There was a process which placed the dots where they are, therefore they were ordered by that process — Metaphysician Undercover

First, of course, is that we may take a collection of dots as given, without stipulating that a particular person placed the dots herself.

Second, let's even suppose that "actual order" is a function of a person placing the dots. Say that Joe places the dots in temporal succession and Val places the dots in a different temporal succession. But that both collection of dots look exactly the same to us. So there's "Joes actual (temporal) order" and "Val's actual (temporal) order", but no one can say which is THE actual order of the collection of dots we are looking at without Joe and Val there to tell us (if they even remember) the different order of placement they used.

This is the magical ideation of crank mathematics. That for all the possible formulas, statements, objects, and states-of-affairs, there are actual people running around creating each of them individually. It's so ludicrous that even a child would know it makes no sense; and it surely does not "correspond to reality".

This is what Tones and I discussed earlier. How can we count a specific number of points without assigning some sort of order to them? To count them we need to distinguish one from the other by some means or else we do not know which ones have been counted and which have not been counted. — Metaphysician Undercover

You are mentioning me yet again, without quotation or context. This is about the fifth time you've done it.

I never claimed that we can count things that are not distinguishable.

without any order — Metaphysician Undercover

You are obfuscating by sliding between adressing "order" and "actual order" (or "inherent order"). That's typical of your intellectual sloppiness.

It is not the case that there are not orderings. The point though is that there is not a single ordering that is "THE actual ordering". There are many orderings and they are actual even though 'actual' is gratuitious.

I have a hunch about cranks in logic and mathematics. It's not something I can prove, but it seems to me to be a plausible narrative:

The crank is not necessarily unintelligent. He (it seems that virtually all cranks have male names if their username does suggest gender) may be adept at peforming complicated mathematical operations, computer programming, applied mathematics, engineering and physics. Some cranks got good grades in high school math and even into college. This was a point of pride for the crank. But when the crank was confronted by more abstraction, there was a breakdown. He cannot understand such things as the empty set, material implication (with the FT and FF truth table rows mapping to T), infinite sets, diagonalization, uncountability, incompleteness, and the unsolvability of the halting problem. So when the crank sees other people understanding what he cannot understand, to avoid feeling inadquate, he lashes out with sour grapes that logic and math are all a bunch of nonsense. And the more you try to help him with information and explanation, the more entrenched he becomes in his own world of "they're all wrong; I'll show them who is right!" Then, for him, not only are logicians and mathematicians wrong, but they are knaves and scoundrels (one crank on another forum didn't just want to defund and abolish univerersity mathematics, but he (seriously) advocated mass executions). Internet discussion forums are where the crank lives out his pathetic agenda, and once he claims his perch, he will howl from it forever.

P.S.

The order which they have is their actual order — Metaphysician Undercover

It's ironic that if you knew any mathematics, you could have given an answer:

<b c> is before <d f> if and only if (b < d or (b = d and c < f)).

That is a linear ordering.

(But it can be called the 'standard ordering' only by convention. It is no more "inherent" than any other ordering.)

What is added or multiplied is the quantity or number of individuals. The number is of the individual, a predication, and what is added or subtracted is the individuals, not the number.
— Metaphysician Undercover

That's just a plain contradiction from one sentence to the next. — Luke

That's a beauty.

I simply don't accept it as a realistic notion of "truth", and don't want to waste my time discussing it. — Metaphysician Undercover

Dollars to donuts that, without copy/paste from Wikipedia, you could not in your own words state the distinction bewteen syntax and semantics and the notion of truth in a model.

Physicists, engineers, and others, applying mathematics in the world have a huge impact on the world in which I live [...] bad mathematics will have a bad effect. — Metaphysician Undercover

The computer you're typing on and the science and technology that makes your world better are enabled by mathematics. What is an example of a bad effect from mathematics?

That people vehemently support and defend fundamental axioms which may or may not be true, refusing to analyze and understand the meaning of these axioms, simply accepting them on faith — Metaphysician Undercover

But in the philosophy of mathematics, which includes many mathematicians themselves, people do investigate, question, and debate the axioms - giving reasoned arguments for and against axioms. It's just that you are ignorant of that.

The order which they have is their actual order, whereas all those others are possible orders. — Metaphysician Undercover

That is one of the best, most risible, evasions of a challenge I've ever read. What is "the order they actually have" as opposed to all the others? Saying that they have the order they "actually" have is not telling us what you contend to be the order nor how other orderings are not the "actual" ordering. You are so transparently evading and obfuscating here.

When confronted with the challenge of points in a plane, a reasonable response by you would be "Let me think about that." But instead you reflexively resort to the first specious and evasive reply that comes to you and post it twice with supposed serious intent. That indicates once again your lack of intellectual curiosity, honesty or credibility.

the term "random" — Metaphysician Undercover

'random' in this context need not have anything other than an informal sense. One could just as well say 'unstated'. You're harping on the word 'random' to evade the heart of the argument against you.

