I don't accept your authority on the argument of authority, guess that means you are wrong.

I may have been off somewhat on appeal to authority, but that is not a reason to sweep aside 100 plus years of history, especially when we are talking about language as how people use language for over 100 years is how language is defined. If people have been calling it a paradox for over 100 years, guess what, it is a paradox.

Also you forgot the link to the OED, which provided the definition of a paradox. Try reading it, as it turns out contradictions can be paradoxes. Why did I use the OED? Because it is an authority. Appeal to authority is not a reason to shrug off a valid authority.

I don't accept your authority on the argument of authority, guess that means you are wrong. — Jeremiah

Lol, you troll (at least for your sake I hope you were trolling). You are trying to substanciate your appeal to authority by refuting your unwarrented assumption of my appeal to authority. Now even if I made such an appeal, you still couldn't use your refusal of that to substanciate your appeal to authority. Two wrongs doesn't make a right.

Now I could provide the actual argument on why you shouldn't be appealing to authority, especially when said authority is questioned. But I think you already know (if I'm wrong about this, please say so and I'll provide the actual argument), you seem smart and educated enough on most instances, you are just abit sloppy/lazy on occasion, wich I like to point out when it happens, since you also seem to have the capabilities to understand me correctly. Perhaps I overestimated you though, time will tell.

Secondly you are conflating the conclusion with the type of argument used. I objected to the argument you used there, not to it's conclusion. It's most unfortunate you have this tendency, since otherwise you could have benefitted from my remarks to your posts to improve the formulations of your positions, instead of picking a fight with an ally.

So my advice, stop guessing, you have demonstrated your logic has more merits than your guesses.

I may have been off on appeal to authority, but that is not a reason to sweep aside 100 years of history, especially when we are talking about language, if people have been calling it a paradox for over 100 years, guess what it is a paradox. — Jeremiah

I didn't object to the conclusion, I objected to the type of argument used. It's rather irrelevant how long people perceive something to be the case or not, what is relevant are the arguments used for or opposed. Now your first paradox we don't disagree on I think. The barber paradox could be more accurately formulated.

Also you forgot the link to the OED, which provided the definition of a paradox. Try reading it, as it turns out contradictions can be paradoxes. Is that the argument I made that you cut out? Was that what I was trying to say with the link? Because it is an authority. Appeal to authority, is not a reason to shrug off an valid authority. — Jeremiah

I didn't even question the authority in this case, so I didn't see any merits in adressing this, but it seemed someone else might have. Hence I objected to the appeal to authority, not it's conclusion. You were doing great untill you made this argument. I saw it as weakening your case, hence I objected.

I didn't mistake anything. — MindForged

You raised a number of interesting points. Before I respond in detail, it would help me to understand your point of view if you could tell me in clear and unambiguous terms what you find different about these two situations.

a) There is no largest prime. Proof: We assume there is a largest prime and derive a contradiction. Hence there is no largest prime.

b) We can't define a set using an arbitrary predicate. Proof: We assume we can define a set using an arbitrary predicate and derive a contradiction. Hence we can not define a set using an arbitrary predicate.

Why is it that in the case of (a) you regard this as a basic mathematical truth; yet in the case of (b) you regard this as a philosophical conundrum perhaps susceptible to attack via paraconsistent logic?

I assume (although you have not confirmed this) that you don't regard the infinitude of primes as being subject to modification or revision based on paraconsistent logic. Why is (b) different?

Could there perhaps be some recency bias? Frege and Russell worked just a little over a century ago; and Euclid's proof is over 2000 years old.

But human nature doesn't change. It's reasonable that there was a contemporary of Euclid, an ur-Frege if you will, who was brilliant and accomplished and who maintained that the primes were finite in number. After all there is a perfectly sensible and compelling heuristic in support of that proposition, namely the fact that the primes get more and more rare the farther out you go; and that there are in fact arbitrarily large runs of consecutive composite numbers.

Perhaps ur-Frege published his masterwork; and right on the eve of publication, Euclid showed that there is no largest prime. Perhaps this caused a big stir back in the day. The historical record is lost; but it's certainly plausable. The fact that Euclid felt the need to write down a proof shows that the question was in the air at the time.

