You mentioned Popper. He considered metaphysics to be important, but just not a science. He considered it be, although not itself a science, indispensable to science. This is because creative imaginative thought is indispensable to science just as much as it is to the arts. — Janus

I am certainly a fan of Popper's epistemic result, and I have completely adopted his seminal publication, "Science as falsification". That does not mean, however, that I would adopt everything that he has ever written.

The same holds true for Socrates, Plato, Aristotle, and Kant. Not everything that they have proposed, has turned out to be equally successful.

I consider the core axis of knowledge to be mathematics, science, and history. However, the reason for this, is neither mathematical, nor scientific, nor historical.

Mathematics deals with patterns that arise in carefully and explicitly constructed abstract, Platonic worlds. Science deals with patterns that arise in the real, physical world. History corroborates witness depositions.

Epistemology is meta-knowledge. It deals with patterns that arise in the abstract world of knowledge as this world of knowledge gradually arises and emerges, while focusing on its essential characteristics, which is, how the knowledge is justified.

Metaphysics, however, has nothing to do with figuring out knowledge justification. It does not revolve around justified (true) beliefs (JtB). It rather tries to rationally question the starting points of knowledge.

I consider that to be nonsensical in, for example, mathematics, because the starting points in mathematics are not the job of anybody else than mathematicians. Seriously, if you have no clue as to what the effect of changes in the starting points will be on the body of theorems that rest on it, then you are clearly doing it wrong.

The problem is clearly of a very similar nature in science and history.

Furthermore, if these starting points had a rational justification, then they would not be starting points. Therefore, the questioning of starting points is mostly in vain, even by people who are aware of the consequences of modifying the starting points. Therefore, I also consider metaphysics to be a futile and often ignorant exercise in infinite regress.

A good example of how all of that goes wrong is Collingwood's "An essay on metaphysics".

An example that Collingwood gives of this concerns the frameworks (he calls them “constellations”) provided by the Newtonian, the Kantian and the Einsteinian modes of scientific enquiry. Each of these assumes a peculiar notion of causation, and within any one of them, this notion cannot be questioned. It is not thought of as true; it is simply taken for granted. It is an absolute presupposition."

So first Collingwood writes that there are three kinds of physics, Newtonian, Kantian, and Einsteinian. That is bullshit. Collingwood's metaphysical views on physics are totally ignorant of how physics operates. His views were already considered absurd by the cognoscenti even back in the 1930ies when he wrote them. There is simply no such thing as "Kantian physics".

His already wrong views on physics then automatically degenerate into an exercise in infinite regress:

"any question involving the presupposition that an absolute presupposition is a proposition such as the questions ‘Is it true?’ ‘What evidence is there for it?’ ‘How can it be demonstrated?’ ‘What right have we to presuppose it if it can’t?’

So, that is typically metaphysics: ignorant and degenerative, infinite regress.

Is it true or not that the Bible claims that human beings have immortal souls? — Daniel C

The resurrection of the dead originates from:

Daniel 12:2. Multitudes who sleep in the dust of the earth will awake: some to everlasting life, others to shame and everlasting contempt.

This eschatological belief in the resurrection of the dead was not generalized in second-temple Judaism, but all its offshoots -- that managed to survive destruction by the Roman empire -- adopted it.

Christianity, just like Rabbinic Judaism and Islam, is an offshoot of second-temple Judaism and inherits the core of its eschatology from then already existing pre-temple-destruction beliefs. The striking new element is the Second Coming of Jesus (both in Christianity and Islam).

The sun will rise tomorrow. — fdrake

if ( time = tomorrow, 5h30 ) then { the sun will rise }

So:

time = tomorrow, 5h30 $\Rightarrow$ the sun will rise

It is still the modus ponens. If you want a counterexample, you will need to find something that looks like:

The sun will rise.

In that case, you do not justify, and then the question becomes ... Why? Why will the sun rise?

A Miracle Cure is defined as lasting remission of a terminal disease that can not be explained in terms of existing medical science. It has to be verified by high calibre medical experts. — Jacob-B

Medical experts have more knowledge about cause-and-effect situations in the human body than people without that training, but their knowledge is still fundamentally limited. They still do not understand the human body to a very important extent.