You were presented with points in a plane, without being given a stated particular ordering. You were asked to say what is the "inherent order". You reply by saying, essentially, that their order is the postions they have. But that is not ordering. Ordering, such as a linear order, is a relation in which each object is determined to be before or after another object. And not necessarily temorally or physically. And you can't say what is the "inherent order" even temporally or physically! You fail.

I looked at the Wikipedia entry — Metaphysician Undercover

And you comprehended nothing from it.

The part that doesn't make sense is when you move deeper into the theory. This is just like mathematics. — Metaphysician Undercover

I want to understand: Are you saying that music theory is wrong just as you say mathematics is wrong? And, by the way, do you know any music theory?

When I was seven years old I had no idea what an abstraction is, or what a concept is. I didn't understand this until much later when I studied philosophy. — Metaphysician Undercover

When I was seven I didn't know about abstraction, but I used abstraction. Later, I didn't have to wait until studying philosophy to know about abstraction. You seem to have a condition that prevents you from grasping the notion of abstraction and therefore to revile it.

This is why mathematics really is like religion. We are required just to accept the rules, on faith, follow and obey, without any real understanding. — Metaphysician Undercover

That is false. It's the opposite. That describes the grade school memorization and regurgitation of tables and rules for basic addition, subtraction, multiplication, and division that you find so suitable. Mathematics though provides understanding of the bases for those rules.

Meanwhile, your own presentation is not by reason but from your own very personal and subjective misundertstandings and dogma. And your use of even common language terms is wildly personal and impossible to negotiate with common meanings. You insist on confused, incoherent, illogical, and self-contradictiory concepts, meanings, and flat out unsupported assertions, expecting that others should accept them while you are ignorant of even the basics of the subject as it has been developed and offered to open scrutiny in a rich peer-reviewed literature.

/

And aside from you specious (essentially vacuous) argument about the points in plane, here are some of the other points still unanswered by you (and these are just the most recent):

What else could demonstrate falsity other than a reference to some form of inconsistency?.
— Metaphysician Undercover

Falsity is semantic; inconsistency is syntactical.

Given a model M of a theory T, a sentence may be false in M but not inconsistent with T. — TonesInDeepFreeze

An axiom is expressed as a bunch of symbols, so it must be interpreted.
— Metaphysician Undercover

Formulas don't have to be interpreted, though usually they are when they are substantively motivated. — TonesInDeepFreeze

If in interpretation, there is a contradiction with another principle then one or both must be false.
— Metaphysician Undercover

It might not be a matter of principles but of framework. Frameworks don't have to be evaluated as true or false, but may be regarded by their uselfulness in providing a conceptual context or their productivity in other ways. — TonesInDeepFreeze

Notice there is an exchange of "equal" and "same"
— Metaphysician Undercover

Even though there is nothing wrong with taking 'equal' to mean 'same', the axiom of extensionality doesn't require such mention.

Az(zex <-> zey) -> x=y.

"=' is mentioned, but not "same". — TonesInDeepFreeze

he teacher insisted no, the numeral is not the number. So it took me very long to figure out that the numeral was not the "number" which the teacher was talking about, and that the number was just some fictitious thing existing in the teacher's mind — Metaphysician Undercover

I didn't need a teacher to make me aware that numerals are not numbers. '2' and 'two' refer to the same thing. But '2' is not 'two'. So whatever they refer to is something else, which is a number, which is an abstraction. Rather than be a benighted bloviating ignoramus (such as you), I could see that thought uses concepts and abstraction and our explanations, reasoning and knowledge are not limited to always merely pointing at physical objects.

If mathematics requires self-deception, then this does not make sense to me, and so I will not proceed. — Metaphysician Undercover

And we are so lucky that people who did actually go on to learn mathematics were not arrested in development as you are. You wouldn't be typing on your computer or enjoying all the other comforts of science and technology if all the mathematicians dropped out of the subject at the mere suggestion that there is a difference between numbers and numerals.

I learned to play a musical instrument, and it always made sense to me, right from the start. — Metaphysician Undercover

Actually it doesn't make initial sense. Moving from one letter to the next is always a whole step, except from B to C and from E to F. And then double flats move you down a letter except from C to Cbb and from F to Fbb, and double sharps move you up a step except from B to B## and from E to E##.

And on some instruments, when you play at note it's called one particular letter, but on another instrument it's called a different letter, and usually with a flat sign too.

And a 7th chord is not actually the 7th of the scale of the key, but rather it is the minor 7th. But we don't call it a 'minor 7th chord' because that's yet another different chord.

And some intervals in the scale that are not minor nor diminished are called 'major' but others are called 'perfect'.

