This means either our math is fundamentally wrong or it is incomplete. — Jeremiah

I'm somewhat confused as to what you're arguing, and I probably go further than you do on this issue. Surely the classical mathematician agrees that naive set theory was fundamentally wrong. Specifically, that it was fundamentally wrong in taking the Unrestricted Comprehension Schema as an axiom. In ZF, the paradox cannot be articulated, because the axiom of separation blocks sets from containing themselves.

I understand that some feel the axiom schema of specification resolves the paradox, I was speaking generally to the notion that if it is nonsense we can ignore it. As that argument seems to repeat itself in each of my threads. Mainly I just wanted a thoughtful reply instead of just a brush off and a straw-man. We can't just dismiss mathematical paradoxes as nonsense, that is not a valid solution.

I understand that some feel the axiom schema of specification resolves the paradox, I was speaking generally to the notion that if it is nonsense we can ignore it. As that argument seems to repeat itself in each of my threads. Mainly I just wanted a thoughtful reply instead of just a brush off and a straw-man. We can't just dismiss mathematical paradoxes as nonsense, that is not a valid solution. — Jeremiah

It's nonsense that you cannot seem to accept the very clear arguments in favour of dismissing some of your purported paradoxes. Just because you're able to state an inherently contradictory sentence, does not mean that expression is a paradox.

Srap was clear:

In the analysis given above, R was not the Russell set, but the set of all Russell sets, and it has been shown to be empty. It does not contain any set that contains all and only sets that do not contain themselves, because there can be no such set.

Therefore if you present the paradox by beginning, "Let S be the set of all sets that do not contain themselves as members," then I will deny the premise. No set can be formed in this way, which is exactly Russell's point. — Srap Tasmaner

"A married bachelor drew a round square" is as meaningful as "the set of all sets that do not contain themselves as members". E.g. nonsense that we can and should dismiss.

I think you are making a mistake. A direct contradiction that doesn't result from a compelling argument (e.g. some bachelor is married) is not the same as a paradox, which does show up as the conclusion of a compelling argument. Even if you reject the paradox, it is not of a kind with any old contradiction.

I think you are making a mistake. A direct contradiction that doesn't result from a compelling argument (e.g. some bachelor is married) is not the same as a paradox, which does show up as the conclusion of a compelling argument. Even if you reject the paradox, it is not of a kind with any old contradiction. — MindForged

So a paradox is just a persuasive contradiction.

Basically. There's a clear difference in reasonability between the contradiction in "This sentence is false" and "It's raining and it's not raining". If one insists, diffusing the former requires at least giving up some reasonable principles, the latter is simply false and doesn't seem to require losing anything reasonable to dismiss it.

The Russell set is not what anybody had in mind when they first had the idea of sorting the world into classes. The Liar is not what anybody had in mind when they first had the idea of communicating a fact about the world to another person. What you're both missing is how perverse the paradoxical cases are. As I've said elsewhere, the reaction of the average layperson will almost certainly be, "Oh, it's a trick." In my part of the world it might be worse: "I always figured math was bullshit -- guess I was right."

Both of these cases reveal the dangers of unrestrained generalization. We find an approach that works for some purpose, see that generalizing it allows us to use the same technique for many purposes, invent mathematics and conquer the world with Francis Bacon proudly leading the way.

But do paradoxes show that abstract thought is fundamentally flawed?

There are two cardinal sins from which all others spring: impatience and laziness. Because of impatience we were driven out of Paradise, because of laziness we cannot return. Perhaps, however, there is only one cardinal sin: impatience. Because of impatience we were driven out, because of impatience we cannot return. — Kafka

We rush ahead with our limited understanding of classes, or of truth, and prematurely declare a principle because the first phases of generalizing worked so well. The paradoxes are a warning not to be less ambitious but to be more careful. They are creatures of the work we've done so far -- this is why they have the peculiar form they do. If they go in a box with "incompleteness", "uncertainty", "undecidability" and all that jazz, the label on that box isn't "Not such a big shot now, are you human?" It's just "Hey, you're not done."

