• Why Democracy Matters: Lessons from History
    Martin Luther said that people were better off being ruled by a smart Turk than a dumb Christian. — BC
    The Emirate of Dubai offers just one product, which is globally tremendously and increasingly popular, i.e. freedom from the aberrations caused by democracy:

    https://en.m.wikipedia.org/wiki/Dubai

    Dubai has been ruled by the Al Maktoum family since 1833; the emirate is an absolute monarchy.

    The city has a population of around 3.60 million (as of 2022).[19] More than 90% of the population are expatriates.

    461,000 Westerners live in the United Arab Emirates, making up 5.1% of its total population.

    People vote with their feet.

    They reject rule by the mob.

    Millions more would want to live in Dubai and be governed by its emir instead of the mob back home, but they cannot afford it financially:

    Dubai is the second most expensive city in the region and 20th most expensive city in the world.

    It is all about supply and demand. Dubai cannot expand fast enough to accommodate demand. Don't try to get into Singapore either. That small island is also full now.

    It is an increasingly expensive privilege to escape the madness of democracy and to get access to a sane alternative.

    The poor cannot afford the government by the emir product, unless they get hired by an employer in Dubai, which is not easy to find. There is simply too much demand.

    That is why the poor must stay where they are and keep governing themselves; which they know to be a living nightmare.

    Democracy is delusional.
  • What can we say about logical formulas/propositions?
    In English, on the other hand, we only say, "If P then Q," when we believe that the presence of P indicates the presence of Q. The English has to do with a relation between P and Q that transcends their discrete truth values. — Leontiskos

    Exactly.

    It represents an entailment A ⊢ B, and not just a simple implication A→B.

    Logic makes all its decisions by only looking at truth values while the English version assumes the existence of a system that also investigates justification.
  • Even programs have free will
    You're confusing determinism with predictability, but I thought we'd already covered this. — fishfry

    According to the page on the subject, determinism and predeterminism are "closely related":

    https://en.m.wikipedia.org/wiki/Predeterminism

    Predeterminism is the philosophy that all events of history, past, present and future, have been already decided or are already known (by God, fate, or some other force), including human actions.

    Predeterminism is closely related to determinism.[1]

    The concept of predeterminism is often argued by invoking causal determinism, implying that there is an unbroken chain of prior occurrences stretching back to infinity. In the case of predeterminism, this chain of events has been pre-established, and human actions cannot interfere with the outcomes of this pre-established chain. Predeterminism can be used to mean such pre-established causal determinism, in which case it is categorised as a specific type of determinism.[2][3] It can also be used interchangeably with causal determinism—in the context of its capacity to determine future events.[2][4] Despite this, predeterminism is often considered as independent of causal determinism.[5][6]

    If you believe that everything has a reason, it does not mean that you also know that reason. Predictability requires indeed both.
  • What is a justification?
    For Jews:

    Mosaic law ⊢ isMoral(behavior)

    For Muslims:

    Islamic law ⊢ isMoral(behavior)

    As far as I know, everyone else does not have enough of a documented shared understanding to be able to define an isMoral(behavior) predicate.

    In that case, if you put 5 people around the table, you will end up with 12 different conclusions. It is simply impossible that they will agree on anything. Everything will be up in the air with maximum confusion.
  • What can we say about logical formulas/propositions?
    let me rephrase: it doesn't match MY intuition, and many other people. — flannel jesus

    It is probably a mixup between the implication, which is just a truth table, and the entailment, a ⊢ b, which means that consequent b necessarily follows from antecedent a.

    (2+2=4) ⊢ (Kamala Harris is a presidential nominee)

    is false, because the consequent cannot be justified from the antecedent.

    So, it is rather about a mixup in vocabulary than about intuition. I guess that many other people do that indeed too.
  • What can we say about logical formulas/propositions?
    (2+2=4) implies (Kamala Harris is a presidential nominee). — flannel jesus

    "(2+2=5) implies (Kamala Harris is prime minister of China)" is also true in classical logic.

    But it doesn't really match our intuition at all.
    It actually does.

    It just means that knowledge as a justified true belief is not only about truth but also about justification.
  • Any objections to Peter Singer's article on the “child in the pond”?
    If that's not what Singer means then he needs to reword his commentary — LuckyR

    Singer wants us to give money to Oxfam, because the girls in Haiti are clamoring for more.
  • What can we say about logical formulas/propositions?
    Eventually, you will even need to add quantifiers (∀ ∃) and predicates to express in logic something as simple as:

    All humans are mortal.
    Socrates is human.
    Therefore Socrates is mortal.

    If you want to express in logic statements about logic itself -- which is a requirement for philosophical statements -- you even need to add support for arithmetic.

    The resulting language is full of issues, collectively known as the foundational crisis in mathematics, which is clearly also a foundational crisis in logic.
  • The Most Logical Religious Path
    1) Religion matters. Agreed. Humans seem to need spirituality as well as a definition for morality.
    2) Some truth can be obtained by practicing religion. Agreed. A claim in morality should correspond to unadulterated human nature.
    3) The person experimenting is not at any risk physically, or mentally (mostly by risk of indoctrination, which can be avoided to an extent by being skeptical). Agreed.

    I use the standard Islamic method to raise the bar and fend off mere indoctrination:

    a) The moral advice or ruling must necessarily follow from scripture through reason.

    b) The moral advice or ruling must enjoy consensus amongst independently judging scholars. These scholars must not be on someone's "payroll".

    I have never had to reject moral advice that satisfies these requirements.
  • Why Democracy Matters: Lessons from History
    In my experience, democracy is a severely flawed system.

    You can easily find a democratic majority for ideas that sound good but that are otherwise surprisingly bad, such as "let's tax the rich". The populace simply does not understand second-order consequences.