And a crank (such as you are crank in logic and math) when first confronted with musical notation could come up with nonsense like "a minor second is supposed to be the most dissonant interval, but it's actually two notes that are the closest! It makes no sense!" And the fact that it's enharmonic with an augmented unison. The crank may say, "augmented unison? it's an oxymoron, like empty set!".

And a crank can say, "The major 6th chord is an inversion of a minor 7th chord, so it's two different things! Can't be both major and minor! Doesn't "correspond to reality"! So I'm not going to learn music - it requires that I decieve myself!"

Etc.

Math is like religion — Metaphysician Undercover

Mathematics is the opposite of religion. In another post I completely demolished the comparison.

I look at truth as corresponding with reality. — Metaphysician Undercover

Whatever your personal meaning of "reality", or lack of meaning, might be.

Meanwhile, you're not even familiar with the distinction between semantics and syntax and the notion of model theoretic truth. So you don't know anything about the mathematics you disdain.

If mathematics requires self-deception — Metaphysician Undercover

It doesn't.

how to count, and simple arithmetic, addition, subtraction multiplication, division made sense to me right from the start. It was only later, when they started insisting that there existed a number, distinct from the numeral, that things started not making sense. — Metaphysician Undercover

Exactly, you could perform calculations by rote adherence, but as soon as you confronted actual abstract thought, you couldn't handle it and dealt with it by reviling it.

If you think you can interpret the rules as we go — Metaphysician Undercover

He didn't suggest anything like that. You're back to one of your favorite tactics again: the strawman argument.

The inherent order is the exact spatial positioning shown in the diagram. — Metaphysician Undercover

No one but you uses the word "order' that way. But it does allow you to evade the challenge in the example.

https://thephilosophyforum.com/discussion/comment/543934

I checked one of Yang's website. It says anyone over 18. But in an interview on Freakonimics today, he said 18-64. But I re-listened by going to a posted sound file of the Freakomics episode, and realize now that when he said 18-64 was in January 2019. So his proposal has changed.

at first, the subject makes no sense — fishfry

That the subject at first makes little sense is probably usually true. And it's true for me for many different subjects. But symbolic logic is one subject that made perfect sense to me immediately. Then, after learning the predicate calculus I found that there is a mathematical analysis of it and theorems not just in the predicate calculus but about the predicate calculus. I was blown away with admiration of the human intelligence that would devise such a calculus but also move on to prove its consistency and completeness.

But when I was a child, I was not interested in having to learn arithmetic by rote stipulations. But I was intrigued by some of the other math I was introduced to when I was 10 years old, including sets, Venn diagrams, base number systems and things like that. Those ideas - the abstraction involved - immediately impressed me as like pure cool water of abstract invention. That's what I most admire about mathematics - that it combines full rigor with free exploration of abstract imagination. Later, I was not interested in high school algebra - again just a bunch of stipulations. But in the back of my mind, I wondered what kinds of problems have algorithms (though I didn't know the word 'algrorithm' then) for solving, even if only in principle. Again, years later when I discovered mathematical logic, I found that this question had been investigated thoroughly, and is still being investigated.

Mathematics is not at all one of my intellectual strengths; I'm really not very good at it. But I love it.

The other area that I understood immediately is jazz. The very first time I happend to put on a jazz record to give it real attention, I loved it, understand it, and embraced it for a lifetime.

Andrew Yang's proposal is for UBI to go to all and only 18-64 year olds. That makes no sense and is shockingly reactionary.

Bezos makes nearly 4k/SECOND. He gets UBI 1k/month.

An elderly person with total monthly income of only a few hundred dollars Social Security, or no Social Security at all, gets UBI 0/month. And bear in mind that Social Security is a retirement investment program to which the recipient contributed his or her entire working life.

But it's fair, because Bezos will get cut off when he turns 65, just like everyone else.

What else could demonstrate falsity other than a reference to some form of inconsistency?. — Metaphysician Undercover

Falsity is semantic; inconsistency is syntactical.

Given a model M of a theory T, a sentence may be false in M but not inconsistent with T.

An axiom is expressed as a bunch of symbols, so it must be interpreted. — Metaphysician Undercover

Formulas don't have to be interpreted, though usually they are when they are substantively motivated.

If in interpretation, there is a contradiction with another principle then one or both must be false. — Metaphysician Undercover

It might not be a matter of principles but of framework. Frameworks don't have to be evaluated as true or false, but may be regarded by their uselfulness in providing a conceptual context or their productivity in other ways.

Notice there is an exchange of "equal" and "same" — Metaphysician Undercover

Even though there is nothing wrong with taking 'equal' to mean 'same', the axiom of extensionality doesn't require such mention.

Az(zex <-> zey) -> x=y.

"=' is mentioned, but not "same".