So just tell me please, what is the difference in your mind between (a) and (b)?

By the way I did not intend to appear patronizing. I carefully walked through these two proofs by contradiction in order to elucidate their structural similarity. Assume the contrary, derive a contradiction, learn a truth.

You see a great difference between these two famous proofs, and I don't see a difference at all, except for the antiquity of one and the recency of the other. If you can clearly explain to me why you see a profound difference, I'd understand your viewpoint better.

If people have been calling it a paradox for over 100 years, guess what, it is a paradox. — Jeremiah

I addressed that point in my earlier response to @MindForged. Naming is generally a matter of historical accident. Is the Axiom of Choice an axiom, Zorn's lemma a lemma, and the well-ordering theorem a theorem? But they are logically equivalent, and often introduced to students in relation to one another. Do you regard the infinitude of primes as a paradox? It's often (though to be fair, not necessarily) proved via contradiction, just as Russell's smackdown of Frege is. Historical names mean nothing. One man's freedom fighter is another man's terrorist. What you call things is not the same as what those things are.

Lewis Carroll and many others have made the distinction between the name of a thing and the nature of that thing. Shakespeare noted that a rose by any other name would smell as sweet. Abe Lincoln used to ask, If you call a tail a leg, how many legs does a dog have? Answer: Four. Calling a tail a leg does not make it a leg.

Even the Beatles made this philosophical point: "Her name was Magill, and she called herself Lil, But everyone knew her as Nancy."

Why is it that in the case of (a) you regard this as a basic mathematical truth; yet in the case of (b) you regard this as a philosophical conundrum perhaps susceptible to attack via paraconsistent logic? — fishfry

Because in the case of A, we have every reason to believe we are in a consistent domain (that of classical mathematics), where proof by contradiction is necessary (on pain of triviality), and we know we can give examples of larger primes . In B, we get a paradox unless we rewrite the rules of naive set theory to get something like ZFC. With A, we have a counter example that let's us dismiss the initial supposition, with B we get a contradiction from what seem like reasonable assumptions on their face. The assumption that there's a largest prime doesn't seem to rest on comparably reasonable principles such as a set being any collection defined by whatever condition you have in mind.

I assume (although you have not confirmed this) that you don't regard the infinitude of primes as being subject to modification or revision based on paraconsistent logic. Why is (b) different? — fishfry

I don't think the infinitude of primes will be much affected by a transition in the logic. Paraconsistent logic dispenses with proof by contradiction and tends to instead rely on proof by non-triviality (these are identical in other logics but not with PLs).

Assume the contrary, derive a contradiction, learn a truth. — fishfry

I suppose the simplest way is to point out there are other concerns that bear on something besides consistency. I can't remember if it was in this thread that I mentioned this, but for example it's just a fact that the early calculus was inconsistent. One had to treat infinitesimals as a non-zero value at one step of proofs and then treat them as having and value of zero at another step of the same proof. This was acknowledged by Newton, Leibniz, criticized by Berkeley, etc., and it remained that way for more than 150 years. Now as far as I can tell, if you really tried to insist on this way of proceeding, you would have been rationally required by your standards to have rejected calculus (and therefore everything learned and built because of it) during that century and a half of it being inconsistent. But that's obviously ridiculous, there are other theoretical virtues besides consistency which made calculus tenable to accept despite the contradictions it required one to adopt.

That's what I'm arguing, sort of. Sure, Russell's paradox is a paradox. That was never the dispute. The issue was always that the principles that gave rise to the paradox in naive set theory seem pretty damn reasonable. So the way out of it was to come up with ad hoc restrictions on what constituted a set. There were extra-mathematical considerations which led to that response, not simply a proof by contradiction because that argument itself relies on already dismissing the possibility of paradoxes, which is the very thing under dispute of you accept Russell's Paradox. There has to be a reason (besides arguing against the conclusion) for why you reject the principles that give rise to the paradox, otherwise it seems like the objection is circular. One can get around it the way ZFC does, but the question is if that is more rational or if it results in a more theoretically virtuous theory. Perhaps it does, but it's certainly not answered by pointing out there's a contradiction.