    For example, the safety of airplanes existentially depends on rigorously preventing the average individual from having any say whatsoever on their design. The same holds true for the legal system. I simply do not want the average individual to have a say on that either.

    I prefer monarchies such as the Emirate of Dubai because I do not want to be governed according to the delusions of a retarded populace.
  • Mathematical truth is not orderly but highly chaotic
    Two opposing opinions. Here is a discussion on Quora. — jgill

    I agree with the following comment:

    In other words, Wikipedia articles tend to be written in technical jargon that is impenetrable to non-initiates.

    I disagree with the following comment:

    Many of those original articles should be scrapped entirely and rewritten by a knowledgable scholar.

    That is only going to make the problem worse.

    In my opinion, it is preferable to mention the scholarly publications in the footnotes. That will allow anybody who is interested in the exact technical details to investigate them there.
  • Mathematical truth is not orderly but highly chaotic
    And your claim about ZF\I is incorrect. ZF\I is not bi-interpretable with PA. Rather, it is (ZF\I)+~I that is bi-interpretable with PA. (Actually, we can simplify to (Z\I)+~I, which is bi-interpretable with (ZF\I)+~I and bi-interpretable with PA.) — TonesInDeepFreeze

    The original article that establishes and proves the bi-interpretability:

    On interpretations of arithmetic and set theory
    Richard Kaye and Tin Lok Wong

    School of Mathematics, University of Birmingham, Birmingham, B15 2TT, U.K.

    Notre Dame Journal of Formal Logic Volume 48, Number 4 (2007), 497-510. doi:10.1305/ndjfl/1193667707

    I have already linked to this original publication in a previous comment. The abstract says the following:

    This paper starts by investigating Ackermann's interpretation of finite set theory in the natural numbers. We give a formal version of this interpretation from Peano arithmetic (PA) to Zermelo-Fraenkel set theory with the infinity axiom negated (ZF−inf) and provide an inverse interpretation going the other way.

    The official name for the set theory that it is about, is ZF-inf.

    As I already have written previously in a previous comment, if is about ZF with the axiom of infinity removed and denied.

    I don't know why you believe that the term ZF-inf would be wrong because that is exactly the term that the original authors use, i.e. Richard Kaye and Tin Lok Wong.

    As far as I know, Wikipedia does not mention this publication anywhere in connection with bi-interpretability. There are a few placeholder pages on the subject but they look very much like a draft at this point.
  • Mathematical truth is not orderly but highly chaotic
    So you didn't write what you meant regarding S and F. — TonesInDeepFreeze

    I did. I wrote:

    "There are sentences that are like this. There are sentences that are like that. Both could exist."

    There's a lot of syntactic noise associated to specifying F.

    You should not say 'logic sentences' in general, since the theorem pertains to sentences in certain languages for certain theories. — TonesInDeepFreeze

    It's related to PA or similar. That is always implied.

    You should generalize over formulas in those languages and not over properties (since there are properties not expressed by formulas). — TonesInDeepFreeze

    Well, it is about properties that have formulas in PA. That was also implied.

    Mentioning all of that, including the above, will make the entire explanation impenetrable.

    I just refer to a link that contains all these details but I can pretty much guarantee that few people will ever read it.

    My rendition is not suitable for a mathematical forum, but I had hoped that it would be for a philosophical one.
  • Mathematical truth is not orderly but highly chaotic
    Disjunction is inclusive, but it is never the case that both of these are true: "P is true and Q is false" and "P is false and Q is true". — TonesInDeepFreeze

    There exist sentences that are true or there exist sentences that are false, or both.

    "Or both" means: Potentially, there exist as well true as false sentences.

    "Or both" is not about an individual sentence. It is about the fact that both existence clauses could be true, i.e. there are true sentences but also false sentences that satisfy the lemma.

    In general, a disjunction 'phi or psi' might not allow 'phi and psi', depending on the content in phi and the content in psi. — TonesInDeepFreeze

    I meant to say:

    ∃ phi or ∃ psi, or both exist.

    The term "or both" emphasizes that the "or" is not exclusive. The default interpretation in natural language for "or" is actually exclusive.

    People can decide for themselves what is too technical or not. — TonesInDeepFreeze

    In my experience, it usually is too technical. The consequence is that nobody reads what I just wrote. I could as well not write it at all ...
  • Mathematical truth is not orderly but highly chaotic
    "S ∧ ¬F(r(#S)" is not the same as "S & ~F".
    "¬S ∧ F(r(#S)" is not the same as "~S & F".     — TonesInDeepFreeze

    I left out that detail because it is obvious. So, with the details:

    (S is true and F(r(#S)) is false) or (S is false and F(r(#S)) is true)

    It is more accurate but also much more impenetrable than:

    (S is true and F is false) and (S is false and F is true)

    The resulting syntactic noise detracts from understanding what exactly it is about. It muddies the explanation.

    and lately, you confuse the predicate F with a sentence. — TonesInDeepFreeze

    I simplified F(r(#S)) to just F, because I thought that it was obvious what it was about.

    And I don't know why you would suppose that people would care about your synopsis of Carnap if they didn't also grasp the mathematical basis. — TonesInDeepFreeze

    If that is truly the case, then the subject may not be suitable for a philosophy forum. I had hoped that it was, but you may be right.

    The metaphysical implications do seem out of reach of philosophical investigation. Apparently, they have been for almost a century.
  • Mathematical truth is not orderly but highly chaotic
    Your quoted characterization did not have the specifications you are giving now. Your quoted characterization was a broad generalization about properties and sentences. — TonesInDeepFreeze

    It has always been an explanation about the diagonal lemma:

    S <-> ¬F(r(#S))

    Meaning:
    (S ∧ ¬F(r(#S)) ∨ (¬S ∧ F(r(#S))

    Meaning:
    (S is true and F is false) or (S is false and F is true)

    Meaning:
    A true sentence that does not have the property, or a false sentence that has the property, or both.