As I've argued in other threads, if we adhere to the law of identity, this is a false use of "same". — Metaphysician Undercover

As you argued unsuccessfully, ignorantly and incoherently.

I had difficulty even in grade school — Metaphysician Undercover

The education system let you down. They should have given you proper cognitve tests to investigate your learning disability.

will you simply assert that mathematics is far superior to philosophy — Metaphysician Undercover

We don't have to assert such a thing. But understanding mathematics is prerquisite to philosophizing about it.

How do you propose that one proceed toward "learning the subject", when the most basic principles in that subject do not make any sense to the person? — Metaphysician Undercover

By the person at least reading the first chapters in a textbook on the subject. If the person cannot comprehend those first basics, then we might have to admit that the person is simply ineducable.

If an axiom is false then the proof is unsound. — Metaphysician Undercover

Which axioms of finite set theory do you think are false?

Sorry, that was a stupid question. You don't know any axioms.

If we assume that a set necessarily has an ordering — Metaphysician Undercover

We prove from axioms.

by what principle can we say that each of these many possible orderings constitutes the same set? — Metaphysician Undercover

"constitutes" is your word.

What type of entity is an "element", such that the identity of a unity of numerous elements is based solely in the identity of its parts with complete disregard for the relations between those parts? — Metaphysician Undercover

An element is an x such that there exists a y such that xey. In set theory, every x is an element of some y.

Isn't this a sort of fallacy of composition? — Metaphysician Undercover

No.

Your questions reflect your complete ignorance of set theory. You could remedy your ignorance by getting a book and reading it.

For sets with cardinality greater than 1:

It's not that sets don't have orderings. It's that sets have many orderings (though in some cases we need a choice axiom or an axiom weaker than choice but still implying linear ordering). So the point is that there is no single ordering that is "the ordering".

/

Typically, in an informal sense, the notion of 'set' is taken as undefined. But just a technical note: In set theory and in class theory, we can formally define 'set' from the primitive 'element'.

Team U will prove its claim possibly in only 1 step. Team S will prove its claim only in n steps.
— TonesInDeepFreeze
I've no problem with that; but to be more precise, we don't know U will prove its claim in 1 step. — InPitzotl

No, I said "possibly".

You're mixing metaphors. — InPitzotl

No, I'm not. I'm moving to a different metaphor.

"Cherries to cherries" sounds more like apples to apples — InPitzotl

Exactly.

George [...] Joe — InPitzotl

You unnecessarily change the names and symbols for the examples. I accepted your Land U and Land S and mentioned Team U and Team S. I'll stick with that, so that I don't have to keep reconfiguring the notation:

(1) If ExUx is true, then Team U will prove its claim and might do so early.
(2) If ExUx is false, then Team U will not prove its claim and it won't fail early.

(3) If ~ExSx is true, then Team S will prove its claim but it won't do so early.
(4) If ~ExSx is false, then Team S will not prove its claim but it might fail early.

(1) Is clearly easier than (3).

(4) is clearly easier than (2).

I said that I don't know how to evaluate both (1) and (2) against both (3) and (4). What I mean is, how to evaluate while preserving the sense that it's a matter of proving not just discovering.

Of course, if we just reduce everything to both teams going step by step through their respective domains, then we may wipe out any difference. But that does not capture the essence of the question of how difficult it is to prove, not just how difficult it is to discover. I repeat myself because your "cherry picking" is not a valid objection to the fact that proving something is the case is different from discovering whether something is the case.

Again, if proposition P is false, and I ask a person, "How difficult is a proof of proposition P?" then he may say that is a nonsensical question, because there is no proof of P.

You don't have to share my framework in this matter, as you prefer a framework that wipes out the distinction. But my is the framework that interests me, and I think it is the framework that usually interests other people when this subject comes up - otherwise people wouldn't correctly emphasize that indeed it is more difficult to prove the negation (refined to consideration of case-by-case examination in a finite domain, which refinement I will continue to leave tacit).

To succeed in proving P (that is, to prove P) is different from succeeding to discover whether P is true or false (that is, to discover whether P is true or false). The question that interests me is "What is the difficulty to prove ExUx compared with the difficulty of proving ~ExSx?" and not "How difficult is discovering whether ExUx is true?"

An analysis that doesn't account for that distinction seems to me to be inadequate.

Again, the comparison that interests me is this:

Suppose ExUx is true and ~ExSx is true, and both have the same number of cases to check. What is the difficulty in proving ExUx compared with the difficulty of proving ~ExSx.

/

There's just the single point I'm uninterested in — InPitzotl

Then I correct my remark to say, "I appreciate your candor in saying you are not interested in that point, which happens to be one of the main points in my remarks".

Meanwhile I note that you still won't recognize that your point about "burden" was irrelevant and that your earlier sarcasm was gratuitous.