You see, a car mechanic can replace the motor in a car and get it work again, because he is in a position that he can possibly understand the construction logic of a car, simply, because it is human-designed technology.

If you chop off someone's arm, a medical expert cannot fix it by attaching a new arm. He has no access to the construction logic of this non-human technology, and just has to fiddle in the margin, without fundamentally understanding how it works. If they truly understood the technology of the human body, they would be able to attach that new arm.

In other words, a high-calibre medical expert knows several orders of magnitude less about the human body than a car mechanic about a car.

The fact that medical science declares a disease to be terminal is merely based on probabilistic black-box heuristics. It is certainly not a fail-safe determination. The fact that something cannot be explained in terms of existing medical science does not necessarily mean that much either.

Do all logical arguments, or syllogisms, of propositional logic need to take a specific form such as modus tollens or modus ponens? — MichaelJYoo

Knowledge as a justified (true) belief (JtB), is a modus-ponens arrow:

justification $\Rightarrow$ knowledge claim

The justification must contain all elements, including definitions, that support the arrow. The knowledge claim must necessarily follow from the justification. The justification itself, however, does not need to be justified.

So, yes, knowledge must always be phrased as a modus ponens.

How can you prove Newton's laws? — Fernando Rios

Unfortunately, you cannot prove anything about the real, physical world, simply, because its construction logic is unknown. The term "proving" means that you demonstrate that a theorem necessarily follows from the construction logic of its world.

Still, Newton's laws were really good at resisting falsification for a long while. Only by looking at quite unusual situations, they were falsified:

Experimentally, Newton's law of gravitation has been falsified due to for example the anomalous perihelion precession of Mercury, which matches instead with Einstein's theory of gravitation.

Still, Einstein's theory is rather used as a refinement and not as a replacement for Newton's laws.

What occurs to me is that you only have rational numbers in your sets. — joshua

Agreed.

The positional notation always expresses rationals, and not even all of them, such as 1/3 or so. Furthermore, if the digit stream for a number is infinitely long, then it will never proceed to matching the next number. This problem looks insurmountable.

How will you represent irrational numbers with a finite number of symbols, especially those that aren't computable? — joshua

Point taken. Can't be done, indeed.

So any subset of those strings is at most countably infinite. — joshua

Yes, I came to realize that now.

We have finally reduced the definite conception of morality to the idea of freedom. — Mww

Yes, it is the law that makes you free. The idea is very old, though:

James 2:12. So whatever you say or whatever you do, remember that you will be judged by the law that sets you free.

The "law" here is the religious law of second-temple Judaism.

and so we find that on just the same grounds we must ascribe to every being endowed with reason and will this attribute of determining itself to action under the idea of its freedom....” — Mww

Yes, agreed. People can decide if they will keep religious law or not. If they do, it will set them free. You will find this idea in one form or another in every offshoot of second-temple Judaism: Rabbinic Judaism, Islam, and Christianity.

The moral theory of Christianity is less, if rationality can come in degrees, rational. Reason/logic is utilized at a very basic level. The structure of Christian morals is by and large an appeal to authority - God. — TheMadFool

The ultimate premisses are not explained on grounds of other premisses. There is no other way to do it than to have ultimate premisses. As Aristotle wrote, if nothing is assumed then nothing can be concluded. So, in what way can any moral philosophy avoid reasoning from such ultimate premisses?

Christianity doesn't provide a logical argument for each of the ten commandments for example. — TheMadFool

Moral philosophy is different. It's an argument-based attempt to prove a system. It appeals to reason rather than authority. — TheMadFool

If moral philosophy made sense, it would name its basic assumptions. If it does not, then it is just an exercise in infinite regress. Naming basic assumptions is very much the same exercise as naming ten commandments. From there on, if you explain these ten commandments, then what do you explain them from? Another set of commandments?

Sorry, that approach is not rational.

solve, say, the Riemann hypothesis — StreetlightX

Hypotheses are not to be solved. In math and logic, they are taken as assumptions, as givens, as accepted as true. — god must be atheist

The Riemann hypothesis is a question that arises under the assumptions of number theory (Dedekind-Peano), or a theory that encompasses it, such as set theory (ZFC).

