of mathematics (remember, he's a Platonist) and yet are decidable - we know what's in the set and what isn't. It turns out we can't.

AJ:

If this is in debate, then how do you state the following, which seems to imply that truth is something other than provability (but what?)?

JAlanKatz:

Next, it simply isn't true that to assert your own truth is to assert your own provability. 2 is incorrect.

Truth is something other than provability. The Platonist sees truth as expressing a proposition that correctly describes things in the forms. I see truth as expressing a proposition that correctly describes things in our world. A formalist sees truth as irrelevant, but does not regard provability as irrelevant, hence must see them as different. (Formalism, as I mentioned earlier, is not really taken seriously since Godel.) Everyone agrees about what provability is, though, relative to a deductive system and set of axioms. To see that provability and truth cannot necessarily coincide, consider the empty set of axioms. Nothing is provable from the empty set of axioms (formally) but certainly some things are true. More generally, changing the axiom set changes what is provable, but cannot change what is true.

AJ:

Even kids understand that "You're not not fat" just means "You're fat" without a proof, and a proof could add nothing to their understanding or certainty. Adults understand more, and seasoned mathematicians even more. If you told a hyper-intelligent being the ZF axioms and certain definitions, and then later told him an advanced theorem based on those axioms and theorems, he might say, "You already told me that." So it seems from point of view of a sufficiently intelligent creature, to speak of a theorem as "provable" would be as irrelevant as telling a kid it's provable that "not not fat" means "fat." In other words, it seems it's only our human limitations that make us use a word like "provable" instead of simply "follows from premises."

Well, sure, to be provable is to follow from the axioms. But the point of Godel's proof is that there are things not entailed by the axioms which are nonetheless true - for any decidable set of axioms of sufficient complexity. Considering your example, either double negation is really true, or it isn't. It turns out it is. In natural language, we use a pretty good deductive system psychologically (it's terrible linguistically though. Lots of languages use double negation to emphasize a point rather than to express a positive. A linguist once said in a speech that while the use of the double negative is split between languages, there is no language in which a double positive expresses a negative. An audience member said "yea, sure.") and so certain things become common sense to us, relative to language. That's not helpful in math.

AJ:

Uh oh, another paradox. I'm addicted to these things lately. Will have to check it out.

This one is a bit harder. There is a theorem in set theory (LST) showing that if a set S of axioms has a model M, then it has a countable submodel N. Yet among the axioms, one could include a statement like "the universe is not countable" which would nonetheless has a countable model. This is called the Skolem paradox. It's not really a paradox, though, in that it isn't problematic once you understand how functions work.

Well, I'm not saying that such a position is correct, but you can at least see where it comes from.

AJ:

Yeah, so like "A and ~A" it's a false statement (I'd say that since it's a contradiction, it doesn't follow from any non-contradictory premises).

Right, and further, any two contradictions are logically equivalent. (To see why, notice that from a contradiction, with standard logic, we can prove anything at all:

A and ~A

A

A or B (anything true can be "or'ed" with anything else

B (since either A or B, and we have ~A)

This is true for either one, so they have the same theory, so they are logically equivalent.

AJ:

In Human Action, Mises wrote something to the effect that all mathematical theorems were already contained in the axioms. For reasons I mentioned above (the point of view of a hyper-intelligent being), it seems that provable just means "follows from premises." So - and this is really what I'm hoping to figure out - if "true" doesn't mean "follows from premises" then what does it mean? I know there are many theories of truth, but if that debate were at play Gödel's theorem wouldn't be so widely accepted.

That's exactly what provable means. True means satisfied by the universe or model or structure under discussion. Consider again your example - it is, let' say, true that a particular program P will not halt, but there is no way to prove it in a finite amount of time (including running it.)

AJ:

If "provable" just means "follows from the premises," then we have

G = "G follows from the premises of T, and G does not follow from the premises of T"

So G is a contradiction and hence false, meaning - I propose - that G does not follow from the premises (since a contradiction can't follow from non-contradictory premises/axioms).

The second translation there is incorrect. G says "G is not provable" so G says "G is true and not provable" so G says "G is true and does not follow from the axioms."

When we describe a formal deduction, we are
talking about something that can be carried out mechanically. .... So
showing that a certain deductive system is complete and consistent is
precisely showing that the rules selected to characterize the system do
the same things as the rules operative in the real world.

