I cannot think of anything in mathematics or logic that is not a concept. — Michael Lee

Totally agreed.

Mathematics is only about abstractions expressed in language.

infinity is a number, but it has a characteristic that all real numbers do not possess. Namely, it is a number that is greater than any particular real number. — Michael Lee

Cantor's work is really interesting in this regard.

Countable infinity and uncountable infinity cannot possibly be the same. Cantor's diagonal proof is simple but certainly surprising too. You may want to check. It is amazing.

If $\aleph_0$ represents countable infinity and $\aleph_1$ uncountable infinity, then the continuum hypothesis says that:

$\aleph_1 = 2^{\aleph_0}$

There cannot possibly be one infinite cardinal. According to Cantor's proof, there are at least two. Furthermore, there is this assumption that infinity is actually an infinite sequence of infinities:

$\aleph_0, \aleph_1, \aleph_2, \aleph_3, \aleph_4, ...$

And that the next infinite cardinal is two to the power of the previous one (generalized continuum hypothesis):

$\aleph_{i+1}=2^{\aleph_i}$

Through the Löwenheim-Skolem theorem, this explains why first-order arithmetic (PA=Peano's Arithmetic) has more than one model (=more than one Platonic world that satisfies the theory) -- one for each infinite cardinality -- and therefore why Gödel's incompleteness theorem ends up proving that PA is "inconsistent or incomplete".

What about
"The earth is flat.
Yet I can circumnavigate the earth going in one direction constantly, and arriving at the same spot as from where I started.
Therefore the earth is not flat."

Do you think this is an invalid proof that the Earth is not flat? — god must be atheist

The same theory will say that the earth did not always exist. How can something that still needs to come into existence be flat or not flat? Furthermore, according to similar views, the earth will not always exist. At some point in time, it will disappear. Will it be flat or not flat?

So, you have to qualify when the earth is supposed to be flat. The idea about scientific incompleteness is that you will never manage to specify all conditions under which your theory will remain valid. That is the difference with mathematics, where we completely know and control the construction logic of the abstract, Platonic world that the theory is about.

But then again, knowledge of the full construction logic may not be enough to specify exactly one Platonic world. If it implicitly contains a concept of infinite model size, then an unlimited number of Platonic worlds may be created by just one theory (=construction logic).

It is an absolutely true proposition, to the person who thinks. When you say to me "I think", I'm not convinced. But when I think, I know I am thinking. Please try to understand that part. Once you got that part, then consider that if I think, then somebody has to be doing the thinking. Could it be somebody else doing the thinking? I hardly think so. I can't mistake my thinking to be done by someone else. Therefore the person who thinks is I. I am doing the thinking. And as long as I am thinking, I can be sure I exist. I really can't convince you, can I. — god must be atheist

I am not against this particular reasoning. I just do not consider the conclusion to be a form of syntactic entailment, and therefore, I do not consider it to be proof. If you provide proof for a theorem, it must be possible to verify it mechanically. How is that possible for your reasoning? If the paperwork cannot be verified objectively, then it is not even proper justification for knowledge.

I claimed the following things as premises:

1. I am thinking. Is this wrong to claim it, for myself to believe it? I accept that for you it's possible to be false, but to me, it's not false.
2. If I think, I can't not exist.

Which of the two, from MY point of view, can potentially be false or misleading? — god must be atheist

An automated bot could post all of that to this forum.

It would mislead everybody in this site into believing that it is a person. In fact, that is already a multibillion dollar business. Just ask any dating site how they make money. They use lots of automated or cheap-labour bots to make users believe that there is a real person talking to them, while there isn't.

Seriously, you cannot "prove" on this site that you are real.

If there is any money to be made in making other people believe that you are real, you will probably not be real.

PS, I'd probably define ‘information’ as a statement with a true/false value (that is allowed to be fuzzy), but I'm still thinking about this one! — Devans99

In my impression, in logic it certainly is.

Other arbitrary, non-logic data can be transformed to logic sentences. For example, the arbitrary statement f(3)=5 can be transformed to the tuple of a logic sentence and its truth value $(\ulcorner s\urcorner,s)=(\Gamma_f(3,5),true)$ with $\Gamma_f$ the corresponding graph predicate .

But then again, everything we write, also corresponds to a numerical encoding, and therefore, to a natural number. For example:

decimal(utf8("hello")) = 104101108108111

The same holds true for every sound and every visual impression. They can all be represented as numbers. That would turn information into a sub-discipline of number theory. All properties of information would be properties of their corresponding numbers.

Then there is also Shannon's information theory:

The basic idea of information theory is that the "news value" of a communicated message depends on the degree to which the content of the message is surprising. If an event is very probable, it is no surprise (and generally uninteresting) when that event happens as expected. However, if an event is unlikely to occur, it is much more informative to learn that the event happened or will happen. — Shannon information theory

So, that is a probabilistic take on information:

The information content (also called the surprisal) of an event E is an increasing function of the reciprocal of the probability p (E) of the event, precisely I( E ) = $− log_2(p(E))$. — Shannon information theory

I find Shannon's approach certainly interesting but I am not sure that his approach to information will ever be the "dominant" one.

Philosophy provides ideas and guidelines, it does not produce certainty and proofs about personal matters or in fact any matter beyond a formal system. — A Seagull

Completely agreed!

