Comments

  • Can the existence of God be proved?
    it took other people to fix the inconsistency in his proof just to then generate further issues in these updated proofs. — Lionino

    Modal collapse is not an inconsistency. Who told you that?

    It just means that the proof reverts to standard non-modal logic.

    Since non-modal logic is the default logic anyway, does that mean that pretty much all proofs in mathematics are inconsistent?

    In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily".

    Since standard logic does not even distinguish between necessary and contingent truth, what is supposedly the big problem?

    Furthermore, Anderson has fixed the issue and removed the modal collapse. This is not essential at all. It is just nice to have and not more than that.

    In fact, it may even be a good thing. It means that the proof works, even without using modal modifiers. So, the proof would be valid, even in plain, standard logic.
  • Mathematical truth is not orderly but highly chaotic
    And everytime when someone makes an universal statement that ought to apply to everything, watch out! — ssu

    That may very well be in violation of Carnap's diagonal lemma:

    "For each property of logic sentences, there exists a true sentence that does not have it, or a false sentence that does."

    But then again, it still needs to be a property of logic sentences. For example, a property of natural numbers can apply to all natural numbers.
  • Mathematical truth is not orderly but highly chaotic
    I think you’re mis-using the word there. If everything were chaotic, nothing would exist, and if everything were perfectly ordered, nothing would change. Existence requires both. Beyond that, I can’t see the point, if there is one. — Wayfarer

    People seem to understand this about the truth in the physical universe. They tend to reject this about the truth in arithmetic. I wanted to point out that the situation is the same.
  • Mathematical truth is not orderly but highly chaotic
    Hippasus, who found irrational numbers was ostracized and when he drowned at sea, it was the "punishment of the Gods". — ssu

    Yes, these people want a kind of certainty that simply does not exist ...
  • Can the existence of God be proved?
    The existential claim carries the onus probandi — above

    Gödel did exactly that. He provided a mathematically unobjectionable proof. Of course, math never does more than advertised. The witness for the existential theorem has successfully been supplied. Next.
  • Can the existence of God be proved?
    Out of interest, what type of believer are you? Muslim or Christian, or something less specific? — Tom Storm

    Originally born a Catholic. In the meanwhile, I came to the conclusions that Christians no longer intend to use the rules in the scripture as a benchmark to assess societal sanity. So, my sympathies are definitely much more Muslim nowadays. So, the problem is not necessarily Christianity but the lack of enthusiasm of the Christians. But then again, they completely mishandled the reformation too. The following was clearly not the solution either:

    Charles V's "Edict of Blood" of 1550 in the Burgundian Netherlands

    No one shall print, write, copy, keep, conceal, sell, buy or give in churches, streets, or other places, any book or writing made by Martin Luther, John Oecolampadius, Huldrych Zwingli, Martin Bucer, John Calvin, or other heretics reprobated by the Holy Church.
    ...
    That such perturbators of the general quiet are to be executed, to wit: the men with the sword and the women to be buried alive, if they do not persist in their errors; if they do persist in them, then they are to be executed with fire; all their property in both cases being confiscated to the crown.

    This approach failed in the Burgundian Netherlands but it actually succeeded in France. After successfully eradicating the reformation in France, the Catholic Church probably thought that they were good to go, only to later on end up with the French revolutionaries who did not even try to reform the religion but got rid of it altogether. Forcing everybody to join your club is clearly not a good idea.
  • Mathematical truth is not orderly but highly chaotic
    Perhaps some can see this as chaotic, but math itself is quite logical and hence quite orderly. Unprovability or uncomputability doesn't mean chaotic. Math is orderly, we just have limitations on what to compute or prove. — ssu

    I started using unpredictability as somewhat a synonym for unprovability because of how Stephen Hawking put it:

    https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

    So, we are sitting on a system that is largely unpredictable because most of its truths are unprovable. A system that is largely unpredictable is deemed chaotic:

    Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]

    This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos.

    The only minor difference between the universe of arithmetical truth and a chaotic system is that there are no "initial conditions" that we can change in order to produce a completely different version of arithmetical truth.

    Even if it obviously c is a natural number and has a precise point on the number line, not some range, we cannot prove c exactly. — ssu

    Imagine that we somehow have the information that c=17. Without additional information, it is not possible to prove it. In that sense, c is true but not provable. It could even be impossible to prove. In the standard model of arithmetic, i.e. in the natural numbers, we can somehow see that c=17 but in various nonstandard models, we can see that c is not 17. In those circumstances, proof is not even possible. Only when c=17 in all models of arithmetic, a proof is possible.