So, in the abstract, Platonic world generated by the axiomatic assumptions of number theory, a particular number pattern emerges. You can manually check it for any arbitrary value. Up till now, nobody has discovered a counterexample.

What is now wanted, is a chain of provable, first-order logic that works its way back to the assumptions, and which therefore proves that the pattern will always occur for any arbitrary value.

This particular pattern about the zeta function was first reported in 1859 by Bernhard Riemann (160 years ago).

Some patterns in numbers are easy to derive from the construction logic of number theory, but other ones have resisted every attempt at bringing them back successfully. You can clearly see these patterns and pick examples to verify, but nobody has found a way to link them to the very construction logic of the world in which they occur.

It is not possible to add the Riemann hypothesis to the axioms of number theory, because it may not be independent of the existing axioms. It could actually be provable from them, but that is exactly what is not known today.

By the way, the Clay Institute pays out $1 million to anybody who figures out the Riemann hypothesis.

You say after a character assassination while having nothing to refute anything. — Gnostic Christian Bishop

What is there to refute in his Powerpoint presentation? It is obviously not even wrong.

It is simply not possible to make that kind of far-reaching claims, merely by brandishing a few Excel sheets.

If you believe it is, then you are obviously not familiar with any real knowledge claims. In other words, what you think you know, is not knowledge. It is mere bullshit. Review everything that you think you know, and search -- possibly in vain -- for something that is not completely worthless.

It is clear that these theories being birthed by professional philosophers trained in logic could answer why1? questions: they were all logical arguments and, if anything, were supposed to help people make moral decisions. The mantle of authority passed from God to Reason. — TheMadFool

This view is based on a complete misunderstanding of rationality.

You see, reason consists of arrows of the type p => q.

You can obviously chain them. Still, if you work your way back, you will end up in starting points that cannot be justified, because justifying the starting points is an exercise in infinite regress. Hence, these syllogistical chains start somewhere and finish somewhere. They are required to be of finite length. This means that there are no arrows out of the starting points that chain further back.

This concept is quite difficult to explain to people who incorrectly believe that they are rational, but who in reality are not. These people are also lousy at other axiomatic subjects such as mathematics, because it works in exactly the same way there too.

Just double check it. Someone who incessantly talks about "reason" is usually someone who can barely handle the very, very basics of arithmetic. How can a person like that believe that he is smart? Seriously, is that intelligence?

In other words, someone who simply fails to understand the basic notions of rationality, is not rational at all, and had better shut up about rationality.

Is eating meat morally permissible? Why or why not? — darthbarracuda

You would first need an agreeable and documented starting point for morality. In the offshoots of second-temple judaism that support axiomatic theology, eating meat is kosher or halal, if the jurisprudential requirements and procedures of religious law have been satisfied, the details of which can be examined and assessed by sufficiently-trained religious scholars.

And having sexual relations with someone who is incapable of consenting to them is wrong, and wrong precisely because they did not consent to those relations. — Bartricks

Having sexual relations with someone is forbidden if the ruling authority would object to that. In the end, that is all that matters, because who else could be in a position to enforce anything to the contrary? There are lots of situations where women are not even asked for their opinion. History is full of them.

Occupation by Soviet troops. When Yugoslav politician Milovan Djilas complained about rapes in Yugoslavia, Stalin reportedly stated that he should "understand it if a soldier who has crossed thousands of kilometres through blood and fire and death has fun with a woman or takes some trifle." On another occasion, when told that Red Army soldiers sexually maltreated German refugees, he reportedly said: "We lecture our soldiers too much; let them have their initiative."

The sexual relations that Djilas complained about were not illegitimate because the ruling authority approved of them. It is not realistic to expect soldiers to be more disciplined than strictly necessary. Furthermore, the ruling authority had better approve of sexual relations whenever practical, because otherwise it will not stay the ruling authority for long.

What distinguishes 'philosophy of religion' from 'theology'? — fresco

Theology, in the religions in which it exists, is axiomatic (reductionist) derivation of propositions from scripture. It is about reasoning from the base documents of religion to reach conclusions. Philosophy of religion is about the ontology ("What is it?") and its epistemology ("How does religion reach conclusions?") But then again, religion has a rational part but also a transcendental one, while the handling of either differs. Religion does not exclusively appeal to the rational mental faculty.