Ah, but a man's reach should exceed his grasp,
Or what's a heaven for? — Browning

The Russell set is not what anybody had in mind when they first had the idea of sorting the world into classes. The Liar is not what anybody had in mind when they first had the idea of communicating a fact about the world to another person. What you're both missing is how perverse the paradoxical cases are — Srap Tasmaner

Why is that relevant? I certainly never said people intended to create such paradoxes. My point is precisely that the existence of such things are what motivates changes in the logic. They are perverse, at least in the sense of being unintended consequences of seemingly reasonable principles (hence the designation "paradox"). The layperson will not understand it if you tell them there are different sizes of infinity but we know it's true in math, but we don't take that as evidence against Cantor's work on infinity.

This is just to say that I'm not suggesting people blindly forge ahead. To make sensible use of Unrestricted Comprehension, you have your use a paraconsistent logic. Classically (and in every other logic), doing so makes the resulting mathematics trivial and therefore useless.

The layperson will not understand it if you tell them there are different sizes of infinity but we know it's true in math, but we don't take that as evidence against Cantor's work on infinity. — MindForged

Ah, no, I definitely wasn't saying that the recherche nature of things like the Liar or the Russell set is some kind of evidence they should be shrugged off.

We broadly agree, I think, that there is something reasonable and something unreasonable at work in producing the paradoxes. If forced to choose, my allegiance is with the LNC rather than naive set theory, but whatever. I do wholeheartedly approve of dialetheist tinkering. It's healthy.

But I am suspicious of a kind of magician's (or conman's) patter you see around these things. "I can have a set of numbers, a set of things that are red, a set of bald men. Perfectly natural, right? A set of cars. A set of cars that are blue. A set of all sets that don't have themselves as members. Most natural thing in the world..." I just want to pause over the "Wait, what?" reaction here, while you're always emphasizing how naturally these things arise. (I do understand that it's the principles not the example that's supposed to be natural.)

I guess emphasizing the weirdness of the counterexamples is holding onto the possibility that the counterexample itself is where the sleight-of-hand takes place, rather than in the principles it exploits. I certainly feel that way about the Liar. (Likewise Gettier, which is a whole lot like a magic act; Fitch's; the Slingshot.) Russell strikes me as something a little different, that absolutely unrestricted comprehension is bizarre and not what anyone needed or wanted and it's no surprise that if you explicitly let in anything at all, you'll get some pathological cases.

As I've said elsewhere, the reaction of the average layperson will almost certainly be, "Oh, it's a trick." In my part of the world it might be worse: "I always figured math was bullshit -- guess I was right." — Srap Tasmaner

I shared this paradox with lots of people, non-math people, not a single one of them reacted that way. I share every paradox I post here with friends and co-workers, it acts as a short conversation piece sometimes; some find them interesting, most don't really care.

Maybe you are underestimating the "average layperson".

Wait, let me get the dead horse for you. . . .

Call it whatever you like, but changing the name is not a real argument.

Let me explain the purpose of these conundrums that I have been posting.

They are here to generate discussion and I don't care if they are paradoxes, contradictions or whatever, just as long as they are meaty enough to get a fruitful discussion going. I also don't care if the side I have aligned myself with is right; I hate it when everyone agrees with one another, it is the most unproductive form of discussion. The purpose of this is to walk away with just a bit more understanding and the paradox is merely the raft we use to cross those waters.

We are at seven pages here now, so I think this one was a good one.

From where I sit, these threads have raised the level of discussion on the forum. Specific problems are way more interesting than a battle of isms.

Appreciate it.

A direct contradiction that doesn't result from a compelling argument (e.g. some bachelor is married) is not the same as a paradox, which does show up as the conclusion of a compelling argument. — MindForged

The bachelor statement is not a direct contradiction. One has to deduce the contradiction by a series of steps, so the only difference between that and the assertion of the existence of a Russell set is the length of the deduction by which one arrives at a contradiction from the statement.

To see this, note that a contradiction is a statement of the form

P and not P

Now a person is said to be a bachelor at time t if at that time they are an adult, male human that has never been married. We can write this as:

bachelor(x, t) <-> adult(x, t) and male(x, t) and human(x, t) and for all t' (t' <= t -> not married(x, t'))

Then the statement 'Paul is a married bachelor at time t' is formalised as:

married(Paul, t) and bachelor(Paul, t)

which is the same as

married(Paul, t) and adult(Paul, t) and male(Paul, t) and human(Paul, t) and for all t' (t' <= t -> not married(Paul, t'))

Now intuitively we feel confident that we will be able to deduce a contradiction of the form

married(Paul, t) and not married(Paul, t)

from this.