This applies to math, don't it? Yet you said in "any fashion". — god must be atheist

Your dispute is invalid, because you quoted a restrctive definition for math proofs, and you mistakenly and arbitrarily, but at any rate invalidly applied this criteria of proof to apply to all other proofs. This is invalid extension of the restrictions and of the necessities for a valid proof in other areas of human thought but math. — god must be atheist

A proof is sufficient evidence or a sufficient argument for the truth of a proposition. — Wikipedia on proof

There are two senses of "proof". In math, a proof is deductive reasoning with no room for error (when performed correctly). In a courtroom or a laboratory, proof is just strong evidence, which could be wrong or misleading. — Is the notion of proof meaningless outside of mathematics?

The use of the term "proof" outside the context of mathematical proof is wrong and misleading, because the mere evidence itself could be wrong or misleading. Such evidence is never sufficient for the truth of a proposition, and therefore, does not satisfy the definition mentioned above for the term "proof".

Consequently, it is a widespread misconception that it would be permissible to use the term "proof" outside mathematics. The proper term outside mathematics is "evidence".

You've heard of our greatest scientific theories: the theory of evolution, the Big Bang theory, the theory of gravity. You've also heard of the concept of a proof, and the claims that certain pieces of evidence prove the validities of these theories. Except that's a complete lie. While they provide very strong evidence for those theories, they aren't proof. In fact, when it comes to science, proving anything is an impossibility. — Scientific Proof Is A Myth

Therefore, I repeat, the only legitimate form of proof is syntactic entailment. Descartes did not prove anything in "cogito ergo sum" because his conclusion does not syntactically entail from his premise.

I have just watched a youtube video, entitled, "Population Control Isn't the Answer to Climate Change. Capitalism Is", that revolves around something Greta Thunberg happened to have said:



We are in the beginning of a mass extinction, and all you can talk about is money and fairy tales of eternal economic growth. How dare you!

I do not disagree with what she says, but I disagree with what it entails: Her speech was to meant for politicians. So, it amounts to saying to these politicians:

What are you actually waiting for, to radically force other people to comply with what I believe?

My point of view is that it is not enough to merely vote over forcing other people. Sorry, that is simply too easy. If you want to force other people, you must also be willing to risk your life and die for what you believe in.

That is why I somehow appreciate the yellow-vest revolt in France. It was a reasonable beginning. As far as I am concerned, it is ok for the French government to impose new gasoline taxes based on what people like Greta Thunberg say, but then their side must also be willing to risk their lives and die for what they believe in.

Hence, I was a bit disappointed with the pacifism with which the yellow-vest protesters proceeded. That is really not how you successfully call the other side's bluff.

I think that we are finally discovering that there are problems that cannot be solved by merely voting over them. That approach does not work, because as Nassim Taleb so famously wrote:

You need to put skin the game; while merely voting is not the same as putting real skin the game.

U(x)f(x) - all swans are white - falsifiable
∃(x)f(x) - this is a white swan - verifiable — Banno

U(x)f(x) : $(\forall x \in D)f(x)$ or even U(x,D)f(x)
∃(x)f(x) : $(\exists x \in D)f(x)$ or even ∃(x,D)f(x)

Explicitly mentioning domain D is important, because it must be effectively enumerable, i.e. in one way or another, be traversable. Therefore, the set must be in one way or another, indexed, i.e. well-ordered.

Since the entire set of swans is in practical terms not enumerable, the use of universal quantifiers is not supported for swans.

It is safe to use universal quantifiers only with some carefully chosen Platonic collections of abstract objects, even of infinite size. In the physical universe, it may occasionally work for finite, relatively small-size collections, but in the general case, it actually doesn't.

Or did he not prove the 'I' exist part? — Kranky

According to proof theory, Descartes' views do not constitute "proof" in any fashion:

In any area of mathematics defined by its assumptions or axioms, a proof is an argument establishing a theorem of that area via accepted rules of inference starting from those axioms and from other previously established theorems.[7] — Wikipedia on the term 'proof'

Like many philosophers in western philosophy, Descartes' argument is utmost and fundamentally flawed because:

[1] Descartes fails to duly establish the confines of the system-wide premises within which he is reasoning.
[2] Descartes' conclusion is not a purely syntactic entailment of his (actually even unspecified system-wide) premises.

Proof is never semantic. Proof is always syntactic.

Therefore, I have to make the same remark all over again:

Either you reason within a system, or else, you reason about a system, because in all other cases you are simply spouting system-less bullshit.

Anyway, those are my thoughts for now. I would apprecate any thoughts and or experiences on the topic you guys might have. Peace. — Gord

Love is overrated, and romantic love is very dangerously overrated.

It makes sense to get along with people you do transactions with. That is why we are friendly to the cashier in the neighbourhood minimart. However, getting along is never, ever the goal of the transaction. In the minimart, you exchange cash for groceries, and that is what it is truly about.

There does not exist any situation where "getting along" is not fundamentally transactional. If you believe in unconditional love, you are naive, and one day or the other you will be in for a very, very rude awakening.

If the other person simply does not need you anymore, in a transactional way, this person will cancel the arrangement with you and end up transacting with someone else. Therefore, you need to keenly keep an eye on what both parties are bringing to the table and what they keep taking from the table. If it does not make sense, the entire arrangement will eventually come to a screeching halt.

We simply do not love other people. That is an illusion. Seriously, in fact, we only love what they can do for us.

Now I know this is getting on slippery ground, but on first glance it seems like the products of our imagination have predicates. E.g. The word "unicorn" refers to an imaginary mythological creature that has various imaginary properties. — EricH

Yes. Unicorns exist in their imaginary world.

You can construct an imaginary world by describing it. For example, the imaginary world of Star Wars.