There exists no black dogs... in Saskatchewan. There exists a black dog... in Uzbekistan. — InPitzotl

I don't want to have to spell or copy/paste those long place names every time in discussion.

Let ExUx stand for "there exists a black dog in Land U" and ~ExSx stand for "there does not exist a black dog in Land S. And let both be true. Let n = the number of cases, and the number of cases be the same for both.

Team U will prove its claim possibly in only 1 step. Team S will prove its claim only in n steps.

cherry pick — InPitzotl

I am comparing cherries (succeeding to prove) to cherries, not a blend of cherries and raspberries (failing to prove) to a different blend of cherries and raspberries. The blends are different because:

(2) If ExUx is false, then Team U will not prove its claim and it won't fail early.[/quote]

and

(4) If ~ExSx is false, then Team S will not prove its claim but it might fail early.

are different.

It is not clear how to compare the blend of (1) and (2) with the blend of (3) and (4).

I take the context to be comparing cherries to cherries. Otherwise it would depart from the ordinary question or disagreement people have about this subject. If we take away the immediate comparison of proof of one of two contradictory claims, then we take out the very thrust of talking about the subject.

I don't unquestioningly rely on Wikipedia for information or explanation, but this article does at least reasonably capture the context we often find:

"The difference with a positive claim is that it takes only a single example to demonstrate such a positive assertion ("there is a chair in this house" is proven by pointing to a single chair), while it is typically harder to demonstrate a negative assertion ("there is no chair in this house" requires a thorough search of the house, including any potential hidden crawl spaces)." https://en.wikipedia.org/wiki/Burden_of_proof_(philosophy)#Proving_a_negative

That is a difference in demonstration, not merely the sameness in discovery.

/

So we'll[you'll] disregard your comment about it, after I've pointed out it was not apropos.
— TonesInDeepFreeze
FTFY. — InPitzotl

I suspect that doesn't say what you meant it to say. If I am quoted as saying 'you' then 'you' would refer to you not me.

Anyway, 'we' was the editorial we. And, yes, your comments about "burden' should be disregarded.

And, you've not recognized that your other sarcasm was gratuitous. I mention that as it would help to know that I'm talking with someone who has the ability to recognize a rhetorical mistake.

I have no interest in what you care about.
— TonesInDeepFreeze
That is clearly false, because you keep replying to me and "merely stating" things directly to me. — InPitzotl

I reply to you, while also for whomever is reading, to express my thoughts, and hopefully to communicate. That doesn't entail that I'm interested in whether you care about any particular matter in the discussion.

The neutrality of the terms has nothing to do with my lack of interest in what you're telling me. — InPitzotl

The neutrality was to emphasize that my reasoning does not depend on particular tropes I used for illustration.

But I appreciate your candor in telling me that you're not interested in what I have to say.

as I argued with TIDF earlier in the thread. There are many ways to determine a quantity without referencing an ascending order. — Metaphysician Undercover

I don't know what specifically MU has in mind that I said, but I have not said anything that could be correctly paraphrased as "There are not many ways to determine a quantity without referencing an ascending order".

"f(0) if f(0) is 1 is less than f(0) if f(0) is 2" is gibberish. — InPitzotl

And it's not a meaningful comparison to what I said.

That's entirely correct. You didn't say anything there about who has "burden of proof". — InPitzotl

So we'll disregard your comment about it, after I've pointed out it was not apropos.

That the burden of proof is the main subject of the thread doesn't entail that you can't also comment on individual points that have arisen. — InPitzotl

So we'll disregard your comment about it, after I've pointed out it was not apropos.

Also, my point stands that your sarcasm about my saying why I referenced a black dog was without basis.

What is meant by declaring "the situation" to be that second thing and not that first thing you don't state — InPitzotl

but there's some implication that you really, really want me to care about that second thing and to not care about that first thing. — InPitzotl

No, there is not. Please stop reading past what I actually posted to jump to your own incorrect conclusions about it. I have no interest in what you care about. I'm merely pointing out that there is a difference between (1) claims of opposing views about facts and (2) mere discovery about facts. You really don't see that?

If you want me to be interested in Team-A-winning — InPitzotl

"situation", "team", "winning", et. al are merely tropes to illustrate. I am not claiming that the discussion here is confined to talking about winning debates or being hired to prove the existence of a picture in a stack or any other particular illustration. I even made this clear when I said (twice) that we can reduce to more neutral terms:

Person A sustains his claim when he proves there is a black dog. Person B sustains his claim when he proves there is not a black dog. — TonesInDeepFreeze

Whether Team C [could end] early depends on whether ExBx is true or ~ExBx is true.
Team A might prove its claim and end early only if ExBx is true.
Team B cannot both prove its claim and end early.
— TonesInDeepFreeze
Sure, but there are symmetric descriptions of each of these things for Team A, Team B, and Team C in all of those scenarios. — InPitzotl

The differences are:

(1) If ExBx is true, then Team A will prove its claim and might do so early.
(2) If ExBx is false, then Team A will not prove its claim and it won't fail early.