But that deduction will take quite a number of steps. For a start we need to get rid of two other conjuncts adult(Paul, t) and male(Paul, t) using 'AND elimination'. We also somehow need to deduce

not married(Paul, t)

from

for all t' (t <= t -> not married(Paul, t'))

That is going to involve using an instance of the axiom schema of substitution as well as the axiom that A -> A or B ('OR introduction').

I expect that deducing the contradiction will require a proof of at least ten steps. SO the contradiction is certainly not direct.

For the contradiction to be direct, we would need to define bachelor(x,t) to simply mean x is married at time t. But that is not how bachelor is defined. Under that definition a divorcee, a newborn, a widow, a frog, a rock and the number 3.45 would all be bachelors.

The same analysis can demonstrate that the statement that x is a square circle is not a direct contradiction, but rather a statement from which a contradiction can be deduced by a series of steps - just like the assertion of the existence of a Russell set.

I shared this paradox with lots of people, non-math people, not a single one of them reacted that way. I share every paradox I post here with friends and co-workers, it acts as a short conversation piece sometimes; some find them interesting, most don't really care. — Jeremiah

It may be telling then that most of your acquaintances are not philosophy students. Most bachelors who had to take a course of Analytical Philosophy will be well informed of the context of Frege's and Russell's correspondance.

If you present Russell's Barber at a someone who doesn't have a formation in philosophy as a problem to resolve, outside of Frege's Begriffsschrift or Cantor's set theory, then you are wasting their time. Outside of the context of Frege's attempt at a lingua universalis following Leibniz, Russell's paradox is just a pure contradiction. It is a paradox (or more accurately, an antinomy) for Frege's Grundgesetze der Arithmetik because of it's acceptation of unrestricted comprehension.

Quite simply, I think you have not put in the work, so to speak, into understanding the context of Russell's argument to Frege. Russell's Barber has no value outside of this context, it's not even an interesting problem.

He was talking about the layperson, you are missing the context. Also set theory exist in mathematics, I am able and trained in reading the notation and employing sets in formal proofs.

No, rather, you have been missing the context ever since you started this thread. You cannot ask (at least, meaningfully) someone to resolve Russell's Barber if that person hasn't shown an acceptation of unrestricted comprehension. The problem doesn't present itself to them then.

Even less for the layperson. 99.99% of the human population doesn't know that you could possibly provide a foundation to mathematics with set theory.

Have you ever studied Frege's Begriffsschrift?

No, rather, you have been missing the context ever since you started this thread. You cannot ask (at least, meaningfully) someone to resolve Russell's Barber if that person hasn't shown an acceptation of unrestricted comprehension. — Akanthinos

I am fairly sure I did, so clearly I can.

Btw, do you know what ad hominem is?

I study math, and I pay very little attention as to where it came from. However, as I always do, I spent time studying this paradox before posting it. I was well aware of the proposed resolution.

I am fairly sure I did, so clearly I can. — Jeremiah

That's a poor troll's answer.

Btw, do you know what ad hominem is? — Jeremiah

I'm not trying to insult you. But you have clearly shown a lack of understanding toward the role of thought experiments in philosophy. This thread and Zeno's Paradox thread show it well. Outside of a very specific context, Russell's Barber is just useless. It's purpose is specifically to tell us that we need Zermelo's restriction.

I study math, and I pay very little attention as to where it came from. — Jeremiah

You should. Context is important. Frege's Begriffsschrift makes no sense outside of Frege's criticism of syllogistic logic. Russell's set theory is written directly in the mouvement of Frege's research. Your paradox is just otiose to everything if you don't put it back into it's historical and theoritical setting.

It's also why it's a bit disengenuous for you to come and claim authority on the subject. I mean, some of us will have had around 40+ hours of courses on this very subject alone.

You should. Context is important — Akanthinos

I get the context I need to understand the underlying concepts and do the math. Honestly do you have any idea how many theories and theorems a student of mathematics has to cover? You are not suggesting a realistic approach. I get a brief history, a full proof and than we go. All I really care about personally is the proof and the notation. As long as I can follow proofs and read notation nothing is out of scope.