    It was a choice not to provide these details because this kind of explanations quickly become impenetrable in a multidisciplinary environment.

    (2) PA doesn't say 'true' and 'false'. — TonesInDeepFreeze

    The meaning of the S above is "a true sentence". PA doesn't say it, but that is what it means, for reasons of first-order logic.

    (4) There are properties not expressed by formulas, so the generalization should be over formulas, not properties. — TonesInDeepFreeze

    In that case, it is not a property in PA, because that would require a predicate in PA. In fact, Tarski's truth is also a property but not one in PA.

    It is possible to precisely state all the conditions that apply, but in that case, the explanation becomes impenetrable. Nobody would be interested in a multidisciplinary forum. In order to keep it readable, there is no other alternative than to leave things out.
  • Mathematical truth is not orderly but highly chaotic
    You said that my counterexample is not in PA. — TonesInDeepFreeze

    P(S) := S <-> S

    Is indeed impossible in PA. However, you can implement it as:

    P(n) := n=n

    The diagonal lemma is still perfectly satisfied for the identity property:

    There exists a true sentence that does not have the identity property or a false sentence that does have the identity property, or both.

    All false sentences have the identity property. Hence, you can always find one to satisfy the lemma. So, in what way is the diagonal lemma not satisfied?
  • Mathematical truth is not orderly but highly chaotic
    (3) C doesn't say anything about 'true'. — TonesInDeepFreeze

    That is just how logic works. Asserting:
    .
    S ∧ L

    Means:

    S is true and L is true.
  • Mathematical truth is not orderly but highly chaotic
    (2) C generalizes over formulas, not over properties. — TonesInDeepFreeze

    That is the same.

    A formula that takes a sentence as argument is a property of that sentence.
  • Mathematical truth is not orderly but highly chaotic
    For certain theories T, for every formula F(x) there is a sentence S such that T |- S <-> F(r(#S)). — TonesInDeepFreeze

    First, we replace F by ¬F. If F is a property then its negation is also a property. So, the following is an equivalent statement:

    For certain theories T, for every formula F(x) there is a sentence S such that T |- S <-> ¬F(r(#S)).

    Next, we replace S <-> ¬F(r(#S)) by the equivalent expression:

    (S ∧ ¬F(r(#S)) ∨ (¬S ∧ F(r(#S))

    Meaning:

    (S is true and F is false) or (S is false and F is true)

    Since ∨ is an "inclusive or", we can add "or both":

    (S is true and F is false) or (S is false and F is true) or both.

    So, it means:

    A true sentence that does not have the property, or a false sentence that has the property, or both.
  • Mathematical truth is not orderly but highly chaotic
    That doesn't mention PA. Rather, it a universal generalization over properties and sentences. — TonesInDeepFreeze

    I wrote that about the diagonal lemma, i.e. Carnap's theorem. Of course, there are conditions for when it applies. The context required, is PA or equivalent.

    Furthermore, the term P(S) is in and of itself ambiguous.

    It is a predicate that seemingly applies to a truth value. At first glance, it always means P(true) or P(false). It's as if it were a predicate with a boolean argument.

    The term P(S) only works if you do not distinguish between the source code of the sentence and its truth value. It requires judiciously swapping between both meanings and second-guessing what exactly S means: the source code of the expression or its truth value? Sometimes this and sometimes that.

    It yields expressions that are in fact not computable. No compiler would ever be able to compile that kind of things.
  • Mathematical truth is not orderly but highly chaotic
    No, I meant what I wrote, I showed you a property of sentences that every sentence has. — TonesInDeepFreeze

    The identity predicate in PA is:

    P(n) := n = n

    It cannot be implemented as:

    P(S) := S <-> S

    You are trying to do something that is not supported in PA.
  • Mathematical truth is not orderly but highly chaotic
    No, I meant what I wrote, I showed you a property of sentences that every sentence has.

    And what you wrote doesn't even make sense. # S is a number not a sentence.     — TonesInDeepFreeze

    In arithmetic theory, the argument n in P(n) must be a natural number. You cannot apply the predicate to S. You can only apply it to its Godel number.

    P(S) is simply not a predicate of PA.
  • Mathematical truth is not orderly but highly chaotic
    Counterexample: Let P be the property: P(S) if and only if S is equivalent with S. — TonesInDeepFreeze

    I guess you meant to write:

    Let P be the property: P(S) if and only if S is equivalent with P(#S).

    In that special case, P is actually Tarski's truth predicate, which is indeed not definable. The conclusion here is that truth is not a legitimate predicate.
  • Mathematical truth is not orderly but highly chaotic
    Where did Carnap write that? — TonesInDeepFreeze

    The diagonal lemma:

    https://en.m.wikipedia.org/wiki/Diagonal_lemma

    Rudolf Carnap (1934) was the first to prove the general self-referential lemma,[6] which says that for any formula F in a theory T satisfying certain conditions, there exists a formula sentence ψ such that ψ ↔ F(°#(ψ)) is provable in T.

    (By the way, there seems to be a mistake in the page: "formula" should be "sentence").

    Equivalently replace F by ¬ F:

    ψ ↔ ¬ F(°#(ψ))

    It is used in this negative variant in Gödel's proof and Tarski's proof.