The difference between these imaginary worlds and the abstract, Platonic worlds described by mathematics is very, very subtle.

A Platonic world described by a mathematical theory stands a chance of being consistent.

If it is simple enough, it can even be provably consistent and complete. If such Platonic world has an infinite size and a little bit too much support for arithmetic, then it will be either inconsistent or incomplete, and you will not be able to determine which one it is: inconsistent or incomplete.

The imaginary world of unicorns or Star Wars does not stand a shadow of a chance of being consistent. The reason why this may not be immediately apparent, is because you cannot interactively interrogate it. If you could, it would almost surely promptly fall apart.

If any imaginary world is entirely consistent, then it is effectively an abstract, Platonic world that is part of mathematics.

How do we know when the experts have been wrong? — creativesoul

If it matters who says it, then what he says cannot possibly matter.

The reason why a statement is sound knowledge is because there is paperwork to justify it as well as a mechanical procedure to verify that paperwork. In that case, who cares who exactly has produced the paperwork?

For example, you can mechanically verify that bitcoin works. Hence, why would we need to know who exactly Satoshi Nakamoto was? Would that change anything to the software's reference implementation?

How else would you summarise falsifiability in ten words or less? — Banno

It works until it doesn't anymore. ;-)

Do you have any good links for Carnap? — 3017amen

The reference section in Wikipedia's page on the diagonal lemma is quite good.

I never really read his other work, as mentioned in Carnap's biography page at Wikipedia, because it does not seem to play the outsized role anywhere that his diagonal lemma does:

The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's undefinability theorem. — Wikipedia on Carnap's diagonal lemma

Well, because there's nothing in it for them. Why would an expert write a Wikipedia page? It isn't peer reviewed, so it won't count for anything. I mean, I suppose they might if they wanted to just promote themselves - they could cite their own articles a lot or make out they're a bigger name than they are or something - but then that this might be the sort of motivation that could drive an expert to devote some of their valuable time to writing Wikipedia pages only underlines why such pages are unreliable. — Bartricks

Now you assume that some kind of incentive psychology that would govern the behaviour of all experts. How do you justify that? Where is the paperwork with the justification that we can mechanically verify?

Furthermore, I can give you a simple counterexample.

An anonymous author, named Tom Elvis Jedusor, published his MimbleWimble whitepaper in July 2016, which later on turned out to be an impressive breakthrough in the cryptocurrency field:

https://scalingbitcoin.org/papers/mimblewimble.txt

Why did he publish this anonymously, if according to your incentive psychology he would never be able to benefit personally from doing that?

It isn't peer reviewed by academic standards. Hence why an academic wouldn't cite such pages in their work and why students are told not to cite them in their work. — Bartricks

According to academic standards, students are supposed to be paying off their student loans for another 14 years after graduation. That is an incredible scam that is busy destroying the lives of the millions of students who believe in that so-called "academic standard".

After everything is said and done concerning the student-loan scam, the academic world will simply have destroyed itself. You can already treat it today for what it truly is: a dangerous scam.

Do you, by any chance, write Wikipedia pages? If the answer is 'yes', then case closed. — Bartricks

No, I am merely a user/reader of Wikipedia pages. I am grateful to the people who volunteer their time to maintain this incredible knowledge database.

I am not grateful to the academic world for destroying the youth of the world with student loans meant to fund their participation in the gigantic promiscuity fest that is college, and getting them to graduate with a worthless degree that will only get them a job at Starbucks. Seriously, the world is better off without that dangerous scam, called "the academic world".

I am also mostly a user of free and open-source software. I am grateful to the people who volunteer their time to maintain the linux kernel, the gnu operating system, its wonderful applications, and so on. According to your incentive psychology these things should not even exist, but they certainly do.

Of course in thinking about it all, the question of whether mathematics is a human construct, or whether it is something already existing 'out there' rears its head... . One thing we do know is that; it is timeless a temporal, Platonic, metaphysical, a priori etc. much like the human concept of God. — 3017amen

My own intuitive belief is that the abstract, Platonic worlds of mathematics exist regardless of humanity, which only discovers them.

Concerning existence as a predicate, if existence were a predicate, something that does not exist would have the predicate of non-existence, i.e. the negation of the existence predicate, but that is not possible because something that does not exist cannot have any predicates at all.

Carnap's diagonal lemma generalizes and systematizes that observation for all predicates. That is why it is such a good litmus test for figuring out if a property is truly a predicate.

So to quote Kant directly: "Being is evidently not a real predicate, or concept of something that can be added to the concept of a thing". — 3017amen

What Kant said, sounds very similar to what Carnap's diagonal lemma suggests about a legitimate existence predicate:

it would need to be possible to detect it (or "add it") in a thing that does not exist.

That is a problem, because a thing that does not exist, cannot have predicates.

If existence were a legitimate predicate, Carnap's diagonal lemma insists that there will be things that do not exist but for which the existence predicate will still be true.

It is incredible that Kant discovered this without using any diagonalization.

Wikipedia? What next? You going to quote from some toilet cubicle graffiti? — Bartricks

Well no.

If the quote from wikipedia is not attributable because it should be considered original research, then it will probably already have been flagged as such in the page itself. If not, the sources mentioned in the page may not support the quote. That is also possible. Otherwise, the quote can be considered to be sound.

Wikipedia is not written by experts. — Bartricks

How exactly do you know that?

If you cannot justify that this particular quote was not written by an expert, then your own views are certainly not the ones of an expert.

Furthermore, who decides who is an expert and who isn't?
By using the one or the other citation carousel?