I mean, some of us will have had around 40+ hours of courses — Akanthinos

I have easily spent more than 40 hours on set theory. It's math and if you speak the lingo I'll understand it. I reviewed all the notation and the underlying concepts for this paradox before posting, I knew before hand the axiom schema of specification would be the main counter point. I also know that several of the arguments presented in this thread don't hack it.

Your lording is not convincing me.

I dont care if the approach is realistic for you or not, there are departments dedicated to carrying out this very research : Philosophy departments. If you dont have the time to develop a semi-complete opinion on an academic subject, then you really shouldnt present whatever you have to say with the authority you adopt.

Math is pretty far back in the background of the context of the logicism of Frege. Its there, and its more relevant than say, natural language paradoxes, but the whole point of Russell's barber is not to try and resolve its apparent contradiction via maths, its what it means for Logicism's project. That is the only important thing about Russell's barber, and clearly you didn't figure that out in your research on set theory, otherwise the OP would be wildly different.

I dont care if the approach is realistic for you or not, there are departments dedicated to carrying out this very research : Philosophy departments. If you dont have the time to develop a semi-complete opinion on an academic subject, then you really shouldnt present whatever you have to say with the authority you adopt. — Akanthinos

I did acknowledge that people are doing this research and that they're serious people. And I simply stated that if I met one of them I'd offer up the prime example and ask them to explain to me why they care about the one and not the other. It's a question I'm trying to understand.

I don't know much about philosophy but I do have a bit of a math background. I try to give my perspective. I'm generally pretty upfront about my areas of ignorance. I can be ignorant yet have an opinion, and people may find it interesting or not. I claim no authority I don't have. Are you referring perhaps to your interpretation of my writing style? If I express an opinion that's my opinion. You don't have to agree and I don't even claim that I'm right, and I never claim to have any authority I don't have, or any at all. I do have opinions. And I do have some knowledge of math that bears on philosophical issues from time to time.

So when you say the "authority you adopt," are you referring to my style of expressing my opinions? Or are you thinking that I have claimed authority I don't have? If the latter, please point these instances out so that I can correct them. But if only the former, you should take into account that that's just my style.

I apologize, fishfry, because your posts are, on the contrary of Jeremiah's, very well thought out. I was replying exclusively to him.

Russell's and Zenon's paradoxes are not riddles to be solved. They would have trivial interest to philosophers if they were, because despite apparences, philosophy is pretty much nothing like riddle solving. They show us why its important to think in some contrarian ways at some time in order to test some otherwise untouchable biases. Zenon's is about the need to think about infinite series. Russell's about Frege's mistake in using unrestrained set theory as a foundation for the concept of number.

I also don't care if the side I have aligned myself with is right; I hate it when everyone agrees with one another, it is the most unproductive form of discussion. — Jeremiah

What's unproductive is taking sides and not accepting valid counterarguments.

it's just a fact that the early calculus was inconsistent. — MindForged

You raised a lot of really good points in this post and it's late so I only want to respond to this one point and I'll aspire to get back to the rest of your interesting post later.

Newton's calculus was never inconsistent in the sense of logic. You are equivocating the word inconsistent. See this is something I do happen to know about Aristotle! That he listed some rhetorical fallacies, one of which was equivocation, using the same word with two different meanings within the same argument.

You mentioned calculus was inconsistent in an earlier post, and I didn't push back on it then, but it's important to clarify this point now.

I hope we can agree that a logical system (some axioms along with some inference rules) is inconsistent if there is a proof (a step-by-step application of the inference rules to the axioms) that results in a proposition P, and also a proof of not-P. I'm certain we agree on that.

I will now argue that Newton's confusion over the nature of (what we now call) the limit of the difference quotient is NOT such an inconsistency.

I believe that if I asked you to name the P for which both P and not-P have proofs, you would say, "dy and dy [in modern notation] are both nonzero and both zero." But that's not really the same thing as I hope I can explain clearly enough to earn your agreement.

So in Newton's calculus (using modern terminology and notation) we have a difference quotient $\frac{\Delta y}{\Delta x}$ where $\Delta y$ is not zero and $y$ is a function of $x$. [It's perfectly legitimate for $\Delta x$ to be zero, as in the case of a constant function].