(3) If ~ExBx is true, then Team B will prove its claim but it won't do so early.
(4) If ~ExBs is false, then Team B will not prove its claim but it might fail early.

(5) If ExBx is true, then Team C will discover that it is true and might do so early.
(6) If ExBx is false, then Team C will discover that it is false but it won't do so early.

(1) compared with (3) gives difficulty more to Team A

The question that most interests me (and as it seems to me to be the question that most interests certain other people) is: What is the comparatively difficulty in proving ExBx and ~ExBx? That is the comparison between (1) and (3). And there's a difference.

But, as I mentioned, in a previous post, I admit that if we also consider (2) and (4) then it is not obvious how to weigh for comparison between Team and Team B. But the need to weigh for that pertained only to certain illustration I gave and which I later said I may need to retract it. And, again, for the question of proving a claim, (2) and (4) are not relevant in the same way that (1) and (3) are. If the proposition is false for a team, then they simply can't prove it so, of course, their difficulty to PROVE is maximal.

We can consider two possible worlds: World A in which ExBx is true and World B in which ~ExBx is true.

With World A , Team A will prove its claim and might do so early.

With World B, Team B will prove its claim but won't do so early.

You may take the subject of this discussion to be whatever you like, but where the sense is taken to be "how difficult is it to prove?", then it seems to me that it is more difficult for Team A.

According to you, I cannot prove my claim if my claim is false. That implies that being able to prove the claim true in the first place requires my claim to be a fact. — InPitzotl

Correct.

What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?
The comparison is meaningless. Convince me otherwise. — InPitzotl

What you're asking requires that I repeat myself.

To prove ExBx, the prover might end early. To prove ~ExBx, the prover cannot end early.

I think burden of proof for claims applies in a wide variety of areas having nothing to do with winning debates. — InPitzotl

Agree. So what? I didn't say it has to be a debate. So, since you bridled at a debate, I also offered it in about as neutral terms as I can:

Person A sustains his claim when he proves there is a black dog. Person B sustains his claim when he proves there is not a black dog. — TonesInDeepFreeze

And I didn't even opine about "burden of proof"; I only commented on comparative difficulty.

I might be corrected on this, but I don't recall making a claim about "burden of proof" in sense of a rhetorical obligation
— TonesInDeepFreeze
But you said:
Rather the situation is:
"Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog. "
— TonesInDeepFreeze — InPitzotl

So what? I didn't say anything there about who has "burden of proof".

Have you been thinking all along that I've been making some kind of argument about who has, or should have, the burden of proof or a greater burden of proof? I have not made such an argument. I didn't claim that the difference of difficulty implies or does not imply anything about who should have a burden of proof.

"Burden of proof" is literally in the title of this thread. — InPitzotl

So what? That burden of proof is the main subject of the thread doesn't entail that I can't also comment on individual points that have arisen. The point I have lately been commenting on has been the difference in difficulty between proving ExBx and proving ~ExBx. That difference was offered by a poster as reason to assign burden; but I have not gone on to claim one way or the other that that difference should be a reason for assigning burden. I only commented on the difference itself.

They're discovering the facts, not claiming what the facts are, as opposed to the Positive claimer and Negative claimer who both are claiming what the facts are.
— TonesInDeepFreeze
They're invoking P and arriving at either a proof of ExBx or a proof of ~ExBx depending on what the state of affairs are. And by our metric they expend the same exact effort Team A or Team B would in proving it. — InPitzotl

Team C is dedicated to discovery of whether ExBx or ~ExBx and in that discovery arises a proof of one or the other. That is not the same as Team A declaring ExBx and then whether they can prove it or Team B declaring ~ExBx and then whether they can prove it.

Whether Team C ends early depends on whether ExBx is true or ~ExBx is true.

Team A might prove its claim and end early only if ExBx is true.
.
Team B cannot both prove its claim and end early.

Those are three different situations.

If the discussion here is only about a Team C that is out to discover which is the case but not at the outset to make a claim one way or the other, then that it is a very different discussion from the one that had been presented here, which is that of opposing views being claimed, not just discovery. Of course, you're welcome to prefer to delve into the implications of a Team C situation, but it's not the situation I have been addressing.

Here's the discussion leading up the black dogs — InPitzotl

So? it doesn't vitiate anything I said nor show a basis for your sarcasm.

the state of affairs is the same — InPitzotl

The facts are the same. But the question is not what the facts are, but what is the difficulty in proving the facts. What is the difficulty in proving ExBx when ExBx is true vs. the difficulty in proving ~ExBx when ~ExBx is true?

Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog.
— TonesInDeepFreeze
Okay, you've made a claim that this is the situation. Back it up. — InPitzotl

You're serious? It's a characterization of the problem if the context were a debate. If you don't like "team" and "win" then:

Person A sustains his claim when he proves there is a black dog. Person B sustains his claim when he proves there is not a black dog.

Tell me what "Team A wins" has to do with negative versus positive claims in relation to burden of proof. — InPitzotl

I might be corrected on this, but I don't recall making a claim about "burden of proof" in sense of a rhetorical obligation (as "burden of proof" is usually meant). Rather, I pointed out that Positive is easier in the particular sense that if Positive were correct then it might be proved earlier than Negative could be proved if Negative were correct.

what about Team C, who just wants to figure things out without making claims? — InPitzotl

They're discovering the facts, not claiming what the facts are, as opposed to the Positive claimer and Negative claimer who both are claiming what the facts are.

Your Team C seems to be a red herring.

I don't see a basis for your sarcasm.
— TonesInDeepFreeze
The basis is that you volunteered that you only talked about it because it was mentioned. — InPitzotl

I really don't get you. I didn't claim that I was "nice" to do that. Only that you said that the question was not "Which is easier to prove: ExBx or ~ExBx ?", so I replied that the existential was the question and I only referred to black dogs in particular because that was being discussed

a claim merely being negative or positive does not tell you which of the two claimants has a burden or what it is. — InPitzotl

I agree.

I don't claim to understand what you intend to say with your chart.
— TonesInDeepFreeze
You replied to it. You said this:
That chart seems to capture discovery not proof. For example, the min in row 4 is 1 only because we discover that there is a black dog and give up trying to prove that there is not one. But that is not the task. The task is to prove there is not a black dog. — InPitzotl

Yes, I said that is what it seems to me. I don't claim to understand nor to represent what you mean by it. Merely, that is what it seemed to me.

So that wasn't so difficult. — InPitzotl

For odd n, row n+1 - min max - is the same as row n. They are the same because, as far as I can tell, they don't capture the difference in the challenge of proof.

The situation is not:

"Team A, discover whether there is a black dog; and Team B, discover whether there is a black dog."

Rather the situation is:

"Team A, you win if you prove there is a black dog; and Team B, you win if you prove there is not a black dog. "

And in that situation, Team A might win early, but Team B cannot win early. Team A might prove their claim early, but Team B cannot prove their claim early. As far as I can tell, your chart doesn't capture that.

one of them is true, and one of them is false. And with the metric/method under consideration, we don't know which is which until either we find the black dog, or we searched all of the dog houses among the single set of dog houses. — InPitzotl

Yes. But that doesn't vitiate anything I've said.

How nice of you — InPitzotl

I don't see a basis for your sarcasm.

but "black dog" only came into the discussion as an example because the discussion started to be about black dog as an example. — InPitzotl

The thread didn't start with "black dog" and went for a while without it. Anyway, I don't know what you're driving at. You said that the question was not as I couched it, so I merely replied that the question indeed was which is easier to prove regarding a "black dog".

What exactly is your problem with my table — InPitzotl

I don't claim to understand what you intend to say with your chart. I can only say that the best I can glean from it is that it shows the difficulty in discovering whether there exists a black dog.

Meanwhile, again, what strikes me are these facts:

If there exists a black dog, then proving there exists a black dog might end early.

If there does not exist a black dog, then proving there does not exist a black dog will not end early.

Two more facts though that I admit that I don't know how to weigh in:

If there does not exist a black dog, then there is no proof that there exists a black dog, and trying to prove that there exists a black dog will not end early.

If there does exist a black dog, then there is no proof that there does not exist a black dog, but trying to prove that there does not exist a black dog might end early.

E = everything — TheMadFool

In set theory, 'everything' doesn't name a thing. Rather, 'everything' is used for quantification.

(1)

Suppose ExAy yex. ("There exists an x such that every y is a member of x")

Let Ay yeU.

So UeU.

'UeU' is not a contradiction (self membership is consistent with ZFC-regularity).

(2) Cantor's paradox

Suppose ExAy yex.

Let Ay yeU.

So PU is a subset U. ("The power set of U is a subset of U")

So Ef f is an injection from PU into U.

So Ef f is a surjection from U onto PU.

Previously proved theorem: Ax ~Ef f is a surjection from x onto Px.

So ~Ef f is a surjection from U onto PU.

So ~EAy yex.

The question was "Which is easier to prove: ExBx or ~ExBx ?"
— TonesInDeepFreeze
No, that's not the question. The question is whether it's easier to prove a negative claim or a positive claim. — InPitzotl

An existential vs its negation.

I used 'black dog' only because it came into the discussion as an example.

Goldbach conjecture — InPitzotl

The juncture in the discussion I have recently been addressing is not of deductive determination, but rather empirical determination, on a case by case basis, in a finite domain.