    That is how you get:

    For any property of logic sentences, there always exists a true sentence that does not have it, or a false sentence that has it, or both

    With "formula" or well-formed formula replaced by "predicate" or "property".
  • Any objections to Peter Singer's article on the “child in the pond”?
    Any solid proof? And not just isolated cases, but a systemic critique: embezzlement... And what about other organizations? — LFranc

    Take for example Oxfam, known for exchanging sex for access to taxpayer-funded aid:

    https://www.npr.org/sections/goatsandsoda/2018/03/16/591191365/after-oxfams-sex-scandal-shocking-revelations-a-scramble-for-solutions

    During focus group discussions, some participants said aid workers would "make sexual advances on women and girls in exchange for goods or services necessary for survival." As a result, some women and girls said they would only go to distribution sites with a chaperone, the report states.

    https://www.theguardian.com/world/2018/jun/15/timeline-oxfam-sexual-exploitation-scandal-in-haiti

    Oxfam is accused of covering up an investigation into the hiring of sex workers for orgies by staff working in Haiti after the 2010 earthquake

    https://www.independent.co.uk/news/uk/home-news/oxfam-child-abuse-haiti-scandal-inquiry-sexual-exploitation-charity-commission-a8953566.html

    Oxfam failed to act on reports its workers were raping girls as young as 12, damning report concludes

    In one case, two emails dated 18 July 2011 and 20 August 2011 – both said to be from a 13-year-old Haitian girl – alleged she and a 12-year-old friend had suffered physical abuse and other misconduct at the hands of Oxfam staff.

    https://curriculum-press.co.uk/blog/the-oxfam-scandal

    - Oxfam allegedly covered up claims that senior staff in Haiti, working after the 2010 earthquake, engaged with prostitutes.
    - Some of the prostitutes involved may have been underage.
    - The director of operations in Haiti, Roland Van Hauwermeiren, supposedly used prostitutes at a villa provided by the charity.
    - There was a subsequent cover-up.

    Additional revelations surfaced, including allegations of bullying, harassment, and "colonial" behaviour within Oxfam.

    - The commission stated that the incidents in Haiti identified in 2011 were not isolated events
    - the use of prostitutes in Chad and 16 serious incidents involving volunteers under the age of 18 in some of Oxfam's UK shops

    Highly-paid expensive expat jobs just for westerners while the locals work for peanuts:

    https://www.theguardian.com/global-development/article/2024/may/07/colonial-mindset-global-aid-agencies-costs-localising-humanitarianism-ngo-

    ‘A colonial mindset’: why global aid agencies need to get out of the way

    A western aid worker in Addis Ababa, Ethiopia’s capital, for example, gets as much as $2,000 (£1,600) a month in addition to their salary, just to spend on housing. That money alone could pay the salaries of “four or five” local NGO workers, says Eyokia.

    the humanitarian aid system is “still characterised by a colonial mindset”

    “We have thousands of international NGOs running programmes, but what has really changed?” asks Gul.

    Fundraisers aggressively pestering elderly and other vulnerable demographics for donations:

    https://www.bbc.com/news/uk-40490936

    'Aggressive' charity fundraisers face fines

    Charities with "extremely aggressive" fundraising practices could be fined up to £25,000 if they do not crack down on nuisance calls, emails and letters.

    Fundraising Regulator chairman Lord Grade said "such terrible practices" could not be tolerated.

    Organisations must comply with new data protection legislation and provide marketing opt-outs from Thursday.

    'Not an isolated case'

    In 2015, the 92-year-old took her own life after receiving 466 mailings from 99 charities in a single year.

    The Fundraising Standards Board found that 70% of the charities who contacted Mrs Cooke had acquired her details from third parties.

    Oxfam staff striking in the UK over poor labor conditions:

    https://www.theguardian.com/world/2023/dec/10/oxfams-first-ever-strike-suspended-after-charity-offers-improved-pay-deal

    Oxfam’s first ever strike suspended after charity offers ‘improved pay deal’

    Hundreds of Oxfam workers began 17 days of strike action last Friday and Saturday, with Unite saying the strike of almost 500 workers would affect offices and 200 Oxfam shops.

    Unite claimed last month that average wages at Oxfam have fallen by 21% in real terms since 2018. This “poverty pay” meant some staff were using food banks or unable to afford to pay their rent.

    Oxfam spouts highly ideological propaganda and demands more taxes supposedly on just the rich:
    https://www.theguardian.com/commentisfree/2021/jan/26/the-worlds-10-richest-people-made-540bn-in-a-year-we-need-a-greed-tax

    The world's 10 richest people made $540bn in a year – we need a greed tax

    As for Bezos’s billions, Oxfam notes he could have paid all 876,000 Amazon employees a $105,000 bonus in September 2020 and remained just as wealthy as he was pre-pandemic.

    Every year, the NGO gets accused of exaggerating and manipulating statistics to stoke outrage. Every year, Oxfam calls for higher taxes on the wealthy and every year it is accused of being “obsessed with the rich”.

    https://www.theguardian.com/australia-news/2018/sep/18/pharmaceutical-companies-avoided-215m-a-year-in-australian-tax-oxfam-says

    Pharmaceutical companies avoiding $215m a year in Australian tax, Oxfam says

    “Oxfam objects to these practices but does not claim they are unlawful or liable to penalties.”

    “We are not accusing these pharmaceutical firms or their Australian subsidiaries of doing anything illegal,” Oxfam said.

    Oxfam wastes lots of money on inefficient and useless internal bureaucracy, whistleblowers complain:

    https://www.youtube.com/watch?v=5QbkGw4wm9I

    Oxfam 'wastes thousands of pounds' says former employee

    Oxfam frequently wastes hundreds of thousands of pounds-worth of money that's been donated by members of the public, a former employee has said.