    The problem rises because we just assume that everything in math has to be provable. — ssu

    Yes, David Hilbert even wanted proof for that. In his view, every true statement must have a proof:

    https://en.wikipedia.org/wiki/Hilbert%27s_program

    Statement of Hilbert's program

    - A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.

    - Completeness: a proof that all true mathematical statements can be proved in the formalism.
    - Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
    - Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
    - Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

    Hilbert believed it so strongly that he insisted that all his colleagues should work on proving the above. A lot of people still believe it. You can give them proof that it is absolutely impossible, but they simply don't care about that. They will just keep going as if nothing happened. You can't wake a person who is pretending to be asleep.
  • Can the existence of God be proved?
    The existential claim carries the onus probandi (generally, existential claims are verifiable and not falsifiable, universal claims are falsifiable and not verifiable), it's not for someone else to disprove. — jorndoe

    Since we are talking about proof, it is the mathematical view on the subject that matters. Everybody else should avoid using the term ¨proof¨. What they produce as justification, is at best "evidence". It is never proof.

    Existential proofs are much easier to produce than impossibility proofs. Gödel successfully produced one. It does require higher-order modal logic, but that is still trivially simple compared to what impossibility proofs typically rest on.

    If you want to prove an impossibility, you need to painstakingly discover and make use of a structural constraint that will successfully reject every possible witness. In absence of such structural constraint, you would need omniscience.

    There are impossibility proofs. For example, Abel-Ruffini theorem rests on the Galois correspondence as a structural constraint, while Fermat's last theorem rests on the modularity theorem. So, it is possible. There are impossibility proofs, but non-trivial ones typically took centuries to discover.

    Therefore, you probably understand now that impossibility is not the default in mathematics. On the contrary, it is the result of centuries of hard work. Gödel successfully did his work and produced an existential proof. Where can we see the commendable mathematical work produced by an atheist in which he supports his impossibility claim?

    By the way, atheists really need to prove that they are not making use of omniscience for their impossibility claim that an omniscient entity does not exist. This burden is on them and not on us.
  • Mathematical truth is not orderly but highly chaotic
    This is a far cry from the point that math can be difficult to put into words. The proof is in the very fact you're able to post online consistently for us to read your posts. That was all capable through math. — Philosophim

    All of this is the result of using just one direction ("soundness"):

    If it is provable, then it is always true.

    That is the only direction that we use in engineering. We never use the other direction:

    If it is true, then it is pretty much never provable. It is a rare exception, if it is.

    In math, we mostly don't even see these unpredictable truths. How would we? In the physical universe, we can definitely see the unpredictable chaos, but we mostly ignore it. Mathematical truth is as chaotic as the truth in the physical universe. In my opinion, there is not much difference. We typically just don't want to know about it.
  • Mathematical truth is not orderly but highly chaotic
    Nevertheless, and to all practical purposes, mathematics enables a very wide range of successful predictions, doesn’t it? The mathematical physics underlying the technology on which this conversation is being conducted provides a high degree of prediction and control, doesn’t it? Otherwise, it wouldn’t work. — Wayfarer

    There are two directions.

    If it is provable, then it is always true. (aka, soundness theorem) In this direction, everything is very orderly. That is the only direction that we really use. That is why works so well.

    If it is true, then it is almost surely not provable. In this direction, everything is very chaotic. We almost never use this direction. In fact, we cannot even see most of these random truths. So, why would we try to prove them?

    It took Gödel all kinds of acrobatics in metamathematics to discover that these unpredictable truths even exist.

    Before the publication of Gödel's paper in 1931, nobody even knew about these random truths. Most mathematicians were actually convinced that if it is true, then it is surely provable. Pretty much everybody on the planet was wrong about this before 1931. They were all deeply steeped in positivism. David Hilbert even asked for a formal proof of this glaring error. In fact, there are still a lot of people who believe this. Almost a century after its refutation, it is still a widespread misconception.
  • Can the existence of God be proved?
    For the indifferent or one who finds the question incoherent it is not a matter of truth value, and that is the point. So, Joshs "none of the above": seems most apt. — Janus

    That point of view is not a problem.