Okay, I'm an atheist, but it seems to me that the quality of discussion on these prolific religious threads falls far short of 'philosophical debate' or even 'coherence' for participants . Even the apocryphal question about 'the number of angels who can dance on the point of a needle', would make better reading than what I have read here ! — fresco

If X is a human endeavour, you will often end up with ontologyOf(X), epistemologyOf(X) and moralityOf(X), which are pillars of the philosophyOf(X). It clearly works for X=religion. I wonder, what exactly is the domain D for X? When will such X have a meta-level?

I've found Google Search results for the philosophy of X=fishing and X=hunting ...

Mind you, when education goes up, beliefs go down. — Gnostic Christian Bishop

Where indoctrination goes up, net assets go down, and even get wiped out by student loans.

You can see these idiots paying inordinate amounts of money to watch live circus monkeys, the kind of which the academia are so exceedingly well endowed, farting bullshit.

Knowledge requires justification, and justification is incredibly hard to pull off, and even notoriously hard to verify. It may take dozens of pages of annotated first-order logic that keeps chaining until it reaches a conclusion that objectively only allows for less than an inch of progress, and all of that about an abstract, Platonic world, because we are not even truly capable of doing that about the real, physical world.

Once in a blue moon, someone pulls it off anyway, and publishes something original that resists extensive testing in the real, physical world. That is newspaper-grade news, because it almost never happens.

So, concerning so-called "education", what they are doing is just not hard enough. That is just one, good reason why their degrees are worthless. Furthermore, their beliefs do not go down, because they still believe that they will find a real job after memorizing phone books full of conjectural nonsense, sponsored by the New York banksters who will also be cashing in on the student loans, while only Starbucks will be willing to conditionally employ them. So, no, the false beliefs do not go down. They go up! And up! And up!

Of course, the United Nations wanted their head quarters to be in New York, not too far away from the scheming bankstering cartel that tells them what to do. It is a gigantic scam fest, the goal of which is to make the false beliefs go up, and up, and up; and money in the idiots' pockets go down, and down, and down. A fool and his money are easily parted. He will get a useless degree instead.

What may seem to be clearly stated to someone with the requisite knowledge of the subject matter may sound like nonsense to someone who is not familiar with the terminology and issues. — Fooloso4

Don't automatically assume that what seems to you like an abstruse post is a sign of "intellectual posturing." — SophistiCat

At least 50% of the arguments on the forum come from people using different meanings for the same words. — T Clark

One final thing I would add here, is the quote from Einstein,"If you can't explain it simply, you don't understand it well enough." — rlclauer

There is a good historical example for the Einstein quote.

When Algoritmi wrote his book in the 9th century, the "Liber Algebrae", he needed hundreds of pages to explain what in modern math is just a one liner of sorts. The big book started circulating in Europe in the 12th century, and caused a stir, because Algoritmi proposed a complete solution for the quadratic (degree=2).

That was something that nobody had been able to pull off till then.

So, it was considered some esoteric magic that came from mythical Arabia with its harems and 101 nights, that contained secret spells. So, "Don't tell anybody else that you now know about the secret!" before they passed on the secret to the next person to be initiated.

It took a long while to solve the cubic (degree=3) and the quartic (degree=4), mostly because the Catholic Church did not like the subject at all. The Medieval Italians who finally unravelled it, had to hide that they were working on it, and did stints in prison while frantically trying to avoid getting burned at the stake.

The Soviet historian I. Y. Depman claimed that even earlier, in 1486, Spanish mathematician Valmes was burned at the stake for claiming to have solved the quartic equation. Inquisitor General Tomás de Torquemada allegedly told Valmes that it was the will of God that such a solution be inaccessible to human understanding.

The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna.

They were first secretly circulating encrypted drafts of parts of the half-finished solution:

The solution to one particular case of the cubic equation had been communicated to him in 1539 by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long dispute) in the form of a poem.

Cardano was arrested by the Inquisition in 1570 for unknown reasons, and forced to spend several months in prison and abjure his professorship.

Do you have better statistics to refute him? — Gnostic Christian Bishop

A Powerpoint presentation with some Excel sheets in it?
Come on, what a circus monkey!