    Oxfam is much more about promoting feminism and the LGBTQ ideology than about helping the poor:

    https://www.theguardian.com/world/2023/mar/31/is-oxfam-language-guide-taking-sides-in-the-culture-war

    there will always be people who get their knickers in a twist over their so-called pride in being white/British/cisgender/heterosexual/relatively wealthy/able bodied etc.

    https://www.theguardian.com/commentisfree/2023/mar/21/oxfam-poverty-culture-wars-inclusive-language-charities

    Perhaps not surprisingly, we were quickly accused of “wokery” of the worst kind, of wasting money, banning words and being ashamed of Britain’s heritage.

    The first complaint seemed to be that producing the guide shows Oxfam is wasting money, and instead we should just get on with fighting poverty.

    Talking about the importance of decolonising aid or about trans-inclusion may not feel popular

    I was perhaps most surprised by the strand of criticism that suggested pronouns don’t matter in the global south and that this obsession is a western creation. There are so many communities around the world in which notions of gender are more nuanced than simple binaries.
  • Mathematical truth is not orderly but highly chaotic
    Ok, so what's the interesting thing with having both addition and multiplication? — ssu

    That is a bit of a mystery. Any simplification to Robinson's arithmetic will make it complete: https://en.wikipedia.org/wiki/Robinson_arithmetic . It just turns out to be like that when you do it.
  • Mathematical truth is not orderly but highly chaotic
    So what's the thing with multiplication? — ssu

    Skolem Arithmetic only has multiplication (no addition) and is also complete. The problem occurs when you try to add both addition and multiplication.
  • Mathematical truth is not orderly but highly chaotic
    Notice in my exchange with Tarskian above, I was quickly led to ask what makes one theory "better" than another. Tarskian claimed the "perfect" model of an abstraction is one which is identical with the abstraction which it models. However, this is clearly incorrect if we consider what actually works in practise. In practise, what makes one specific model of an abstraction better than another is some principle of usefulness, and this is not at all a principle of similarity. — Metaphysician Undercover

    My initial interpretation of the term "better" was "more faithful", but indeed, this doesn't necessarily make an abstraction more useful. That does indeed depend on what you are going to use it for.
  • Mathematical truth is not orderly but highly chaotic
    Ordinarily I would not give it much thought, but this thread seems to focus on math truth beyond virtue of proof. You seem to know what that is all about. Can you provide a very simple definition of this sort of truth in math? — jgill

    Noson Yanofsky's paper, "True but unprovable", is about arithmetical truth, also called "true arithmetic":

    https://en.m.wikipedia.org/wiki/True_arithmetic

    In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers.This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms.

    The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation symbols, and a constant symbol for 0. The (well-formed) formulas of the language of first-order arithmetic are built up from these symbols together with the logical symbols in the usual manner of first-order logic.

    The structure is defined to be a model of Peano arithmetic as follows.

    - The domain of discourse is the set ℕ of natural numbers,
    - The symbol 0 is interpreted as the number 0,
    - The function symbols are interpreted as the usual arithmetical operations on ℕ,
    - The equality and less-than relation symbols are interpreted as the usual equality and order relation on ℕ.

    This structure is known as the standard model or intended interpretation of first-order arithmetic.

    A sentence in the language of first-order arithmetic is said to be true in if it is true in the structure just defined. The notation ⊨ φ is used to indicate that the sentence φ is true in .

    True arithmetic is defined to be the set of all sentences in the language of first-order arithmetic that are true in , written Th(). This set is, equivalently, the (complete) theory of the structure .

    I have used the term "mathematical truth" instead of "arithmetical truth" because alternative foundational theories of mathematics such as set theory have large fragments that are bi-interpretable with arithmetic and therefore have the same properties.

    Yanofsky points out that only a very small part of Th(), i.e. arithmetical truth, is provable. The remainder of Th() is unpredictable and chaotic. Most of Th() is even ineffable.
  • Any objections to Peter Singer's article on the “child in the pond”?
    There's a big difference between saving a child from drowning in a pond and giving money to people who say that they will be saving drowning children everywhere.

    The first thing is commendable while the second thing is naive, gullible, and fuels the lucrative business model of turning deception into real dollars.

    That is clearly what Peter Singer really wants:

    Oxfam, Against Malaria Foundation, Evidence Action, and many other organizations are working to reduce poverty ... If these organizations had more money, they could do even more, and more lives would be saved.

    If it could be a scam, then it is a scam.

    Murphy's law makes it impossible to outsource charity.

    The organizations that he mentions, are known to be professional scammers.

    Peter Singer is fuelling the online charity scam business model. He is just better at it than other con artists. For all I know, he might even be getting a commission for that.
  • A tough (but solvable) riddle.
    I've had a go at this one:

    $ ./doors-artists.js
    final number of complete solutions:1
    ----
    door_index hair_color nationality musical_style door_color profession

    1 black Brazilian classical white oil_painter
    2 grey Indian jazz pink sculptor
    3 blonde Australian hiphop teal dig_painter
    4 brunette Canadian reggae orange waterc_painter
    5 red Kenyan elec_dance purple photographer

    For the aficionados, the javascript program:

    #!/usr/bin/env qjs

//The following is what the French would call "le référentiel"

var properties={
    "hair_color":["black","brunette","grey","red","blonde"],
    "nationality":["Indian","Brazilian","Canadian","Australian","Kenyan"],
    "musical_style":["classical","elec_dance","jazz","reggae","hiphop"],
    "door_color":["teal","pink","purple","orange","white"],
    "profession":["photographer","sculptor","oil_painter",
                    "dig_painter","waterc_painter"]
};