    Only a 'yes' or 'no' answer constitutes a real commitment.

    For 'yes' answer, you need to locate a constructive witness. This is possible. Gödel did exactly that. For 'no' answer, the default situation is that you generally need omniscience.

    In fact, impossibility proofs do exist. They are not completely impossible. However, they typically require discovering a structural constraint that could never be satisfied by any possible witness.

    A good example is the Abel-Ruffini theorem. There is no solution in radicals to general polynomial equations of degree five or higher. It took centuries to prove this because at first glance it requires omniscience. It required discovering the Galois correspondence as a structural constraint that any solution would violate. Fermat's last theorem is another good example. Without the modularity theorem, it would also require omniscience to prove this impossibility. It took over 350 years to pull off the proof.

    Where is the structural constraint that makes a "no" answer to the "Does God exist?" question viable without requiring omniscience? Proving an impossibility is substantially harder than locating a suitable witness for a theorem. That is why a proof for atheism is several orders of magnitude more unlikely than a proof for religion.
  • Mathematical truth is not orderly but highly chaotic
    I believe Chaitin made a similar point. He has a proof of Gödel's incompleteness theorems from algorithmic complexity theory. I believe he says that mathematical truth is essentially random. Things are true just because they are, not because of any deeper reason.

    This sounds related to what you're saying.     — fishfry

    Yes, Yanofsky's paper also mentions Chaitin's work:

    http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

    Gregory Chaitin described an innovative way of finding true but unprovable statements. He started by examining the complexity of the axioms of a logical system. He showed that there are certain statements that are much more complex than the axioms of the system. Such statements are true but cannot be proven by the axioms of the logical system. The following motto is sometimes used to explain this:

    “A fifty-pound logical system cannot prove a seventy-five-pound theorem.”

    In particular, basic arithmetic is a logical system that has a level of complexity and so there are certain types of statements that are true but too complex to be proven using basic arithmetic. The main point for our story is that within basic arithmetic we can always find more complicated statements of a certain type. Hence, there are infinitely many true but unprovable statements.

    Cristian Calude extended Chaitin’s findings. He demonstrated that provable statements are actually very rare within the space of all true statements. In a sense, he showed that in the space of all true statements, every provable true statement is surrounded by many unprovable true statements.

    This means that most (but not all) mathematical truth is essentially random.

    Yanofsky's paper mentions an even larger class of random mathematical truth: unprovable because ineffable ("inexpressable"). There is no way to prove truths that cannot even be expressed in language. Because in that case, how are you going to express the proof? That class of random truths is even larger than Chaitin's random truths.

    But then again, there exists a small class of true and provable statements.

    In fact, nature of mathematical facts is quite similar to the nature of facts in the physical universe. Mostly random but with a relatively small class of facts that is still predictable. Unlike what most people believe, math is not more orderly than the physical universe itself.
  • Mathematical truth is not orderly but highly chaotic
    In Godel, it appears consistency is assumed — tim wood

    No, Gödel does not assume consistency. In Gödel's theorems, consistency is exactly the question. In mathematics we implicitly assume consistency. In metamathematics, we don't.
  • Mathematical truth is not orderly but highly chaotic
    Why should we suppose that natural languages are only countably infinite? — Banno

    You can enumerate every sentence in natural language in a list. Therefore, it maps one to one onto the natural numbers. Therefore, their set is countable.

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

    In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.[a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

    For natural language to be uncountable, you must find a sentence that cannot be added to the list. To that effect, you would need some kind of second-order diagonal argument.
  • Mathematical truth is not orderly but highly chaotic
    Could you say a little more about what makes an unprovable mathematical proposition true? — Joshs

    The fact that we can prove that it exists.

    Let's start from Carnap's diagonal lemma. In the context of Peano arithmetic (PA), for each property φ(n) accepting one natural number n as input argument, there exists a true sentence S that does not have the property or a false sentence S that does have it:

    PA ⊢ ∀ φ ∃ S ( S ⇔ ¬φ(ÍSÎ ) )

    This is, in fact, the only hard part in Gödel's proof. The proof for the lemma is very short but it is widely considered to be incomprehensible:

    https://proofwiki.org/wiki/Diagonal_Lemma

    Say that Bew(ÍSÎ) is a property in PA that is true if it proves S and false when it doesn't. In that case, the lemma applies:

    PA ⊢ ∀ φ ∃ S ( S ⇔ ¬Bew(ÍSÎ ) )

    There exists a true sentence that is not provable or a false sentence that is provable. Hence, PA is incomplete or inconsistent. Let's denote this sentence as G:

    PA ⊢ G ⇔ ¬Bew(ÍGÎ ) )

    So, now we have a sentence that is (true or unprovable) or (false and provable). In fact, G is also a truly constructive witness for the theorem. But then again, we do not even need this particular sentence, because in the meanwhile, we also have Goodstein's theorem that is true but unprovable in PA:

    https://en.wikipedia.org/wiki/Goodstein%27s_theorem

    It is very hard to discover this kind of true but unprovable sentences. But then again, we also know that they massively outnumber the true and provable sentences. True but unprovable is the rule while true and provable is the exception. This is the paradoxical situation of the truth in PA. The truth in PA is highly chaotic but it is very hard for us to see that.
  • Mathematical truth is not orderly but highly chaotic
    He attributes to Godel this idea:
    :“'Basic arithmetic cannot prove a contradiction.'     — tim wood

    The paper actually says:

    As a bonus, Gödel described another interesting statement in the language of basic arithmetic. He was able to formulate a statement in basic arithmetic that says:

    “Basic arithmetic cannot prove a contradiction.”

    So, it only insists that the sentence "PA cannot prove a contradiction" can be expressed in PA itself.

    In the following paper, containing a version of the proof, the author expresses it by reifying the truth value for falsehood (⊥):

    http://sammelpunkt.philo.at/id/eprint/2676/1/Bagaria.pdf

    page 12:

    Let CON(T) be the sentence ¬BewT(Í⊥Î). Thus, CON(T) says, via coding, that T is consistent.

    In another paper, with another version of the proof, the author insist that it is enough to express the unprovability of any arbitrary falsehood. No need to reify truth values:

    http://www.sfu.ca/~kabanets/308/lectures/lec11.pdf

    We say that a proof system P is consistent if P does not prove both A and ¬A for some sentence A. That is, a consistent proof system cannot derive a contradiction A ∧ ¬A. In the case of a proof system P for arithmetic, we get that P is consistent iff P does not prove the sentence “1 = 2” (since 1 6 = 2 can be derived in P by the usual axioms (of Peano arithmetic) for the natural numbers).

    ConsP : “the sentence “1 = 2” is not provable in P ”

    So, in that case, let's consP be the sentence ¬BewP(Í1 = 2Î).

    But then again, it is also perfectly possible to express the notion of consistency in full -- straight from its definition -- that PA does not prove both A and ¬A for all sentences A of PA:

    PA ⊢ ∀ A ( ¬Bew(ÍA ∧ ¬AÎ ) )

    In all cases, regardless of how you express consistency of PA, the proof for the second incompleteness theorem always proceeds by considering the first incompleteness theorem:

    PA ⊢ ∃ A ( A ⇔ ¬Bew(ÍAÎ ) )

    The above means: There exists a sentence A that is (true and not provable) or (false and provable).

    Say that G is such sentence:

    PA ⊢ G ⇔ ¬Bew(ÍGÎ )

    If PA can prove its consistency, then it can obviously also prove that G is consistent:

    PA ⊢ ¬Bew(ÍG ∧ ¬GÎ )

    By using the Hilbert-Bernays rewrite rules -- with a few more steps -- we can then prove that this expression leads to the following contradiction about G:

    PA ⊢ ¬(G ⇔ ¬Bew(ÍGÎ ))

    There are many ways to formulate the consistency of PA, i.e. Cons(PA), but proving it will always lead to a contradiction. Therefore, Cons(PA) is unprovable. According to the first incompleteness theorem, the following sentence is true:

    Cons(PA) ∨ Incompl(PA)

    So, PA is inconsistent or incomplete. However, we do not know if Cons(PA) is true. We can only come to that conclusion by proving it, but how are we supposed to do that? So, I disagree with the author when he writes:

    It turns out that this statement is also true but unprovable.

    This statement is only unprovable.

    Of course, we can use Gentzen's equiconsistency proof with PRA but that does not prove PA's consistency. It just proves that it is equiconsistent with PRA (primitive recursive arithmetic). Who says that PRA is consistent? We don't know that. Other authors sometimes write that we can prove PA's consistency from within ZFC. Fine, but who says that ZFC is consistent?