That gentleman headed up a whole university department and is an expert demographer. — Gnostic Christian Bishop

Of course, I am sure that there is a state-run university that will pay him good money for spreading his highly ideological views. The ruling elite invests quite a bit of money in their lies. When a manipulative lie suits the New York banksters, then all the money in the world will be available for it, simply, because it is them who print the dollars.

You see, the very first thing to understand about knowledge, is that it does not matter who says it. The only thing that matters, is its justification. The reverse is also true. If it matters who says it, then what he says cannot possibly matter.

Look at the difference with something like the 1905 special relativity theory. It does not matter that Einstein said it. If John Doe had said it, it would have made no difference. In fact, before 1905 Albert Einstein was still some kind of John Doe.

Or in a more recent example, look at the MimbleWimble white paper. Just like for the Bitcoin white paper, the author chose to remain anonymous. It caused a stir, because the math is provable and compelling.

That so-called "head of a whole university bla bla ... expert bla bla" is just a monkey pulling prostitution tricks. Seriously, I spit on him. He is just a filthy whore of the New York banksters.

https://www.youtube.com/watch?feature=player_embedded&v=ezVk1ahRF78 — Gnostic Christian Bishop

The video shows exactly what is wrong with his views.

It takes over thirty pages of annotated first-order logic to prove just one single, knowledge claim such as the Banach-Tarski paradox, while the speaker in the video spouts claim after claim in a 13-minute talk, each of which would be several orders of magnitude harder to provide evidence for. He still does not hesitate to call all of that science. So, my question becomes: Has he ever seen any real science?

The audience is obviously not capable of assessing that. When the ignorant populace listens to someone like him, how are they supposed to verify justifications that he does not even provide for his long string of conjectures? Just like the speaker, they are also not familiar with legitimate knowledge justification. Of course, nobody asked him how he has experimentally tested any so-called scientific claim he mentioned, simply, because nobody in the audience even knows that this is the question they should ask him.

His talk nicely shows what is wrong with the state-run education system. What he claims, are not justified beliefs -- because serious justification is hard stuff -- but it is exactly what the schools teach, and what they ask their hapless students to memorize, not because these things would be legitimate knowledge, but because it is what the ruling elite wants them to believe.

Behind every lie of the United Nations you will find a New-York bankster.

Only people who can resist the onslaught of indoctrination will keep reproducing. Everybody else will slowly but surely drift to a fertility rate of zero.

Reproduction decision usually have more than one issue, so no, I do not agree. — Gnostic Christian Bishop

Why do some belief communities reproduce while other ones do not? I think that the question is rather interesting. If you believe the wrong things, you will not reproduce. If you believe the right things, then you will. So, listen to people whose sexual reproduction is falling apart, and then don't believe them. Take note of what they say, because it is wrong.

It is the best "How Not To Do It" manual.

On the other hand, it is preferable not to convince them of the fact that what they believe, is wrong, because the problem is clearly solving itself.

Note how prolific Muslims are ... — Gnostic Christian Bishop

Yeah, they do not believe it, and they have lots of kids.
Everybody who believes it, does not.
So, let's conclude that not believing it, is a prerequisite for successful sexual reproduction.

Did you have something specific in mind? — Gnostic Christian Bishop

Societies who in all practical terms are no longer making children are at a distinct disadvantage in spreading the very ideas that cause their problem. Why would anybody else believe a word they say? Seriously, that is what explains why they don't. There is little need for further analysis because these things are getting increasingly obvious. Just look around you, and then confirm that the mess will not be fixed any time soon, if ever.

What about leaders? Every global movement is headed by someone. Could we tell him/her to set his/her house in order before attempting to lead the flock? Nobody's perfect I guess. — TheMadFool

Well, after negotiating with the Taliban for over a year in Doha, Qatar -- the deal was finally going to be announced this week -- the Donald still managed to flake out last minute.

He had first said that he was going to do it, before backpedalling and not doing it, because hey, "I have changed my mind and I am not going to do it!".

To tell you the truth, I knew that he was going to do something like that.

The Donald never really liked the idea of making a deal, but he wanted to do the negotiations because there is nothing to lose by merely sitting at the negotiation table. Still, when push came to shove, I knew that he was going to bail out, because that was the plan from the very start; and now he did.

