If that's it, then you're not philosophizing, as I see it, just misusing (e.g. reifying) logic. I prefer to use reality instead of existence (just as I prefer mind to consciousness / mindbody to subject) because the latter tends to be less dynamic and less contingent than the former.

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.

Without basic beliefs, reason is not possible.

Therefore, there is no such sharp distinction between reason and faith. — Tarskian

Yet when you reason, you can change your beliefs. Naturally we do start from our premises, the things we assume to be true. But if by reasoning we come to the conclusion that our starting assumptions were wrong, we change them.

With something like faith, and to love something, that's not so.

Accepting a truth without evidence is faith. Therefore, an axiom represents faith. — Tarskian

Faith requires belief despite the evidence. Evidence is the Devil's doing.

As someone with actual training in philosophy, what do you make of arguments?

Well,

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

looks a bit... overstated.

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.

@TonesInDeepFreeze?

No, I have not and it doesn't. That the conclusions follow from the premises can be said about every fiction book — and yet we can't cast Avada Kedavras. If you had actually read the "article" you linked, you would know that Gödel's original axioms are inconsistent — the solution to that exists, but I will let you scurry for it instead of giving it for sophists to abuse.

Instead of reading through some web article anyone can edit and that no academic uses for research for its extremely poor quality, I have gone through the original papers for the proofs. There are positive, non-arbitrary, reasons why we may want to reject the axioms of the argument, even in its consistent form.

You abuse every source you can get your hands on to support the conclusion you started with before researching the arguments, like a politician would. If it is not some dumb ontological argument from the Middle Ages or some toddler-ish question like "Uh where does the big bang come from", the religious sophists will latch onto whatever they can find next.
In that sense, I suggest you go spill your drivel somewhere other rather than a philosophy forum, which is not a debate forum, but perhaps the other users will be eager to waste time on your sophistry.

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

Because there is no argument. It is crankery and abuse of philosophy/logic worse than PL's.

"You might very well think that; I couldn't possibly comment"

See also https://plato.stanford.edu/entries/ontological-arguments/#Gdel , and the concluding observations.

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).

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.

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.

Some thing that I noticed in the article is the trouble around "positive properties". A comment in K. Gödel, Appx.A: Notes in Kurt Gödel’s Hand, 144–145 says that "positive property" is to be interpreted in a moral-aesthetic sense only — which by itself is troublesome. Nonetheless, the argument may be rejected for other, better, reasons. Contrary to what the weekly sophist implies, choice of axioms is not arbitrary.

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. — Tarskian

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

In terms of logic, we have: yes, no, maybe. The view you describe is a maybe. In my opinion, that is perfectly fine. — Tarskian

What you are missing is the possibility that an atheist, having no disposition towards theism at all, may not take up any of the positions you characterize by "yes, no, maybe" that is they may not believe, disbelieve or suspend judgement in relation to the question, but simply give it no thought whatsoever, perhaps on account of not acknowledging it is as a question, thinking of it as incoherent or a non-question, or perhaps due simply to a complete lack of interest.

:up:

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.

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.

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

So you would have 'don't care' mapped to unknown?

So you would have 'don't care' mapped to unknown? — Tom Storm

Or ‘none of the above’.

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!

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.

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.

, you mentioned existential justification, yet I don't think a vague unknown is of much concern.

Rather, contenders include, say, the Vedic Shiva, the Avestan Ahura Mazda, the Biblical Yahweh, and a few others, where adherents/believers go by rituals, commands/rules, fate designations, speak of divine intervention/participation, etc. These are of concern to the various adherents/believers of course, and also to others due to proselytizers indoctrinators discrimination conquerors (concerted organized efforts), their political influences, and impact on societal affairs (other peoples' lives).

Shouldn't be difficult to find people with a laissez-faire (or "who knows") sort of attitude towards the former (vague unknown), and an attitude of disbelief towards whatever deities of the latter. It's a difference that makes a difference.

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. Upon repeated failure, expect disregard/dismissal of the claim (until further notice perhaps). Though not deductive, it's a rational, reasonable response just the same, happens all the time.

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.

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. — Tarskian

I’ve not met many atheists who would argue this. How would we know? Atheism is as botched and bungled as any religion in its range of strident and moderate advocates. I’ve met atheists who believe in ghosts, fairies and Bigfoot. Perhaps be a bit more cautious about your characterisation of atheists. I don’t consider all theists to be stupid rubes.

Out of interest, what type of believer are you? Muslim or Christian, or something less specific?

This OP title would have benefitted from a single-word response:


YES?

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.

Existential proofs are much easier to produce than impossibility proofs. — Tarskian

A model just requires a counter-example and it's out.
If existential proofs are the easier parts, then why do less than half the world's population believe that the Biblical Yahweh is real?

But ...

vague unknown [...] contenders [...] a difference that makes a difference

The existential claim carries the onus probandi [...] Upon repeated failure, expect disregard/dismissal of the claim — above

... was the main point. Ball's still in your court.

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.

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. — Tarskian

Thanks for the background. I thought as much. You're definitely interesting, even if we disagree about many things. I appreciate your generally good nature and politeness. Some folks get pretty abusive on here sometimes.