You're trying to tell me that you can compare the proof of a true thing being false to the proof of it being true — InPitzotl

No, I am not.

or maybe that simply not knowing whether you're comparing the proof of a true thing being false to the proof of it being true or you're comparing the proof of a false thing being true to the proof of it being false makes sense out of it somehow. — InPitzotl

That's not what I've said.

set theory is regarded as the basis for number theory, no? — Wayfarer

Set theory is one way to axiomatize mathematics.

In set theory, numbers are sets.

0 = the empty set
1 = {0}
2 = {0 1}
etc.

This is not a claim that numbers are "really" sets (whatever "really" might mean as pertains to abstract objects), but rather that they are treated definitionally that way in set theory.

Doghouses don't hurt, but they're not necessary.

The question was "Which is easier to prove: ExBx or ~ExBx ?"

The only way that question makes sense is to compare ExBx when it is true vs. ~ExBx when it is true, because if ExBx is false then there's no proof of it and if ~ExBx is false then there is no proof of it.

If ExBx is true, then it is possible that it might be proven quickly. But if ~ExBx is true, then it can only be proven by showing all possible cases.

suffering from malnourishment during World War I.

That doesn't say that he died by self-imposed starvation.

So if ‘everything’ includes ‘every possible set’ - which it must do, otherwise it would be incomplete - then there couldn’t be such a set, because it would have to include itself. — Wayfarer

Self-inclusion is not in itself paradoxical.

However, three ways to derive a contradiction from a claim that there exists a set whose members are all and only the sets are Russell's paradox, Cantor's paradox, and the Burali-Forti paradox.

a 2007 BBC Documentary called Dangerous Knowledge. It's about four great and controversial mathematicians - Cantor, Boltzmann, Godel and Turing - all of whom died by suicide — Wayfarer

What's controversial about Godel and Turing?

What source does that film provide for its claim that Cantor died by suicide?

The original paper is in Jean van Heijenoorts's 'From Frege To Godel'.
— TonesInDeepFreeze

[,,,] I'm wondering if you could summarize. — fishfry

It's too many technicalities to easily summarize. Roughly speaking, primitives:

2-place operation
(x y)
"pairing"

2-place operation
[f x]
"value of the function f at argument x"

constant
A

constant
B

predicate
I-object
"is a function"

predicate
II-object
"is an argument"

predicate
I-II-object
"is a function that is itself also an argument"

Then a lot of axioms with those.

Did von Neumann anticipate the categorical approach — fishfry

It doesn't seem to me to be a precursor to category theory, but I don't opine.

https://thephilosophyforum.com/discussion/comment/535447

I'm having second thoughts about this and I might need to retract that particular argument.

The table of outcomes is:

ExBx is true, and you can prove it and possibly go home early.
ExBx is false, and you can't prove it, and you won't go home early.

~ExBx is true, and you can prove it but you won't go home early.
~ExBx is false, and you can't prove it but possibly you can go home early.

Now it occurs to me that actually it is not clear how even to summarize those into one single advantage for ExBx side.

The only direct comparison is between the sides when it is true for that side. And that goes back to the point I made earlier.

if ExBx is true, then it is easier for the ExBx side than it is for the ~ExBx side when ~ExBx is true.

Yet, I still can't shake the intuitive feeling that, given that it is equally likely whether ExBx is true or ~ExBx is true, choosing to prove ExBx would be a better choice.

https://thephilosophyforum.com/discussion/comment/539584

In all cases, it seems to me that, since we are concerned with finding the shortest proof, we only need to consider the finite part of the signature that occurs in the theorem, and we don't need to consider alphabetic variants.

(1) I think 'Yes'. There are only finitely many strings of a given finite length from a finite set of symbols, and only finitely many axioms to check whether a string is an axiom.

(2) I think 'Yes'. There are only finitely many strings of a given finite length from a finite set of symbols, and it is algorithmic to check whether any one of them is an axiom.

(3) I didn't check the details of the argument at

https://math.stackexchange.com/questions/1002618/how-to-find-the-shortest-proof-of-a-provable-theorem

that purports to show that the answer is 'No', but the poster seems to be knowledgeable.

(4) I didn't check the details of the argument at

https://math.stackexchange.com/questions/1002618/how-to-find-the-shortest-proof-of-a-provable-theorem

that purports to show that the answer is 'No', but the poster seems to be knowledgeable. And, aside from his argument, it does seem to make sense that if there are infinitely many axioms, then there are infinitely many different possible proof lines, so the algorithm could go on indefinitely before arriving at an answer.

while we flitter about, inconsequential moths circling your flame. — jgill

If a flame be a dumpster fire.

The winner takes first place and the runner up takes second place — fishfry

Or, as Jerry Seinfeld reminds us, taking Silver in the Olympics just means you're the best of all the losers.