    Hence, we can only assume PA's consistency. We cannot just state Cons(PA) to be true. This cannot be done.
  • Can the existence of God be proved?
    So you would have 'don't care' mapped to unknown? — Tom Storm

    Well, how many additional truth values do we need to invent before all our needs for additional truth values will have been completely satisfied?

    Seriously, it is a slippery slope. We are going to end up with more truth values than genders!
  • Can the existence of God be proved?
    perhaps due simply to a complete lack of interest — Janus

    If someone is not interested in the issue, fine, but then his answer should still get mapped to the truth value unknown/maybe.

    There is no need for an additional truth value to reflect this.

    Again, the answer unknown/maybe is perfectly fine. Unlike the answer "no", it does not reflect a problem of omniscience.
  • Can the existence of God be proved?
    Contrary to what the weekly sophist implies, choice of axioms is not arbitrary. — Lionino

    That is clearly a straw man. You are attacking an argument that I did not make. You are using Don Quichotte tactics. Who exactly is the sophist here?

    As previously stated, you have not read the article you yourself linked. Congrats. — Lionino

    That is classical non sequitur. Again some word-salad nonsense.

    Godel flawlessly proved the equiconsistency between his theorem and the axioms from which it follows. Godel's proof is therefore mathematically unobjectionable. Of course, Godel did not prove the axioms themselves. But then again, he is not even supposed to.

    Your arguments amount to just a bit of black mouthing and shit talking. That says much more about you than about Godel's work.
  • The essence of religion
    But religion is certainly not about this. It is about ethics. What is ethics? — Constance

    What are sound ethics? We take a snapshot of a sane society as well as an inventory of its rules. This is a suitable benchmark for ethical sanity. Benchmarking sane rules when it was still possible is what the scriptures implicitly do.

    That is why the scriptures had to be transmitted as soon as possible, in order to front run the degeneracy that would inevitably follow later on. That is also why it is no longer possible to transmit new scriptures. It would simply describe the depravity of our contemporary society and not be suitable as a benchmark. Prophetic times are over.

    Thanks to the scriptures we still know what we are supposed to be and how we are supposed to behave. It is a fantastic tool against the manipulative narrative of the ruling mafia. They handsomely benefit from growing depravity. We don't.
  • Can the existence of God be proved?
    Well,

    PA is a chaotic complex system without initial conditions. — Tarskian

    looks a bit... overstated.     — Banno

    I have just found an interesting paper that elaborates on why the overwhelming majority of true statements in arithmetic are unprovable -- and therefore unpredictable. In fact, most truth in PA is simply ineffable.

    http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

    We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

    Most mathematical truth is unpredictably chaotic.
  • Can the existence of God be proved?
    That the conclusions follow from the premises can be said about every fiction book — Lionino

    Well no. You need to be quite sure that the book is about a sane society. You cannot just invent one. It needs to have historically existed.

    If you had actually read the "article" you linked, you would know that Gödel's original axioms are inconsistent — Lionino

    They are not inconsistent. There may be an issue of modal collapse but Curtis Anderson proposed a fix for that. It is not a major problem.
  • Can the existence of God be proved?
    Aren't these the "initial conditions"...? These are the Peano axioms:
    Zero is a natural number.
    Every natural number has a successor in the natural numbers.
    Zero is not the successor of any natural number.
    If the successor of two natural numbers is the same, then the two original numbers are the same.
    If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
    It's far from obvious what this has to do with chaotic systems.

    I'm not following Tarskian's argument at all.     — Banno

    A chaotic system is one that follows a seemingly random path albeit deterministic. If you repeat the path with exactly the same initial conditions, it will follow exactly the same path.

    Example:

    Initial condition: "hello world"
    sha256 hash: b94d27b99...
    sha256 hash: 049da0526...
    and so on (you keep feeding the output as new input)

    If you change one letter to the initial seed, the path will change completely.

    This is a chaotic complex system. Its facts look random. If you don't know the initial seed, it is for all intents and purposes effectively random.

    Since most facts in arithmetic (PA), i.e. the arithmetical truth, are unprovable from the axioms, it has similar characteristics to the example system.