Now as $\Delta x$ gets very close to zero, it may be the case that the quotient $\frac{\Delta y}{\Delta x}$ seems to get very close to some number, which Newton called the fluxion and that we now call the derivative. The derivative can be naturally interpreted as, for example, the instantaneous velocity of a moving particle. So whether we can mathematically formalize it or not, it's clearly an important concept in need of elucidation.

[Just as with the proof of the infinitude of primes, I'm going over familiar territory in detail just to make sure everyone's on the same page].

So we can sort of think of what the quotient does "when both $\Delta x$ and $\Delta y$ are zero," yet we know that this does not actually make any mathematical sense because the expression $\frac{0}{0}$ is not defined, and can not be defined consistent with the usual laws of arithmetic. So it's a puzzler. Berkeley's "ghosts of departed quantities" is a great line, a rhetorical zinger that shines the spotlight on Newton's problem.

For what it's worth, Newton himself perfectly well understood the problem and struggled over the rest of his career to try to explain it, but without success. It did take 150 years, as you mentioned earlier, to develop the concept of a limit; and it was well into the twentieth century before we saw the complete path from ZF to calculus.

What would we call Newton's problem? It's not an inconsistency in the sense of being able to prove both P and not-P. At no time did Newton ever say that $\Delta x$ and $\Delta y$ are both nonzero AND they are both zero. Newton knew better than to say that. We do NOT have a logical inconsistency in the formal sense.

What we have is something that clearly works, but we haven't got the vocabulary to express it mathematically. That's a mental state familiar to everyone who's ever had to construct a proof. We get to the point where we can SEE what's going on, but we can't mathematically SAY what's going on. That's where Newton and the mathematicians of the 18th and 19th century got stuck till they finally worked out a proper formalization.

I hope you can agree that this is not a case of a system that can derive a proof of some proposition P and not-P. That was not the case in Newton's calculus. Rather, Newton just saw a truth that he could not formalize, either with existing concepts or even by inventing new ones.

So there is an equivocation between

* Inconsistency as a formal proof of both P and -P; and

* Inconsistency as in getting to a point where it's intuitively obvious what's true yet you can't figure out how to formalize it properly.

Calculus was never inconsistent, just un-formalizable for a couple of centuries.

I do take your point that it's noteworthy that mathematicians kept at it till they developed a conceptual and symbolic framework to explain calculus. But that's not exactly the same as keeping at it to resolve a direct P and not-P contradiction as in the case of Russell's demonstration.

On the other hand I see that in both cases, we are keeping at it in order to get to the bottom of some antinomy in which we perceive a larger truth that we can't properly express. I will grant you that much. The Newton difference quotient isn't an actual inconsistency, but it's still a pretty thought-provoking datapoint for your case.

What do you think?

Thank you @Srap Tasmaner for the MathJax pointer.

I was replying exclusively to him. — Akanthinos

Oh sorry I didn't realize that. Thanks for clarifying.

The OP clearly did exactly what I wanted it to do. It was chosen for its value as a center of conversation, that was the defining reason.

The only thing you have really expressed is your disproval of me, then you made some lame subjective argument of how you think I should have approached this. Your opinion has been noted, as an opinion.

I disagree, obviously.

What is oddly backwards about your entire argument, is that I am using these paradoxes to generate discussion, which is their central reason for engagement. While you are pretending they belong to your fantasy club of those you consider your fellows. You seem to think there is a specific context, under certain terms and with select people these matters should be discussed. I, on the other hand, discuss these topic and much more complicated notions with everyone and anyone. I have had conversations about multi-variable calculus with people who have no greater than high school algebra, because I assume that if I can figure it out and understand it then they can as well.

Yes, I can be a troll, I am well aware of that and fully admit it, but clearly you are also not without your egoistical hang ups as well.

Jesus Bloody Christ.

I'm not making an elitist argument. What I'm saying is that you are losing all philosophical value to these historical/theoritical artefacts if you do not present them as such, but as riddles to resolve. You may not have realized this because you have not been discussing this mainly with philosophers or philosophy students.

Anyone with a Analytical Philosophy course completed, upon being asked "who shaves Russell's Barber" out of context, will simply look the interlocutor with amused pity. Just like I would look at someone if he were to tell me that he's trying to derive quantum physics from Democrite's atomism.