//constraints_type_1: 
//It specifies that a particular property value must always
//coexist with another property value

var constraints_type_1 = [
    {"cix":1,"property_needle_1":"hair_color","value_needle_1":"black",
        "property_needle_2":"musical_style","value_needle_2":"classical"},
    {"cix":2,"property_needle_1":"musical_style","value_needle_1":"elec_dance",
        "property_needle_2":"profession","value_needle_2":"photographer"},
    {"cix":4,"property_needle_1":"door_index","value_needle_1":3,
        "property_needle_2":"door_color","value_needle_2":"teal"},
    {"cix":5,"property_needle_1":"profession","value_needle_1":"sculptor",
        "property_needle_2":"door_color","value_needle_2":"pink"},
    {"cix":9,"property_needle_1":"hair_color","value_needle_1":"red",
        "property_needle_2":"door_color","value_needle_2":"purple"},
    {"cix":10,"property_needle_1":"profession","value_needle_1":"dig_painter",
        "property_needle_2":"hair_color","value_needle_2":"blonde"},
    {"cix":11,"property_needle_1":"profession","value_needle_1":"waterc_painter",
        "property_needle_2":"nationality","value_needle_2":"Canadian"},
    {"cix":12,"property_needle_1":"profession","value_needle_1":"oil_painter",
        "property_needle_2":"door_index","value_needle_2":1},
    {"cix":13,"property_needle_1":"door_color","value_needle_1":"orange",
        "property_needle_2":"musical_style","value_needle_2":"reggae"},
    {"cix":14,"property_needle_1":"nationality","value_needle_1":"Australian",
        "property_needle_2":"musical_style","value_needle_2":"hiphop"}
];

//check constraints of type 1

function isValidForType1(solution) {
    for(let assignment of solution) {
        for(let constraint of constraints_type_1) {
            let property_needle_1=constraint["property_needle_1"];
            let value_needle_1=constraint["value_needle_1"];
            let property_needle_2=constraint["property_needle_2"];
            let value_needle_2=constraint["value_needle_2"];
            //the assignment must have both properties assigned
            //for a constraint violation to even be possible
            if(!assignment.hasOwnProperty(property_needle_1)) continue;
            if(!assignment.hasOwnProperty(property_needle_2)) continue;
            //check for: value1 correct but value2 wrong
            if(assignment[property_needle_1]==value_needle_1 && 
                    assignment[property_needle_2]!==value_needle_2)
                return false;
            //check for: value2 correct but value1 wrong
            if(assignment[property_needle_2]==value_needle_2 && 
                    assignment[property_needle_1]!==value_needle_1)
                return false;
        }       
    }    
    return true;
}

//constraints_type_2
//It specifies that a particular property value must always coexist
//with a property value of a neigbor

var constraints_type_2 = [
    {"cix":3,"property_needle":"nationality","value_needle":"Indian",
            "property_neighbor":"musical_style","value_neighbor":"classical"},
    {"cix":7,"property_needle":"nationality","value_needle":"Brazilian",
            "property_neighbor":"musical_style","value_neighbor":"jazz"},
    {"cix":8,"property_needle":"hair_color","value_needle":"grey",
            "property_neighbor":"profession","value_neighbor":"oil_painter"}
];

// utility function: find neighbor of assignment by door index

function findNeighborByDoorIndex(solution,door_index) {
    for(let assignment of solution) {
        if(assignment["door_index"]==door_index)
            return assignment;
    }
    //not found
    //This should never happen. Maybe throw an exception of sorts?
    return null;
}

//check constraints of type 2

function isValidForType2(solution) {
    for(let constraint of constraints_type_2) {
        let property_needle=constraint["property_needle"];
        let value_needle=constraint["value_needle"];
        let property_neighbor=constraint["property_neighbor"];
        let value_neighbor=constraint["value_neighbor"];
        for(let assignment of solution) {
            //the assignment must have the property assigned
            if(!assignment.hasOwnProperty(property_needle)) continue;
            //the assignment must have the value assigned to the property
            if(assignment[property_needle]!==value_needle) continue;
            //retrieve door_index from assignment
            let door_index=assignment["door_index"];
            //we assume that we will not find a valid neighbor
            let foundValidNeighbor=false;
            //check left neighbor, if applicable
            if(door_index>1) {
                let assignmentLeftNeighbor=findNeighborByDoorIndex(solution,door_index-1);
                if(!assignmentLeftNeighbor.hasOwnProperty(property_neighbor)) continue;
                if(assignmentLeftNeighbor[property_neighbor]==value_neighbor) {
                    foundValidNeighbor=true;
                }
            }
            //check right neighbor, if applicable
            if(door_index<5) {
                let assignmentRightNeighbor=findNeighborByDoorIndex(solution,door_index+1);
                if(!assignmentRightNeighbor.hasOwnProperty(property_neighbor)) continue;
                if(assignmentRightNeighbor[property_neighbor]==value_neighbor) {
                    foundValidNeighbor=true;
                }
            }
            if(!foundValidNeighbor) {
                return false;
            }
        }        
    }
    return true;
}

//check constraints of type 3
//cix=6, Special case.
//this function only checks:
//"The red head is the right-hand neighbor of the Brunette."

function isValidForType3(solution) {
    for(let assignment of solution) {
        //the assignment must have the property assigned
        if(!assignment.hasOwnProperty("hair_color")) continue;
        //the assignment must have the value assigned to the property
        if(assignment["hair_color"]!=="brunette") continue;
        //we found the brunette
        //check that there is a right-hand neighbor
        var door_index=assignment["door_index"];
        //Brunette cannot be assigned to door 5
        if(door_index==5) return false;
        //find right-hand neighbor
        var assignmentRightNeighbor=findNeighborByDoorIndex(solution,door_index+1);
        //The right-hand neighbor must be the red head
        if(assignmentRightNeighbor["hair_color"]!=="red") return false;
    }
    return true;
}

//validate solution

function isValid(solution)
{
    let validForType1=isValidForType1(solution);
    if(!validForType1) return false;
    let validForType2=isValidForType2(solution);
    if(!validForType2) return false;
    let validForType3=isValidForType3(solution);
    if(!validForType3) return false;
    return true;
}