Every single one of these threads I have made someone jumps out and goes, "Oh it is not a paradox, therefore paradox resolved." It gets old and I get tired of going back and forth on that point. I mean it is actually moot whether it is officially a paradox or not, the conundrum doesn't fade away just because someone decided not to call it a paradox. So it is easier just to tell people it widely recognized as a paradox, or something along those lines and I am not lying, these are well known paradoxes.

So call it an appeal to authority if you like. I don't really think it falls as neatly in those lines as you do, but either way it is an effective approach to move the discussion off a moot line of discussion.

You can't resolve a paradox but simply stating that it is not a paradox. A paradox by any other name is still a conundrum.

You can't resolve a paradox but simply stating that it is not a paradox. A paradox by any other name is still a conundrum. — Jeremiah

Do you regard the proof by contradiction that there's no largest prime a conundrum or paradox? Why or why not?

In other words: The assumption that there's no largest prime leads to a contradiction. so we conclude that there's no largest prime. The assumption that you can define a set with an arbitrary predicate leads to a contradiction, therefore we have a powerful paradox that must be addressed by philosophers. I simply do not understand the difference except as a manifestation of psychological recency bias.

@MindForged You raised some good points that I"m taking some time to think about.

I need a good explanation of why you think that is relevant before I spend my time on it. I am not trying to evade or be rude, I just don't have a lot of free time and I need to hear a good justification of the parallels.

I just don't have a lot of free time — Jeremiah

Uh ... LOL. That made me chuckle.

It's because the form of the two proofs is identical:

* Assume there's a largest prime.

* Derive a contradiction.

* Conclude there's no largest prime.

versus

* Assume you can form a set from an arbitrary predicate.

* Derive a contradiction.

* Conclude that you have a deep paradox that must be addressed or resolved.

I don't see the difference. In the 20th century the smartest mathematicians in the world regarded these two patterns as the same. In the case of Russell's smackdown of Frege, everyone realized that you CAN'T always make a set from a predicate, hence the need for better rules of set formation.

So myself, I don't see the difference between the two proofs. If your assumption leads to a contradiction, you ditch the assumption. That's exactly what all the mathematicians did.

It wouldn't make any sense to say, "Oh Euclid's proof by contradiction shows there's a terrible paradox." Rather, Euclid's proof shows that there's no largest prime. And Russell's proof shows that we can't form sets from arbitrary predicates. It's as simple as that.

And -- admittedly an argument from authority -- every mathematicians agrees with me

Now of course that doesn't make me right, that's just an argument from popularity or authority. But it does place the burden of argument on you to say why everyone's wrong and you and @MindForged are right.

I actually already addressed this argument of yours.

I actually already addressed this argument of yours. — Jeremiah

Link please, I didn't see it. But it wasn't an argument, since I'm merely stating what every single mathematician agrees with. I'm asking you a question. Why do YOU find the two cases so radically different? Two proofs by contraction but only one is a paradox in your viewpoint.

@MindForged has the same opinion and he gave a longer post that I'm working through before I respond. If you did respond to this question, just point me at the response please.

https://thephilosophyforum.com/discussion/comment/183352

Proof by contradiction would lead us right back to Russell's Paradox.

It seems you have another contradiction on your hands. — Jeremiah

That didn't even make sense. I do remember reading it now. I don't follow your point at all.

We assume there's a largest prime and derive a contradiction, so we conclude there's no largest prime.

We assume we can form sets out of arbitrary predicates and that leads to a contradiction, so we conclude we can't form sets out of arbitrary predicates.

This seems perfectly sensible to me. And (argument by authority and popularity) every mathematicians in the world agrees. That doesn't mean they're right, but you have to make a much stronger argument, which you haven't done.

By the way, are you and/or @MindForged making some kind of constructivist or intuitionist argument that rejects the law of the excluded middle and/or proof by contradiction? That would at least make some sense, but intuitionists aren't trying to resurrect naive set theory, as far as I know. The modern neo-intuitionists have given up on set theory entirely and are working with some flavor of type theory. Type theory was Russell's own solution to his discovery.

It makes complete sense and I am not interested in your straw-man.