Many immoral thinkers have. Gays and women have been targets for the immoral who do not like the notion of equality for many years. — Gnostic Christian Bishop

Long-term survival is the overriding concern here. Few people would deny that the religious communities are at a very visible advantage on that matter. For everybody else, long-term survival is increasingly turning into an futile exercise in squaring the circle. In other words, you have to solve your own problems first before trying to advise anybody else.

It is futile to argue with you, as you are a fanatic. This is another opinion I formed on you ... I suggest the mods would remove you as a major troll. But it's their call, not mine. — god must be atheist

Well, from what you are saying, it is not clear who exactly is the "fanatic", and a "major troll", and so on.

What has this got to do with anything we are talking about? Law has nothing to do with faith. — god must be atheist

You would benefit from reading Nassim Nicolas Taleb's article, titled "We Don’t Know What We Are Talking About When We talk about Religion".

This article is exactly about your misconception and your ignorance, the type of which is incredibly widespread. Nassim is a Lebanese Orthodox-Greek Christian, and from his background and experience growing up in the Middle East, he really does understand the difference between the various Abrahamic religions.

You absolutely do not. You know absolutely nothing about religion, but you think you do. That is a Dunning-Kruger problem: Stupidity is defined as not knowing when you do not know. Let me put it straight for you:

You do not know what you are talking about when you talk about religion.

That is why you desperately need to read Taleb's article.

This is an excerpt:

People rarely mean the same thing when they say “religion”, nor do they realize that they don’t mean the same thing. For early Jews and Muslims, religion was law. Din means law in Hebrew and religion in Arabic. For early Jews, religion was also tribal; for early Muslims, it was universal.

((In fact, it still is.))

Let me repeat. People like you are very, very stupid. The worst problem about stupidity is that stupid people have no self-knowledge, simply, because they are too stupid for that. So, you fail to understand your own limitations. That is what makes you stupid. It is not the lack of knowledge on the subject. It is the fact that you are convinced that you know, while you know f_ckall.

Here you go — StreetlightX

I've run into the following search result, somewhere at the top: The Best Philosophy Books. I haven't read many of the books listed in that page, but I did read parts of:

• The Dialogues (Gorgias, Meno, Theatetus, Sophist, Symposium, Phaedrus, Timaeus, The Republic) – Plato.
• Physics, Ethics, Poetics, Metaphysics, Categories, On Logic, On the Soul – Aristotle.

I have only read the parts that have epistemic significance.

• Critique of Pure Reason – Emmanuel Kant --> superb but dense
• The Prince – Niccolò Machiavelli --> incredibly funny
• The Art of War – Sun Tzu --> good read
• Analects – Confuscius --> very bad stuff with an ugly hidden agenda

I hate the Analects. The following sentence is my summary of what it is about:

The sky is high and the sun is hot, and there are numerous fish swimming in the sea, but most of all, the subjects of the empire should obey to the emperor like children obey to their parents.

Confucius is obnoxiously and very uncritically statist. Even though I do acknowledge the necessity of having a government reduce gang violence inside the territorial perimeter of its authority, I believe that people should be highly skeptical about what else such goverment should be sticking its nose into. Furthermore, if the regime or its ruling elite misbehave, there should be no hesitation whatsoever to replace it. Hence, I utterly dislike Confucius' message of unconditional obedience to the (Chinese) emperor and his naive idea that the emperor would to his subjects some kind of good father to his children. The emperor very often isn't, and there should be no expectation that he would be. Better safe than sorry.

You uptake new information rapidly but have a hard time staying focussed. Is that a good guess? — fishfry

It's complicated. If there is nobody is waiting for me to finish some work, I know from experience that I will abandon ship within one month, because I will get distracted by some other shiny toy that I will have run into.

My longest time on a project has been 2.5 years up till now. Still, it was very interactive, with lots of feedback, and with people who were waiting for the job to be done, and in a situation where I would have felt embarrassed to flake out on my commitment to get something going.

So, yes, serious levels of perseverance are for me often about not losing face and not letting key people down.