You seem to be implying that there is no pure math, or that any math
system will necessarily be limited to the creator's biases. Can't one
just create a random set of axioms, check for consistency, and then
have a new field of mathematics (though perhaps not useful or
interesting)?

JAlanKatz:

I see truth as expressing a proposition that correctly describes things in our world.

Again, what if the math is totally fanciful and corresponds to
nothing (known) about the world? (Although in the first place I'm not
sure that any math describes the real world - for instance, I've never
seen the number "5" / but then again, what do we mean by "real world" if not our mental models based on what we are sensing.)

JAlanKatz:

This one is a bit harder. There is a
theorem in set theory (LST) showing that if a set S of axioms has a
model M, then it has a countable submodel N. Yet among the axioms, one
could include a statement like "the universe is not countable" which
would nonetheless has a countable model. This is called the Skolem
paradox. It's not really a paradox, though, in that it isn't
problematic once you understand how functions work.

Thanks, that sounds interesting.

JAlanKatz:

Nothing is provable from the empty set of
axioms (formally) but certainly some things are true. More generally,
changing the axiom set changes what is provable, but cannot change what
is true.

... But the point of Godel's proof is that there are things not entailed by the axioms which are nonetheless true

... That's exactly what provable means. True means satisfied by the universe or model or structure under discussion.

... The second translation there is incorrect. G says "G is not provable" so G says "G is true and not provable"

In each of the above, you're using the idea that truth is different from what follows from the premises. Of course true just means whatever the speaker intends it to mean, but for example I might ask how you know what the world is like, other than by making assumptions based on what sensations present themselves to you. Since those assumptions would be premises as well, I arrive at the notion that true would be most aptly used to describe just what follows from the premises.

JAlanKatz:

Right, and further, any two contradictions are logically equivalent. (To see why, notice that from a contradiction, with standard logic, we can prove anything at all:

Yeah, agreed.

JAlanKatz:

Consider again your example - it is, let' say, true that a particular program P will not halt, but there is no way to prove it in a finite amount of time (including running it.)

I do think the notion makes sense for (real-world) mechanical operations. I am just not sure how you mean to say that math corresponds to the real world, or how Gödel can be a Platonist given that one could make, for example, two different geometries with conflicting axioms.

You seem to be implying that there is no pure math, or that any math
system will necessarily be limited to the creator's biases. Can't one
just create a random set of axioms, check for consistency, and then
have a new field of mathematics (though perhaps not useful or
interesting)?

I don't see where I've implied that. Everything I've said about mathematics proper is entirely consistent with Platonism, which is the ultimate in a "pure math" is it not? On your last question, checking for consistency is not so easy. In fact, Godel's second theorem is that we cannot operate within a system to prove its own consistency. So what we do is relative consistency proofs - assuming ZFC is consistent, so is ZFC with CH, for instance.

AJ:

Again, what if the math is totally fanciful and corresponds to
nothing (known) about the world? (Although in the first place I'm not
sure that any math describes the real world - for instance, I've never
seen the number "5" / but then again, what do we mean by "real world" if not our mental models based on what we are sensing.)

Such math can exist, although I'm not quite sure just what would help you to set up the axioms. In such a system, we'd have plenty of provable statements, but I maintain they would not be true.

AJ:

In each of the above, you're using the idea that truth is different from what follows from the premises. Of course true just means whatever the speaker intends it to mean, but for example I might ask how you know what the world is like, other than by making assumptions based on what sensations present themselves to you. Since those assumptions would be premises as well, I arrive at the notion that true would be most aptly used to describe just what follows from the premises.

You've said many times that in math, true means follows from the premises. I've told you that it does not, but your reply has continued to be "yes it does." I really don't know what else to say on the subject. I am a practicing mathematician, not a philosopher, so I don't regularly delve into philosophical issues, although they interest me. I can tell you, as a working mathematician, that almost none of us think this. If that doesn't help, consider reading any text on philosophy of math, or mathematical logic. I recommend Leary's A Friendly Introduction to Mathematical Logic to start with.

AJ:

I do think the notion makes sense for (real-world) mechanical operations. I am just not sure how you mean to say that math corresponds to the real world, or how Gödel can be a Platonist given that one could make, for example, two different geometries with conflicting axioms.