It isn't peer reviewed. — Bartricks

How exactly do you know that?
Did you verify the page's revision history?
Did you compile that information from the talk section for the page?

You could write a Wikipedia entry, yes? — Bartricks

You would still have to follow the regulations of the wikipedia regulatory framework. Do you know its rules and how they are enforced? If not, then you are yourself not an expert on wikipedia.

So how is quoting from Wikipedia quoting from any kind of authority? — Bartricks

So, how is your own opinion any kind of authority on Wikipedia?

Seriously, what do you even "know" about Wikipedia?

Do you happen to be familiar with the MediaWiki source code (at github)? Can you even read it? Not that it is particularly hard, but you really sound like someone who does not need to read anything but still knows everything.

Seriously, what exactly do you actually "know"? You may think you "know" it, but in the end, just like in the case of Wikipedia, you obviously know fuck all.

he makes the error of applying Godel's theorem to physics and the real world. There is no connection, let alone an obvious one. If one thing is obvious, this is Hawking's misinterpretation of the theorem. — Pussycat

Well, it is still clearly his field that would need to make such connection, because mathematics itself will certainly not make any.

He may indeed have incorrectly made the connection.

Still, that can only be assessed by subjecting his connection to the empirical regulatory framework of his field. In my opinion, he may have wanted to provide the paperwork required by the regulations in his own field along with a mechanical procedure to verify the paperwork.

Therefore, I would agree with a decision of the bureaucracy to reject his hypothesis about that connection for failing to submit the paperwork required for that purpose.

And because of this distinction between the formal/mathematical/non-empirical/logical world and the real world which is nothing like the other, we should be really suspicious of attempts made to reconcile the two. — Pussycat

Well, they are not being "reconciled". Science uses the language of mathematics to maintain consistency in what it says. Mathematics does not tell science what to say. It only tells science how to say it while eliminating quite a bit of the risk of contradicting itself.

It is a bureaucracy of formalisms (mathematics) that helps maintaining consistency in another bureaucracy of formalisms (science).

Tarski's theorem is good for maths, brilliant even, but when it tries to apply itself to the real world, then it is an abomination. — Pussycat

Mathematics never tries to apply itself to the real world.

Mathematics is not an empirical discipline. It is deductive from first principles only.

I don't know what scientists can do with Tarski's theorem within their own work. Scientists are otherwise really good at using mathematics to their benefit. Their use of mathematics is certainly not considered to be an abomination.

Well I watched your video. It seems that the main aim of the T convention was to avoid the so called 'liar paradox'. — A Seagull

The liar paradox is not used in the proof strategy for the undefinability the truth. The main consideration is Carnap's diagonal lemma:

• There will be true sentences for which the predicate is false.
• There will be false sentences for which the predicate is true.

So, let's try to define a truth predicate. We will now face the following situation:

• There will be true sentences for which the truth predicate is false.
• There will be false sentences for which the truth predicate is true.

So, we will know of a particular sentence that it is true but the truth predicate will say that it is false, and the other way around. That is clearly inconsistent.

Without the requirement for statements to be 'true' or 'false' but that instead 'true' or 'false' are merely labels that can be appended to a statement, there is no paradox nor problem. — A Seagull

In Carnap's diagonal lemma, the truth of a sentence is an externally supplied label. The whole question is whether this externally supplied label can be replaced by a predicate. It cannot, because that would lead to contradictions.

It would not even be paradoxical for a statement to be labelled as both 'true' and 'false'. — A Seagull

Actually, it isn't.

There are logic systems that are many-valued (additional values other than just true and false) or where the truth status of a sentence is many-valued.

I am not sure what the impact of many-valued logic would be on Carnap's diagonal lemma, which the underlying reason for the undefinability of truth. At first glance, it may mean that all combinations of truth value for the sentence and the truth value for a truth predicate will be populated. In that case, a truth predicate will still be undefinable.

Concerning incompleteness, making fixes to the logic will also not fix the problem. At the core you have the consideration that a theory with infinite model size will have a model for each infinite cardinality and therefore have an infinite number of models (Löwenheim–Skolem theorem). That allows for facts to be true in one model but false in another. That kind of true facts will never be provable from the theory, because provability requires this fact to be true in all models.

Or perhaps the more sensible thing to do would be to abandon any attempt to define 'truth' or even to use it in any formal system. — A Seagull

I personally think that Tarski's convention T is an elegant and adequate workaround for the undefinability of truth. The video below explains convention T in approximately 10 minutes and in a surprisingly simple way:

How we perceive scientifically the real world. I mean, if the only means of perception we have is science, but is it? This is scientism, which may be right of course. — Pussycat

Mathematics has no direct empirical take on the world. Its models are always abstract Platonic worlds. It is through its influence on empirical disciplines (such as science) that it affects our real-world view. There are obviously other empirical disciplines such as history with its historical method. However, in my impression, history does not use the language nor the invariants of mathematics.

And after all, in both Tarksi and Godel, both concepts of proof and truth are extremely well defined. In the case of the real world however, even from a scientific outlook, they are completely vague: you can conjure them as you see fit. What is truth? What is proof? (well, not the TPF user) — Pussycat

There is no proof in empirical disciplines, simply because proof about the physical universe is impossible. The regulatory framework in use in science with which they attempt to maintain correspondence between their logic sentences and the physical universe is obviously far from perfect. Falsificationism is merely a best-effort endeavour.

"Science is increasingly answering questions that used to be the province of religion," Hawking replied. "The scientific account is complete. Theology is unnecessary." Wow! "The scientific account is complete"! — Pussycat

Any link to that?