    However, there is no initial seed in PA. The chaos in PA is caused by another phenomenon. Provable statements in PA are not merely true in the model/universe of the natural numbers. They are also true in an unlimited number of nonstandard models/universes of arithmetic. Most of its true facts are, however, not true in all its models/universes. That would be a precondition for their provability/predictability. That is why most facts in arithmetic are not predictable/provable from theory.

    https://en.m.wikipedia.org/wiki/Non-standard_model_of_arithmetic

    In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

    This phenomenon explains why PA is incomplete (i.e. having unprovable truths) or inconsistent (i.e. provable falsehoods) or even possibly both.

    Hence, the nature of the majority of facts in arithmetic is chaotic, i.e. unpredictable (unprovable).
  • Can the existence of God be proved?
    might have been done by any number of fanatics (Castro, Hitler, Putin, whoever) — Tom Storm

    Hitler tried and failed.

    The effectiveness of math can be demonstrated through its consistency and predictability. — Tom Storm

    Concerning the consistency of any theory such as PA (Peano arithmetic theory), it is merely an assumption. Gödel's second incompleteness theorem proves that if a mathematical system is capable of proving its consistency, it is necessarily inconsistent.

    Therefore, the consistency of PA is based on faith alone.

    Of course, we use PA to maintain consistency in downstream applications, and it works surprisingly well, but it is certainly not a provable property of PA.

    Concerning the predictability of PA, whenever there exist true but unprovable theorems in a system, they massively outnumber the provable ones.

    Hence, PA is mostly unpredictable.

    According to Stephen Hawking, the unpredictability of the universe is tightly connected to the unpredictability of PA:

    https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

    PA is predictable in one direction, with provable implying true. However, when you look at the universe of true facts in PA, it is not predictable, because true rarely implies provable. PA is highly chaotic, albeit in a deterministic way.

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

    Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]

    PA is a chaotic complex system without initial conditions.

    even within the single religion. It is unpredictable and inconsistent.

    PA is also mostly unpredictable and its consistency is at best a statement of faith.
  • Can the existence of God be proved?
    What do you mean by the term "existence"? — 180 Proof

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

    Existential quantification

    In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)").
  • You build the machine, or you use the machine, because otherwise you are trying to be the machine
    You are unhappy with students being taught the state of the art in their field. — fishfry

    It is not the state of the art in their field.

    The president of the United States draws a paycheck. — fishfry

    Not because he wants to. There are so many people willing to pay a million dollars just for an 15-minute appointment with him, but he is not allowed to accept the money. He could easily put it up for auction. So, the paycheck is just part of a dog-and-pony show. Money does not matter to the people who print the money.

    All the 18 year olds are apprenticed out to people who will pay them even though they're completely ignorant? You can't be serious. What are you talking about? — fishfry

    That is how it used to work. They would become apprentices at the age of 14 and learn a job. This makes much more sense than keeping them in holding pens like cattle. There was no youth unemployment in past times.

    So maybe you're against large organizations. — fishfry

    Not necessarily.

    I have worked as a contractor and done lots of consulting work at large organizations.

    I would never have wanted to be staff, though. When we talk about "bottom line", the only one that mattered to me was my own "bottom line". I was not interested in selflessly "sacrifice" myself for someone else's bottom line. I cannot identify with the profit of the company. I can only identify with my own profit. I understand that C-level execs somewhat care, since they receive payments for when the stock goes up, but the other salaried office drones? Seriously, why would anybody else care?
  • Can the existence of God be proved?
    We don’t prove existence ... We might be able to prove a god wouldn’t struggle, or a god wouldn’t need sleep, but we can’t prove that struggle-free, always awake god exists. — Fire Ologist

    https://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof

    Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God.

    The proof[8][10] uses modal logic, which distinguishes between necessary truths and contingent truths.

    Gödel's proof is considered mathematically unobjectionable. That is why the only mathematical criticism is that it merely proves equiconsistency between the theorem and its axioms:

    Most criticism of Gödel's proof is aimed at its axioms.

    Furthermore, we can certainly prove existence or non-existence.

    Existence. Kakutani's fixed-point theorem proves the existence of a fixed point. So does Brouwer's fixed-point theorem.

    Non-existence. Abel-Ruffini theorem proves the non-existence of a solution in radicals for quintic polynomials or higher degrees. Fermat's Last Theorem proves the non-existence of particular three-tuples of natural numbers.
  • Can the existence of God be proved?
    the difference between ~b(G) and b(~G) — 180 Proof

    It depends on b() whether they are different or the same. It is a similar situation as whether a ⊕b is equal to b ⊕ a. It depends on the properties of ⊕.