By the way, are you and/or MindForged making some kind of constructivist or intuitionist argument that rejects the law of the excluded middle and/or proof by contradiction? — fishfry

I think I've already articulated my position without recourse to intuitionism. Once you think it over (need not agree obviously) let me know what you think.

I'll admit, I'm something of a logical pluralist so it's not like I'm advocating a wholesale abandonment of standard maths. Honestly, I actually wonder what mathematicians who think about this sort of thing believe (rare-ish to see it done in depth, most don't bother with the foundations of maths these days). Really, it seems like Gödel's Incompleteness Theorems in particular and the death of Logicism (using classical logic) seems to have killed foundationalism in the eyes of mathematicians and logicians, so I wonder if they're pluralists of a sort?

I think I've already articulated my position without recourse to intuitionism. — MindForged

Ok. Just wanted to make sure you accept law of excluded middle and proof by contradiction.

Once you think it over (need not agree obviously) let me know what you think. — MindForged

Will do. I got in trouble once around here when I deferred responding to someone's long and complex posts while responding quickly to other people's short posts. The poster whose long posts I was trying to give serious and considered thought to, got more and more impatient and finally abusive. Just wanted to be clear that I'm deferring my thoughts till I have a block of time tomorrow.

I'll admit, I'm something of a logical pluralist so it's not like I'm advocating a wholesale abandonment of standard maths. Honestly, I actually wonder what mathematicians who think about this sort of thing believe (rare-ish to see it done in depth, most don't bother with the foundations of maths these days). Really, it seems like Gödel's Incompleteness Theorems in particular and the death of Logicism (using classical logic) seems to have killed foundationalism in the eyes of mathematicians and logicians, so I wonder if they're pluralists of a sort? — MindForged

I think Category theory and homotopy type theory are getting most of the foundational work these days. Homotopy type theory as I understand it actually relates to the resurgence of intuitionism. And the set theorists study large cardinals and are still hard at work on CH. You can Google names like Woodin and Hamkins to see what the set theorists are up to. But nobody worries about Russell's paradox because there's nothing to worry about. It just shows that we can't use unrestricted set comprehension. And I still don't know why you think people should be concerned about a run of the mill proof by contradiction. Sure it ruined Frege's day, but it revealed a mathematical truth about the nature of sets. But that's what we're talking about so I'll try to respond to your specific points soon.

Proof by contradiction does not actually resolve this paradox, it still exist if you apply proof by contradiction to the question: Is R a member of itself? In fact it only proves the paradoxical nature of the question under the assumption: Let R be the set of all sets that are not members of themselves

Also, in neither math or logic are straw-mans valid methods of proof.

Ok. Just wanted to make sure you accept law of excluded middle and proof by contradiction. — fishfry

Yea, I don't have much issue with Excluded Middle. Then again, I've only passing familiarity with intuitionism. ;)

Hamkins to see what the set theorists are up to. But nobody worries about Russell's paradox because there's nothing to worry about. It just shows that we can't use unrestricted set comprehension. And I still don't know why you think people should be concerned about a run of the mill proof by contradiction. Sure it ruined Frege's day, but it revealed a mathematical truth about the nature of sets. — fishfry

I'm not talking about Russell's Paradox in that bit, I'm talking about the general outlook regarding mathematics post-Incompleteness Theorems. ZFC's development was intentionally practical: we need to get on with the business of doing sensible maths but classical logic cannot function sensibly with an inconsistent set theory. Once it became clear that there was no strict necessity in picking one formalism over another (i.e. no privileged set of indubitable axioms), it seems like mathematicians and logicians became a bit more cavalier about the whole thing. Rightly so, in my view, the interest shifted to the virtues of particular formal systems applied in specific domains, particularly when such systems are fruitful.

Like from the Incompleteness Theorems, we know you can (for systems expressive enough to articulate arithmetic truths) either have an inconsistent but complete mathematics (Paraconsistent mathematics) or you can have a consistent but incomplete maths (Classical math, Intuituonistic math, etc.). Classical logic is so preferred because of its wide usability, but there are known issues and domains where it's questionable (quantum mechanics, representing human reasoning, databases, some evidence paraconsistent logic operations are faster to compute, etc).

So I wonder if this modern openness to more or less any non-trivial logic/math indicates some kind of pluralism. What do you think?