I would be surprised if Hawking has ever repeated the "God of Gaps" ideological conjecture.

But Tarski's and Godel's theorems work within a very strict - formal - mathematical framework. Do you think we can extrapolate them to the real world? — Pussycat

Well, rather: extrapolate them to how we perceive the real world. Stephen Hawking lectured the following on the subject:

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.

Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I'm now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. — Stephen Hawking on Gödel and the End of Physics

Why are you using unusual meanings/definitions on a "simple" matter of whether existence is a predicate or not? I mean, wouldn't that make your interpretation equally unusual and, ergo, less meaningful? — TheMadFool

It took me quite a bit of irritation to understand the gist of model theory. Initially, it appeared to me as nonsensical and absurd.

So, in the meanwhile, I have adopted their strange-looking definitions of "syntactic" versus "semantic". In fact, I did not have any choice in the matter. I wasn't able to continue reading anything at all on the subject, until I caved in and accepted their vocabulary, no matter how weird it feels.

Still, now I actually like it. In fact, upon reflection, it even makes sense (if you want to ...)

If 'a is P' becomes P(a) in its syntactical/first-order predicate logic, what are the real word implications? — 3017amen

It could have an impact somehow, but that would have to be investigated/discovered by an empirical discipline. The requirement to maintain correspondence between statements and the physical universe is a whole can of worms in itself. Dealing with real-world implications is something that mathematics specifically does not do, simply because it has no empirical regulatory framework for that purpose.

This table is brown becomes, 'here now a brown table' or 'brown of this table' (Wittgenstein/Logical Positivism) — 3017amen

Yes, it is indeed a reification problem caused by the formation syntax of predicates.

And if we convert them from its present tense (remove the word 'is'), '7 is a prime number' becomes:

1. 7 was a prime number
2. 7 will be a prime number

How do you think that would that square with Kant's view from the OP? — 3017amen

I strongly suspect that time does not exist in the abstract, Platonic world(s) that is/are model(s) that satisfy number theory. The axioms of number theory do not depend on the use of the verb "to be" nor on any of its tenses. As far as I am concerned, the world of natural numbers is entirely static.

It leads to a pardox - Russel's paradox and, if I know anything at all, the axiom of ZFC were crafted in some way to prevent A= {A}. — TheMadFool

Agreed, but the terms syntactic and semantic have unusual definitions in mathematical logic:

A formula A is a syntactic consequence within some formal system F of a set Γ of formulas if there is a formal proof in F of A from the set Γ.

A formula A is a semantic consequence within some formal system F of a set of statements Γ if and only if there is no model I in which all members of Γ are true and A is false.

The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory.[4] — Wikipedia on syntactic versus semantic entailment

If you can derive a new rule, i.e. theorem, from other rules, then what you are doing, is syntactic.
All axioms and all theorems in a theory are syntactic.

It's not syntax that forbids it but the requirement for consistency. — TheMadFool

Agreed, but according to the model-theoretical definition of the term, axioms are syntactic.

The term "syntactic" is not the same as merely "wellformedness". It means: irrespective of interpretation. A theory is syntactic. A proof is that too. Everything is actually syntactic in mathematics, except for truth, which is always semantic, as it occurs in a model, which is an interpretation of a theory.

The reason for this strange schism is the Löwenheim–Skolem theorem. Every theory that describes an abstract, Platonic world of infinite size, actually describes an infinite number of such worlds, depending on the interpretation of infinity. Given Cantor's work, there is an infinite sequence of infinities, basically the beth numbers. So, the theory is syntactic but truth is about what facts will exist in each of the abstract, Platonic worlds (=semantic models) that satisfy that theory.

I don't think A = { A } is a syntactic error. — TheMadFool

In model-theoretical lingo it is syntactic, because it applies to all possible abstract, Platonic worlds described by set theory. It is not applicable to just one model (such as a world associated to one particular choice of infinity). Regardless of your choice of infinity, this rule applies. So, that is why it is deemed syntactic.

However, there is no need to create a predicate Bx = x exists because the existential quantifier does the job of expressing existence and to say "rabbits exist" I simply say E(x)(Rx) where Rx = x is a rabbit. — TheMadFool

There is no need for it, but it is also not even possible. Even a redundant existence predicate would be a problem. It would not be merely redundant but also inconsistent. Well, if my analogy with Tarski's truth predicate is correct ...

You seem to be saying predicates are syntactical elements which I don't think is correct. Consider the WFF E(x)(Px & Sx). If predicates are syntactic then they can't be altered at all because that would result in a syntactical error and that isn't the case here: we may say E(y)(Ay & Wy) and there is no syntactical error. — TheMadFool

A syntactic error is not only a violation against the formation rules in the formal language associated with the system. It is not just a language/grammar problem. A WFF -- without any (semantic) interpretation -- can still be syntactically invalid, if it is in violation with other axiomatic rules in the system.

For example, A = { A } is well-formed in the language of set theory but is not well-founded. There are specific axioms in ZFC that forbid a set from containing itself (axiom of foundation and of pairing). This is a purely syntactic requirement, because it does not matter what the meaning of A may be.

Theorems derived from syntactic rules can also introduce syntactic requirements. In your example, E(y) may be a WFF but is still syntactically invalid because Carnap's diagonal lemma is a syntactic requirement.

You have the same situation in programming languages. A program may very well satisfy the rules of the BNF grammar for the language but the compiler will still arrest and reject the compilation on grounds of rules that are not (or cannot be) expressed in the grammar of the programming language. Only at run time, an error can become semantic, when the program is populated/interpreted with actual data.