Coq and machine-assisted proofs in general are an interesting subject, but let's hopefully not get sidetracked into that. To that end, you should definitely read the Mephist I linked earlier. He knows a lot more about Coq and constructive math than I do if he's around. — fishfry

I have always had it somewhere in the back of my head that I would want to figure out and elaborate on Coq and/or Isabelle. Still, if there is nobody else who really needs my Coq scripts, I will probably fail to keep pursuing it.

Run-off-the-mill software scripting projects automatically include that kind of incentives, because there is someone paying for the result that he is waiting to see. Or at least there is an open-source team waiting for the job to be done. Otherwise, I will indeed just be flaneuring from one topic to the other. There needs to be a deliverable to be shipped to someone, for me to decisively lock in on a target. This is just a hobby, really.

If you are interested, the key buzzphrase is: "The free group on two letters has a paradoxical decomposition." That phrase leads to a web of interrelated Wiki pages that are very good. You'll halfway understand B-T from those. The heart of the proof is about formal language. It's about all the finite strings you can make using two symbols and their inverses. It is the most syntactic thing you ever saw. It's a very natural construction that contains a surprising paradox. — fishfry

That sounds interesting! Looking into it.

These days logic is geometry. It's all come out via the mysterious categorical point of view. Buzzword: "topos theory." — fishfry

Ok, I'm reading up on "free group on two letters" and ""topos theory". Hopefully the vocabulary pyramid will not insurmountable!

I have answered the question. If you insist that the question is unanswerable, demonstrate that by showing my answer to be false. — Bartricks

I have never said that I think that your answer is false. I am saying that I am absolutely sure that your answer is not even false.

I regard the paradox as a pseudo-problem since an omnipotent being is a hypothetical, but I do not think the problem of logical contradiction here is a language problem. — Fooloso4

The Creator of the real, physical world cannot be existentially contained in it. So, it is not a question of about the real, physical world. The question then becomes: Can human knowledge even reach outside the universe in order to answer questions about what we would observe there?

Furthermore, how are we supposed to ascertain that any purported answer really is the answer? We cannot try to inspect anything outside the universe. This objection would indeed involve the semantics of the question. But then again, if there are already syntactic issues, why even involve semantics?

I do, however, think the problem is compounded when one attempts to solve it on the basis of an abstract symbolic system. — Fooloso4

Yes, the abstract symbolic system will already fail on the bureaucracy of formalisms involved. They have limitations which prevent us from answering a whole range of questions about abstract, Platonic worlds, i.e. mathematical ones. Inasmuch as theories about the real, physical world -- in this case, even outside its boundaries -- make use of this bureaucracy of formalisms in order to maintain their own consistency, they will also start failing.

That is why the question, "Is it uberhaupt possible to address that kind of questions?", will automatically get propelled to the forefront. I wonder how anybody could take decidability/computability for granted, knowing that there is an entire field of investigation about just that issue?

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem.

This entire field would not even exist, if all possible questions were solvable ...

I will not attempt to answer these questions, for any answer is based on certain assumptions that are not held in common by those who offer a contrary view. — Fooloso4

Well, yes, the ontology of logic and of mathematics are up in the air, and will probably remain so for the foreseeable future.

I was only referring to what causes the problem of undecidable questions in logic. According to Gödel incompleteness theorems, they are not caused by the axioms, or lack of axioms, and cannot be fixed by the axioms (including the axiomatic inference rules).

It is the language of logic itself that causes the issue. It occurs when propositional logic gets extended to first-order logic, by adding existential quantifiers. The decidability damage is caused by introducing just two symbols: ∀ and ∃.

That is one concept of logic, but certainly not the only one. — Fooloso4

Yes, agreed.

It is a formal language along with transformation/rewrite/inference rules. The EBNF grammar of a particular language of logic does not capture its rewrite rules, but can/must be used to express them. So, a logic is indeed a complete formal system. There are, of course, numerous variants of different expressive power.

Still, the problem of computability/decidability is not caused by these inference/rewrite rules. These rules are just axioms, while it gradually (historically) became clear it is not the axioms causing (or solving) the problems. They are caused by the excessive, expressive power of universal quantifiers; which is a language problem.

Fast forward to present day neuroscience and find your answer. — I like sushi

The true understanding of neuroscience must be considered limited to the problems that they can really solve. Neuroscience can obviously solve some problems, but in my impression, you may seriously be overestimating its ability to solve arbitrary problems phrased in the language of neuroscience. My own impression is that they are many more questions that they cannot answer than questions that they can.