//permutator

const permutator = (inputArr) => {
  let result = [];

  const permute = (arr, m = []) => {
    if (arr.length === 0) {
      result.push(m)
    } else {
      for (let i = 0; i < arr.length; i++) {
        let curr = arr.slice();
        let next = curr.splice(i, 1);
        permute(curr.slice(), m.concat(next))
     }
   }
 }
 permute(inputArr)
 return result;
}

//for debugging purposes

function output(label,structure) {
    console.log(label+":"+JSON.stringify(structure));
}

//initial solution space

var solutionSpace=[
    [{"door_index":1},{"door_index":2},{"door_index":3},
               {"door_index":4},{"door_index":5}]
];

//iterate over the properties
for(let property of Object.keys(properties)) {
    let propertyValues=properties[property];
    let permutations=permutator(propertyValues);
    let newSolutionSpace=[];
    //iterate over the permutations of the property values
    for(let permutation of permutations) {
        //cartesian multiplication of existing solutions with
        //new permutations
        for(let solution of solutionSpace) {    
            let newSolution=[];
            for (let i = 0; i < solution.length; i++) {
                let assignment=solution[i];
                let newAssignment={...assignment};
                newAssignment[property]=permutation[i];
                newSolution.push(newAssignment);
            }
            //verify if the solution satisfies all constraints
            if(isValid(newSolution)){
                newSolutionSpace.push(newSolution);
            }
        }
    }
    //the new solution space now replaces the existing one
    solutionSpace=newSolutionSpace;
}

console.log("final number of complete solutions:"+solutionSpace.length);

//output the solutions

function pad(str){
    let pad=Array(15).join(' ');
    return (str + pad).substring(0, pad.length);
}
console.log("----");
let headerPrintedAlready=false;
for(let solution of solutionSpace) {
    for(let assignment of solution) {
        let line="";
        for(let key of Object.keys(assignment)) {
            line=line+pad(assignment[key]);
        }
        if(!headerPrintedAlready) {
            let header="";
            for(let key of Object.keys(assignment)) {
                header=header+pad(key);
            }
            console.log(header);
            console.log("");
            headerPrintedAlready=true;
        }
        console.log(line);
    }
    console.log("----");
}
  • Mathematical truth is not orderly but highly chaotic
    Hence Wheeler’s conjecture of the One Electron Universe — Wayfarer

    That was an interesting phone call:

    proposed by theoretical physicist John Wheeler in a telephone call to Richard Feynman in the spring of 1940.

    It still generates a flurry of new articles in 2024!

    But these articles don't particularly say anything new. They do not seem to make much progress in investigating the matter.
  • Mathematical truth is not orderly but highly chaotic
    This place is not real. — Lionino

    The existence of the multiverse is a Pythagorean belief.

    It is not possible to prove that the physical universe is part of a multiverse, simply because it is not possible to prove anything at all about the physical universe.
  • Mathematical truth is not orderly but highly chaotic
    It's structurally similar because what constitutes "a group" is artificial, just like what constitutes "a set" is artificial. So you are just comparing two human compositions, the conception of a group and the conception of a set.. — Metaphysician Undercover

    The notion of group may indeed be an abstraction, a way of perceiving things, but there are still five people, which are physically there.

    How would your proposed computer simulation provide a "better" replica of the universe? — Metaphysician Undercover

    Fewer differences.

    You could measure a random sample of these differences, add up their squares, take the root, and rank replicas according to their sampled "deviation" from the original. This is actually done very routinely. It is even standard practice.

    what would make an abstract world the perfect abstract world? Do you see what I mean? — Metaphysician Undercover

    A perfect map of an abstract world is the abstract world itself. Perfect means "isomorphic" in this case.

    According to the structuralist ontology, an abstraction consists of only structure. An abstraction is structure, turtles all the way down.

    Hence, an isomorphic mapping of a structure is equivalent to the structure itself:

    https://en.m.wikipedia.org/wiki/Isomorphism

    The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism.

    Since two isomorphic abstractions have the same properties, they are essentially identical:

    https://en.m.wikipedia.org/wiki/Law_of_identity

    Wilhelm Wundt credits Gottfried Leibniz with the symbolic formulation, "A is A".[4] Leibniz's Law is a similar principle, that if two objects have all the same properties, they are in fact one and the same.

    Two abstraction are not truly identical. They are identical up to isomorphism.

    For example, the symbols "5" and "five" are identical up to simple translation (which is in this case an isomorphism). Two maps can also be isomorphic. In that case, they are "essentially" identical.

    The number "5" is an abstraction and is therefore not truly unique. It has numerous isomorphisms, such as "2+3" or "10/2" that represent essentially the same abstraction.

    Abstraction are never truly unique.
  • Mathematical truth is not orderly but highly chaotic
    If we don't differentiate between objects sensed and ideas grasped by the intellect. then there is nothing to prevent us from believing that the universe is composed of numbers. This is known as Pythagorean idealism, and often called Platonism. — Metaphysician Undercover

    I do not believe that the universe is composed of numbers.

    What I believe is limited to the idea that the arithmetical multiverse is structurally similar to the physical multiverse.

    For example, if there are five people in a group, this situation is structurally similar to a set with five numbers. It does not mean that a person would be a number.

    You could conceivably make a digital simulation of the entire universe and run it on a computer. This simulation of the universe would consist of just numbers. What you would see on the screen will be an exact replica of what you would see in the physical world. It would still not mean that this collection of numbers would be the universe itself.