    In this case, they are clearly the same. The expression: 
    not belief("God exists")
    is equivalent to 
    belief("God does not exist")
    . There is no difference.

    True, False, Not-True — 180 Proof

    Ok, Heyting logic does indeed work like that, with (true,false,not-true) truth values, while the Kleene, Priest, and Łukasiewicz logics stick to (true,false,uinknown). After having spent decades fiddling with SQL and years fiddling with javascript, I subconsciously tend to revert to (true,false,null). I wonder if Heyting logic is even implemented anywhere? Is there a programming language that uses it?
  • Can the existence of God be proved?
    An atheistic can quite coherently say they see no reason to believe in a god, and yet that they cannot rule it out. — Janus

    In terms of logic, we have: yes, no, maybe. The view you describe is a maybe. In my opinion, that is perfectly fine.
  • Can the existence of God be proved?
    Math axioms can be shown to work. — Tom Storm

    No, they can't. There is no justification for axioms. If an axiom can be justified, it is not a legitimate axiom.

    Religion cannot demonstrate gods. — Tom Storm

    Math cannot demonstrate its axioms either.

    I would call that evidence of religion's disfunction. — Tom Storm

    No, it is its stated goal. The goal of religion is not what you would want it to be. That is wishful thinking.

    Or do you think the supposed truths held by Marxists — Tom Storm

    Marxism has collapsed. Some religions are unsustainable. Nobody urges you to choose one of those.

    We know that people can be galvanized by deception and undemonstrated beliefs. — Tom Storm

    Undemonstrated beliefs are the foundation of all knowledge. That is exactly what Aristotle pointed out in Posterior Analytics. If nothing is assumed then nothing can be concluded. Furthermore, the ability to galvanize is exceptionally meritorious. Motivating people is not easy. It is no small feat if you manage to do it. It is also the biggest failure of any manager, if he cannot galvanize his people. Humanity is a social species, which means that leadership is a pressing requirement. If you can galvanize other people, then you can achieve greater things. If you are critical about that, you simply do not understand how human organizations work.
  • Can the existence of God be proved?
    If the difference between faith and reason isn't obvious to people — ssu

    Knowledge is fundamentally foundationalist:

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

    Identifying the alternatives as either circular reasoning or infinite regress, and thus exhibiting the regress problem, Aristotle made foundationalism his own clear choice, positing basic beliefs underpinning others.

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

    Beliefs therefore fall into two categories:

    - Beliefs that are properly basic, in that they do not depend upon justification of other beliefs, but on something outside the realm of belief (a "non-doxastic justification").

    - Beliefs that derive from one or more basic beliefs, and therefore depend on the basic beliefs for their validity.

    Without basic beliefs, reason is not possible.

    Therefore, there is no such sharp distinction between reason and faith. Reason allows us to reach derived beliefs. However, their ultimate justification can only be found in properly basic beliefs
  • Can the existence of God be proved?
    We accept science and math because they work — Tom Storm

    I accept religion, also because it works.

    We take a snapshot of a presumably ane society along with its rules and call that our scripture. Now we have a benchmark to compare our own society to, as well as where it is heading. Next, we threaten the government to stop overruling the laws of nature and of a sane society, and make it cave in.

    What is there about religion that does not work? In my opinion, the tool is perfectly suitable for purpose.

    My point is that math demonstrates its utility — Tom Storm

    Religion also demonstrates its utility. The government fears us more than the result of its elections. So, the tool achieves its goal.

    You see, when the Taliban unceremoniously deported NATO from Kabul airport, they achieved something that nobody else was able to do. Or do you think that you can do that too?
  • Can the existence of God be proved?
    We still can't demonstrate that there are any gods. We can demonstrate that math works. We seem unable to get past this point. — Tom Storm

    You need to compare apples to apples:

    - We can demonstrate that math works.
    - We can demonstrate that religion works.

    That is the fair comparison. Or even:

    - We still can't demonstrate that there are any gods.
    - We still can't demonstrate that Peano's successor function exists as mentioned in the axioms of arithmetic.

    That is another fair comparison.