I refer you to my earlier answer. You're not addressing the question, or realizing that you're not addressing the question. — Bartricks

I am pointing out the issue of the addressability of the question. You seem to have pre-1936 views on this matter. Unlike what you seem to believe, disregarding the entire issue of decidability/computability will not help solving your question.

By the way, I am not answering Russell's and Mirimanoff's questions either. These problems were not solved by answering the question but by declaring them unanswerable.

We're talking at different levels. my question is about whether or not an omnipotent agent would have control over logic. What you're doing is talking about the content of logic. What you're talking about it is irrelevant. Whatever you say about the content of logic, my point is that an omnipotent being isn't bound by it.

If you say no question can be given a yes/no answer, the omnipotent being can give you a definitive answer to any question you ask. And so on. — Bartricks

Well, everything you are saying, is simply not decidable, unless you manage to point out a purely mechanical procedure that calculates your conclusion as its conclusion.

This requirement is simply part of the limitations of formal knowledge. You have never demonstrated that your yes/no question would fall within the boundaries of Turing-complete decidability/computability. If you do not demonstrate this convincingly, the only conclusion left, is that it falls outside these boundaries, and that formal knowledge-justification methods cannot reach it.

I appealed to reason. — Bartricks

So, do you believe that Alan Turing's and Alonzo Church's answer to David Hilbert's question is not reasonable? On what grounds would you then believe that?

And I answered the question and gave my reasons for the answer. Those reasons justify that answer. You seem to be thinking they don't - why? — Bartricks

Where is the purely mechanical procedure that would answer your question?

You see, a machine must be able to come to the same conclusion as you do. So, the very first part of the answer must consist in the description of an algorithm. I am just applying Alan Turing's and Alonzo Church's conclusions here, with regards to David Hilbert's question concerning the decidability of yes/no questions.

You're just expressing beliefs about the content of logic. It's beside the point. I am asking whether being omnipotent involves having control over the content of logic. You can't provide any insight into the answer by just telling me more and more about the content of logic. — Bartricks

Well, I am only pointing to the history of "smartass" questions in mathematics.

After adding a rule to fix Bertrand Russell's "smartass" question, simply by making it impossible to ask (by introducing the axiom of restricted comprehension), another genius, Dmitry Mirimanoff, discovered a new way of asking "smartass" questions, because hey, "look, the system is full of bugs", and "look at this", and "look at that", and "Is this normal?", "Didn't I tell you so!?" ...

So, between 1917 and 1920, Mirimanoff published his endless rant, in a long series of articles, showing that there are "non-well-founded" sets that can cause all kinds of mischief in set theory. It is actually simple. The expression:

A = { A }

will easily cause havoc, because it means that you can replace A by { A }. Therefore:

A = {{ A }}
A = {{{ A }}}
A = {{{{ A }}}}
...
... ad nauseam ...
...

So, what did they do to get rid of Mirimanoff "smartass" questions?

Well, they (=Zermelo and Fränckel) introduced a new rule, titled "the axiom of regularity", which explicitly forbids asking this kind of "smartass" questions by making it impossible to do so. Problem solved.

Well, not really.

In 1928, Hilbert asked: If we forbid asking "smartass" questions, simply by outlawing them, can all yes/no questions be answered with a yes or a no?

Das Entscheidungsproblem. As late as 1930, Hilbert believed that there would be no such thing as an unsolvable problem.

In 1931, the first blow came with Kurt Gödel's incompleteness theorems. It is the language of logic itself that is the problem and that allows for asking "smartass" questions. So, fixing the system, by adding new rules, will not help. In 1936, Alan Turing and Alonzo Church then independently proved that a yes/no question is answerable only if there exists a purely mechanical procedure that can answer it.

Given the history of "smartass" questions in mathematics, and the 1936 conclusion, my position is:

I do not know of a purely mechanical procedure that can answer your yes/no question. Unless you can point out the existence of such procedure, your question must be deemed unanswerable.

Note: Computability, computation, and computer systems propel epistemology to the forefront. Knowledge-justification methods are truly key now.