What do you think? — MindForged

This is exactly how I got in trouble last time. Conversating back and forth while deferring responding to the important earlier post. But a few thoughts ...

I'm not talking about Russell's Paradox in that bit, I'm talking about the general outlook regarding mathematics post-Incompleteness Theorems. ZFC's development was intentionally practical: we need to get on with the business of doing sensible maths but classical logic cannot function sensibly with an inconsistent set theory. — MindForged

I don't see why. Classical logic goes back to Aristotle. And even math doesn't need set theory. There wasn't any set theory till Cantor and there was plenty of great math getting done before that. Archimedes, Eudoxus, the medieval guys Cardano and so forth, Newton, Gauss, Euler, Cauchy, and all the rest. None of them ever heard of set theory and did fine without it.

If set theory were discovered to be inconsistent tomorrow morning, the foundationalists would get busy patching it and nobody else would care. As an example, how would group theory change? The group axioms and their logical consequences would still be the same.

As far as incompleteness, that's already been verified and sliced and diced via computer science, information theory, and almost another century of study. Gödel published in 1931, that's almost a century already. Incompleteness is literally a classical result now. Everyone's moved past it. So we can't use the traditional axiomatic method to determine what's true. If anything, that's perfectly sensible. We have to find other paths to truth. That's exciting, not worrisome I think.

Once it became clear that there was no strict necessity in picking one formalism over another (i.e. no privileged set of indubitable axioms), it seems like mathematicians and logicians became a bit more cavalier about the whole thing. Rightly so, in my view, the interest shifted to the virtues of particular formal systems applied in specific domains, particularly when such systems are fruitful. — MindForged

I don't think that's completely true. People don't study random sets of axioms. See Maddy's great articles Believing the Axioms parts 1 and 2, in which she works through the axioms of ZFC and discusses the philosophical reasons why they have gained mindshare. I really don't believe that incompleteness is any kind of nihilistic disaster. Interesting math is being done every day.

Like from the Incompleteness Theorems, we know you can (for systems expressive enough to articulate arithmetic truths) either have an inconsistent but complete mathematics (Paraconsistent mathematics) or you can have a consistent but incomplete maths (Classical math, Intuituonistic math, etc.). Classical logic is so preferred because of its wide usability, but there are known issues and domains where it's questionable (quantum mechanics, representing human reasoning, databases, some evidence paraconsistent logic operations are faster to compute, etc). — MindForged

Right. All of this is thrilling intellectual stuff. It's not the end of the road for reason. On the other hand, perhaps it's related to postmodernism and the reaction against reason. Reason has given us better ways to wage war and promote economic and social inequality. There are good reasons (!?) to distrust reason.

So I wonder if this modern openness to more or less any non-trivial logic/math indicates some kind of pluralism. What do you think? — MindForged

Pluralism. Yes. Crisis = opportunity. Something new is coming. Hilbert's program failed, but that doesn't lead to people being cavalier as you put it. Alternatives are being explored. I think 100 years from now all this will be more clear. Reason and logic are going through some kind of revolution that we can't see the outlines of yet. Computers and the computational way of looking at things. We're in some kind of transitional period.

Hamkins has something called the set-theoretic multiverse. It's (to the extent I understand it, which isn't much) the consideration of all possible set theories considered as a whole. The worlds where CH is true, where CH is false, and so forth. There's no one true set theory, they're all part of some grand structure. These are my words, not any claimed description of what Hamkins is thinking.

Here's his "popular" exposition, which isn't what I'd call elementary or comprehensible. But for what it's worth, contemporary set theorists are already way past Gödel. By the way (rambling on now), I think the really big breakthrough wasn't Gödel. It was Cohen, who showed how to cook up nonstandard models. That's when things really started getting crazy in the set theory business.

http://jdh.hamkins.org/the-set-theoretic-multiverse/

learn a truth — fishfry

The axiom schema of specification blocks Russell. Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification?