    Pythagorean idealism is actually a widespread fallacy:

    https://en.m.wikipedia.org/wiki/Map%E2%80%93territory_relation

    The map–territory relation is the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Mistaking the map for the territory is a logical fallacy that occurs when someone confuses the semantics of a term with what it represents. Polish-American scientist and philosopher Alfred Korzybski remarked that "the map is not the territory" and that "the word is not the thing", encapsulating his view that an abstraction derived from something, or a reaction to it, is not the thing itself. Korzybski held that many people do confuse maps with territories, that is, confuse conceptual models of reality with reality itself.

    A map of the world can help us understand the world. The map will, however, never be the world itself.

    Now, if it is about an abstract world, then the perfect map of such abstract world is indeed the abstract world itself. There is no difference between a perfect simulation of an abstract world and the abstract world itself.

    That is why an abstraction cannot be truly unique. An abstraction can only unique up to isomorphism.

    Physical objects, on the other hand, can be truly unique in this physical universe (but almost never in the physical multiverse).

    Not everything that Pythagoras said was necessarily correct. The same for Plato. The same for Aristotle. It is just that they have managed to also say things that are amazingly insightful.
  • Mathematical truth is not orderly but highly chaotic
    Except it doesn’t allow for the iunreasonable effectiveness of mathematics in the natural sciences. — Wayfarer

    Viewing mathematics as just string manipulation highlights a different aspect of the same thing. The same holds true for structuralism. You can see mathematics as mostly templates with template variables. There are circumstances in which an alternative ontological view is actually the most inspiring one.

    As I said above, the reason the most people won’t defend platonism is because they don’t understand or can’t live with the metaphysical commitment it entails. Myself, I have no such difficulty. — Wayfarer

    I intuitively believe that arithmetical truth and physical truth are structurally similar. This explains why it is unreasonably effective in a physical context. For exactly the same reason, it should also be unreasonably effective in a metaphysical context.

    I fully endorse Pythagoras' view on the matter:

    https://en.wikipedia.org/wiki/Pythagoreanism

    Pythagoras, in his teachings focused on the significance of numerology, he believed that numbers themselves explained the true nature of the Universe. Numbers were in the Greek world of Pythagoras' days natural numbers – that is positive integers (there was no zero).

    In modern lingo, arithmetical theory, i.e. the theory of the natural numbers (PA), and the unknown theory of the physical universe exhibit important model-theoretical similarities.

    For example, the arithmetical universe is part of a multiverse. I am convinced that the physical universe is also part of a multiverse.

    The metaphysics of the physical universe is in my opinion nothing else than its model theory.

    Model theory pushes you into a very Platonic mode of looking at things. In my opinion, it is not even possible to understand model theory without Platonically interpreting what it says.
  • Mathematical truth is not orderly but highly chaotic
    Any observations on the arguments for or against mathematical platonism as outlined in this post? — Wayfarer

    I subscribe to the following take on Platonism:

    https://en.m.wikipedia.org/wiki/Philosophy_of_mathematics

    Davis and Hersh have suggested in their 1999 book The Mathematical Experience that most mathematicians act as though they are Platonists, even though, if pressed to defend the position carefully, they may retreat to formalism.

    In my opinion, you cannot actively do mathematics if you do not believe that its objects are real while you are doing it.

    Godel also thought that talent for Platonism is a prerequisite for being successful at mathematics:

    Kurt Gödel's Platonism postulates a special kind of mathematical intuition that lets us perceive mathematical objects directly.

    It is, however, mentally very easy to switch to formalism.

    You can simply switch off the lights and declare that it is all just meaningless symbol manipulation and about nothing at all, which it actually is, if you take the time to think about it.
  • Mathematical truth is not orderly but highly chaotic
    So you don't accept that 7=7? — Wayfarer

    In that regard, Victoria Gitman writes the following alarming statement:

    https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html

    In particular, a nonstandard model of arithmetic can have indiscernible numbers that share all the same properties.

    Even though the law of identity is certainly applicable in the standard model of the natural numbers, it may fall apart in nonstandard models of arithmetic.

    So, ω+7 ¬= ω+7 may be true in a nonstandard context, with ω the infinite ordinal representing the order type of the standard natural numbers. If it is false in any other nonstandard context, then this statement is even true but unprovable. I am not sure if this can be the case.

    Victoria Gitman points to the following publication for a more elaborate explanation on what's going on:

    R. Kossak and J. H. Schmerl, The structure of models of Peano arithmetic, vol. 50.

    Unfortunately, the publication is not available online. It can be ordered in paper-based format for $180 from Oxford University Press:

    https://global.oup.com/academic/product/the-structure-of-models-of-peano-arithmetic-9780198568278?cc=us&lang=en

    So, we already had ineffable numbers. Now we also have indiscernible ones. What other monstrosities are they going to discover in the melted plutonium core of Chernobyl reactor number four?
  • Mathematical truth is not orderly but highly chaotic
    Besides, formalism is not an ontology of mathematics, it is an approach to foundations. — Lionino

    Apparently, other people call formalism also an ontology:

    https://tomrocksmaths.com/2023/10/20/an-introduction-to-maths-and-philosophy-platonism-formalism-and-intuitionism/

    Mathematical Formalism is a theory for the ontology of mathematics according to which mathematics is a sort of game of symbols and rules, where new theorems are nothing more than new configurations of said symbols by said rules.

    Platonism and intuitionism are in his opinion the other main ontologies:

    Broadly speaking, Mathematical Platonism (deriving from Plato’s broader theory of ‘forms’) is an ontology of mathematics according to which mathematical objects are abstract, timeless entities existing objectively independent of the circumstances of the physical universe in a separate, abstract realm.

    Another crucial tenet of Intuitionist Ontology is a recognition of the temporal nature of our progression of mathematical knowledge over time.

    So these are the three big ontologies of mathematics – most other positions, like Empiricism, Psychologism, or Logicism can be more or less categorized as combinations and variants of the primary three.
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