Even in physical reality, we have examples of Riemann and Euclidean geometries, so I don't see why there couldn't be both flat and curved surfaces in the mathematical universe. Reaching further, though, certainly a Platonist has to admit that one can make axioms (or rules of inference) that do not relate to what happens in the universe of mathematics. When you do that, you make a game that has no relation to anything true, and your conclusions are "provable from the axioms" yet not true. I don't see why this is problematic. This, in fact, is what I've said throughout - that one can make useful (true) axioms, or not. This is the point of the debate between Hilbert and Frege. References on that debate are easily available, if you're interested, and might shed light on some of these questions.

All right, I think I see what you're saying now. It looks like there is a broader philosophical question underlying this, and that my tentative position on the matter disagrees with most mathematicians working in the field. I would be very interested to read more on this subject. I will check out the Leary book for starters.

JAlanKatz:

Reaching further, though, certainly a Platonist has to admit that one can make axioms (or rules of inference) that do not relate to what happens in the universe of mathematics. When you do that, you make a game that has no relation to anything true, and your conclusions are "provable from the axioms" yet not true.

My contention is that everything is a "game" so to speak, in the sense of "the map is not the territory," so I can see now that this is a deeper question touching on epistemology and/or a theory of mind, and why it might not be of much interest to a practical mathematician. Still, this conversation made the issue much clearer for me. Thank you.

The reason I have taken an interest in this is mostly because of Adam Knott's attempt to extend the strict version of Mengerian/Misesian praxeology, wherein it became apparent to me that there are some far-reaching philosophical issues that may need to be resolved. Can I ask what your specialization is, and how pure/applied your focus is?

JAlanKatz:

References on that debate are easily available, if you're interested, and might shed light on some of these questions.

All right, I think I see what you're saying now. It looks like there is a broader philosophical question underlying this, and that my tentative position on the matter disagrees with most mathematicians working in the field. I would be very interested to read more on this subject. I will check out the Leary book for starters.

I may be a bit out of my depth, as I said, because I explore these issues as a mathematician, not a philosopher, although I have some background in philosophy. I do appreciate that there are issues and so forth, but also, remember that I myself have positions on larger philosophical issues which I am not making explicit here - because they are also the positions that most mathematicians hold, and so to describe what mathematicians do, I have no need to open that can of worms. The Leary book is a great place to start to understand how mathematicians actually work with and view these concerns. Another book you might consider is Frege's Foundations of Arithmetic. It's not at all how we think, but it shows where we started. Reading that before Leary would be good - Leary covers Hilbert through Godel, roughly (although he puts in more computer science) so seeing the first attempt first, to understand the difference between logicism and formalism, would be good. The Oxford book on philosophy of math is also full of readings. You can buy this book for under $40, and it's available at most college libraries. It includes Russell's letter to Frege, as well as the Frege/Hilbert letters I mentioned.

AJ:

My contention is that everything is a "game" so to speak, in the sense of "the map is not the territory," so I can see now that this is a deeper question touching on epistemology and/or a theory of mind, and why it might not be of much interest to a practical mathematician. Still, this conversation made the issue much clearer for me. Thank you.

I'm pretty sure I don't understand your contention. That the map is not the territory, I think, is exactly what I've been saying, and I thought you were arguing opposite that point.

AJ:

The reason I have taken an interest in this is mostly because of Adam Knott's attempt to extend the strict version of Mengerian/Misesian praxeology, wherein it became apparent to me that there are some far-reaching philosophical issues that may need to be resolved. Can I ask what your specialization is, and how pure/applied your focus is?

I have three areas of interest - chaos/complexity, math education, and logic. My current research program is in logic, more narrowly reverse mathematics, and more narrowly still, the reverse mathematics of certain theorems in set theory. Reverse mathematics, as the name suggests, is math in reverse. Rather than beginning with axioms and deriving conclusions, we ask people in other fields - either within math or in applied fields - what theorems they need to do their work, and then we attempt to discover minimal axiom sets that make this true. Philosophically, we are asking just how far from ideal a structure/universe can be and still exhibit nice properties. My interests are largely pure, in the sense that I do not look at how math is used in other fields. On the other hand, my goal is application, but of a different sort. Rather than looking for "how can this formula be used here?" I look at how other structures - biological, economic, political, etc. exhibit patterns that can be seen in mathematical structures, then look for isomorphisms so that math can be used to give some answers over there. This is still considered pure math.

AJ:

That would be great if you have them handy.

I don't, actually I do but they're password protected. They can be found in the Oxford book, or a google search.