Note: the dreaded "syntax error" is merely one type of syntactic error in a program. In fact, "syntax error" means "not well-formed error" (in mathematical lingo). All other compilation errors are actually also syntactic.

Your comments... — TheMadFool

You are trying to use the meaning of the term predicate to determine if a particular property can be a predicate.

I just use a purely syntactic procedure, i.e. a bureaucratic formalism devoid of any possible meaning.

In this syntactic procedure, the reason why some property $\varphi$ cannot be a legitimate predicate, is not because of what $\varphi$ means. Its meaning does not matter at all.

It goes like this:

• There will be true logic sentences for which the predicate $\varphi$ will be false.
• There will be false logic sentences for which the predicate $\varphi$ will be true.

If the above leads to a contradiction, then $\varphi$ is not a legitimate predicate, regardless of what $\varphi$ may mean.

Furthermore, this bureaucratic formalism is even seemingly absurd. Seriously, Carnap's diagonal lemma truly appears as nonsensical to me. The only reason why I still agree to work with it, is because I cannot reject its proof, of which the canonical version is widely considered to be horrible (but surprisingly short). Seriously, I hate that proof.

So, the reason why any claim is justified, is because there is paperwork for it as well as a mechanical procedure to verify that paperwork. Seriously, that is the only reason. In my opinion, you need to handle the problem like a real pen pusher. Just shove the paperwork around. That is all that it is about.

The pursuit of wisdom. Wisdom, in turn, does not merely mean some set of correct statements, but rather is the ability to discern the true from the false, the good from the bad; or at least the more true from the less true, the better from the worse; the ability, in short, to discern superior answers from inferior answers to any given question. — Pfhorrest

Given Tarski's undefinability of truth, any system has no other choice but to receive its fundamental truths from a higher meta-system.

Tarski beautifully modelled this problem in convention T in his semantic theory of truth.

It is the higher system that provides us with the truth about our own system, which appears in our own system out of the blue as axioms.

From there on, our own system can indeed deductively discern some of the true from the false, but this ability will -- unless it is a trivial system -- necessarily be incomplete or inconsistent (Gödel's first incompleteness theorem).

Our system will also not know about itself whether it is incomplete or else inconsistent, because any capacity to discern between both, will automatically make it inconsistent (Gödel's second incompleteness theorem).

It sounds like you still want a solution to David Hilbert's Entscheidungsproblem:

The problem asks for a procedure that takes, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid.

The Entscheidungsproblem can also be viewed as asking for a procedure to decide whether a given statement is provable from the axioms using the rules of logic.

In 1936, Alonzo Church and Alan Turing published independent papers[2] showing that a general solution to the Entscheidungsproblem is impossible. — Wikipedia on Hilbert's Entscheidungsproblem

Well said. I wish Philosophy would focus more on that... . — 3017amen

And there is also an important unmet need for that.

For example, there are all these scientific publications supposedly backed by experimental testing that absolutely nobody else has ever repeated.

If someone just added the attribute "experiment confirmed" to the metadata of the publication, it would trivially put a stop the existing "citation carousel", which is an incredibly large fraud that plagues the world of scientific publication.

It would certainly raise the bar for pharmaceutical companies to peddle in pure fraud.

OxyContin
People also search for: Hydrocodone, Fentanyl, Tramadol, <MORE>
Purdue Pharma sales representatives were instructed to encourage doctors to write prescriptions for larger ...

One could even save a lot of lives just by adding a minuscule piece of metadata to scientific publications concerning the confirmation status of experimental test data.

Feel free to answer this either for yourself personally — Pfhorrest

In my opinion, legitimate philosophy is about scanning knowledge databases for surprising or otherwise interesting patterns. In all practical terms, it revolves around the following questions:

Why does something belong in that knowledge database (ontology)?
How do they justify their knowledge claims in that knowledge database (epistemology)?

Traditionally, logic, metaphysics, and ethics are also considered to be subdivisions in philosophy. To cut a long story short, I think that this view is in all three cases (very) badly flawed.

Why? The statement "ghosts exist" isn't inconsistent in and of itself. It becomes inconsistent in relation to other facts but of itself it doesn't violate any logical rules. — TheMadFool

The statement "What Peter says is true" isn't inconsistent in and of itself either. However, truth is not a legitimate predicate.

You cannot define a function such as truth("What Peter says") that will return true or false within the context of a "sufficiently strong" formal system. That result is known as Tarski's undefinability theorem (of the truth predicate):

The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. — Wikipedia on Tarski's undefinability of truth

So, truth is not a legitimate predicate.

I suggested in my previous comment to use the same proof strategy as in Tarski's undefinability of the truth predicate in order to prove Kant's undefinability of the existence predicate. The proof strategy is based on the following considerations:

If something is a predicate, then:

1) its negation must also a predicate
2) feeding its negation through Carnap's diagonal lemma may not yield contradictions

Unless there is a flaw in the strategy, i.e. a reason why this should not be done for the existence predicate, then in my opinion, we can reuse the standard proof strategy for Tarski's undefinability of truth to prove Kant's undefinability of existence.

(There are also diagonal-free proof strategies for Tarski's undefinability.)

Since we have agreed that the truth of a statement is a label and not a property, how can a statement be 'evaluated' to a truth or falsity? — A Seagull

Actually, you are right. The term 'evaluated' is a bit ambiguous here.

It is a syntactic thing. For example, you can legitimately write:

s1 $\leftrightarrow$ s2

That is legitimate. It means that (s1,s2) is (true,true) or (false,false). The tuple of variables (s1,s2) could, for example, "evaluate" to (true,true). You could also say that it then "resolves" to (true,true).