I don't see why. Classical logic goes back to Aristotle. And even math doesn't need set theory. There wasn't any set theory till Cantor and there was plenty of great math getting done before that. Archimedes, Eudoxus, the medieval guys Cardano and so forth, Newton, Gauss, Euler, Cauchy, and all the rest. None of them ever heard of set theory and did fine without it. — fishfry

No, Aristotle created Syllogistic. Classical logic was invented in the 1870s by Frege. These are not the same system, Classical Logic validates a different set of arguments than Syllogistic, it has logical connectives and quantifiers that Syllogistic lacked, and funnily enough, Syllogistic was a type of paraconsistent logic since according to Aristotle you cannot derive anything from a contradiction. Without set theory, we wouldn't understand a lot of maths, it came as part of the program to understand how various kinds of numbers were defined and related to each other. The "classical" in "classical logic" is misleading, if not propagandistic, heh.

Incompleteness is literally a classical result now. Everyone's moved past it. So we can't use the traditional axiomatic method to determine what's true. If anything, that's perfectly sensible. We have to find other paths to truth. That's exciting, not worrisome I think. — fishfry

I didn't say it was worrisome, I was just pointing out a consequence of the theorems. You can use the traditional axiomatic methods, you'll just have an inconsistent theory.

I really don't believe that incompleteness is any kind of nihilistic disaster. Interesting math is being done every day. — fishfry

Now I certainly didn't say it resulted in nihilism, and I don't deny good math is being done. As I said, I don't reject standard math.

On the other hand, perhaps it's related to postmodernism and the reaction against reason. — fishfry

Eh, I don't think it's Po-Mo at all. It's just that the landscape of possible formal systems worthy of mathematical investigation turned out to not be so limited.

Thanks for the link, looks interesting.

The axiom schema of specification blocks Russell. Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification? — Srap Tasmaner

Well that's the conventional wisdom, pretty much universally accepted.

But I wouldn't say that we don't need unrestricted comprehension (I don't know why they use the word comprehension, I'd just say "set formation by predicates"). We simply discovered that set formation by arbitrary predicates leads to a contradiction. So we are FORCED to abandon it, reluctantly.

I do agree that this is psychologically or intuitively unpleasant. We want to think of sets as Cantor originally did:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

That's how we teach school children about sets. It's how we think of sets. The collection of things that satisfy a predicate. But Cantor's definition fails. It leads to a contradiction. So we learn our lesson, we move on, we abandon naive set theory.

I do empathize with those who are troubled by Russell's refutation of naive set theory. But I don't agree with anyone who gets stuck on their intuition so firmly that they can't move past it. It was John von Neumann who said that we don't understand math, we just get used to it. That's a great insight.

No, Aristotle created Syllogistic. Classical logic was invented in the 1870s by Frege. These are not the same system, — MindForged

Sorry, I overstepped my knowledge. I don't know anything about Aristotle. Poor Frege, such a brilliant and original thinker, forever remembered for his big mistake.

I better leave this be for tonight. Now I'm two posts behind you.

Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification? — Srap Tasmaner

(I know this wasn't directed at me but I can't resist)

Depends on what you mean by "need". If you aim to prove Logicism then you do need unrestricted comprehension. Without it, we end up with a lot of unprovable hypotheses. For example, it has been demonstrated that naive set theory + a paraconsistent logic lets you prove the Continuum Hypothesis is false. However, in standard maths it's unprovable.

This isn't to say that because one formalism can solve a problem the other can't that we should ditch one for the other. It's just that there are extra-mathematical considerations to what we pick (i.e. assessing theories for their worth/virtue). Most mathematicians prefer working in a consistent system (and a fruitful one at that) so they privilege one which is consistent but lacks a bit over one where some contradictions are provable.

it has been demonstrated that naive set theory + a paraconsistent logic lets you prove the Continuum Hypothesis is false. — MindForged

Ah, you must be working from knowledge of paraconsistent logic that I lack. Reference for the above fascinating factoid?

Sure! (This won't hold in standard math obviously)

"It should also be noted that Brady’s construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.
(...)
Even more radically, Weber, in related papers (2010), (2012), has taken the inconsistency to be a positive virtue, since it enables us to settle several questions that were left open by Cantor, namely, that the well-ordering theorem and the axiom of choice are provable, and that the Continuum Hypothesis is false (2012, 284)."

https://plato.stanford.edu/entries/mathematics-inconsistent