    What you are doing, is comparing apples to oranges.
  • Can the existence of God be proved?
    He hasn't. Read the Reddit article you yourself linked. — Lionino

    You misunderstand what the Wikipedia page on the matter says. Godel has perfectly demonstrated the equiconsistency between his theorem and the axioms that he used. What else does any proof do, if not exactly that?
  • Can the existence of God be proved?
    Yes, it can be but that formulation is not popular – though it's formerly my preferred position (while quite reasonable, it's too narrow in scope): — 180 Proof

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

    Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there are no deities.

    In my opinion, the difference between "absence of belief" and "disbelief" is just language engineering.

    It implies that the position could also be indeterminate. However, we already have a term for that position, i.e. agnosticism.

    Why would there be a need to create that ambiguous overlap between atheism and agnosticism? It merely mixes up the underlying truth values. A logic sentence is true, false, or indeterminate. Why deviate from standard logic. To what benefit?
  • Can the existence of God be proved?
    I'm not a foundationalist. — Tom Storm

    In that case, you will need to reject mathematics as it is staunchly foundationalist, i.e. axiomatic. Since science is not viable without math, you will also need to reject science.

    Even animals use some basic arithmetic for reasons of survival. Hence, an anti-foundationalist animal cannot survive.

    Again, every living creature needs to have at least some faith in order to survive.

    But then again, with the birth rate collapsing, atheist populations are in the long run not surviving. Indeed, why would they? In the end, you still need some faith to believe that it would be meaningful to begin with. There is no compulsion in religion. Therefore, they are indeed at liberty to die out.

    As usual, the proof will be in the pudding. Atheism will disappear. Only religion will survive. That is how it has always been. Nothing new there.
  • Can the existence of God be proved?
    We do not say there is no god, that would be making a positive claim. — Tom Storm

    There are three possibilities concerning the belief in God: true, false, indeterminate. Religion believes it is true. Atheism believes that it is false. Agnosticism is indeterminate.

    Atheism is defined as a positive claim. It is agnosticism that refuses to make a claim. While agnosticism makes perfect sense, atheism doesn't.

    For many, atheism is about belief not knowledge. — Tom Storm

    If we look at the JTB account for knowledge, then knowledge is defined as a particular kind of belief:

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

    The JTB account holds that knowledge is equivalent to justified true belief; if all three conditions (justification, truth, and belief) are met of a given claim, then we have knowledge of that claim.

    There is no knowledge without belief. Furthermore, at the foundationalist core of knowledge you always find necessarily unjustifiable beliefs. Rejecting the foundation of unjustifiable beliefs amounts to rejecting the entire edifice of knowledge. If you can't have faith, you cannot know either.
  • Can the existence of God be proved?
    The quesion we are addressing is - is there good reason to belive in god the way there are good reasons to believe in math? — Tom Storm

    The reasons are similar. The belief in Peano's axioms allows you to use arithmetic theory and maintain consistency in downstream applications. The belief in religion creates a common understanding between billions of people that constitute a political counterweight to prevent governments from overruling the laws of nature. Different tools for different purposes.

    Religion all over the world behaves like a political party - theism being incidental to its machinations — Tom Storm

    Politics is unavoidable. The government is essentially a monopoly on violence. There needs to be a mechanism to suspend this monopoly when the government abuses it:

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

    The Mandate of Heaven (Chinese: 天命; pinyin: Tiānmìng; Wade–Giles: T'ien1-ming4; lit. 'Heaven's command') is a Chinese political ideology that was used in Ancient China and Imperial China to legitimize the rule of the king or emperor of China.[1] According to this doctrine, Heaven (天, Tian) bestows its mandate[a] on a virtuous ruler. This ruler, the Son of Heaven, was the supreme universal monarch, who ruled Tianxia (天下; "all under heaven", the world).[3] If a ruler was overthrown, this was interpreted as an indication that the ruler was unworthy and had lost the mandate.[4]

    The so-called democratic voting circus was advertised as being capable of achieving this but it has now become obvious that it has failed at doing so. We are now effectively in the long run of all the past short-termism.
  • Can the existence of God be proved?
    Dunning-Kruger is in full effect. — 180 Proof

    Dunning-Kruger is about people who think that they know but in fact they don't. Since atheism requires omniscience while faith in God does not, doesn't Dunning-Kruger rather describe atheists and not religious people?
  • Can the existence of God be proved?
    WTF are you talking about, kid? — 180 Proof

    Ha ha ah! You have just made my point!
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