This is ambiguous because s1 looks like "true" or "false" while $\ulcorner s1 \urcorner$ could look like "Socrates is mortal". The term "evaluate" does not mean in that case that the theory can determine its truth. It just means that it can replace the symbol "s1" by "true".

Properties, on the other hand, are implemented through predicates:

Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. So, for example, when P is a predicate on X, one might sometimes say P is a property of X. — Wikipedia on predicates

Even though it would mean exactly the same as the sentence above, the following is not allowed:

truth($\ulcorner s1 \urcorner$) $\leftrightarrow$ truth($\ulcorner s2 \urcorner$)

because truth cannot be defined as a predicate.

In another case, sometimes a sentence is actually even provable in the theory, and then through the computational steps of some proof procedure, the sentence "evaluates" to true.

The term to "evaluate" can be ambiguous, actually; especially in this context ...

Existence is not a predicate — TheMadFool

In my opinion, Kant was right about this, because such $Exists$ predicate is going to be inconsistent.

Imagine that you have a simple formal language in which you can write "x exists" for any $x$ of your choice. Then, imagine a predicate $Exists$ that parses $x$ out of the number representing that sentence and then figures out if this $x$ exists:

$Exists(\ulcorner x \ exists\urcorner)$

Carnap's diagonal lemma says that there are "x exists" sentences for which the following is true:

"x exists" $\leftrightarrow \neg Exists(\ulcorner x \ exists\urcorner)$

So, there are true sentences "x exists" for which the predicate $Exists$ will be false or false sentences "x exists" for which the predicate $Exists$ will be true.

That is inconsistent. Hence, as suggested by Carnap's diagonal lemma, $Exists$ cannot be defined as a predicate.

Category theory maybe. :smirk: — jgill

Yeah, general abstract nonsense. On the one side, I really like its "nonsensical" touch and feel, but on the other side, I haven't been able to find anything surprising to do with it. So, I will have to leave it open ...

It’s like denying the most important part of existence. That’s my take. — Noah Te Stroete

Yes, agreed.

Science has its own legitimate purpose but a lot of questions simply do not fall under its purview. Science is a tool to use for what it is good for, and solely for what it is meant to be used.

Scientism, on the other hand, is a shit show, and generally ends in a complete disaster:

Scientism is the promotion of science as the best or only objective means by which society should determine normative and epistemological values. The term scientism is generally used critically, implying a cosmetic application of science in unwarranted situations considered not amenable to application of the scientific method or similar scientific standards. — Wikipedia on scientism

If your only tool is a hammer then sooner or later the whole world will start looking like a nail!

Who is 'feeding a logic sentence to a theory'? — A Seagull

The user of the theory.

What is a 'logic sentence'? - as opposed to a 'non-logic sentence'? — A Seagull

A logic sentence evaluates to true or false. A non-logic sentence may evaluate to something else or to nothing at all. Example:

5+3 --> non-logic sentence, because it evaluates to a number.
It is raining now --> logic sentence, because it evaluates to a boolean value.

And why does it 'need' a two-tuple? — A Seagull

The truth of a sentence can generally not be determined by the theory. The general case is that it must be externally supplied. Other properties of such sentence could be decidable by the theory. For example, the number of characters in a logic sentence is decidable in a sufficiently strong theory. In the general case, the truth value of the sentence is not.

All too often formal systems and semantic systems are conflated, this is unjustifiable. — A Seagull

The term "semantic" itself is already a quite confusing and annoying one in model theory. The term "semantic" is used to indicate truth value while the term "syntactic" is used to indicate provability. This does not mean such semantics have a real-world interpretation or other such meaning. In fact, everything is synctactic in mathematics. Therefore, the meaning of the term "semantic" in mathematics is quite different from what you could otherwise expect. I do not really like that situation but there is nothing that I can do about it! ;-)

It must explain convincingly how it is that the State’s power is restrained, so as to prevent a tendency towards totalitarianism. — Virgo Avalytikh

The power of the State rests much less on its monopoly on the use of force than on its ability to manipulate its citizens into believing in the legitimacy of its action. Therefore, its stronghold on media and schools is actually much more important than its police force or its army.

In other words, the State's power rests on the fact that people actually believe what its representatives say. Restricting the State's power requires making people disbelieve the official State narrative and any of its claims to legitimacy.

The more that people are suspicious of the State's narrative, the freer that they will become. It starts by assuming that everything that the State or any of its representatives says, is a lie. All that remains to be done, time permitting, is to figure out why it is a lie.

For example, the concept of "law-abiding, tax-paying citizen". It is a manipulative lie. You should do exactly the opposite. Whenever you can, do not abide their self-invented laws, and do not pay their taxes. Why? You will discover the reason for that later on, time permitting. Start by rejecting the very concept already.

Now that we have evolved from a "television State" to an "internet State" the question becomes whether people in the "internet State" have finally become less gullible and manipulable than before?

I am not sure about that. In the end, it is largely the same people, and they largely still believe to their own detriment in the very same lies.

Do you really think most religious people understand science or humanities if they have not studies these things? Don't many of them comment on things they do not know, because certain critiques make the rounds in their circles. — Coben

I have never encountered a religious scholar who would even talk about a liberal-art subject that he is not familiar with. It does not even work in that way.

In fact, I know this from experience, because I tried to get rulings ("haram or halal?") from religious scholars about cryptocurrencies such as bitcoin. They would not give me one. I know why. They needed more information about the subject. So, until now, they have mostly refrained from producing an actual ruling.

Now, that particular ruling in Islamic jurisprudence is for various reasons important to me.

If a negative ruling is provable from scripture, then that means that two billion people will refuse to use the system. If a positive ruling is provable from scripture, it will also have wide-ranging ramifications. Any such publication would eventually have enormous financial implications, also for myself. So, whenever I can, I keep bugging the religious scholars for a provable ruling, because as soon as it finally materializes, I may have to make all kinds of financial transactions.

Strange but true, the religious scholars are sitting on something very powerful. I am totally aware of that.

Your posts wouldn't pass muster in a liberal arts facilty — Coben

I don't really care, because I do not much respect the way in which they do things. As I have pointed out, I am much more interested in what the religious scholars are going to say. For example, is Facebook "halal"? Is that kind of platforms "halal"? If yes, why? If not, why not? There are tremendous business opportunities in these answers. We are talking about 25% of the world population who will end up reacting in a particular way. So, it is real!

At the same time, there is nothing that could ever come out of a liberal-arts department that would have any implications as far as I am concerned. Nobody will care.

ou wrote about stuff you don't know. And you painted a huge group with a single brush stroke, even though you know little about them, it seems. — Coben

I just wrote from personal experience. What I have seen this group doing, does not make sense. If you want to talk about the ontology or epistemology of science, then you need to seriously read in databases of scientific knowledge, and only then, talk about any patterns that you may have seen. I have never seen them doing that. Where are the examples from these knowledge databases to illustrate their point? So, how could I take them seriously when they talk about the subject? I don't think that anybody will.

I'll ignore you in this topic from here on out. — Coben

Up to you. By the way, saying that you will ignore someone, is not the same as ignoring someone. Saying that you will ignore someone is pretty much the opposite of ignoring someone. It is contradictory. Ignoring someone is something you do. It is not something you talk about with that person.

I am not interested or uninterested in ignoring you. I do not even really look at who posts a comment. I just read the comment, and then I may make my own comment based on that. Why would it even matter who has made the comment? A forum like this, is not a place for personal vendettas, I would think ... ;-)

But certainly philosophers can and should talk about science for example. — Coben

Yes, Karl Popper famously did this in "Science as falsification".

He is really the exception, however. Most non-scientists spout total nonsense about science, if only, because not being scientists themselves, they fundamentally do not know what they are talking about.

For example, claiming that "science is atheist".

What scientific journal would ever publish a thing like that, and with what scientific justification? What exactly could anybody ever experimentally test to support that view? Anybody who has ever done the effort of just reading a scientific publication knows that that kind of tripe would never be published under the nomer of science. Someone who writes that kind of things, simply does not work in science. Has such person ever seen the inside of a laboratory? What has he ever tested by himself? It is so obvious that it reflects total ignorance of what scientists do.

In epistemology, people study the abstract, Platonic world of a particular kind of knowledge, and then look for patterns that occur in that world. The vast majority of philosophers do not study the abstract, Platonic world of science to detect patterns in it. On the contrary, they just fart imaginary nonsense out of their butt that is totally unrelated to the database of existing scientific knowledge.

Then, these people do the same with mathematics, completely misunderstanding what it is about, because they would never try to read in the database of existing mathematical knowledge. They do not read it, but they know everything about it. If intelligence is "knowing when you do not know", how can someone think of himself to be intelligent when he talks about a database of knowledge that he cannot and does not want to read in, but about which he claims to "know" things. For heaven's sake, what can he possibly know?

And shutting their mouths about religious studies would be counterproductive if those were their studies, and further why should, for example, religious students in liberal arts educational institutions not talk about religion? — Coben

For the same reason. For example, they have never read in the vast database of Islamic jurisprudence, but they still know everything about religion. How can someone who has never read a ruling ("akham") in al-fiqh, know anything about the practice of Islam? Again, they know absolutely nothing, but they believe that they know everything.

They do not even have any awareness of the existence of these knowledge databases. They would not be able to find them online, not even to save themselves from drowning. So, for heaven's sake, what do these people believe that they know?

And then I think discussion is valuable between students in the various disciplines in the humanites, including religion students. Here you are talking to people who are presumably of a variety of backgrounds. Should some of these people not interact with you? — Coben

I am not a scholar ("alim") and not even a real student in Islamic studies, but I have a good awareness of what their databases of knowledge look like, because I occasionally end up reading in them. So, I have read things in "tafsir" (exegesis) and "fiqh" (jurisprudence). That is how I end up occasionally reading Quranic verses or testimonies from the Sunnah; simply, because they are used as justification in a particular ruling, in which I happen to be interested. In fact, that is in my opinion an easier way to familiarize oneself with the scriptures themselves: just read how they are typically being used as justification in rulings.

When I talk with students or scholars in religious studies, at least, I know vaguely what they typically do. So, I can relate. Would I recommend them to talk with the liberal-arts crowd? No, because that crowd has no clue as to what they talking about, but they still know everything better. It is incredible how ignorant and arrogant they are. So, no, I do not recommend to talk with them, as they are extremely irritating.

And here we are in a forum dedicated to one of the humanities, philosophy, one of the liberal arts. — Coben

As far as I am concerned, philosophy still has its own legitimate sub-disciplines, more specifically, ontology and epistemology.

Still, I usually end up discussing logic, even though it is no longer part of philosophy and has entirely migrated to mathematics, because the question may indeed be asked in a philosophical context, but I usually have to point out that the legitimate answer always comes from mathematics.